Keywords: error correction cycle, preparation, 3 Optimized readout scheme

a quadrature gate . . . . . . . . . .

2.4

Emergent dynamics entailed by repeated preparation rounds . . . . . . . 3

The realization of quantum computing devices is one of the pressing challenges in today's research and, although the last decades showed impressive advances in controlling and manipulating quantum systems, key obstacles on this path are still to overcome. Progress on both the experimental and theoretical side is necessary to reduce the error rates of the underlying hardware to make useful and programmable devices possible. In general, quantum systems need to be well isolated as they rapidly loose their coherence due to the interaction with the environment. However, they can not be entirely shielded from their surroundings, since a coupling to an ancillary system is necessary for readout and to perform operations. Techniques to protect quantum information encoded in quantum bits (qubits) from decoherence are referred to as Quantum-Error-Correction (QEC). The concept is based on encoding quantum information in a larger Hilbert space, allowing to detect and correct for errors. One main route consists in encoding a logical qubit into several physical qubits, such that noise only induces transitions from a code state to states outside of the code manifold.

These spurious transitions are detected without revealing the logical qubit by measuring the error syndromes of the code, allowing to correct for errors. The objective lies in constructing logical qubits with significantly longer coherence times than their physical constituents. An implementation of such error-correcting codes, targeting sufficiently low logical error rates to perform useful computations however require a vast number of physical qubits [START_REF] Austin G Fowler | Surface codes: Towards practical large-scale quantum computation[END_REF]. In recent years, a more hardware-efficient approach, called bosonic QEC, has been developed, in which the logical qubit is encoded in the infinite-dimensional Hilbert space of a harmonic oscillator. Recent experiments showed promising results for the Binomial code [START_REF] Hu | Quantum error correction and universal gate set operation on a binomial bosonic logical qubit[END_REF] and the Cat code [START_REF] Ofek | Demonstrating quantum error correction that extends the lifetime of quantum information[END_REF], [START_REF] Leghtas | Confining the state of light to a quantum manifold by engineered twophoton loss[END_REF].

In 2001, Gottesman, Kitaev, and Preskill (GKP) introduced a different bosonic encoding, in which the code states are Gaussian-weighted superpositions of displaced squeezed states. In this approach, the quantum information is encoded non-locally in a grid-shaped pattern in phase space and equal protection against logical bit-and phase flips. This encoding is considered to be a promising approach, especially since all Clifford operations can be performed by Gaussian operations and homodyne detection. However, the code state preparation remained an experimental challenge until recent experimental advances. The group of Jonathan Home demonstrated the preparation and error correction of GKP states in the motional degree of freedom of a trapped ion [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF], [START_REF] Flühmann | Encoding a qubit in a trapped-ion mechanical oscillator[END_REF] and similar experiments were performed with superconducting circuits in the group of M. Devoret [START_REF] Campagne-Ibarcq | Quantum error correction of a qubit encoded in grid states of an oscillator[END_REF]. The error correction scheme relies in both instances on repeated measurements of the error syndromes of the code via conditional interactions with a two-level system and feedback displacements. Bit-flip errors of the two-level system during the interaction cause displacement errors of the state that can lead to flip errors of the logical qubit. The code manifold can therefore not be stabilized in a fault-tolerant manner, as errors from the ancillary system lead to direct logical errors. Multiple schemes have been proposed to solve this key problem, including controlling the state of the harmonic oscillator with a multiple-level ancilla [START_REF] Rosenblum | Fault-tolerant detection of a quantum error[END_REF], a flag qubit [START_REF] Noh | Fault-tolerant bosonic quantum error correction with the surface-gottesman-kitaev-preskill code[END_REF], or by using a noise biased qubit with a suppressed bit-flip error rate [START_REF] Puri | Bias-preserving gates with stabilized cat qubits[END_REF].

This manuscript presents another route that is based on using a GKP state as an ancilla, that is tailored and prepared in an asymmetric manner with respect to its quadratures. Our proposal does not require any new experimental tools, except for a tune-able quadrature-quadrature interaction between the target and ancilla oscillator. This type of interaction, that consist of a balanced sum of a Beam-splitter and a Two-mode squeezing Hamiltonian has not been experimentally realized to this date. This chapter gives an overview of the basic concepts and notations on the evolution of quantum systems and quantum error correction that are relevant for this manuscript. Further we introduce the GKP code in its infinite-and finite-energy form.

Chapter 2, reviews the GKP code state preparation and error-correction with an ancillary two-level-system, that has been demonstrated in recent experiments [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF], [START_REF] Campagne-Ibarcq | Quantum error correction of a qubit encoded in grid states of an oscillator[END_REF]. We present the key measurement circuits and feedback strategies and outline the main limitations, that we aim to solve in the succeeding chapters. In chapter 3, we present and study the dynamics of the error-correction circuit with an ancillary GKP state.

An asymmetric preparation of the ancilla state paired with a coupling of a single ancilla quadrature to the target oscillator is presented, that allows to limit the error propagation to the target oscillator. The error-correction scheme is adapted for the finite-energy code states in chapter 4, in which a specific limit of the GKP ancilla state is chosen that coincides with a squeezed cat state. We give numerical simulation results for state-of-the-art error rates that are realistic in near-term experiments.

A quantum bit and its environment

We give a brief introduction to main concepts in Quantum Information theory that are relevant in this manuscript. One of the main cornerstones in quantum physics is that a physical system can be in a linear superposition of d states labeled by |k⟩, that are elements of a Hilbert space H,

|Ψ⟩ = d k=1 c k |k⟩ , ( 1.1)
with complex amplitudes c k , which fulfill k |c k | 2 = 1. In the field of quantum information science, these physical systems of dimension d are generally referred to as qu-dits, which can encode information in the coefficients c k . A general quantum state, that is a statistical ensemble of l pure states weighted by p l , is described by a density operator,

ρ = l p l |Ψ l ⟩ ⟨Ψ l | . (1.
2)

The most simple, non-trivial quantum system consists of only two distinct states, which is referred to as a quantum bit (qubit) in analogy with classical information theory or a two-level system (TLS). We denote the two basis states as |0⟩/|1⟩ or occasionally as |g⟩/|e⟩, indicating the ground and excited state of the physical system. The state of a TLS is often expressed in terms of two angles θ and φ |Ψ⟩ = cos( θ 2 ) |0⟩ + e iφ/2 sin( θ 2 ) |1⟩ , (1.3) that allow to describe the TLS states using the Bloch sphere representation. A global phase of the TLS, e iϕ |Ψ⟩, is irrelevant as it does not alter the probability am-plitudes. Any single hermitian qubit operator O = a I I + a x σ x + a y σ y + a z σ z can be decomposed in terms of the identity I and the 2 × 2 Pauli operators

σ x = |0⟩ ⟨1| + |1⟩ ⟨0| σ y = i |1⟩ ⟨0| -i |0⟩ ⟨1| σ z = |0⟩ ⟨0| -|1⟩ ⟨1| . (1.4)
The Pauli operators σ i fulfill the commutations relations [σ i , σ j ] = 2iϵ ijk σ k , where 2 (I + ⃗ n⃗ σ), with ⃗ σ = (σ x , σ y , σ z ) and the Bloch vector ⃗ n = (⟨σ x ⟩ , ⟨σ y ⟩ , ⟨σ z ⟩), that encodes the state information. A rotation by an angle φ around the σ i axis of the Bloch sphere is given by e i φ 2 σ i = cos( φ 2 )1 + i sin( φ 2 )σ i .

ϵ
An observable in quantum mechanics is given by a hermitian operator O that can be expressed as a sum of projection operators O = i λ i P i weighted with the respective with eigenvalue λ i . For an initial state, described by a density matrix ρ, the conditional state after the measurement, the post-measurement state, is given by

ρ + = P i ρP i p i , ( 1.5)
where p i = Tr(ρP i ) is the probability to find the eigenvalue λ i . In order to extract information about the physical system without performing a direct projective measurement, it can be further coupled to an ancillary system (ancilla). Before the interaction, the composite state is described by |Ψ s ⟩ = |ψ a ⟩ ⊗ |θ b ⟩, with |ψ a ⟩ being the state of the target and |θ b ⟩ of the ancilla system. We consider a general coupling of strength χ of an operator A on the target and an operator B on the ancilla system, described by the Hamiltonian H int = -χA ⊗ B. The corresponding unitary operator for a time t is given by

U int (t) = k e iχtb k A ⊗ |k b ⟩ ⟨k b | , ( 1
ρ a+ = M k ρaM † k p k , ( 1.8)
where the probability to find the ancilla in the eigenstate |k b ⟩ is given by p k = Tr(M k ρ a M † k ). The set of measurement operators M k defines a Positive Operator Valued Measurement (POVM). Measurements of a target observable described by an operator O a , which do not alter the eigenstates of O a , are called Quantum Non-demolition (QND) measurements. In this case O a and U int (t) commute, such that

O a U int (t)(|µ a ⟩ ⊗ |θ b ⟩) = µU int (t)(|µ a ⟩ ⊗ |θ b ⟩), (1.9)
where |µ⟩ is an eigenstate of the operator O a with eigenvalue µ. In the remaining chapters of this thesis, the symbol of the tensor product for composite systems is omitted.

These kind of operations are included in the most general description of the evolution of a quantum system, by a completely-positive trace-preserving (CPTP) map

ρ ′ = M[ρ].
The Kraus representation of a CPTP map is given by

ρ ′ = y M y ρM † y , (1.10)
and in the case of measurements of a continuous variable y (for instance, a homodyne detection) by ρ ′ ∝ dy M(y)ρM † (y). The set of Kraus operators {M y } need to fulfill the completeness relation y M † y M y = 1 (respectively dy M † (y)M(y) ∝ 1), ensuring that the trace is preserved through this channel. In particular, in this thesis, we will consider the case where a measurement corresponding to a set of Kraus operators M y is followed by a feedback control based on the outcome y, corresponding to the unitary evolution operator R y .

Flip errors of a qubit induced by noise can be described by a CPTP map. The evolution of an idling qubit corrupted by a bit-flip with probability p is modeled by the map

M 0 = 1 -p I M 1 = √ p σ x (1.11)
A similar formula holds for phase-flip errors, by replacing σ x with σ z , where with probability p a Pauli error σ i and with probability 1 -p no error occurs.

The Lindblad master equation equivalently describes a quantum system interacting with the environment and can be derived from the Kraus operator representation (as for instance found in [START_REF] Haroche | Exploring the quantum: atoms, cavities, and photons[END_REF]).

dρ dt = -i[H, ρ] + k (L k ρL † k - 1 2 {L † k L k , ρ}) (1.
12)

The first term represents the unitary evolution given by the Hamiltonian operator H. Whereas the other terms describe possible transitions the system undergoes due to the interaction with the environment.

Concepts of quantum error correction (QEC)

The main objective of quantum error correction (QEC) lies in the protection of quantum information in the presence of noise. The general spirit is to encode a qubit -often referred to as the logical qubit -into a subspace of a larger Hilbert space, allowing to detect and correct for errors

α |0⟩ + β |1⟩ → α |0 L ⟩ + β |1 L ⟩ .
The logical states |0 L ⟩ / |1 L ⟩, can for instance represent states of a composite system of many physical qubits equipped with a multi-qubit code (i.e. Shor-, Steane-, Repetition-, Surface-code, etc.) or in the realm of bosonic codes, states in the infinite Hilbert space of a harmonic oscillator.

An error correction scheme in classical error-correction is based on encoding information in additional (redundant) copies that allow to verify if a bit-flip error occurred.

For instance, by encoding 0 → 000 and 1 → 111 respectively, a single bit-flip error can be distinguished by comparison with the remaining, un-flipped bits. This is referred to as error-correction by repetition, or the Repetition code. A similar strategy is not directly applicable for encoding information in quantum bits. Indeed, a classical bit can be copied easily, as its state can be inferred by a direct measurement, without a back-action perturbing the information encoded in the state. A similar measurement of a TLS along an axis σ i , projects the qubit in one of the eigenstates of σ i . Such a measurement yields solely a single classical bit and is insufficient to reconstruct the quantum state |Ψ⟩ = α |0⟩ + β |1⟩. In fact, it is fundamentally impossible to generate an exact copy of a quantum state, which has been coined the No-Cloning theorem [START_REF] William | A single quantum cannot be cloned[END_REF].

In the field of quantum error correction, a generic code needs to fulfill the Knill-Laflamme condition [START_REF] Wk | A single quantum cannot be cloned[END_REF]. Considering a quantum system described by a state ρ, subject to a noisy channel for which the density operator evolves according to the Kraus map ρ ′ = y E k ρE † k , where the E k describe the error processes. Any correctable error needs to satisfy the Knill-Laflamme condition [START_REF] Wk | A single quantum cannot be cloned[END_REF] for any codeword |j⟩ and |k⟩ ,

⟨j| E † m E n |k⟩ = a mn δ jk , (1.13)
with a mn being coefficients and δ jk the Kronecker Delta. This guarantees that the codewords remain orthogonal under the action of any operator in the set {E k } and for any linear combination of E k . If this condition is satisfied, there exists a unitary recovery map R[ρ ′ ] = l R l ρ ′ R † l to bring the state ρ ′ back to the code space .

3 Encoding quantum information in a harmonic oscillator Bosonic codes exploit the infinite Hilbert space of a harmonic oscillator to encode a logical qubit. This approach is commonly praised as a hardware-efficient path towards realizing fault-tolerant computations, as a logical qubit can be robustly encoded and corrected in a single harmonic oscillator.

A quantum harmonic oscillator

The general Hamiltonian of a quantum harmonic oscillator with the position Q and momentum operator P, a generalized mass m and frequency ω is given by

H QHO = P 2 2m + mω 2 2 Q 2 = ℏω(a † a + 1 2 ), (1.14)
which is factorized above using the so-called creation and annihilation operators

a = 1 √ 2 (q + ip) and a † = 1 √ 2 (q -ip). By defining Q 0 =
2ℏ mω and P 0 = √ 2ℏmω the reduced quadrature operators read q = Q Q 0 and p = P P 0 with the commutator -i[q, p] = 1. The eigenstates of the position/momentum operator, q |q⟩ = q |q⟩ respectively p |p⟩ = p |p⟩ form each an orthogonal basis of the Hilbert space. These states are orthogonal, but as Fock states. In the remainder of this thesis, the reduced quadrature operators q 3. Encoding quantum information in a harmonic oscillator and p and the convention ℏ = 1 is used.

non-normalizeable, ⟨q|q ′ ⟩ = δ(q -q ′ ) (⟨p|p ′ ⟩ = δ(p -p ′ )),
In usual physical implementations of oscillators, the dominant error channel is energy decay over time. The Kraus map representation for this process for a small time interval dt (see for instance in [START_REF] Haroche | Exploring the quantum: atoms, cavities, and photons[END_REF]) is given by

ρ(t + dt) = M k ρ(t)M † k M k = (1-e -κdt ) k k! e -κdt 2 a † a a k (1.15)
In numerical simulations, the series is often truncated to consider only single deexcitation events during short timescales dt

M 0 = 1 -κdt 2 a † a M 1 = √ κdt a, (1.16)
where M 1 describes the decay event with probability κdt and M 0 the no-decay evolution. Similarly, the energy gain process can be formulated by replacing a with a † .

Coherent states and squeezed coherent states

An important family of states are the eigenstates of the annihilation operator a, the so-called coherent or Glauber states [START_REF] Roy | Coherent and incoherent states of the radiation field[END_REF] a |α⟩ = α |α⟩ (1.17)

In contrast to the Fock states, they belong to the Gaussian state family, as they can be equivalently defined as the vacuum state |0⟩ displaced in phase space by an amplitude α (|α⟩ = D( √ 2α) |0⟩). The displacement operator is defined as D(γ) = e -i(Re(γ)p-Im(γ)q) . Displacements in phase space generally do not commute

D(γ)D(β) = D(β)D(γ) e (γβ * -γ * β)/2 , ( 1
a s = S(ζ) a S † (ζ) = a cosh(ζ) -a † sinh(ζ), (1.19)
where the squeezing operator is given by S

(ζ) = e - 1 √ 2 (ζa 2 -ζ * a † 2 )
with ζ being the squeezing strength. The quadrature operators transform accordingly as

q s = S(ζ) q S † (ζ) = q e -ζ p s = S(ζ) p S † (ζ) = p e ζ .
(1.20)

Cat codes

In this section, we review the the 2-component cat code, which is based on the superposition of two coherent states in phase space with opposite phase. The even and odd photon number cat states are +1/-1 eigenstates of the photon number parity operator P = e iπa † a , given by

|C + α ⟩ = 1 √ 2 (|-α⟩ + |α⟩) |C - α ⟩ = 1 √ 2 (|-α⟩ -|α⟩). (1.21)
The properties of the cat states can be made more apparent in their respective q and p representation

|C + α ⟩ ∝ dq ′ e -q ′2 /2σ 2 (|q = -α + q ′ ⟩ + |q = α + q ′ ⟩) ∝ dp ′ e -p ′2 σ 2 /2 cos(αp ′ ) |p = p ′ ⟩ (1.22) |C - α ⟩ ∝ dq ′ e -q ′2 /2σ 2 (|q = -α + q ′ ⟩ -|q = α + q ′ ⟩) ∝ dp ′ e -p ′2 σ 2 /2 sin(αp ′ ) |p = p ′ ⟩ ,
with α being real. In its q-representation, the wavefunctions are superpositions of two Gaussian states with variance σ 2 spaced by 2α. In the conjugate quadrature p, the respective wavefunction is represented by a cosine/sine with frequency α and a

Gaussian envelope with variance 1 σ 2 . The parameter σ signifies possible squeezing and equals 1, for the ordinary cat states. In fact, the wave function of the Schroedinger cat states are very particular as their support overlap in the q-distribution and are distinct by π 2α in p. Small noise processes, inducing small displacements can lead to transitions between the two states and therefore they are very suitable as sensor states for large values of α [START_REF] Gilchrist | Schrödinger cats and their power for quantum information processing[END_REF]. Whereas their symmetric superposition states

1 √ 2 (|C + α ⟩ ± |C - α ⟩ ≈ |±α⟩
) have distant support in the q-quadrature. The following convention is used for the logical code states and its superposition states

GKP code

|+ L ⟩ = |C + α ⟩ |0 L ⟩ = 1 √ 2 (|+ L ⟩ + |-L ⟩) = |α⟩ + O(e -2|α| 2 ) (1.23) |-L ⟩ = |C - α ⟩ |1 L ⟩ = 1 √ 2 (|+ L ⟩ -|-L ⟩) = |-α⟩ + O(e -2|α| 2 ).
Since |C + α ⟩ / |C - α ⟩ have an even/odd photon number they are exactly orthogonal ⟨C + α |C - α ⟩ = 0, whereas their superpositions have a non-zero overlap (⟨0 L |1 L ⟩ = O(e -|α| 2 )). However, for large α this overlap becomes negligible. Given that the wave-function support of the |0 L ⟩ / |1 L ⟩ states are very distant, the Cat code is shown to have exponential suppression in the number of photons n = α 2 against bit-flip errors, whereas the phase flip rate increases only linearly [START_REF] Mirrahimi | Dynamically protected catqubits: a new paradigm for universal quantum computation[END_REF]. Given this robustness in one quadrature, Cat states have recently attracted interest as candidates for noise-biased qubits [START_REF] Puri | Bias-preserving gates with stabilized cat qubits[END_REF], [START_REF] Guillaud | Repetition cat qubits for fault-tolerant quantum computation[END_REF]. In the next section, we introduce the GKP code that can correct for small displacements in phase space.

GKP code

In 2001, Gottesman, Kitaev, and Preskill introduced an encoding designed to correct for small shift errors in both quadratures of phase space [START_REF] Daniel Gottesman | Encoding a qubit in an oscillator[END_REF]. In this section, the code states are introduced as well as their corresponding stabilizer and logical Pauli error operators. In the stabilizer framework, the code space is spanned by the simultaneous eigenspace of two commuting displacement operators S q and S p = 0 with eigenvalue +1,

S q = e i 2π α q = D(i 2π α ) S p = e -i2αp = D(2α). (1.24)
S q and S p are displacement operators in phase space and encode the value of q mod α (q m ) and p mod π α (p m ). Since the stabilizer operators commute, the modular quadratures q m and p m are simultaneously measurable. Bit flips (X L : |0 L ⟩ ↔ |1 L ⟩) and phase flips (Z L : |+ L ⟩ ↔ |-L ⟩) in the GKP encoding are displacements by half of the period compared to the stabilizer operators (X L 2 = S p and Z L 2 = S q ),

X L = e -iαp = D(α) Z L = e i π α q = D(i π α ) Y L = e i( π α q-αp) = D(α + i π α ). (1.25)
The area of the lattice cell 2α • 2π α , spanned by the stabilizers amounts to 4π such that the Pauli errors anti-commute with themselves and commute with the stabilizer operators. It should be stressed that for a square unit cell α

= π α = √ π, Y L = iZ L X L
corresponds to a displacement along the diagonal in the grid, whose length differs from the other Pauli operators by √ 2.

The eigenstates |+Z L ⟩ and |-Z L ⟩ of the Pauli Z L operator with eigenvalues ±1 span the code space. This condition dictates the position variable of the logical states to be a multiple of 2α. The logical |-Z L ⟩ is derived from the action of the bit-flip operator

(|-Z L ⟩ = X L |+Z L ⟩).
The non-normalized code states read

|+Z L ⟩ = ∞ t=-∞ |q = 2tα⟩ |-Z L ⟩ = ∞ t=-∞ |q = (2t + 1)α⟩ . (1.26)
In the conjugate quadrature p, the codewords are represented by

|+Z L ⟩ = ∞ t=-∞ (|p = 2t π α ⟩ + |p = (2t + 1) π α ⟩) (1.27) |-Z L ⟩ = ∞ t=-∞ (|p = 2t π α ⟩ -|p = (2t + 1) π α ⟩).
|+X L ⟩ = 1 √ 2 (|+Z L ⟩ + |-Z L ⟩) = ∞ t=-∞ |p = 2t π α ⟩ (1.28) |-X L ⟩ = 1 √ 2 (|+Z L ⟩ -|-Z L ⟩) = ∞ t=-∞ |p = (2t + 1) π α ⟩ .
and expressed in the position q basis as

|+X L ⟩ = ∞ t=∞ (|q = 2tα⟩ + |q = (2t + 1)α⟩) (1.29) |-X L ⟩ = ∞ t=-∞ (|q = 2tα⟩ -|q = (2t + 1)α⟩).
The |±X L ⟩ states have an analogous representation in q (p) as the |±Z L ⟩ states in p (q).

Error detection and decoding

Given the distant support of the GKP-codewords in both q and p, small shift errors are correctable. A measurement of the stabilizer operators, corresponding to a measurement of q mod α or p mod π α , captures the modular peak shift information without revealing the logical state information. Assuming a GKP code state |Ψ L ⟩ is affected by an error of the form |Ψ L ⟩ → |Ψ ′ L ⟩ = e -iϵp |Ψ L ⟩ (as depicted in Fig. 1.3), the action of the stabilizer S q = e i 2π α q reads 2α . The shift error is simply corrected by applying a feedback displacement D(-θα 2π ). A similar argumentation is valid for errors in the momentum quadrature e iϵq by the application of the stabilizer operator S p = e -i2αp . A displacement error that exceeds half the distance of a Pauli error, is wrongly decoded and a feedback dis-placement leads to a logical error. Therefore, errors are correctable in the GKP code if the magnitude of the displacement in phase space is smaller than α 2 along q and π 2α along p respectively. A shift error ϵ along the q-quadrature can be detected by measurements of the stabilizer operator S q of the code. Displacements that exceed α 2 , highlighted by the yellow region, lead to a wrong decoding.

S q e -iϵp |Ψ L ⟩ = e 2π α ϵ[q,p] e -iϵp e i 2π α q |Ψ L ⟩ = e i 2π α ϵ e -iϵp |Ψ L ⟩ , ( 1

Finite-energy GKP code states

The logical states of the GKP encoding can be understood as coherent superpositions of infinitely squeezed states, which are not realistic given that they have infinite energy and are unbound in phase space. The finite-energy GKP states |Ψ ∆ ⟩ are related to the infinite-energy version |Ψ ∞ ⟩ by the non-unitary operator

|Ψ ∆ ⟩ = e -∆a † a |Ψ ∞ ⟩ [19],
where ∆ sets the envelope width in q and p. Further, the finite-energy stabilizer operators are derived from this relation

S ∆ q = e -∆a † a S q e ∆a † a = e i 2π α (cosh(∆)q+i sinh(∆)p) (1.31) S ∆ p = e -∆a † a S p e ∆a † a = e -i2α(cosh(∆)p-i sinh(∆)q) .
which still fulfill the commutation relation [S ∆ q , S ∆ p ] = 0. The finite energy states are superposition of squeezed coherent states (squeezing parameter ζ = ln( 1 ∆ )) with a bounded envelope function. The +1-eigenstates of these stabilizer operators have equal peak and envelope variance in the quadratures q and p. Here, we define the code states with a peak variance σ 2 in q ( 1 ∆ 2 in p) and envelope variance ∆ 2 in q ( 1 σ 2 in p).

GKP code

|+Z ∆ L ⟩ ∝ t e -1 2∆ 2 (2tα) 2 dq e -1 2σ 2 (q-2tγα) 2 |q⟩ ∝ t e -σ 2 2 (t π α ) 2 dp e -∆ 2 2 (p-tγ π α ) 2 |p⟩ (1.32) |-Z ∆ L ⟩ ∝ t e -1 2∆ 2 (α(2t+1)) 2 dq e -1 2σ 2 (q-(2t+1)γα) 2 |q⟩ ∝ t e -σ 2 2 (t π α ) 2 dp e -∆ 2 2 (p-tγ π α ) 2 |p⟩ with γ -2 = 1 -σ 2 ∆ 2
being chosen for a symmetric description with respect to q/p. Equivalently the code states can be expressed in terms of a continuous envelope function for which the peak and envelope positions are decoupled. In this representation, the states are of the form

|+Z ∆ L ⟩ ∝ dq e -q 2 2∆ 2 t e -γ 2 2σ 2 (q-2t α γ ) 2 |q⟩ ∝ dq e -σ 2 p 2 2 t e -γ 2 ∆ 2 2 (p-t π γα ) 2 |p⟩ . (1.33) |-Z ∆ L ⟩ ∝ dq e -q 2 2∆ 2 t e -γ 2 2σ 2 (q-(2t+1) α γ ) 2 |q⟩ ∝ dq e -σ 2 p 2 2 t e -γ 2 ∆ 2 2 (p-t π γα ) 2 |p⟩ .
In chapter 4, will treat GKP states with a non-Gaussian normalizing envelope, that are defined by the number of q-peaks n weighted by a Binomial distribution. In the next chapter, we review error-correction schemes for the finite-energy code states using Rabi-type interactions with an ancillary two-level system, to realize the necessary modular measurements of the phase space quadratures. The first experiment [START_REF] Campagne-Ibarcq | Quantum error correction of a qubit encoded in grid states of an oscillator[END_REF] to stabilize GKP states were based on measurements of the GKP error-syndromes through a Ramsey-type sequence in which the target oscillator is coupled to an ancillary two-level system, as seen in figure 2.1. By the Hamiltonian H r = -γ rσ z with an controlable interaction rate γ, the value of the modular position or momentum operators r m = q m , p m is mapped to the TLS. ). It then interacts with the oscillator via a conditional displacement gate (orange and purple blocks) by which the value of the modular coordinate q m or p m is mapped to the TLS phase. The TLS is then measured along the axis σ x , σ y in the equatorial plane (by a TLS readout in the σ z basis preceded by a π 2 pulse with the appropriate phase). Feedback displacements D(ϵ)/D(iϵ) conditioned to the TLS detection outcomes are applied to the oscillator state to correct for the shift errors along q/p.

|A⟩ 2n+1 ∝ n t=-n dq 2n n+t e -1 2σ 2 (q-2tα)
The detailed measurement sequence is represented in figure 2.1. The TLS, initially in |g⟩, is first prepared in |+⟩ = (|e⟩ + |g⟩)/ √ 2 by a π 2 rotation around σ y . The interaction Hamiltonian is then activated for a time T to perform the evolution described by the operator U r (T ) = e -iγT rσz = |g⟩ ⟨g| e +iγT r/2 + |e⟩ ⟨e| e -iγT r/2 .

(2.1)

U r (T ) results in a TLS-state-dependent displacement of the target state along the

r ⊥ quadrature |Ψ⟩ = 1 √ 2 (|g⟩ e iγT r/2 + |e⟩ e -iγT r/2 ) |ψ⟩ , (2.2)
for a general state |ψ⟩ of the harmonic oscillator. Conversely, this evolution can be seen as a rotation of the TLS of phase θ = γT in the equatorial plane (σ x -σ y ) of the Bloch sphere (see figure 2.2). The interaction strength/duration is chosen to be γT q = π α (γT p = α), to realize a measurement of q mod α (p mod π α ). Thereby, the TLS phase encodes the stabilizer information (⟨S r ⟩ = ⟨σ x -iσ y ⟩). A second Ramsey pulse (a): Preceding the interaction, the GKP state is displaced by a shift error δ along q and the TLS is initialized in |+⟩. (b) During the interaction (e -iγT qσz ), the TLS Bloch vector precesses in the σ x -σ y plane conditioned on the q-value of the GKP state (indicated by a coloured Bloch vector associated to the respective peak). The interaction strength is chosen such that the corresponding Bloch vectors refocus to the same angle encoding solely the value of δ of the modular coordinate without revealing more information on q.

Accordingly, the Kraus operators for the TLS measurement outcomes |g⟩ / |e⟩ are 1. Feedback strategies q-Error correction p-Error correction (2.4)

M g = sin( π α q + ϕ R 2 ) N g = sin(αp + ϕ R 2 ) M e = cos( π α q + ϕ R 2 ) N e = cos(αp + ϕ R 2 ).
In this thesis, we will often refer to these measurement sequences of the stabilizer operators based on a Ramsey-type sequence with a TLS as TLS-S r stabilizer measurements that are denoted by R. A single measurement sequence allows to extract a single classical bit of information of the stabilizer phase. In order to acquire a more accurate value of the shift error, multiple measurement rounds are necessary, a procedure known as phase estimation [START_REF] Yu | Quantum measurements and the abelian stabilizer problem[END_REF]. In the next section, different feedback strategies conditioned on the outcomes of the TLS measurements are discussed.

Feedback strategies

An exact readout of the stabilizer phase (peak shift error), corresponding to an infinitely long bit-string, requires an infinite number of rounds. Terhal et al. [START_REF] Terhal | Encoding a qubit into a cavity mode in circuit QED using phase estimation[END_REF] have proposed optimized phase-estimation schemes in finite-time by adjusting, at each round the phase ϕ r of the TLS measurement and/or of the applied feedback displacement length ϵ. The probabilities for a measurement of the TLS along σ x (ϕ R = 0) are given by P ±σx = 1 2 (1 ± ⟨Re(S q )⟩) and along σ y (ϕ R = π 2 ) by P ±σy = 1 2 (1 ± ⟨Im(S q )⟩).

In the case of non-adaptive phase-estimation, n • σ x and n • σ y measurements are performed, to equally probe the real and imaginary part of the stabilizer operator.

The Kraus operator M k,l , after n cycles for k • σ +x and l • σ +y outcomes followed by a feedback displacement ϵ k,l for a q-error correction cycle is given by

M k,l ∝ D(ϵ k,l )[ n k cos( π α q) k sin( π α q) n-k n l cos( π α q + π 4 ) l sin( π α q + π 4 ) n-l ]. (2.5)
The feedback length ϵ k,l is applied to maximize ⟨Re(S q )⟩, or other figure of merits that have been introduced specific to the GKP code.

In the adaptive phase estimation, the measurement axis ϕ R is varied in each round l as a function of the previous measurement settings [START_REF] Yu | Quantum measurements and the abelian stabilizer problem[END_REF], resulting in the Kraus operator

M l ∝ D(ϵ) n l=1 cos l ( π α q + ϕ R [l]). (2.6)
Terhal et al. computed an analytical formula for the conditional feedback displacement ϵ after n rounds, which is found in [START_REF] Terhal | Encoding a qubit into a cavity mode in circuit QED using phase estimation[END_REF]. Maximal information is extracted within one round for the measurement axis being chosen along the angle ϕ R [l] = θ[l] + π 2 , exactly orthogonal to the value θ[l] of q m estimated from the outcomes of rounds prior to l.

Figure 2.3: Comparison of different feedback strategies for GKP state preparation. 1 -⟨Re(S q )⟩ as a function of S q measurement rounds n with an ancillary TLS. Family of curves depict nonadaptive, adaptive phase estimation and Markovian feedback for ideal displacement (chosen to maximize Re(S q )) and constant feedback as a fraction of the lattice parameter (α = a).

An alternative, equivalent strategy is to keep the measurement angle fixed, but to apply a feedback displacement by ±ϵ l = -q ± l , where q ± l is the estimated modular position after the measurement round l that yielded an outcome ±. Thus after the feedback displacement has been applied, the estimated modular position before the round l + 1 begins is then q l+1 = 0. It follows that the optimal measurement angle is

ϕ R = π
2 for all rounds. This strategy is referred to as Markovian since the feedback displacement is proportional to the outcome ±1 of the measurement round.

In figure 2.3, we represent the evolution of 1-⟨Re(S q )⟩ under n rounds of adaptive/non-adaptive phase estimation and adaptive Markovian feedback, from an initial broad Gaussian state 1 . In all three cases, 1 -⟨Re(S q )⟩ decreases, indicating that the oscillator state converges toward the GKP manifold. Unsurprisingly, the adaptive methods yield equal results, slightly surpassing the non-adaptive method. However, computing the values θ l in the adaptive methods require non-trivial Bayesian filtering based on the whole measurement record of previous rounds. In [START_REF] Campagne-Ibarcq | Quantum error correction of a qubit encoded in grid states of an oscillator[END_REF], a simpler strategy, in which the feedback displacement length is fixed, was employed. In figure 2.3 we observe, in the non-adaptive Markovian case, longer feedback displacements allow a more rapid initial decrease of 1 -⟨Re(S q )⟩, which however saturates to a smaller value with respect to shorter feedback displacements.

[22], [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF] recently proposed that the Markovian feedback D ±ε can be applied autonomously, without the need of TLS detection (Fig. 2.4). By extending the unitary operator in eq. 2.1, by e -iϵpσ Y

U coh = e -iϵpσ Y e i π α qσ Z (2.7)
the feedback is applied in a coherent manner. The TLS needs to be reset after the sequence in order to re-use it2 .

Figure 2.4: Equivalence of Ramsey-type measurement sequence of the stabilizer S q with TLS measurement-based feedback and the autonomous sequence in eq. 2.7, in which the feedback is applied via e -iϵpσ Y .

Error correction for finite-energy code states

A similar error correction scheme applies to the finite-energy code states. However, since the corresponding finite-energy stabilizers are non-unitary operators, an analogous measurement scheme would necessitate the implementation of a non-hermitian Hamiltonian. Recently, [START_REF] Royer | Stabilization of finite-energy gottesman-kitaev-preskill states[END_REF], [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF], [START_REF] Hastrup | Improved readout of qubit-coupled gottesman-kitaev-preskill states[END_REF] have proposed (and experimentally realized in [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF], [START_REF] Vv Sivak | Real-time quantum error correction beyond break-even[END_REF]) sequences to effectively perform measurements of the finite-energy stabilizers. In this section, these sequences are discussed in detail as they are of relevance in chapter 4.

A measurement of the infinite-energy stabilizer operator S q (S p ), described in the first part of this chapter, entails a conditional displacement by D ±i π α (D ±α ), that leads to an expansion of the envelope to higher photon numbers. In the experiment by [START_REF] Campagne-Ibarcq | Quantum error correction of a qubit encoded in grid states of an oscillator[END_REF], the envelope has been stabilized by interleaving stabilizer measurements with measurements of the modular operator q, p, mod 2π η with much longer period in phase space (η << 1). This measurement is simply performed via the unitary U η = e iηrσz . The corresponding Kraus operators for Markovian feedback are

M η -= D -α sin(ηq + π 4 ) N η -= D -i π α sin(ηp + π 4 ).
(2.8)

M η + = D +α cos(ηq + π 4 ) N η + = D +i π α cos(ηp + π 4 ).
with a conditional feedback D ±α (D ±i π α ) to recenter the oscillator state. These displacements by half the GKP lattice period induce deterministic flips of the logical qubit and have to be accounted for in software. An intuitive understanding of the correction mechanism is shown in Fig. 2.5.

[22] and [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF] introduced an alternative method in which the correction of the envelope is performed simultaneously as the stabilizer measurement. In other words, one directly performs a measurement of the finite-energy stabilizers S ∆ q and S ∆ p introduced in eq. 1.31. They consider the following sequence of two conditional displacements to measure S ∆ q (the measurement of S ∆ p is analogous by a rotation of the quadratures by an angle π 2 , q ⊥ = p, p ⊥ = -q)

U η = e i π α qσ Z e -iηpσ Y . (2.9)
Note, that the first conditional displacement is conditioned on the Pauli operator σ y of the TLS. One obtains such an altered gate by flanking a σ z conditional displacement with π 2 andπ 2 TLS rotations around σ x . We can rewrite this expression as Corrective feedback displacements by a multiple of ± π α alpha along p (respectively ±α along q) have to be applied to recenter the state envelope. These displacements by half the GKP lattice period induce deterministic flips of the logical qubit.

U η = e i π α qσ Z e -iηpσ Y = + e -i π α q cos(ηp) |g⟩ ⟨g| + e i π α q cos(ηp) |e⟩ ⟨e| (2.10) + e -i π α q sin(ηp) |g⟩ ⟨e| -e i π α q sin(ηp) |e⟩ ⟨g| .

Applying this operator to a general oscillator state |ψ⟩ and a TLS prepared in

1 √ 2 (|g⟩ + |e⟩), the composite state results in |Ψ⟩ = (e -i π α q cos(ηp -π 4 ) |g⟩ + e i π α q cos(ηp + π 4 ) |e⟩) |ψ⟩ . (2.11)
For the particular initial TLS state, this evolution is equal to that resulting from the application of the non-unitary operator

O C = e -i π α q cos(ηp -π 4 ) |g⟩ ⟨g| + e i π α q cos(ηp + π 4 ) |e⟩ ⟨e| . (2.12)
At first order in η (indicated by ≃)

O C = 1 √ 2 [e -i π α q (cos(ηp) + sin(ηp)) |g⟩ ⟨g| + e i π α q (cos(ηp) -sin(ηp)) |e⟩ ⟨e|)] (2.13) ≃ 1 √ 2 (e -i π α q e ηp |g⟩ ⟨g| + e i π α q e -ηp |e⟩ ⟨e|) = e ηπ 2α √ 2 e -i( π α q+ηp)σz ,
where the last line is equivalent to a Rabi-type gate with the operator e -∆a † a q e ∆a † a = cosh(∆)q + i sinh(∆)p that has been introduced in chapter 1, Sec 4.2 on the finite energy GKP states. Here with tanh(∆) = π ηα , realizing an effective measurement of the finite energy stabilizer S ∆ q .

Analogously to the measurement of the infinite stabilizer S q , that was probed via the unitary operator e i π α qσz , this sequence effectively maps information from S ∆ q onto the TLS, which is measured to obtain up to one bit of information. [START_REF] Royer | Stabilization of finite-energy gottesman-kitaev-preskill states[END_REF] and [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF] then considered a Markovian feedback strategy, in which the TLS is measured along σ y , before applying proportional feedback by ±ϵ, which gives the Kraus operators

M -= D -ϵ [sin( π α q + π 4 ) cos(ηp) + i cos( π α q + π 4 ) sin(ηp)] (2.14)
M + = D +ϵ [cos( π α q + π 4 ) cos(ηp) -i sin( π α q + π 4 ) sin(ηp)].
U SBSq = e -iϵpσ Y e i π α qσ Z e -iηpσ Y . (2.15)
and a reset of the TLS after the interaction. The optimal value of ϵ is found to be equal to η, which maximizes the state purity in steady state (under repeated rounds of the sequence for the q and p quadrature). Royer et al. [START_REF] Royer | Stabilization of finite-energy gottesman-kitaev-preskill states[END_REF] coined 4.21 the Small-Big-Small (SBS) sequence indicating the length of the displacements. Further, they derived an additional sequence

U BSBq = e i π α qσ Z e -i4ηpσ Y e i π α qσ Z , (2.16)
equivalently stabilizing the finite energy states, referred to as the Big-Small-Big (BSB) sequence. The sequence for the conjugate quadrature is simply found by rotating the quadratures (q ⊥ = p,p ⊥ = -q). As for the SBS sequence, the control parameter η sets the size of the GKP grid envelope in phase space. Note that, when considering single-photon dissipation of the oscillator, the envelope size, and therefore η, needs to be optimized to maximize the logical qubit lifetime. Intuitively, the envelope size should be wide enough for the grid peaks to be narrow and well separated from one another. We recall that for a pure GKP state, the envelope function along one quadrature is the Fourier transform of the peak function on the conjugate quadrature. On the other hand, an increasing grid state envelope, increases its average photon number, and therefore photon loss induces a stronger distortions of the code state. For a given oscillator lifetime, there is an optimal envelope size, that is set by η in the SBS sequence. In Fig. 2.7, the lifetime of the logical qubit is plotted for different oscillator lifetimes T cav as a function of η = ϵ, showing a maximum that is shifted towards higher values of ϵ for shorter cavity lifetimes.

Phase-swap sequence for an infinitely squeezed cat state

In this section, we show that the BSB-sequence introduced in the earlier section, described through the unitary operator

U BSBp (ϵ) = e iαpσ Z e -iϵqσ Y e iαpσ Z ∝
|A⟩ 2n+1 σ ∝ n t=-n dq 2n n+t e -1 2σ 2 (q-2tα) 2 |q⟩ ∝ dp e -σ 2 p 2 2 cos 2n (αp) |p⟩ (2.18) |C⟩ 2n σ ∝ n+1 t=-n dq 2n+1 n+t e -1 2σ 2 (q-(2t-1)α) 2 |q⟩ ∝ dp e -σ 2 p 2 2 cos 2n+1 (αp) |p⟩ .
We use a short hand notation for an infinitely squeezed cat state |C⟩ 2 σ=0 = |q = -α⟩+ e iϕ |q = +α⟩, with an initial phase e iϕ and consider the TLS being prepared in |+⟩ initially,

U BSBp |+⟩ |C⟩ 2 0 ∝ |g⟩ e -iαp (cos(ϵq) e -iαp + sin(ϵq) e iαp ) (|-α⟩ + e iϕ |+α⟩) (2.19)
+ |e⟩ e iαp (cos(ϵq) e iαpsin(ϵq) e -iαp ) (|-α⟩ + e iϕ |+α⟩)

= |g⟩ e -iαp [cos(ϵq)(|-2α⟩ + e iϕ |0⟩) + sin(ϵq) (|0⟩ + e iϕ |2α⟩)] + |e⟩ e iαp [cos(ϵq)(|0⟩ + e iϕ |2α⟩) -sin(ϵq) (|-2α⟩ + e iϕ |0⟩)]
By choosing ϵ = π 4α , a conditional ∓π pulse is applied for the population centered at |±2α⟩ that results in

U BSBp |+⟩ |C⟩ 2 0 ∝ |g⟩ e -iαp [e iϕ |0⟩ + e iϕ |2α⟩] + |e⟩ e iαp [|0⟩ + e iϕ -|-2α⟩] (2.20) =(|g⟩ e iϕ + |e⟩)(|-α⟩ + |+α⟩).
In the case of a cat state with infinite squeezing, the BSB sequence realizes a perfect phase swap. For finite squeezing, the sequence generates minor populations as well at the positions |±3α⟩. sequence between an infinitely squeezed cat state and a TLS. Initially the TLS and the infinitely squeezed cat state with a phase µ = e iϕ are in a separable product state. Schematic of wave function representation Ψ(q) depicts the TLS state corresponding to the respective q-peak position for each step of the BSB sequence. For a value of ϵ = π 4α in the conditional displacement gate e -iϵqσ Y , a conditional ∓π pulse is performed that results in a destructive interference of populations at the positions |q = ±3α⟩ when the last gate e iαpσ Z is applied. At the end of the sequence a perfect phase swap is realized for infinitely squeezed peaks. In the case of finitely squeezed peaks the sequence generates minor populations as well at the positions |q = ±3α⟩.

Error propagation from the TLS

We now analyze the impact of the two types of TLS errors, namely bit-flips and phaseflips, during the error correction protocol. For simplicity, we focus on the case of infinite-energy GKP states and expect qualitatively similar results for the finite-energy case.

Bit-flips of the TLS

We consider bit-flips induced by absorption or emission of the TLS energy in the environment, at respective rates Γ + and Γ -. Their effect on the density matrix of the system is modeled by Lindblad dissipators

√ Γ + D[σ + ] and √ Γ -D[σ -],
where σ + and σ -are respectively the raising and lowering operators of the TLS. Each dissipator D[L] yields, over an infinitesimal time-step dt, an evolution of the density matrix

dρ = dtD[L](ρ) = dt LρL † - 1 2 (L † Lρ + ρL † L) . (2.21)
We focus on the case Γ + = Γ -= Γ 1 /2 and briefly describe the most general case Γ + ̸ = Γ -at the end of this section. Note that this particular case of equal rates of TLS excitation and de-excitation applies to current superconducting circuits experiments.

Indeed, the control sequence employed in these experiments to generate conditional displacements includes regular flips of the TLS on a timescale much shorter than its energy relaxation time [START_REF] Campagne-Ibarcq | Quantum error correction of a qubit encoded in grid states of an oscillator[END_REF][START_REF] Vv Sivak | Real-time quantum error correction beyond break-even[END_REF]. As a result, TLS relaxation events, which are natively much more frequent than excitation events (Γ + ≪ Γ -≃ Γ 1 ), effectively induce transitions from any eigenstate of the TLS to the other with equal probability over the course of a conditional displacement.

We consider the effect of bit-flips during the application of a conditional displacement gate U CD q b = e i π 2α qσz along the p quadrature of an oscillator-the calculation is directly adaptable to the case of a conditional displacement along q. We assume this gate to be performed by the application of a Rabi-like Hamiltonian-also known as longitudinal coupling Hamiltonian

H CD q = -χqσ z (2.22)
with constant rate χ = π 2αT CD over the gate duration T CD (the coupling Hamiltonian is then turned off until the following gate). In the weak noise limit Γ 1 T CD ≪ 1, we unravel the effect of bit-flips as stochastic collapses onto the ground state |g⟩ or the excited state |e⟩ [START_REF] Howard | Quantum measurement and control[END_REF], each occurring with probability Γ 1 dt/2 in a time-interval of duration dt around any given flip time 0 < t err < T CD 3 . From an arbitrary initial state described by the density matrix ρ and when such a flip occurs, the non-normalized density matrix of the system after the gate (endpoint of the quantum trajectory) is

ρ ± terr = O ± terr ρO ± † terr ,
where the non-Hermitian evolution operators O ± terr read

O ± terr = Γ 1 dt 2 e iχ(T CD -terr)qσz P ± e iχterrqσz = Γ 1 dt 2 e ±i π 2α (1-2τerr)q P ± , (2.23)
with P -= |g⟩⟨e| and P + = |e⟩⟨g| the operators respectively modelling a collapse onto |g⟩ or |e⟩, and τ err = 1 -t err /T CD defined as in Fig. 1a. O ± terr thus collapses the TLS and displaces the oscillator state along p by

± π 2α (1 -2τ err ) ∈ I = [-π 2α , π 2α ].
In the protocol described in this work, conditional displacement gates are immediately followed by a measurement of the σ y Pauli operator of the TLS-yielding a random outcome when the TLS has been collapsed onto |g⟩ or |e⟩-whose outcome controls a feedback displacement by ±ϵ along q. Then recombining all trajectories to model our proportional (memoryless) feedback strategy and tracing out the TLS, the oscillator density matrix reads

ρ′ = (1 -p BF )U CD q ρU CD † q + p BF ρ I (2.24)
where p BF = Γ 1 T CD ≪ 1 is the total bit-flip probability during the gate and ρ I denotes the density matrix ρ randomly displaced by ±ϵ along q and by a length uniformly sampled in I along p.

In the general case, in which the TLS excitation and de-excitation rates are not equal, one also needs to account for the partial collapse of the TLS state during the no-flip evolution. This evolution commutes with the conditional displacement gate and only unbalances the relative amplitude of probability of the two conditionally displaced copies of ρ, thereby reducing the contrast of the subsequent TLS measurement similarly to the phase-flips of the TLS described in the next section.

Phase-flips of the TLS

By comparison with bit-flips, phase-flips of the TLS are simpler to model. Indeed, in the quantum trajectory approach described above, they correspond to σ z gates randomly applied to the TLS over any time-interval of duration dt with probability

Γ ϕ 2 dt
, where Γ ϕ is the TLS pure dephasing time. Since σ z commutes with the interaction Hamiltonian, phase-flips are equivalently modeled as a σ z gate applied after the gate with probability p P F = Γ ϕ 2 T CD (in the weak noise limit). By flipping the sign of the subsequently measured σ y Pauli operator, this error results in an erroneously applied feedback displacement. We set p P F = p BF /2, typical of superconducting circuit experiments, in all simulations performed in this work.

Note that TLS readout errors have an impact similar to phase-flips, but may cause more damage when the TLS is actively reset based on the measurement outcome, yielding a TLS erroneously prepared in | -x⟩ for the subsequent ancilla preparation round (see Fig. 2b). Experimentally, such reset errors may be mitigated by repeating the reset procedure in |g⟩, assuming the measurement to be Quantum Non Demolition for the |g⟩ state [START_REF] Ristè | Feedback control of a solidstate qubit using high-fidelity projective measurement[END_REF][START_REF] Campagne-Ibarcq | Persistent control of a superconducting qubit by stroboscopic measurement feedback[END_REF]. Readout errors are not modeled in this work.

In conclusion, phase-flips of the TLS lead to errors in the measurement of the stabilizer operators of the oscillator. Nevertheless, when stabilizing infinite energy states, the measurement is Quantum-Non-Demolition, in the sense that the oscillator state is not impacted by the measurement (in the absence of applied feedback). Thus, measurement errors induced by phase-flips can be robustly suppressed by repeating the sequence (TLS preparation -quadrature gate -TLS readout) and performing a majority vote on the TLS readout outcomes before applying a corrective feedback displacement on the oscillator. On the other hand, bit-flips of the TLS during the conditional displacement gate directly propagate as random shifts of the oscillator, yielding logical errors with probability 1 2 . This is the major bottleneck in progressing further towards fault-tolerant quantum computation with GKP qubits directly stabilized with a TLS ancilla. Indeed, while evidence exists that logical errors stemming from any intrinsic noise channel of the oscillator are exponentially suppressed when the noise strength decreases with respect to the repetition rate of correction rounds [? ? ], the rate of propagated errors decreases at best linearly with respect to the probability of a bit-flip of the ancillary TLS during conditional displacement gates4 . Various strategies were proposed [START_REF] Ma | Path-independent quantum gates with noisy ancilla[END_REF], [START_REF] Puri | Bias-preserving gates with stabilized cat qubits[END_REF] to mitigate this advert effect, but unleashing the full potential of GKP qubits will require to suppress propagating errors at a level beyond the reach of state-of-the-art hardware [START_REF] Rosenblum | Fault-tolerant detection of a quantum error[END_REF][START_REF] Grimm | Stabilization and operation of a kerr-cat qubit[END_REF]. This chapter is dedicated to the error-correction of infinite-energy states with a second bosconic mode. We focus on the Steane-type error correction protocol (proposed for GKP states in [START_REF] Daniel Gottesman | Encoding a qubit in an oscillator[END_REF] ) as illustrated in figure 3.1. The scheme was outlined in the seminal paper by Gottesman, Kitaev and Preskill [START_REF] Daniel Gottesman | Encoding a qubit in an oscillator[END_REF], followed by abbreviated schemes that are more adapted to Quantum Optics experiments [START_REF] Glancy | Error analysis for encoding a qubit in an oscillator[END_REF] and protocols based on state-teleportation have been introduced recently [START_REF] Fukui | All-optical long-distance quantum communication with gottesman-kitaev-preskill qubits[END_REF], [START_REF] Mikkel V Larsen | Fault-tolerant continuous-variable measurementbased quantum computation architecture[END_REF]. In this scheme, the stabilizer phase of the target mode -encoding the shift error information -is transferred to the ancilla via a quadrature-quadrature coupling. The ancilla stabilizer information is then measured by a quadrature readout, and the outcome controls a corrective feedback displacement that is applied to the target mode.

| Error correction for infiniteenergy states with a GKP ancilla

Commonly, a discussion on the GKP ancilla state preparation is widely omitted in a variety of these proposals and often assumed to be ideal. In this respect, the challenge of preparing/stabilizing GKP states is transferred to the ancilla oscillator.

For instance, if the ancilla is prepared from vacuum through a series of TLS-based measurements of its stabilizer operators, bit-flips of the TLS induce long shifts of its state, which propagate through the quadrature gate as long shifts to the target oscillator, causing logical errors. In our work, we consider such a TLS-based preparation of the ancilla, but tailor it in order to robustly suppress the error propagation to the target mode.

The ancillary mode b is prepared by repeated TLS-stabilizer measurements (see chapter 2), each followed by a feedback displacement proportional to its outcome (Markovian feedback strategy, see chapter 2 Sec. 1). We refer to the TLS preparation that consist of the sequence {Conditional displacement gate, TLS readout, Proportional feedback} as a single preparation round. We label the preparation round for the respective stabilizer S q b /S p b by R q b and R p b .

The stabilizer operators of the target a and ancilla b mode with the according lattice parameters α and β are given by

S qa = e i 2π α qa = D(i 2π α ) S pa = e -i2αpa = D(2α) (3.1) S q b = e i π β q b = D(i π β ) S p b = e -i2βp b = D(2β)
With this choice, the ancilla oscillator acts as sensor to displacements of the target oscillator and does not necessarily need to encode a whole logical qubit. Indeed, the unit cell for the ancilla is chosen to be 2π, and there exists a single +1 eigenstate of the stabilizer operators S q b and S p b ,

| ⟩ b = n∈Z |q b = n2β⟩ b .
The Hamiltonian of the twomode quadrature-quadrature interaction to transfer the target shift error information to the ancilla is given by a quadrature-quadrature coupling of the form

H = γ r a r b = g 2 (ab † e -iϕ ∆ + a † b e iϕ ∆ + ab e -iϕ Σ + a † b † e iϕ Σ ) (3.2)
where

r i = q i cos(ϕ i ) + p i sin(ϕ i ), ϕ Σ = ϕ a + ϕ b and ϕ ∆ = ϕ a -ϕ b .
The interaction consists of a balanced sum of a Beam-Splitter (H BS = ab † e -iϕ ∆ + a † b e iϕ ∆ ) and a Twomode squeezing interaction (H T M SQ = ab e -iϕ Σ + a † b † e iϕ Σ ). The corresponding unitary

r a → r a r ⊥ a → r ⊥ a + θr b r b → r b r ⊥ b → r ⊥ b + θr a , ( 3.3)
where r ⊥ i denotes a rotation of r i by an angle of π 2 (q ⊥ i = p i and p ⊥ i = -q i ). The interaction parameter θ = γT = √ π β for a square target lattice α = π α = √ π, with T being the interaction time, is chosen in order to maintain the lattice structure of both modes after the gate. Subsequent to the quadrature-quadrature interaction, a shift error of the target mode along r a is encoded in r ⊥ b quadrature and vice versa. More precisely, the stabilizer phase information is transferred from the target to the ancilla mode

S r ⊥ b → S r ⊥ b S ra .
The error syndrome of the target oscillator can therefore be extracted by a readout of the ancilla r ⊥ b quadrature, which can be of destructive nature, such that typically a homodyne measurement is considered. Shift errors of the ancilla in r ⊥ b perturb the readout of the target stabilizer information. In this case, or for imperfect readout, the stabilizer phase information can be faithfully reconstructed by repeated error correction cycles with a fresh ancilla state and by averaging the measurement outcomes. Subsequently, a feedback shift is applied to the target state to bring it back to the code manifold. We call this sequence consisting of {the preparation of the ancilla state, the quadrature gate, ancilla readout and conditional feedback on the target oscillator} an r a -error correction cycle, labelled by C ra . Given the symmetry of the quadrature gate, the back-action on the target state in terms of stabilizer operators is given by S r ⊥ a → S r ⊥ a S 2 r b . Shift errors of the ancilla along the quadrature r b , propagate to the target r ⊥ a quadrature, potentially resulting in flip errors of the logical qubit.

In this chapter, we will further describe how ancilla errors, either stemming from intrinsic noise or from long displacements induced by bit-flips of the the TLS, propagate to the target mode. Focusing on periodic, infinite energy oscillator states (chapter 4 is dedicated to finite-energy states), we derive an ancilla preparation scheme that suppresses these propagating errors. We optimize the parameters of our errorcorrection protocol and analyze its performance in the case where both the target and the ancillary oscillator are affected by quadrature noise. Quadrature noise induces uniform diffusion of the oscillator state in phase-space with a given diffusion constant κ and is commonly considered when analyzing performances of GKP error-correction schemes [START_REF] Daniel Gottesman | Encoding a qubit in an oscillator[END_REF], as it induces random displacements of grid states but preserves their periodicity.

Propagation of GKP-ancilla errors

In this section, the propagation of ancilla shift errors for the error correction gates of the form e iθrar b is discussed in more detail. Ancilla shift errors e iδr b along the r ⊥ b quadrature, commute with the quadrature gate ([e iγT rar b , e iδr b ] = 0) and perturb the subsequent r ⊥ b -quadrature readout. Therefore, these errors propagate indirectly via an incorrect feedback applied to the target mode. These kind of errors can be mitigated by an adapted feedback strength and repetitions of the error-correction cycle. In this sense, they propagate similarly as the phase flip errors of the TLS (discussion in chapter 2, Sec 1). On the other hand, shift errors µ in the r b quadrature before or during the interaction at a time instant 0 < t f < T propagate to the target mode in the following manner,

e iγ(T -t f )rar b e iµr ⊥ b e iγt f rar b = e iγ(T -t f )µra e iµr ⊥ b e iγT rar b . (3.4)
with r i = {q i , p i } and r ⊥ i = {p i , -q i }. This error propagation is the unavoidable back-action of the quadrature gate, which symmetrically maps target r a errors to to the ancilla r ⊥ b quadrature, allowing the target error detection via the ancilla. Indeed, the length of ancilla displacement errors caused by intrinsic noise channels does not depend on β, so that propagated displacement errors scale as 1 β . If these propagated displacement errors are short enough, they can be corrected in a subsequent C r ⊥ a cycle. However, bit-flip errors σ x during the TLS-stabilizer measurement rounds R p b result in random displacements along the ancilla q b quadrature sampled from the interval [-β, β]. They propagate to the target as displacements in the whole interval

[ √ π, √ π] along r a during a C r cycle.
The strategy for the ancilla preparation in this thesis is based on two key observations. First, the measurements of the target stabilizer operators can be performed via the gates e iθraq b , where we exploit the liberty to solely couple to the ancilla via the q b quadrature. Therefore, the ancilla state properties and preparation can be tailored with respect to the q b quadrature, as displacement errors along q b are the only source of directly propagating errors to the target oscillator. In other words, the ancilla can be asymmetrically prepared, with a focus on a very sharp and reliably centered peak distribution in q b -⟨S q b ⟩ close to 1 -at the expense of a less resolved peak distribution in p b . Admittedly, displacement errors of the ancilla state along p b lead to an inaccurate phase-estimation of the target stabilizer value ⟨S ra ⟩ after a single cycle of the Steane error-correction scheme, but as previously noted, this cycle can be repeated and the measurement outcomes averaged to mitigate these kind of errors. Second, in the TLS-based preparation of the ancillary oscillator state, bit-flip errors of the TLS only induce long displacement errors along q b if they occur during a R p b correction round. On the other hand, during a R q b correction round, as far as the q b distribution is concerned, they solely result in an inaccurately extracted information and can similarly mitigated by repetitions of the R q b round. In the next section, we detail the ancilla state-preparation that is tailored with respect to the choice of the oscillatoroscillator interaction.

GKP-ancilla preparation

A core idea of our proposal is to prepare the ancilla state by a large number of R p b correction rounds N p followed by a large number of R q b correction rounds N q , as depicted in figure 3.2. With this ordering, large shift errors in the q b distribution, originating from TLS bit-flip errors during R p b rounds are corrected by the subsequent R q b rounds. In the following, we analyze and numerically simulate the ancilla state evolution under such a sequence of preparation rounds. We make use of the state periodicity in the quadratures q b and p b -preserved through the evolution as detailed below -and of the fact that the modular quadrature operators q b mod 2β and p b mod π β commute to describe the states with periodic, classical probability distributions Q b and P b . These distributions are the diagonal elements of the state density matrix expressed in the Zak basis.

Zak basis

The dynamics of our system is conveniently described in the Zak basis [START_REF] Zak | Finite translations in solid-state physics[END_REF] of the oscillators, which is the basis formed by displaced GKP states within one GKP unit cell of each oscillator GKP lattice. Equivalently, the Zak basis for each oscillator can be seen as the joint eigenbasis of the GKP code stabilizers. Zak states are defined as

|u, v⟩ a = e -iupa+ivqa | + Z⟩ = e i 2 uv n∈Z e i2nvα |n2α + u⟩ qa |u ′ , v ′ ⟩ b = e -iu ′ p b +iv ′ q b | ⟩ = e i 2 u ′ v m∈Z e i2mv ′ β |2mβ + u ′ ⟩ q b (3.5)
where we use the convention

α = √ π, u ∈ [-α, α], v ∈ [-π α , π α ] , u ′ ∈ [-β, β] and v ′ ∈ [-π β , π β ].
We will later use the following properties : where u + w and v + w are to be considered as modular coordinates (respectively modulo 2α or 2β and modulo π α or π β ).

Momentum basis representation |u, v⟩ a = e -i 2 uv n∈Z e -inu π α |n π α + v⟩ pa |u ′ , v ′ ⟩ b = e -i 2 u ′ v ′ m∈Z e -imu ′ π β |m π β + v ′ ⟩ p b . ( 3
We now show that, if the target mode is initialized in the | ± Z⟩ logical basis, the states of both the ancillary and target modes are described by diagonal density matrices in their respective Zak bases throughout the ancilla preparation and Steane-type error correction. Therefore, they can be represented by periodic probability distributions. Moreover, these distributions are separable between the two parameters of each Zak basis:

ρ a (t) = u v Q a (u)P a (v)|u, v⟩⟨u, v| a ρ b (t) = u ′ v ′ Q b (u ′ )P b (v ′ )|u ′ , v ′ ⟩⟨u ′ , v ′ | b (3.8)
We also give evolution rules for these distributions throughout correction rounds and cycles, on which the numerical simulations used in this thesis are based.

State evolution through a preparation round

For simplicity, we focus on the evolution during R q rounds, and the results can be directly adapted to the case of R p rounds (subscripts are dropped for simplified notation). A R q round labeled by j (N p + 1 ≤ j ≤ N p + N q ) starts with a TLS initialization in the +1 eigenstate of its Pauli operator σ x , followed by a conditional displacement gate U CD q = e iζqσz where ζ = π 2β . The TLS is then measured along σ y , and a feedback displacement by ±ϵ j is applied along q depending on the outcome (as shown in detail in chapter 2, Sec. 1). We recall the Kraus operators corresponding to the two possible outcomes are

M -= e -iϵ j p cos(ζq + π 4 ) M + = e +iϵ j p cos(ζq - π 4
).

(3.9)

If no TLS flip occurred during the gate, which happens with probability p N F = 1 -p P F -p BF (where p BF and p P F are the respective bit-and phase-flip probabilities during a conditional displacement gate, that are supposed to be much smaller than 1), the non-normalized conditional probability distributions for the two initial q/p-

probability distributions Q j-1 (u)/P j-1 (v) read Q ±,N F j-1 (u) = p N F 1 2 ± 1 2 sin π β (u ± ϵ j ) Q j-1 (u ± ϵ j ) P ±,N F j-1 (v) = p N F P j-1 (v).
(3.10)

As detailed in chapter 2, phase flips of the TLS during the gate, occurring with probability p P F , lead to a erroneously applied feedback displacement, yielding the non-normalized conditional probability distributions

Q ±,P F j-1 (u) = p P F 1 2 ± 1 2 sin π β (u ∓ ϵ j ) Q j (u ∓ ϵ j ) P ±,P F j-1 (v) = p P F P j-1 (v), (3.11)
while bit-flips of the TLS during the gate, occurring with probability p BF , yield no measurement back-action, but a randomly applied feedback displacement along q b and a long random displacement along p b . The corresponding non-normalized conditional probability distributions read

Q BF j-1 (u) = p BF 2 Q j-1 (u + ϵ j ) + Q j-1 (u -ϵ j ) P P F j-1 (v) = p BF β π . ( 3

.12)

After recombining all conditional probability distributions to model the memoryless feedback strategy, the probability distributions are given by

Q F B j-1 (u) = 1 2 + 1 -2p P F -p BF 2 sin π β (u + ϵ j ) Q j-1 (u + ϵ j ) + 1 2 - 1 -2p P F -p BF 2 sin π β (u -ϵ j ) Q j-1 (u -ϵ j ) P F B j-1 (v) =(1 -p BF )P j-1 (v) + p BF β π . (3.13)
At the end of the round, we apply an effective quadrature noise channel, which convolves the probability distributions with periodic normal distributions G q and G p , re-

spectively defined on [-β, β] and [-π 2β , π 2β ],
both with variance σ 2 = κT round . At the beginning of the following round, we thus get a state of the form (3.8) with probability distributions

Q j (u) = Q F B j-1 * G(u) P j (v) = P F B j-1 * G(v). (3.14)
The evolution of the ancilla state through a R p round is simply obtained by the exchange q ↔ p in the above formulas. Overall, repeated R q and R p rounds preserve the ancilla state periodicity in both quadratures q b and p b . Note that in simulations, we assume the ancilla density matrix in the Zak basis to be the identity at the start of the preparation (uniform Q b and P b distributions), but this hypothesis can be lifted since the state after the preparation depends negligibly on the initial conditions for all considered preparation parameters (see Sec. 3.1).

Effective modelling quadrature noise channel

Quadrature noise at rate κ is modeled by two Lindblad dissipators √ κD[q] and √ κD [p] yielding, over an infinitesimal time-step dt, an evolution of the oscillator density matrix

dρ = dtD[L](ρ) = dt LρL † - 1 2 (L † Lρ + ρL † L) (3.15)
Its effect can equivalently be modeled by the application of stochastic evolution operators

U q dt = e i √ κdWqq U p dt = e i √ κdWpp (3.16)
where dW q and dW p are independent Wiener processes characterized by dW q = dW p = 0 and dW 2 q = dW 2 p = dt

Effective noise channel after a conditional displacement gate

We here consider a conditional displacement gate applied on the q quadrature of the ancillary oscillator reading

U CD q = e iθqσz , ( 3.17)
where θ = π 2β and, for simplicity, we dropped the subscript b to designate the ancillary oscillator quadrature. It is straightforward to adapt the following calculation to the case of a conditional displacement along the p quadrature.

When the gate is applied in finite time T CD and in presence of quadrature noise, we use Trotter decomposition over N = T CD dt steps to write the stochastic evolution over a single trajectory

ŨCD q = N j=1 e i θ N qσz e i √ κdW j
q q e i √ κdW j p p (3.18) where all Wiener processes dW j q , dW j p are independent. Using Baker-Campbell-Hausdorff formula, we reorder this product to put the noise terms in front

ŨCD q = N j=1 e -i √ κdW j p jθ N σz N j=1 e i √ κdW j q q e i √ κdW j p p N j=1 e i θ N qσz (3.19)
The second and last products correspond to a quadrature noise channel applied for a duration T CD after an error-free conditional displacement gate U CD q . Our effective noise model neglects the first term, which rotates the TLS Bloch vector around the σ z axis conditioned on the stochastic trajectory of the oscillator state due to q noise during the gate. Its physical interpretation is clear: random displacements of the oscillator that occur at the beginning of the gate (j → N ) leave an imprint on the TLS phase similarly to displacements having occurred before the gate, while displacements of the oscillator that occur toward the end of the gate (j → 1) impact negligibly the TLS phase. By discarding this information in our effective noise model and applying a quadrature noise channel after the gate, we thus expect to underestimate the accuracy of the re-centering feedback displacement controlled by the TLS readout outcome since this feedback displacement would partially correct for shift errors having occurred during the gate in the case of exact noise modeling. This approximation should negligibly impact the ancillary oscillator preparation and phase-estimation in the limit κT CD ≪ 1.

Effective noise channel after a quadrature gate

We follow a similar reasoning for the quadrature gate

U quad qa = e iθqaq b (3.20)
where θ = √ π/β. Here, for simplicity, we consider q a and q b quadrature noise only-p a where all Wiener processes dW j pa , dW j p b are independent. Using Baker-Cambpbell-Hausdorff formula, we reorder this product to place the noise terms in front

Ũquad q = e iϕ N j=1 e i √ κdW j pa pa e -i √ κdW j p b jθ N qa N j=1 e i √ κdW j p b p b e -i √ κdW j pa jθ N q b N j=1 e i θ N qaq b (3.22)
where ϕ is an irrelevant global phase that can be neglected. We see two new noise terms having appeared form this reordering. First, an extra p a quadrature noise term, correlated to the ancillary mode q b noise during the gate. Since, in our protocol, we do not measure the S q b stabilizer following the gate, it can be modeled as random displacements of the target state along p a , with zero mean value and variance

( √ κθ N j jdW j p b ) 2 → κ θ 2 3
T quad , where the last limit is taken for N → ∞. We account for this term by renormalizing the p a quadrature noise term during the gate following

κ → κ(1 + θ 2
3 ). Note that its effect could be partially mitigated by measuring the S q b stabilizer at the end of the cycle and decoding the information it contains. Second, an extra p b quadrature noise term, correlated to the target mode q a noise during the gate.

In analogy to the conditional displacement gate detailed in the previous section, it is interpreted as partial information on the noise-induced displacements of the target oscillator during the gate, mapped to the ancillary mode stabilizer S p b . We neglect this term in our simplified model, and thereby expect to slightly underestimate the performances of our protocol.

Emergent dynamics entailed by repeated preparation rounds

In absence of noise, the dynamics in these modular coordinates entailed by repeated R b rounds is captured by a classical random walk along r b , whose steps are biased toward r b = 0 mod r 0 (q 0 = 2β and p 0 = π β ) as seen in eq. 3.10. In the limit of short feedback displacements ϵ ≪ r 0 , the evolution of the periodic probability distribution R b of the ancilla r b -coordinate is governed by a Fokker-Planck equation with r b -dependent drift velocity v 0 (r b ) = ϵ T round sin(2π r b r 0 ) and uniform diffusion constant D 0 = ϵ 2 2T round . As far as R b is concerned, the noise model we consider preserves this structure and only slightly renormalizes these parameters in the weak noise limit. Indeed, quadrature noise at rate κ ≪ 1/T round adds a small contribution to the total diffusion constant

D = D 0 + κ 2 .
Bit-flips of the TLS -occurring with small probability p BF during each conditional displacement gate -and phase-flips occurring with small probability p P F during each round -lead to erroneously applied feedback displacements that decrease the drift velocity to v = p N F v 0 where p N F = 1 -p BF -2p P F . We mitigate this advert effect by varying the feedback displacement length ϵ j (with 1 ≤ j ≤ N p for R p rounds, N p + 1 ≤ j ≤ N p + N q for R q rounds), starting with long displacements to evacuate the R b population lying near |r b | = r 0 2 and ending with short displacements of order ϵ min . The feedback value ϵ j interpolates in-between these extreme values with a 1/j law. With this strategy, we observe in numerical simulations that the Q b distribution displays both a minimum-width central peak and tails exponentially suppressed for increasing values of N q (see Fig. 3.3, right bottom). Thereby a robust protection of the target oscillator from shift errors that have propagated through the quadrature gate is ensured. We note that all numerical results presented are obtained by directly computing the evolution of the periodic distributions of the oscillators -encoded as vectors of N Fourier coefficients, with N ∼ 60 -120 (see Sec 4) -and not the evolution of the quantum state of the system, which drastically reduces the simulation time. The only approximation made, is the modeling of noise as an effective quantum channel applied in-between perfect gates, which leads to a negligible underestimation of the error-correction performance.

As the R b distribution is being 'sharpened' by repeated R r rounds, random displacements triggered by bit-flips of the ancilla uniformize the R ⊥ b distribution along the conjugate quadrature and quadrature noise deflates its central peak. In figure 3.3, the former have no impact on the final Q b distribution, which is prepared last and assumed to be uniform before the ancilla preparation. On the other hand, they have a dramatic effect on the P b distribution as N q increases, since the probability 1 -(1 -p BF ) Nq of at least a single bit-flip having occurred during the R q b rounds approaches 1, yielding a uniform P b distribution. Therefore, N q cannot be arbitrarily large for the prepared ancilla state to be a useful resource for Steane-type error-correction cycles. This is all the more true when correcting against intrinsic noise of the target oscillator, which requires to minimize the total preparation time (N p + N q )T round . We turn now to the dynamics of the probability distributions of the target oscillator.

Target mode error-correction in presence of quadrature noise

This ancillary state is now considered as a resource for a Steane-type error correction protocol, that consists of alternating error-correction cycles C ra with r a = q a , p a .

For simplicity we perform the calculation for a q a -error correction cycle C q , but it is completely analogous for the conjugate quadrature p a . Subsequent to the quadrature gate U quad qa = e iθqaq b where θ = √ π β , phase-estimation of the ancilla S p b stabilizer (see next section) is performed, corresponding to a partial measurement in the Zak basis of the ancilla. A feedback displacement f (m) along q a is applied based on the outcome

m ∈ [-π 2β , π 2β ].
The preparation of the ancilla for the next cycle concludes the cycle. We suppose the target state to be of the form in eq. (3.8) when the j th cycle begins. After the ancilla preparation, which also yields a state of the form (3.8) with probability distributions Q b,Np+Nq+1 and P b,Np+Nq+1 , abbreviated to Q b and P b for simplicity, the joint state of the system reads

ρ 0 j-1 = u v u ′ v ′ Q a j-1 (u)P a j-1 (v)Q b (u ′ )P b (v ′ ) |u, v⟩⟨u, v| a |u ′ , v ′ ⟩⟨u ′ , v ′ | b dudvdu ′ dv ′ (3.23)
After the quadrature gate, the state reads

ρ 1 j-1 = Q a j-1 (u)P a j-1 (v)Q b (u ′ )P b (v ′ ) |u, v+θu ′ ⟩⟨u, v+θu ′ | a |u ′ , v ′ +θu⟩⟨u ′ , v ′ +θu| b dudvdu ′ dv ′ , ( 3.24)
where for simplified notation the integral signs have been combined and the sub-

ρ m a j-1 = Q a j-1 (u)P a j-1 (v)Q b (u ′ )P b (m -θu) |u, v + θu ′ ⟩⟨u, v + θu ′ | a dudvdv ′ . (3.25)
After a feedback displacement by f (m) and summing over all outcomes m to model the memoryless feedback strategy, we get

ρ F B a j-1 = Q a j-1 (u)P a j-1 (v)Q b (u ′ )P b (m -θu) |u + f (m), v + θu ′ ⟩⟨u + f (m), v + θu ′ | a dudvdv ′ dm. (3.26)
Given that the probability distributions are periodic functions and that the integrals are defined over their whole domains, we find that this state is of the form in eq. (3.8)

with probability distributions

Q F B a j-1 (u) = m Q a j-1 (u -f (m))P b m -θ(u -f (m)) dm (3.27) P F B a j-1 (v) = u ′ P a j-1 (v -θu ′ )Q b (u ′ ) du ′ . (3.28)
Finally, we apply an effective noise channel accounting for quadrature noise affect-ing the ancilla during the quadrature gate and the following N q + N p re-preparation rounds of the ancilla. This effective channel convolves the probability distribution Q a with a periodic normal distribution G a of variance σ 2 = κ T quad + (N q + N p )T round , and the probability distribution P a with a periodic normal distribution Ga with slightly larger variance to account for the renormalized quadrature noise κ → κ during the quadrature gate ( see section 2.3.2). We thus get the target state at the beginning of the following cycle (of form in eq. (3.8)) with probability distributions

Q a j (u) = Q F B a j-1 (u) * G a (u) P a j (v) = P F B a j-1 (v) * Ga (v). (3.29)
The evolution of the target oscillator state during a C p error-correction cycle is derived through similar calculations, inverting the role of Q a j and P a j . Therefore, if the target is initialized in a state of form eq. (3.8), e.g. when prepared in | + Z⟩, it remains so indefinitely. In order to extract the logical flip rate under a particular set of error-correction parameters, one simply needs to compute the evolution of Q a through successive C q and C p cycles.

After some number of cycles N c , the logical qubit is decoded and its z Bloch sphere coordinate reads

z(N c ) = u Q a Nc (u)Θ(u)du (3.30)
where Θ is a step function with value +1 on [-

√ π 2 , √ π
2 ] and -1 elsewhere. By fitting the decay of z(N c ) with an exponential function, one extracts the logical flip rate κ log . In section 4, we present a more efficient method to extract this same rate.

We note that with the Zak basis we chose, constructed from the logical | + Z⟩ basis states, we cannot directly simulate the decay of other logical Pauli operators. One could do so by considering alternative Zak basis definitions. However, the square GKP code symmetry properties ensure that the three components of the logical Bloch vector decay with respective rates κ z = κ x = κ y / √ 2 = κ log .

Phase-estimation of the ancilla

In the previous section, we considered the phase-estimation of the S p b stabilizer as perfect and instantaneous. Since this measurement can be destructive for the ancillary oscillator state, homodyne detection is typically considered. However, the time the ancillary field takes to leak out of the resonator to be detected is at least a few 1/κ. This is not a viable option for error-correction, which requires that κT cycle << 1. This problem could be partly circumvented by mapping the value of p b to a supplementary, low-Q resonator via a quadrature gate [START_REF] Gao | Programmable interference between two microwave quantum memories[END_REF], [START_REF] Flurin | Superconducting quantum node for entanglement and storage of microwave radiation[END_REF], but we found that, for a quadraturequadrature interaction strength of the same order as for the one between the target and ancillary oscillators, the operation would similarly limit the error-correction cycle duration. Moreover, combined photon collection and homodyne detection efficiencies are in practice limited to η ⩽ 1/2 across all experimental platforms, which would result in a too low phase-estimation accuracy. Alternatively, we consider estimating the phase of S p b through repeated TLS-based measurement rounds. Conveniently, the outcome of the R p rounds preparing the ancilla state for the following cycle can be straightforwardly decoded to reconstruct the phase of S p b at the end of the current cycle, with sufficient accuracy for error-correction.

In order to justify this approach and estimate the phase-estimation accuracy, we first suppose that the ancillary oscillator is in a Zak-diagonal state of the form (3.8)

with a P b probability distribution given by a Dirac-peak in p 0 , whose value is to detect. Over a number N p of R p preparation rounds, this distribution is on average shifted and broadened by the feedback displacements {±ϵ j } 1≤j≤Np applied at the end of each round. Denoting m = {±m j } 1≤j≤Np a particular measurement record and ϕ(m) = Np i=1 m j ϵ j the total applied displacement, the ancilla p-distribution after repreparation reads

P m p 0 (p) = δ(p -p 0 -ϕ(m)). ( 3

.31)

Averaging over all possible measurement outcomes, the ancilla distribution after repreparation reads

P p 0 (p) = m P p 0 (m)δ(p -p 0 -ϕ(m))dm = ϕ Π p 0 (ϕ)δ(p -p 0 -ϕ)dϕ (3.32)
where P p 0 (m) is the probability of the measurement record m and the distribution

Π p 0 (ϕ) = m P p 0 (m)δ(ϕ -ϕ(m))dm becomes smooth when N p is large.
We simply propose to estimate p 0 with p m = -ϕ(m) for a given measurement record m. The accuracy of the S p b phase-estimation is characterized by an error distribution

E p 0 (τ ) where τ = p m -p 0 E p 0 (τ ) = ϕ Π p 0 (ϕ)δ(-ϕ -p 0 + τ )dϕ = P p 0 (τ ) (3.33)
The last equality simply signifies that the phase-estimation accuracy is as good as the ancilla re-preparation.

Crucially, we observe that for all the re-preparation sequences used in this work, the error distribution E p 0 (τ ) depends negligibly on p 0 (see Fig. 3.4). This is simply understood as the long feedback kicks ϵ j applied during the first few R p rounds quickly erase the memory of its prior state. Note that this justifies a posteriori the hypothesis made in the previous section that the ancilla density matrix is the identity prior The feedback displacements applied after each round and the ancilla rectangularity parameter are the ones returned by gradient ascent (see section 4) to minimize the logical error rate for p BF = 2p P F = 0.005 and κT round = 2.10 -5 . We pick these example parameters-in particular the small number of preparation rounds-as the a priori less favourable situation for the prepared ancilla state not to depend on the initial condition p 0 . We observe that the final distributions corresponding to different initial states do not differ significantly, justifying our approach to using phaseestimation.

In our reasoning, we omitted shifts of the ancilla distribution entailed by flips of the TLS and intrinsic quadrature noise of the ancilla. The former only entails shifts of the ancilla Q probability distribution during R p rounds. The effect of the latter is a blurring of the P distribution as it is being measured and re-prepared. We model it by including quadrature noise in the numerical computation of P Np , by which we expect to slightly underestimate the accuracy of the phase-estimation. Indeed, by supposing that P Np is solely broadened by the total feedback displacement distribution Π(ϕ), we overestimate the spread of Π(ϕ) and thus the spread of E = P Np .

Target state evolution through an error correction cycle

To highlight the suppression of TLS error back-propagation to the target mode, we first consider the case of a noiseless ancilla and target oscillator (κ = 0). Since the cycle duration is irrelevant in that limit, we allow the ancilla to be prepared with a large number N p → ∞ of R p rounds. Following the reasoning that has been outlined in the previous section, the ancilla periodic distribution along p b is, at this stage, infinitely narrow for the appropriate choice of p b feedback displacements lim j→Np ϵ j = 0. Subsequent to the following N q rounds, it reads P b (p b ) = p no flip δ(p b ) + (1 -p no flip )β/π, where p no flip = (1-p BF ) Nq is the probability that no bit-flip of the TLS occurred during the R q rounds. Moreover, we assume the S p b phase-estimation succeeding the quadrature gate to be perfect -it can consist of an infinite sequence of non-destructive TLSbased phase-estimation rounds. For a given sequence of q b -feedback displacements {ϵ j } Np+1≤j≤Np+Nq applied after each R q preparation round and feedback-law f governing the length of r a -feedback displacements applied after each error-correction cycle, we simulate the evolution of the target oscillator periodic distribution Q a -compactly encoded in a N -Fourier coefficient vector -over alternating C q and C p cycles.

We showed that the distribution quickly converges from an arbitrary to a metastable state with two peaks centered at q a = 0 and q a = √ π, as expected from a state close to the GKP code manifold. A slow dynamics then comes into play, for which the respective amplitude of the two peaks equilibrate as the logical qubit relaxes to the fully mixed logical state. We extract the convergence rate Γ conv toward the code manifold -in units of T -1 cycle , with T cycle = T quad + (N p + N q )T round -and the logical z-error probability per cycle p log by spectral analysis of a N × N evolution matrix (see Sec. 4).

These values are represented in figure 3.5 as a function of N q , for various values of the TLS bit-flip probabilities per preparation round p BF (the phase-flip probability is set to p P F = p BF /2). For each value N q , the length of the ancilla feedback dis-placements {ϵ j } Np+1≤j≤Np+Nq and the feedback function f are determined by gradient ascent to minimize the logical error probability (see Sec 4.4). When N q → 0, the Q b -distribution is widely spread, so that the back-action of the C p cycles entails long random shifts of the target state along q a , increasing p log . In the opposite limit N q → ∞, P b is a near-uniform distribution as p no flip → 0, resulting in a strong blurring of the error-syndromes extracted from the target oscillator. As a result, the amplitude of the optimal feedback function f drops to 0 (see Sec 4.4), and thereby the convergence rate to the code manifold (see inset in figure 3.5). For N q = ∞ (infinitely narrow Q b distribution) and f = 0, one expects the target state to remain unchanged through both the C q and C p rounds, so that the logical error rate is strictly speaking 0. However, this is not a regime that is considered in our logical error estimate by spectral analysis, which assumes a slow exponential decay of the logical qubit, after a fast convergence toward the code manifold (see Sec 4 for details). For the optimal preparation round number, p log decreases exponentially as p BF decreases. Moreover the corresponding convergence time 1/Γ conv remains of the order a few T cycle , so hat spurious shift errors of the target oscillator are corrected within a few error-correction cycles.

Logical error rate in presence of quadrature noise

By considering quadrature noise on both oscillators with rate κ, the number N p of R p preparation rounds, their respective feedback displacements and the S p b phaseestimation strategy need be optimized to limit the total cycle duration. As in the case of no dissipation, the evolution of Q a is simulated and the logical error rate extracted and the control parameters are optimized by gradient ascent -except for N p and N q which are swept. In figure 3 of R q preparation rounds in each cycle, in absence of intrinsic oscillator noise and for various TLS flip probabilities per round (encoded in color, with p P F = p BF /2.). We allow N p → ∞ and assume perfect S p b phase-estimation in this dissipation-less case. For each value of N q , remaining control parameters are optimized by gradient ascent. The minimum value of p log appears to be exponentially small as p BF decreases. Inset: convergence rate toward the code manifold, decreasing with the probability of a single TLS flip to have occurred over the R q preparation rounds. b) Logical error rate κ log in units of T round as a function of the oscillators quadrature noise rate κ and TLS flip probabilities encoded in color as in (a). Phase-estimation is performed by decoding the measurement outcomes of the R p rounds and its finite accuracy is accounted for in simulation. For a round number N p = N q swept from 20 to 100-N p and N q were varied independently for a few noise values, not leading to significant improvement of performances-we optimize remaining control parameters by gradient ascent, and report the minimum value of κ log as a function of the preparation round number.

Efficient numerical estimate of the logical error rate

Computing the evolution of the ancilla and target states under the form of classical probability distributions Q a , P a , Q b and P b as detailed in the previous sections greatly reduces the cost of numerical simulation compared to a description in terms of density matrices in the Fock basis. Typically, one keeps track of the distributions as two vectors of length N 0 = 1000. In this section, we further reduce simulation costs by representing the probability distributions in Fourier domain, as vectors of length N ∼ 60 -120 Fourier coefficients. The evolution of Q b and P b over each preparation round is performed by the application of a distinct N × N matrix, and the evolution of Q a and P a over each correction cycle is performed by the application of a single N × N matrix. The logical decay rate over a given parameter set is then extracted by spectral analysis of the latter matrix. Moreover, we efficiently compute the gradient of this rate with respect to the continuous parameters of the protocol (length of the feedback displacements {ϵ j } 1≤j≤Np+Nq applied after each R p and R q round, Fourier coefficients

{f k } 1≤k≤N f ≤N of the feedback function f , rectangularity R = π 2 1
β of the ancilla GKP lattice), which greatly facilitates their optimization.

In order to simplify calculations, we consider in the following the re-scaled periodic distributions defined over [-π, π]

Π qa (ϕ) = 1 √ π Q a ( ϕ √ π ) Π q b (ϕ) = β π Q b ( βϕ π ) Π p b (ϕ) = 1 2β P b ( ϕ 2β ) (3.34)
and define the Fourier coefficients of a 2π-periodic distribution g as g (k) = 1 2π π -π g(ϕ)e -ikϕ dϕ. The evolution of the P a probability distribution is not computed as it is equivalent to that of Q a . We also consider the re-scaled feedback shifts e p j = 2ϵ j β for 1 ≤ j ≤ N p , e q j = ϵ j π β for N p + 1 ≤ j ≤ N p + N q , phase-estimation outcomes at the end of each cycle ψ = m2β and the feedback function

F (ψ) = √ πf ( ψ 2β ).

Ancilla preparation

We revisit the evolution described in Sec. 4.1 to write it in a form adapted to the encoding in the Fourier domain. The ancilla is in the identity state before preparation, with Fourier coefficients Π (k)

p b ,0 = Π (k) q b ,0 = δ k /(2π)
, where δ is the Kronecker symbol.

During the j-th R p round, the Π p b distribution evolves after TLS readout and appli-cation of a feedback displacement following Eq. (3.13), which in rescaled coordinates reads

Π F B p b ,j-1 (ϕ) = 1 2 + 1 -p F 2 sin Φ + e j Π p b ,j-1 (ϕ + e j ) + 1 2 - 1 -p F 2 sin ϕ -e j Π p b ,j-1 (ϕ -e j ) (3.35)
where we used the shorthand notation p F = p BF + 2p P F Expanding this expression in powers of e j , we get

Π F B p b ,j-1 (ϕ) ≃ 1 2 nmax n=0 e n j n! ∂ n Π p b ,j-1 (ϕ) ∂ϕ n (1 + (-1) n ) + ∂ n (Π p b ,j-1 (ϕ)sin(ϕ)) ∂ϕ n (1 -(-1) n ) (3.36)
Note that for n max = 2, one recovers a Fokker-Planck equation, with drift velocity e(1 -p F ) sin(ϕ)/T round and diffusion constant e 2 /(2T round ) as described in Sec. 2.2. In Fourier domain, this expression translates to

Π F B (k) p b ,j-1 = n even (ike j ) n n! Π (k) p b ,j-1 + n odd (1 -p F ) (ike j ) n n! 1 2i (Π (k-1) p b ,j-1 -Π (k+1)
p b ,j-1 ).

(3.37)

The distribution is then convolved with a Gaussian kernel modeling the effect of quadrature noise (see eq. (3.14)). In Fourier domain, it reads

Π (k) p b ,j = Π F B p b ,j-1 (k)e -1 2 κpT round k 2 (3.38)
where κ p = 2πκ/R 2 is the rescaled quadrature noise rate. After N p rounds, the phaseestimation error function E is inferred from the distribution Π p b ,Np (see Sec. 4.1). The Π q b ,Np distribution is still uniform at this stage.

Through the sequence of R q rounds, the Π p b evolves due to quadrature noise and random displacements induced by bit-flips of the TLS as

Π (k) p b ,Np+Nq+1 = (1 -p BF tot )Π (k) p b ,Np+1 e -πNqκpk 2 + p BF tot δ k 2π , ( 3.39)
where p BF tot = 1 -(1 -p BF ) Nq is the probability for at least one bit-flip to have occurred. As for the Π q b distribution, it evolves through R q rounds following the same rules as Π p b through R p rounds (eqs. (3.37, 3.38)), albeit with a re-scaled quadrature noise strength κ q = 2πκR 2 for the Gaussian kernel convolution.

Overall, we thus compute the prepared ancilla state under the form of two Ndimensional vectors of Fourier coefficients (-

N -1 2 ≤ k ≤ N +1
2 ), and obtain the phaseestimation error-function under the same form. Moreover, it is straightforward to compute the gradient of each vector with respect to each feedback displacement length e j , as well as with respect to the grid rectangularity parameter R, by taking the derivative of the formulas given above and applying chain rules.

Target oscillator dynamics in Fourier domain

We revisit the target oscillator evolution over a pair of C qa /C pa cycles, labeled j and j +1, described in Sec. 4 to translate it in Fourier domain. The ancilla distributions before the quadrature gate are Π q b ,Np+Nq (abbreviated to Π q b ) and Π p b ,Np+Nq , as computed in the previous section. As detailed in Sec. During the C qa cycle, the initial target oscillator distribution Π qa,j-1 is first evolved with the left expression in (3.27) modelling the quadrature gate followed by phase estimation of S p b whose outcome controls a feedback displacement applied to the target oscillator. In re-scaled coordinates, this evolution reads

Π F B qa,j-1 (ϕ) = π -π Π qa,j-1 ϕ + F (ψ) Π p b ψ -2(ϕ + F (ψ)) dψ (3.40)
We now expand this expression in powers of the re-scaled feedback displacement F (ψ) applied to the target oscillator, and truncate the series at n T (n T = 30 for all simulations performed in this work). We then get

Π F B qa,j-1 (ϕ) ≃ π -π n T n=0 F n (ψ) n! ∂ n ∂ϕ n Π qa,j-1 (ϕ)Π p b ψ -2ϕ dψ = n T n=0 1 n! ∂ n ∂ϕ n D n (ϕ)Π qa,j-1 (ϕ) (3.41)
where we defined the generalized Fokker-Planck coefficient functions

D n D n (ϕ) = π -π F n (ψ)Π p b (ψ -2ϕ) = (F n * Π p b )(2ϕ) (3.42)
(we use that Π p b is even in the last equality). In Fourier domain, this translates to

Π F B (k) qa,j-1 = n T n=0 (ik) n n! N l=-N D (k-l) n Π qa,j-1 (l)
(3.43) Figure 3.6: Eigenvectors of the stochastic evolution matrix M . For κT round = 2/10000, p BF = 1/1000, N p = N q = 60 and all other parameters optimized by gradient ascent, we represent the inverse Fourier transform of the eigenvectors of M with largest eigenvalues λ 0 = 1 and λ 1 = 1 -1.3 × 10 -5 , respectively labeled byΠ 0 and Π 1 . Rescaled to a unit L 1 norm, Π 0 is the probability distribution of the target mode steady-state under error-correction (Π qa,j with j → ∞ in Sec. 4). This state is close to the code manifold, with narrow peaks centered at ϕ = 0 mod π and is decoded as the fully mixed state of the GKP qubit. Π 1 has a null L 1 norm, and is here re-scaled to the same L ∞ norm as Π 0 . Given that λ 1 is close to 1 and that a gap exists with the next largest eigenvalue (λ 2 = 0.55), a general state converges in a few correction cycles to a probability distribution Π 0 + ζΠ 1 , where ζ is an excellent approximation of the z-component of the GKP qubit Bloch vector when the peaks of Π 0 and Π 1 are sufficiently narrow.

through a cycle and through a round, and apply chain rules to obtain the derivative of the evolution matrix M with respect to a given parameter x. Each component of the gradient is then given by ∂λ

1 ∂x = P L 1 • M • P R 1 P L 1 • P R 1 (3.49)
where (•) denotes the matrix product and P L 1 and P R 1 are respectively left and right eigenvectors of M for the eigenvalue λ 1 .

Optimizing continuous parameters by gradient ascent

For a given set of noise values κT round , p BF and p P F (p P F = p BF /2 throughout this work) and a givenamount of preparation round numbers N q and N p , we optimize all remaining correction parameters by gradient ascent to maximize the value of λ 1 . In detail, we consider the vectors

A = { ∂λ 1 ∂e j } {1≤j≤Np} B = { ∂λ 1 ∂e j } {Np+1≤j≤Np+Nq} C = { ∂λ 1 ∂F (k) s } {1≤k≤kmax } D = { ∂λ 1 ∂R } (3.50)
where we defined k) )/(2i). This choice constrains the feedback function F to the odd sector, ensuring that the target probability distribution remains symmetric at all time. We limited the number of free Fourier coefficients of F to N ′ = 10 < N to limit aberrations entailed by Fourier series truncation during the convolution step (3.44). Pushing N ′ to larger values-and increasing N accordingly to avoid aberrations-did not lead to significant improvement in error-correction performances.

F (k) s = (F (k) -F (-
At each step l of the gradient ascent-for a total number of steps L = 100-we update the parameter values in the following manner

{e j } l+1 {1≤j≤Np} = {e j } l {1≤j≤Np} + a A |A| ∆ l {e j } l+1 {Np+1≤j≤Np+Nq} = {e j } l {Np+1≤j≤Np+Nq} + b B |B| ∆ l {F (k) s } l+1 {1≤k≤kmax} = {F (k) s } l {1≤k≤kmax + c C |C| ∆ l {R} l+1 = {R} l + d D |D| ∆ l ,
(3.51) Figure 3.7: Optimization of error-correction parameters by gradient ascent. The continuous control parameters of our protocol are optimized from an initial guess (black dashed line) by gradient ascent to minimize the logical error probability (represented as a function of the ascent step number l in bottom panels). We represent, in rescaled coordinates, the feedback displacements e j applied after each R p preparation round (j ≤ N p , first line), after each R q preparation round (j > N p , second line) and the feedback law F controlling the displacements that are applied to the target oscillator as a function of the phase-estimation outcome ψ (third line) returned by the gradient ascent algorithm for N q = N p = 60 rounds. We vary the TLS flip probability (p BF = 2p P F encoded in color in the left column, κT round = 10 -5 is fixed) and the quadrature noise strength (κT round encoded in color in the right column, p BF = 2p P F = 5.10 -4 is fixed). Compared to the initial guess of e j ∝ 1/j, gradient ascent favours a more rapid decrease to short displacements after a number of rounds of the order of 10.

where ∆ l is a linear function of l decreasing from 1 to 1/20 from l = 1 to l = L, and we use as initial guess parameters

e 0 j = e i e f e f + (e i -e f ) j Np for j ≤ N p e 0 j = e i e f e f + (e i -e f ) j-Np Nq for j > N p F (k) s 0 = -f 1 δ k-1 R 0 = 1.
(3.52)

The initial guess for the ancilla feedback displacements {e j } is a truncated 1/j function with initial value e i of order 2π to suppress the tails of the Π p b and Π q b distributions, and final value e f of the order of (κT round ) 1/2 to minimize the width of the distributions central peak (see Sec. 2.2). The 1/j power law was chosen to maximize the reduction rate of the distributions central peak width, while ensuring that this width reaches 0 when n → ∞, in absence of intrinsic noise of the oscillator. The initial guess for the feedback law f is a simple sine function of amplitude f 1 . In practice, we empirically adjust the parameters e i = π/2, e f = 0.05 and f 1 = 0.2 to minimize the logical qubit decay rate after gradient ascent. Indeed, we observe that the final value of λ 1 -and the correction parameters-returned by the gradient ascent algorithm depends slightly on the initial guess, indicating the existence of multiple local minima of λ 1 (not shown). The rugged aspect of {e j } l+1 {Np+1≤j≤Np+Nq} observed after gradient ascent for some noise values (see Fig. 3.7) tends to confirm this complex structure. More refined gradient ascent techniques may avoid these issues, but were not attempted in this work. We represent the optimal control parameters returned by the gradient ascent algorithm-except for the number N q = N p of ancilla preparation rounds which is swept to obtain the optimal value N min q -for the noise figures corresponding to the data shown in figure 3.5. We represent only the average value of the feedback displacements applied to the ancilla after R p rounds (e j min for j ≤ N p ) and R q rounds (e j min for j > N p )both increasing with quadrature noise as the targeted variance of the distributions central peak increases-and the first Fourier coefficient of the function F controlling the feedback displacement applied to the target oscillator. In chapter 3, the Steane-type error correction scheme for infinite-energy states and quadrature noise affecting both the target and ancilla state has been considered. This simplified model allows a drastic reduction in the cost of the numerical simulations. In this chapter, we adapt the error-correction scheme for the realistic, finite-energy code words and tailor the ancilla preparation and readout according to realistic hardware requirements. First, we derive the evolution for an error-correction cycle, in which the target interacts with the ancilla oscillator via a quadrature gate and is then destructively measured by a homodyne detection. We demonstrate that this idealized error-correction cycle stabilizes the finite-energy code in the target mode and how the target state properties are inherited from the ancilla state. Second, we present an optimized ancilla preparation and readout scheme, based on conditional displacements with a TLS only. that is merged with the asymmetric state-preparation sequence introduced in chapter 3, to guarantee the suppression of non-correctable errors propagating from the ancillary system. Further, we demonstrate that a TLSreadout fares better than an ideal homodyne detection for a very asymmetric limit of the GKP ancilla state with solely two peaks in the q-quadrature, in terms of errorcorrection performance. Finally, we extract by numerical simulations the coherence time of the logical qubit encoded in the target oscillator, when stabilized by repeated error-correction cycles. By optimizing the ancilla preparation, we demonstrate that the coherence time of the logical qubit can be extended beyond the break-even point, for state of the art parameters.

| Error-correction with finite energy states

Ancilla readout by homodyne detection

In this section, the Kraus map corresponding to the quadrature gate exp(i √ π β r a q b ) and subsequent homodyne detection of the ancilla p b quadrature is computed for an ancilla prepared in a pure finite-energy grid state with a Gaussian envelope. The ancilla state |∅⟩ b is given in the continuous envelope representation, which is more convenient for the following analysis since the peak and envelope displacements are decoupled

|∅⟩ b ∝ dq e -q 2 2d 2 q n e - (q-2nβ) 2 2s 2 q |q b = q⟩ ∝ dp e - s 2 q p 2 2 n e - d 2 q 2 (p-n π β ) 2 |p b = p⟩ , ( 4.1)
with g -2 = 1 -

s 2 q d 2 q
. For simplified notation, we consider the corrections of the ancilla (as well for the target) lattice cell to be negligible in this chapter, g = 1, and the final result can be re-scaled at the end if necessary. Subsequent to the quadrature gate, β q a q b ) performs a displacement of the ancilla state along p b conditioned on q a . Here, the ancilla and target states are entangled, with quantum correlations loosely represented by colored separated wavefunctions. In contrast to the periodic infinite energy GKP states, the displaced ancilla copies corresponding to the respective q a peaks, only partially overlap (highlighted by the multicoloured ancilla peaks). This leads to a controlled collapse of the target state envelope. Given the symmetry of exp(i √ π β q a q b ), the ancilla information is analogously encoded in the target distribution.

⟨p b = y| e iθraq b |∅⟩ b |Ψ a ⟩ ∝ dr dp Ψ b (p) ⟨p b = y|p b = p + θr⟩ Ψ a (r) |r a = r⟩ (4.3) ∝ dr Ψ b (y -rθ)Ψ a (r) |r a = r⟩ . ( 4.4) (4.5)

The corresponding Kraus operator is given by

M y ∝ n e - s 2 q 2 (raθ-y) 2 e - d 2 q 2 (raθ-y-n π β ) 2 . ( 4.6)
.

For simplicity, the measurement outcome is re-scaled y θ → y, such that the respective Kraus operators for a q a /p a -error correction cycle (C qa /C pa ) read accordingly

M yq ∝ n e - s 2 q θ 2 2 (qa-yq) 2 e - d 2 q θ 2 2 (qa-nα-yq) 2 (4.7)
M yp ∝ n e - s 2 q θ 2 2 (pa-yp) 2 e - d 2 q θ 2 2 (pa-n π α -yp) 2
.

The important parameters controlling the dynamics are πd 2 q β 2 and πs 2 q β 2 , respectively. In absence of intrinsic oscillator errors, a square ancilla lattice cell (β = √ π) can be considered for simplicity since for different values of β are unchanged if one adapts the quadrature gate parameter θ and uses the re-scaled variances d 2 q θ 2 → d 2 q and s 2 q θ 2 → s 2 q . The full Kraus map describing a single C ra -error correction cycle for an initial target state ρ a , is given by

K[ρ a ] = dy D(γ y )M y ρ a M † y D † (γ y ), (4.8)
with a feedback displacement D(γ y ) conditioned on the outcome y (subscript of y omitted).

Given that the expression of the Kraus operators in equation 4.7 are of similar Gaussian-comb shape as the code state distributions, we present in Appendix A an efficient formalism, describing the dynamics of the target q a /p a -probability distributions under the Kraus map in eq. 4.8. We show that for an initial target r-probability distribution P 0 (r) (subscript a is dropped for simplified notation),

P 0 (r) = n e - (r-ϵ) 2 ∆ 2 r e - (r-n2α-ϕ) 2 σ 2 r := G[∆ 2 r , σ 2 r , 2α, ϵ, ϕ], (4.9)
that is defined by the peak/envelope variance σ 2 r /∆ 2 r , the shift error ϕ/ϵ in r and by the lattice spacing α = √ π. In this representation, the two logical target states are identified for r = q, by the parameter choice {ϵ = 0, ϕ = 0} (|+Z L ⟩) and {ϵ = 0, ϕ = α} (|-Z L ⟩) respectively. For an outcome y of the homodyne detection at the end of a single

C r cycle (M r 2 =: 3 λ=0 G[s -2 q , d -2 q , 2α, y, y +α λ 2 ]
, see Appendix A), the resulting conditional target probability distribution is given by

G[∆ 2 r , σ 2 r , 2α, ϵ, ϕ] 3 λ=0 G[s -2 q , d -2 q , 2α, y, y + α λ 2 ] = 3 λ=0 k f k,λ G[ ∆2 r , σ2 r , 2α, ε, φ - (2k- λ 2 )ασ 2 r σ 2 r +d -2 q ] (4.10) with f k,λ = e - (ϵ-y) 2 ∆ 2 r +s -2 q e - (α(2k- λ 2 )-(ϕ-y)) 2 σ 2 r +d -2 q and ∆2 r = ∆ 2 r s -2 q ∆ 2 r +s -2 q , σ2 r = σ 2 r d -2 q σ 2 r +d -2 q , ε = y∆ 2 r +ϵs -2 q ∆ 2 r +s -2 q , φ = yσ 2 r +ϕd -2 q σ 2 r +d -2 q
. The probability distribution is that of a mixture of grid states, with the same reduced peak/envelope variance σ2 r / ∆2 r and displaced by φ -

(2k- λ 2 )ασ 2 r σ 2 r +d -2
q and ε respectively. For sufficiently narrow target and ancilla p-peaks and assuming that the initial grid state is close to a code state (ϕ mod α << 1) , the grid state labeled by (k = 0, λ = 0) (respectively (k = 0, λ = 2)) in the sum of eq. 4.10 referred to as dominant copy, dominates over the others when ϕ is close to 0 (respectively when ϕ is close to α). Given a homodyne detection outcome y, a proportional feedback γ i is applied to recenter the ϕ and ϵ distribution. The feedback for the ϕ-distribution is given by

γ ϕ = -yσ 2 r σ 2 r +d -2 q
and for the ϵ-distribution by

γ ϵ = -y∆ 2 r ∆ 2 r +s -2 q
. The new position after feedback of the dominating peak is at φ mod α, with φ =

ϕ 1+σ 2 r d 2 q , and ε = ϵ 1+∆ 2 r s 2 q .
Since | φ mod α| < |ϕ mod α| and | ε| < |ϵ| the dominant copy is re-centered.

Note that the spurious copies of the main grid are not re-centered by this feedback displacement, and may lead to logical errors. Thus, when stabilizing finite energy states, logical errors occur in the absence of any intrinsic error channels of the oscillator. This is expected for any error-correction scheme since the grid peaks have a finite width and extend beyond the interval [-α 2 , α 2 ] around their center of mass. These errors occur with vanishingly small probability for very narrow target and ancilla ppeaks σ r , s p → 0.

We now neglect the spurious copies and focus on the dominant features of the target grid state. At the end of the C r cycle, this state has reduced envelope and peak variances σ2 r / ∆2 r compared to the initial state. During the error-correction cycle in the conjugate quadrature C r ⊥ a , the target r a -distribution is convolved with the ancilla q b -distribution, transforming the variances as

σ2 r = σ2 r + s 2 q = σ 2 r 1+σ 2 r d 2 q + s 2 q ∆2 r = ∆2 r + d 2 q = ∆ 2 r 1+∆ 2 r s 2 q + d 2 q . (4.11)
while ϕ and ϵ are not affected. By repeating n C ra and C r ⊥ a cycles, the target distribu-tion converges toward a centered grid state (since φ mod α and ε decrease in absolute value) with variances that quickly converge toward a limit cycle (steady state found from equation 4.11).

σ 2 ±r∞ = s 2 q 2 ( 1 + 4 s 2 q d 2 q ± 1) ∆ 2 ±r∞ = d 2 q 2 ( 1 + 4 s 2 q d 2 q ± 1), (4.12)
hereafter denoted as 'breathing oscillations'. The ± signifies the difference between the minimum and maximum of the target peak and envelope variances of the quadrature r as they expand by s 2 q and d 2 q , respectively due to the convolution during the C r ⊥ a cycle. Given that the probability distributions along q and p breath in phase opposition, the target average photon number (⟨n a ⟩ = 1 2 (⟨q 2 a ⟩ + ⟨p 2 a ⟩ -1)) in steady state is constant and given by

n∞ = 1 2 (∆ 2 +r∞ + ∆ 2 -r∞ -1) = d 2 q 2 1 + 4 s 2 q d 2 q -1 2 . (4.13)
In order to test the predictions of this simplified Gaussian comb model, we exactly compute the evolution of the target oscillator density matrix under repeated C q and C p cycles (using the Kraus map defined in eq. 4.8) and represent the evolution of ⟨q 2 a ⟩ and ⟨n a ⟩ in figure 4.2. In these simulations we vary the variance d 2 q of the ancilla envelope (encoded in color), and adapt the variance

s 2 q = d 2 q n4 ∞ -d 4
q /4 to target a fixed photon number of n∞ = 7.5, according to eq. 4.13. The results match our simplified model predictions (dotted lines) quantitatively.

Further, we estimate a confinement rate onto the breathing steady state with the following reasoning. We consider the case in which, after a large number of cycles that brought the target oscillator state to the breathing steady state, we displace it by ϕ << α, for instance along q. We show that this offset is reduced after each following C q cycle by a factor

1 1+σ 2 r d 2 q
, yielding a confinement rate

Γ conf = ln(1 + σ 2 r d 2 q )/T cycle .

Choice of ancilla parameters

We give now a qualitative discussion on the choice of the ancilla grid state parameters considered in the following sections. The variances σ 2 r± of the target peaks increase with the ancilla peak variance s 2 q (eq 4.11). Thus in absence of photon dissipation, s q should be chosen as small as possible in order to reduce σ r and thus suppress logical errors. However, the mean photon number ⟨n a ⟩ diverges when s q , → 0 as d p → ∞.

When considering photon dissipation, a too low value of s q results in a large sensi- tivity to dissipation. Thus the optimal choice of s q results from a trade-off that we numerically investigate in Sec. 3. The optimization also needs to take into account the duration of the ancilla state preparation, which is more time-consuming for small values of s q , as detailed in Sec 1 in chapter 2.

We now turn to the ancilla envelope variance d 2 q . For a given value of σ r+ , the confinement rate Γ conf onto the breathing steady state increases with the ancilla qenvelope variance d 2 q . Intuitively, a large q-envelope ancilla state (narrow p-peaks) allows to extract more information per cycle, and thus enforces a faster convergence toward the code space. Such a large confinement rate is desirable when considering intrinsic error channels of the target oscillator as errors are corrected faster. On the other hand increasing d q too far results in a large photon number in the target oscillator (eq. 4.13). This results again in an enhanced sensitivity to photon loss. We thus also expect the optimal choice of d q to result from a trade-off between the two effects, which will be analyzed in a future work.

One can partially evade this trade-off by considering, instead of a centered ancilla grid state

| ∅ ⟩ b ∝ dq e -q 2 2d 2 q n e - (q-2nβ) 2 2s 2 q |q b = q⟩ dq→0 -→ dq e -q 2 2s 2 q |q b = q⟩ , ( 4.14)
which reduces to a squeezed state when d q → 0, yielding a null confinement rate, an off-centered state

|∅⟩ b ∝ dq e -q 2 2d 2 q n e - (q-(2n+1)β) 2 2s 2 q |q b = q⟩ sq<<dq<<1 -→ dq e -q 2 2s 2 q (|q b = q -β⟩ + |q b = q + β⟩), (4.15)
which reduces to a squeezed cat state for s q << d q << 1. Intuitively, in the limit s q → 0, the squeezed cat behaves as an ancillary TLS that can encode up to one bit of information about the target mode stabilizers after the quadrature gate (see next section). We thus expect a non-zero confinement rate onto the code manifold as in the case of an actual TLS ancilla (numerical simulations in section 2 of chapter 4). The breathing oscillations of the target oscillator are minimal in this regime, as the target r-envelope expands by √ π during a C r ⊥ cycle.

Besides s q and d q , the remaining parameter that has not been discussed so far, is the ancilla grid state rectangularity, which is encoded in the interaction parameter θ = √ π β . In chapter 3, this parameter has been tuned by gradient-ascent based optimization. These optimized simulations were made by a simplified error model, in which the oscillators were subject to quadrature noise only. Similar efficient simulations of photon loss is the subject of future work. An adaption of θ, effectively re-scales the ancilla q-peak and envelope width as sq = θs q and dq = θd q respectively and all the previous results need to be re-scaled accordingly. In numerical simulations presented in chapter 3 and 4, the rectangularity is optimized for a minimal average photon number in the ancilla oscillator ([36]). As stressed earlier, a homodyne readout detection is limited by the finite detection efficiency and duration. In the following section, we present how the target peak and envelope error information can be detected via the additional TLS, that is in place for the preparation of the ancilla state, to circumvent these limitations.

Ancilla preparation and readout by TLS-based measurements

In the previous section, we demonstrated that the Steane-type error correction scheme with homodyne detection and appropriate feedback, stabilizes the finite-energy code manifold in the target oscillator, whose properties are set by those of the finite-energy ancilla state. In this section, the 2-peak (cat state) ancilla is considered, in order to limit the average photon number in the target and ancilla oscillator.

Peak-shift error detection of the target oscillator

This section focuses on the retrieval of the peak-shift error which has been mapped to the phase of the infinite-energy stabilizer operator of the ancilla S p b . Since the width of the ancilla grid-state p b -envelope, d p = 1 sq , is irrelevant to this discussion, we assume it for now to be infinite (s q → 0) for simplicity. In this limit, we write the non-normalized By applying the phase-swap/BSB p sequence,

|Ψ ′′ ⟩ = U BSBp ( π 8β ) |Ψ ′ ⟩ = 1 √ 2 (|g⟩ e +i √ πra + |e⟩ e -i √ πra ) (|q b = -β⟩ + |q b = +β⟩) |ψ⟩ a (4.

17)

.

with U BSBp (η q ) = e iβp b σ Z e -i2ηqq b σ Y e iβp b σ Z , the modular operator e +i2 √ πra is transferred from the ancilla state to the TLS state that now encodes the stabilizer information (⟨σ x -iσ y ⟩ = ⟨S ra ⟩). By rewriting the expression in terms of the states

|± ϕ R ⟩ = 1 √ 2 (|e⟩ ± e iϕ R |g⟩)
, where ϕ R sets an angle in the σ x -σ y plane of the Bloch sphere, one receives

|Ψ ′′ ⟩ = 1 2 (|+ ϕ R ⟩ cos( √ πr a + ϕ R 2 ) + |-ϕ R ⟩ sin( √ πr a + ϕ R 2 )) |ψ⟩ a (|q b = -β⟩ + |q b = +β⟩) (4.18)
After measuring the TLS in the |± ϕ R ⟩ basis, and tracing out both the ancilla and the TLS, the post-TLS measurement states are

ρ a,+ = cos( √ πr a + ϕ R 2 )ρ a cos( √ πr a + ϕ R 2 ) ρ a,-= sin( √ πr a + ϕ R 2 )ρ a sin( √ πr a + ϕ R 2 ). ( 4

.19)

The corresponding Kraus map is equivalent to a direct single TLS-stabilizer measurement of the target oscillator along the axis ϕ R . As discussed in section 1, for small displacement errors of the target state, a measurement along the σ y -axis corresponds to extracting maximum information. We refer to this sequence (BSB p + TLS measurement along σ y ) as the BSB+ p readout. In comparison with the Kraus map modeling of the homodyne ancilla detection, the BSB+ p readout is formally equivalent to a homodyne detection with post-selection on the outcomes y ± = ± π 4β + 2πl, with l ∈ Z.

In figure 4.3, we compare the preparation of a GKP code state in the target oscillator, by performing a homodyne detection versus the BSB+ p readout of the ancilla.

Here, we solely numerically simulate the evolution of the probability distributions of a periodic target state, using the techniques that have been introduced in chapter 2, Sec. 1. The figure of merit 1 -⟨S qa ⟩ is visualized as a function of the number of error correction cycles. For simplicity, the optimal feedback is chosen to maximize ⟨Re(S qa )⟩ for each possible measurement outcome for both the homodyne and TLSbased detection. β r a q b ), the ancilla oscillator is measured by a homodyne detection or by performing the BSB+ p sequence with a TLS that is subsequently measured along the σ y axis. A feedback displacement is applied that is chosen to maximize Re(S qa ) in both cases.

Given that a measurement of the TLS along σ y yields more information than along any other angle, the BSB+ p readout fares better than homodyne detection. In the remaining part of this chapter, the TLS-based detection will be the preferred choice of readout, since it outperforms homodyne detection even for unit efficiency and does not project the ancilla p b distribution onto a single peak. In the next section, we focus on the envelope correction, that is similarly performed via the TLS coupled to the ancilla oscillator and multiple rounds of TLS-based phase estimation.

Envelope-shift error detection of the target oscillator

We consider now the 2-q b -peak ancilla state for finite squeezing values, which we refer to as (squeezed) cat ancilla. In Sec 1, we showed that after a quadrature gate with the target, a measurement of the ancilla p b quadrature sharpens and re-centers the target peak and envelope position. In this section, we substitute the homodyne detection with a sequence of TLS-based measurements of the ancilla, which simultaneously re-prepares the ancilla for the following error-correction cycle. In chapter 3, we illustrated that a sequence of TLS-S p b measurements followed by feedback displacements, does not only re-prepare the ancilla in a sharply distributed state, but further allows to estimate the value of p b mod β prior to the re-preparation, by decoding the records of the TLS measurements. More generally, by performing repeated TLS-e iηp b measurements, one prepares a sharp distribution of the modular variable p b mod 2π η and accesses its value prior to the sequence. If 2π η is chosen much larger than the A series of TLS-S q b measurements are necessary to suppress these errors that would directly propagate to the target oscillator in the following error correction cycle C r , as demonstrated in chapter 3. In this section, an optimized readout scheme is presented that is merged with the re-preparation of the ancilla state, enabling a faster errorcorrection cycle.

For this optimized readout-scheme, adaptations of the finite-energy stabilizer measurement sequences proposed by [START_REF] Royer | Stabilization of finite-energy gottesman-kitaev-preskill states[END_REF], [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF] are considered, that have been discussed in detail in chapter 2. We demonstrate how the target information encoded in the ancilla oscillator can be recovered by multiple measurement sequences of the finite stabilizer operators S ∆ p b and S ∆ q b , that simultaneously prepare the ancilla state for the next error-correction cycle. We recall the respective SBS r and BSB r sequences

U SBSr = e -i δr 2 r ⊥ b σ Y e iωrr b σ Z e -i ηr 2 r ⊥ b σ Y (4.21) U BSBr = e iωrr b σ Z e -i2ηrr ⊥ b σ Y e iωrr b σ Z ,
where ω q = π 2β and ω p = β. We remind that the last conditional displacement in the U SBSr (U BSBr ) sequence can be replaced by a direct TLS detection along σ y (σ z ) and an active feedback displacement of e ∓i δr 2 r ⊥ b (e ±iωrr b ) conditioned on the TLS detection outcome. We will further use the latter sequences based on active feedback and refer to them as SB+ r and BS+ r . A key difference of the two sequences is that the TLS measurement after the SB+ r sequence yields information on the stabilizer value ⟨S r b ⟩ and for the BS+ r information on ⟨e i2ηrr ⊥ b ⟩. These represent exactly the operator values that are necessary to extract the target peak and envelope shift information as presented in the previous section. The corresponding Kraus operators for the two respective sequences are

J ± SBr (η r ) = 1 √ 2 [e -iωrr b sin( η r 2 r ⊥ b + π 4 ) ± i e iωrr b cos( η r 2 r ⊥ b + π 4 )] (4.22) J ± BSr (η r ) = 1 √ 2 [cos(2η r r ⊥ b ) e ±iωrq b ∓ sin(2η r r ⊥ b ) e ∓iωrr b ]
In the previous section we have seen that in the case of an infinitely squeezed cat ancilla state, the BSB+ p sequence extracts the target stabilizer information and fares better than the homodyne readout. However, since this sequence is based on two conditional-displacements of length β along q b , it is more favourable to use the We consider in this circuit the 2-peak ancilla limit, for which a single TLS-S ∆ p b measurement suffices to readout the target peak shift error. This allows to merge the readout sequence with the asymmetric preparation sequence for a robust ancilla error suppression, resulting in a faster error correction cycle. However, interleaving the ancilla readout and re-preparation is not entirely straightforward, as the control parameters for the optimal feedback strategies that have been considered in chapter 3 are not compatible with the bounds of η q in the BS+ q sequence, for the optimal readout of the p b envelope position (see figure 4.5). In the next section, we present numerical simulation results for this error correction scheme and a detailed discussion on the feedback strategies and control values for the ancilla preparation. β q a q b ). In the case of a squeezed cat ancilla state, the target peak shift error is detected via a single TLS-S ∆ p b measurement round. The envelope shift error is detected by multiple TLS-S ∆ q b measurement rounds. Ancilla TLS-S ∆ r b measurement rounds R r b are depicted by the generic icons defined in figure 4.6 and either represent a SB+ r or BS+ r sequence. The feedback displacements applied to the target and the ancilla mode are conditioned on the TLS detection outcomes. Simultaneously, the readout sequence consitutes the core of the asymmetric re-preparation of the ancilla state, allowing it to be reused for the following p a -error correction cycle C pa .

SB+
4 Error-correction beyond the break-even point with state of the art superconducting circuits A principle sketch of our experimental proposal is visualized in Fig. 4.8, in which a non-linear element mediates the quadrature-quadrature interaction between two har-monic oscillators, illustrated as 3D microwave (post-)cavities. The non linear element can represent a three wave mixing element, for instant a SNAIL device or a four-wave mixing element [START_REF] Arne | Quantum error correction with the gottesmankitaev-preskill code[END_REF]. The ancilla oscillator is further coupled to a TLS -depicted by a transmon and a readout resonator -that is required for the ancilla state preparation.

In this sense, the ancilla mode b, can be interpreted as a buffer between the target mode a and the noisy TLS, preventing the propagation of non-correctable errors. [START_REF] Arne | Quantum error correction with the gottesmankitaev-preskill code[END_REF]) that mediates the quadrature-quadrature interaction. An additional TLS (transmon + readout resonator) is coupled uniquely to the ancilla b mode for the ancilla state preparation and readout. The Wigner function depicted in the a/b mode highlights, the choice of a symmetric/asymmetric GKP state (squeezed cat state).

In this section, we give a step-by-step description of a full error-correction cycle C ra , that presents and justifies the exact choice of of the control parameters in the TLS-stabilizer sequences R r b for the squeezed cat ancilla state, supported by numerical simulations (performed with the QUTIP library [START_REF] Robert Johansson | Qutip: An open-source python framework for the dynamics of open quantum systems[END_REF]). In part, the numerical optimization for the ancilla re-preparation is preformed by Lindblad simulations on the ancilla oscillator + TLS subsystem 1 .

As introduced in the previous section, the target stabilizer phase information ⟨S ra ⟩ is recovered via the SB+ p sequence and a feedback displacement on the target by ε conditioned on the TLS detection outcome is applied (J1 ± SBp = e ±iεra J ± SBp ( π 2β )). The optimal value of η S p = π 2β is found in numerical simulations, as visualized in figure 4.9 (right panel), that depicts the probability for a ⟨σ y ⟩ = ±1 TLS detection outcome for a 1 Full Lindblad simulations of the composite system (2 harmonic oscillators + 1 TLS) is computationally costly and untenable for sufficiently large values of the Fock space truncation of the harmonic oscillators. Therefore, the time-discrete Kraus map representation is used to simulate the dynamics of our error-correction scheme, to avoid including the TLS directly. For this reason, the corresponding Kraus operators have been given for the specific gate sequences throughout this thesis. The joint quantum state of the two harmonic oscillators is represented by a N 2 × N 2 density matrix, with the Fock space truncation N = 120. In all the following numerical simulations, the two SB+ p sequences are fixed and referred to as the S ∆ p b readout. The residual optimization focuses on the control parameters and the number of TLS-S ∆ q (BS+ q ) rounds. The control parameter η q sets the ancilla q b -peak (p b -envelope) variance and can be varied under repeated BS+ q rounds for an optimal readout of the p b -envelope position information. First, we consider the dynamics in absence of TLS and intrinsic oscillator errors, to benchmark the control parameters and number of BS+ q rounds that are necessary for the optimal envelope size of the target state. Conditioned on the measurement records of all the n BS+ q rounds, a feedback displacement e ±i2m

2 ], and are therefore non correctable. The integral over the ancilla q b -probability distribution |Ψ n (q b )| 2 outside this interval gives the logical error probability

A 0 (n) = 1 - β/2 -β/2 |Ψ n (q b )| 2 dq b .
(4.23)

However, shift errors close to the border of the interval are highly likely to result in a non-correctable error within the next error-correction cycle, if no or not an adequate corrective feedback has been applied. By acknowledging repeated measurements and propagating errors within subsequent rounds, we define further the quantities

A k (n) = 1 - β/2 -β/2 (|Ψ n (q b )| 2 ) * k dq b . (4.24)
where the respective moment A k (n) corresponds to the probability that a logical error is induced, subsequent to k successive r a -error detection gates exp(i √ π β r a q b ), in absence of any error correction measure for the accumulating shifts in the r ⊥ a quadrature. By choosing large values of η q , the contribution of directly propagating error contribution expressed by A 0 , is suppressed more rapidly, However at the cost of increasing A k for higher values of k, since the target peaks are convolved by broader ancilla q b -peaks in each cycle C r and accumulate to result in logical errors.

In the case of long TLS lifetimes (left column of Fig. 4.11), A 0 is suppressed with the number of BS+ q rounds, while the next higher A k moments are not impacted substantially, whereas it is the case for shorter TLS lifetimes (right column of Fig.

4.11)

. As stressed throughout this thesis, the additional BS+ q rounds perturb the ancilla p b distribution, that will encode the target error information in the succeeding error-correction cycle, as visualized in the probability distribution of the ancilla state P (p). To note, the BS+ q rounds generate minor peaks at |q b ± 3β⟩ (visible in Fig. 4.11) in the ancilla q b -distribution, which are suppressed by the TLS-S ∆ p b sequence in the succeeding error-correction cycle.

In summary, given that the optimal control values for a fast and efficient readout of the target envelope information are small compared to the ancilla lattice constant β, a large number of BS+ q rounds are necessary to correct errors in the q b -distribution of the ancilla state, preventing a rapid error correction cycle C r . In principle, once the envelope information is recovered, large values of η q can be applied, to decrease A 0 at the cost of increasing higher A k moments. However, for the final BS+ q rounds, η q needs to be decreased to small values as it dictates the peak variance of the q b -distribution. In order to reduce the number of TLS-S ∆ q b rounds, we propose to use the SB+ q sequence with the Kraus operators J ± SBq = e ±i δq 2 p b J ± SBq (η q ), in addition right before the BS+ q rounds. The freedom to choose the value δ q much larger and independently of η q is exploited, allowing large feedback displacements without measuring strongly the ancilla p b -envelope position as it is the case in the BS+ q sequence. It should be strongly emphasized, that this sequence is introduced for a faster correction of errors in the q b -distribution, at the expense of perturbing slightly the envelope information contained in the ancilla p b -distribution. In figure 4.12, we depict the impact of the additional SB+ q rounds (for δ q = η q ) on the average photon number of the ancilla and target mode in steady state, that signifies the information loss of the target envelope position. In numerical simulations, we find the optimum of η q in the SB+ q sequence (for minimal perturbation of the envelope position information) corresponds to the value used in the succeeding BS+ q rounds, that sets the q b -peak variance for the next error correction cycle. In summary, the error correction scheme is modified by applying a first batch of M SB+ q rounds with large feedback displacements, followed by N -M BS+ q rounds, after the TLS-S ∆ p b readout. .12: Analysis on the impact of additional SB+ q rounds on the steady state envelope size. Numerical simulations of the error correction scheme in figure 4.7 for M SB+ q rounds, where the parameters of the sequence are chosen to be δ q = η q = 0.06 followed by a fixed number of 6 BSB q rounds with η q = 0.06. The average photon number in steady state is plotted versus the number of SB+ q rounds M.

The optimal value of the feedback displacement δ q (m) at each round m are found by an optimization algorithm (BFGS-algorithm, scipy library [START_REF] Virtanen | SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python[END_REF]), in which the different A k moments are used as the objective to be minimized. The number of rounds M are chosen such that residual TLS-induced errors during the SB+ p rounds are balanced with the intrinsic oscillator and TLS errors that accumulate during the SB+ q and BS+ q rounds. Moreover, M can not be taken too large, as the measurement back-action slightly perturbs the envelope information and prevents an efficient stabilization of the target envelope (see figure 4.12). The R q rounds are partitioned in M SB+ q and N -M = 6 BS+ q rounds (separated by the vertical dashed line). The top row shows the optimized feedback values with respect to the moments A 0 and A 4 for each round n. For clarity, the control parameter η q = 0.12 during the BS+ q rounds is kept constant. The lower row depicts the behaviour of the first A k moments over the rounds n, for the two different feedback strategies A 0 /A 4 , that are distinguished in dotted/dashed linestyles.

In the end of this chapter, we turn to the numerical simulations of the errorcorrection circuit in presence of TLS errors (σ x , σ z ) and photon loss. Since a full sweep over all free parameters is numerically exhaustive, we fix the number of N R q b rounds and use the optimal values for the ancilla preparation optimized with regard to a respective A k as an input. For illustration, we give an example in figure 4.13 for a fixed number of M = 6/10 SB+ q rounds with varying δ q (m) and N -M = 6 BS+ q rounds with fixed value of η q = 0.12, to show the optimal feedback values and the suppression of the first A k values. The first M rounds with large feedback displacements allow a rapid correction of large shift errors along q b (reducing A 0 ) and the successive BS+ q rounds extract the envelope information and simultaneously further correct the ancilla peaks in q b . More qualitatively, the SB+ q rounds suppress A 0 faster at the cost of increasing the higher A k moments temporarily. The successive BS+ q rounds with small η q values reduce the latter contributions, to minimize the propagation of errors.

In the end, the strength η q in the BS+ q sequence, that is kept fixed for clarity in these curves, can also be decreasingly varied (in a small interval, for the optimal extraction of the envelope information) to improve the preparation of ancilla-q b distribution even further.

In figure 4.14, we depict the lifetime of the logical qubit, as a function of different feedback strategies that have been individually optimized on the different A k values.

An approximate error model is used to include the impact of TLS errors during the R b rounds, with which we update the density matrix of the composite system ρ a,b .

Considering a R b round on the ancilla system only (for instance for state preparation), for a single TLS error that happens with probability p l , the ancilla state results in We demonstrate an substantial increase with respect to the bare lifetime of the harmonic oscillators and of the TLS for this set of values. In this parameter regime the feedback strategy for high values of k in A k is optimal, since the large feedback The explicit values of the feedback displacements for M = 10 SB+ q rounds δ q (m) and for the L = N -M = 6 SB+ q rounds η q (l) are plotted on the left. The dashed vertical line separates the two batches. (Right) Extracted logical qubit lifetime is plotted versus the different feedback strategies that have been optimized on the respective A k values.

strengths for k being small are too large for that given round number M to reinitialize the narrow ancilla q b -peak distribution.

| Conclusion

In this thesis, we addressed the issue of error propagation in GKP error-correction schemes and presented a hardware-efficient proposal to robustly suppress logical errors that are induced by the noise of the ancillary system. To this date, experiments that demonstrated state-preparation and correction were based on a two-level-system ancilla [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF], [START_REF] Flühmann | Encoding a qubit in a trapped-ion mechanical oscillator[END_REF], [START_REF] Campagne-Ibarcq | Quantum error correction of a qubit encoded in grid states of an oscillator[END_REF], [START_REF] Vv Sivak | Real-time quantum error correction beyond break-even[END_REF]. We have extensively reviewed and presented the strengths and roadblocks of this approach in chapter 2. In particular, bit-flips of the TLS during the instrumental conditional displacement gates entail uncorrectable errors. This problem can be circumvented by error-correction with a bosonic ancilla prepared in a GKP-state. However, in previously proposed schemes [START_REF] Blayney W Walshe | Continuous-variable gate teleportation and bosonic-code error correction[END_REF], [START_REF] Noh | Fault-tolerant bosonic quantum error correction with the surface-gottesman-kitaev-preskill code[END_REF], the ancilla preparation has been mostly omitted. In this thesis, we showed that by exploiting the liberty to couple to a single quadrature of the ancilla in the Steane error-correction circuit and tailoring the ancilla preparation accordingly, the logical errors induced by noise propagation from the TLS can be robustly suppressed. In chapter 3, we demonstrated that for infinite energy states both TLS errors and quadrature noise on the target and ancillary oscillators, the logical qubit coherence time can be extended by more than an order of magnitude beyond that of the embedding hardware for values within reach of state-of-the-art superconducting circuit experiments.

In chapter 4, we adapted this protocol to finite energy states. In that case, the finite squeezing of the ancilla will contribute to uncorrectable errors in the target state.

However, we focused on the limit of a very asymmetric ancilla state, which coincides with a squeezed cat state. We demonstrated that a readout of the target stabilizer information encoded in the ancilla via a TLS outperforms a homodyne readout, even for unit efficiency. Further we presented an optimized readout sequence of the target error-syndromes that is merged with the asymmetric ancilla preparation. This allows a significant reduction of the error-correction cycle duration which is crucial to efficiently correct errors induced by intrinsic noise of the target oscillator. For hardware parameters that are realistic in the domain of superconducting circuits, we demonstrate that the logical qubit lifetime surpasses substantially the break-even point. and ω ϕ = σ 2 γσ σ 2 +s 2 .

Therefore, the probability distributions in ϕ and ϵ evolve as

P (ϕ) = dϕ 0 P (ϕ 0 ) k e -1 σ2 (ϕ-η ϕ ϕ 0 -2kαω ϕ ) 2
(A.12)

P (ϵ) = dϵ 0 P (ϵ 0 )e -1 ∆2 (ϵ-ηϵϵ 0 ) 2
The error correction cycle in p, realized through e iθrraq b , represents for the qprobability distribution a convolution with the q-distribution of the ancilla. Solely, the variances of both the peaks and envelope are updated accordingly .13)

∆2 = ∆ 2 + d 2 2 σ2 = σ 2 + s 2 2 (A

1 √ 2 ( 1 √ 2 (

 1212 ijk is the Levi-Civita symbol. The common convention, in which |0⟩ / |1⟩ are chosen as the eigenstates |±z⟩ with eigenvalue ±1 of the σ z operator, is used. The eigenstates of the remaining Pauli operators are |±x⟩ = |-z⟩ ± |+z⟩ and |±y⟩ = |-z⟩ ± i |+z⟩. The density operator can be similarly expressed in terms of the Pauli operators by ρ = 1

. 6 ). 7 )

 67 where B = k b k |k b ⟩ ⟨k b | has been expressed in its diagonal form with eigenstates |k b ⟩. The composite state after the interaction for a time T readsU int (T ) (|ψ a ⟩ ⊗ |θ b ⟩) = k (e iχT b k A |ψ a ⟩ ⊗ |k b ⟩) = k (M k |ψ a ⟩ ⊗ |k b ⟩).(1Upon a measurement of the ancilla system in one of the eigenstates |k b ⟩ and tracing it out, the density operator of the post-measurement state of the initial target state ρ a = |ψ a ⟩ ⟨ψ a |, is given by

Figure 1 . 1 :

 11 Figure 1.1: A sketch of a quantum harmonic oscillator, indicating the ladder operators a and a † for the evenly spaced energy levels.

1 √

 1 where δ(y) is the Dirac delta distribution. The basis states are related by the Fourier transform |q⟩ = 2π dp e -iqp |p⟩ . From the commutation relations [a, a † ] = 1, [H, a † ] = ℏωa † and [H, a] = -ℏωa, one immediately deduces that the eigenstates form an evenly spaced ladder by the transition energy ℏω. The eigenstates of the Hamiltonian H |n⟩ = E n |n⟩ are commonly referred to

. 18 )

 18 solely if the geometric phase φ = -i 2 (γβ * -γ * β), accumulated along the closed phase-space trajectory entailed by D(γ)D(β)D(-γ)D(-β) amounts to multiples of 2π. Another family of Gaussian states which posses a smaller variance in one quadrature at the expense of the other, are called squeezed states. They are eigenstates of the transformed mode operator

Figure 1 . 2 :

 12 Figure 1.2: The q/p wavefunction representation (Ψ(r) in figures represent the real part of the wavefunction, the imaginary part is zero) for the GKP code states |±Z L ⟩ (left), |±X L ⟩ (right)

. 30 )

 30 where the Baker-Campbell-Hausdorff formula (e X e Y = e Y e X e [X,Y] , if X and Y commute with [X, Y]) and the invariant action of the stabilizer on |Ψ L ⟩ has been used. A measurement of the phase θ = 2π α ϵ ∈ [-π, π] unambiguously reveals the error ϵ if |ϵ| < π

Figure 1 . 3 :

 13 Figure 1.3: Sketch of wave function representation of logical GKP code states |±Z L ⟩.A shift error ϵ along the q-quadrature can be detected by measurements of the stabilizer operator S q of the code. Displacements that exceed α 2 , highlighted by the yellow region, lead to a wrong decoding.

Figure 1 . 4 :

 14 Figure 1.4: Wave function representation (Top: Ψ(q), Down: Ψ(p)) of the Binomial GKP states (eq. 2.18) as a function of the number of q peaks n. (Left) Wavefunction representation of |A⟩ 2n+1 states and (Right) |C⟩ 2n states .

Figure 2 . 1 : 1 √ 2 (

 2112 Figure 2.1: Error syndrome detection circuit: Measurement sequence of the stabilizer operators S q /S p realized using the gates e i π α qσz /e iαpσz and a Ramsey type-readout of the ancillary TLS. The ancillary TLS initialized in |g⟩, is prepared in the state |+⟩ = 1 √ 2 (|g⟩ + |e⟩) via a first

π 2 -

 2 pulse (exp(i π 4 σ y )

π 2 1 √ 2 (Figure 2 . 2 :

 21222 Figure 2.2: Peak shift error detection via the TLS. Sketch of the composite state: q-probability distribution distinct in colour) and representation of the TLS in the Bloch sphere σ x -σ y plane.(a): Preceding the interaction, the GKP state is displaced by a shift error δ along q and the TLS is initialized in |+⟩. (b) During the interaction (e -iγT qσz ), the TLS Bloch vector precesses in the σ x -σ y plane conditioned on the q-value of the GKP state (indicated by a coloured Bloch vector associated to the respective peak). The interaction strength is chosen such that the corresponding Bloch vectors refocus to the same angle encoding solely the value of δ of the modular coordinate without revealing more information on q.

Figure 2 . 5 :

 25 Figure 2.5: State-envelope correction: (a) Sketch of initial finite-energy GKP state (for simplicity only 3 peaks) and TLS prepared in |+x⟩ represented in the σ x -σ y plane of the Bloch sphere. (b)The unitary e iηpσz rotates the Bloch vector in the σ x -σ y plane conditioned on the p value of the GKP state (shaded regions are associated to the spread of Bloch vectors corresponding to the respective coloured peak). (c) Bloch vector copies, corresponding to the orange peak have a higher probability to lead to an outcome |+y⟩ and therefore the measurement-back-action leads to a suppression of the population of the green peak. (Intuition highlighted by bright and dark half-discs) (d) Similar argument for a TLS outcome |-y⟩, for which the population of the orange peak is suppressed. Corrective feedback displacements by a multiple of ± π α alpha along p (respectively ±α along q) have to be applied to recenter the state envelope. These displacements by half the GKP lattice period induce deterministic flips of the logical qubit.

Figure 2 . 6 :

 26 Figure 2.6: Measurement sequence for finite-energy stabilizer operators with TLS measurement and manual feedback is equivalent to a coherent feedback and reset of the qubit.

Figure 2 . 7 :

 27 Figure 2.7: We simulate the evolution of the oscillator state under repeated SBS correction rounds (see eq. 4.21) along the q and p quadrature, and extract the logical qubit lifetime T z , plotted against the control parameter η = ϵ for several oscillator single-photon lifetimes T cav (encoded in color). The optimal value of ϵ depends on T cav .

Figure 2 . 8 :

 28 Figure2.8: Equivalence of the finite-energy GKP stabilizer sequence U BSBp and a phase-swap sequence between an infinitely squeezed cat state and a TLS. Initially the TLS and the infinitely squeezed cat state with a phase µ = e iϕ are in a separable product state. Schematic of wave function representation Ψ(q) depicts the TLS state corresponding to the respective q-peak position for each step of the BSB sequence. For a value of ϵ = π 4α in the conditional displacement gate e -iϵqσ Y , a conditional ∓π pulse is performed that results in a destructive interference of populations at the positions |q = ±3α⟩ when the last gate e iαpσ Z is applied. At the end of the sequence a perfect phase swap is realized for infinitely squeezed peaks. In the case of finitely squeezed peaks the sequence generates minor populations as well at the positions |q = ±3α⟩.

Figure 3 . 1 :

 31 Figure 3.1: (a) Measurement circuit of the Steane-type type error correction sequence. Subsequent to the quadrature gate e iθqaq b , the target error information is detected by a measurement of the S r ⊥ b

Figure 3 . 2 :

 32 Figure 3.2: Schematic displaying the circuit for the asymmetric ancilla state preparation by a large number of R p b correction rounds N p followed by a large number of R q b correction rounds N q .

. 6 )

 6 Displacements (for Zak states of either mode) e -iwp |u, v⟩ = e -i 2 wv |u + w, v⟩ e +iwq |u, v⟩ = e i 2 wu |u, v + w⟩.(3.7)

and p b

  noise terms commute trivially through the gate-and decompose the noisy gate over N =

Figure 3 . 3 :

 33 Figure 3.3: Periodic probability distributions of a square ancilla state (β =

  script dropped. The phase-estimation of the ancilla S p b stabilizer yielding an outcome m, is modeled by the application of the Kraus operator M m = du ′ |u ′ , m⟩⟨u ′ , m| b . After tracing out the ancilla, the un-normalized target oscillator density matrix conditioned on the outcome m reads

  to re-preparation: any initial state would yield the same prepared state. As for the finite-accuracy of the S p b phase-estimation this method yields, it can be modeled by an ideal phase-estimation preceded by a convolution of the ancilla P b probability distribution with the error function E = P Np , where P Np is the distribution describing the ancilla state prepared by a number N p of R p rounds as detailed in Sec. 4.1 (used that P Np is an even distribution).

Figure 3 . 4 :

 34 Figure3.4: Phase-estimation of the ancilla. We compute the P p 0 distribution of the ancilla (plain lines) prepared by a number N p = 20 of R p rounds from a narrow Gaussian distribution (standard deviation σ ∼ 0.1) centered at p 0 (dashed lines, p 0 encoded in color). The feedback displacements applied after each round and the ancilla rectangularity parameter are the ones returned by gradient ascent (see section 4) to minimize the logical error rate for p BF = 2p P F = 0.005 and κT round = 2.10 -5 . We pick these example parameters-in particular the small number of preparation rounds-as the a priori less favourable situation for the prepared ancilla state not to depend on the initial condition p 0 . We observe that the final distributions corresponding to different initial states do not differ significantly, justifying our approach to using phaseestimation.

Figure 3 . 5 :

 35 Figure3.5: a) Logical error probability p log per correction cycle as a function of the number N q of R q preparation rounds in each cycle, in absence of intrinsic oscillator noise and for various TLS flip probabilities per round (encoded in color, with p P F = p BF /2.). We allow N p → ∞ and assume perfect S p b phase-estimation in this dissipation-less case. For each value of N q , remaining control parameters are optimized by gradient ascent. The minimum value of p log appears to be exponentially small as p BF decreases. Inset: convergence rate toward the code manifold, decreasing with the probability of a single TLS flip to have occurred over the R q preparation rounds. b) Logical error rate κ log in units of T round as a function of the oscillators quadrature noise rate κ and TLS flip probabilities encoded in color as in (a). Phase-estimation is performed by decoding the measurement outcomes of the R p rounds and its finite accuracy is accounted for in simulation. For a round number N p = N q swept from 20 to 100-N p and N q were varied independently for a few noise values, not leading to significant improvement of performances-we optimize remaining control parameters by gradient ascent, and report the minimum value of κ log as a function of the preparation round number.

  3.1, we model the inaccuracy of the pb detection by convolving Π p b ,Np+Nq with an error distribution E = Π p b ,Np -which is a simple vector multiplication in Fourier domain-and denote the resulting distribution by Π p b .

Figure 3 . 8 :

 38 Figure 3.8: Supplemental data to figure 3.5. We represent the optimal control parameters returned by the gradient ascent algorithm-except for the number N q = N p of ancilla preparation rounds which is swept to obtain the optimal value N min

Contents 1

 1 Ancilla readout by homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 1.1 Choice of ancilla parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2 Ancilla preparation and readout by TLS-based measurements . . . . . . . . . . . . . . . . . . 69 2.1 Peak-shift error detection of the target oscillator . . . . . . . . . . . . . . . . . . . . . 69 2.2 Envelope-shift error detection of the target oscillator . . . . . . . . . . . . . . . . . . . 71 3 Optimized readout scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4 Error-correction beyond the break-even point with state of the art superconducting circuits . . 77 4.1 Suppression of TLS-induced errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

  the composite state for an initial ancilla state |∅⟩ b = dp Ψ b (p) |p b = p⟩ and a pure initial target state |Ψ a ⟩ = dr Ψ a (r) |r a = r⟩ (r a = q a or r a = p a ) is given by e iθraq b |∅⟩ b |Ψ a ⟩ ∝ dr dp e iθrq b Ψ b (p) |p b = p⟩ Ψ a (r) |r a = r⟩ (4.2) ∝ dr dp Ψ b (p) |p b = p + θr⟩ Ψ a (r) |r a = r⟩ , with θ = √ π β . The readout of the ancilla p b -distribution, performed by an ideal homodyne detection with outcome y, is modelled by the projection on a p b eigenstate |p b = y⟩,

Figure 4 . 1 :

 41 Figure 4.1: Sketch of target and ancilla wavefunction, depicted as infinitely narrow peaks for simplicity. For clarity, the target peaks are coloured individually. (Top) Initial target state q a -distribution with a peak shift error ε and displaced envelope. The ancilla peaks and envelope are assumed to be perfectly centered in p b . (Bottom) The quadrature-quadrature interaction exp(i √ π

Figure 4 . 2 :

 42 Figure 4.2: Numerical Fock-simulation on the target state dynamics under the Kraus map in eq. 4.8. The expectation values ⟨q 2 a ⟩ and ⟨n a ⟩ are plotted over multiple error correction cycles for different values of d q = 1 0.25 , 1 0.30 , 1 0.35 , 1 0.40 for a target average photon number of n∞ = 7.5.

raq b [ 1 √ 2 ( 1 √ 2 (

 1212 ancilla state as |∅⟩ b = (|q b = -β⟩ + |q b = β⟩) ∝ dp cos(βp) |p b = p⟩. The Kraus map on the target density matrix ρ a for an ideal homodyne detection, directly found from equation 4.8, is given by K[ρ a ] ∝ π β π β dy D(γ y ) cos( √ πr a -βy) ρ a cos( √ πr a -βy) D † (γ y ),for an outcome y and a proportional feedback displacement by D(γ y ). This is formally equivalent to a single round of phase estimation with an ancilla TLS being measured along an axis defined by the stochastic angle ϕ = -βy mod 2π in the σ x -σ y plane of the Bloch sphere (see chapter 2, Sec. 1). Therefore, at most one bit of information is extracted about the target stabilizer value ⟨S ra ⟩.Instead of a homodyne detection, we consider now the phase-swap/BSB sequence between an infinitely-squeezed cat state and a TLS, that has been presented in chapter 2, Sec 2.8. We consider the target oscillator to be in a pure state |ψ⟩ a for simplicity (the result can be easily generalized to a general state ρ a ). The composite state |Ψ⟩ of the two oscillators and the TLS after the quadrature gate e |g⟩ + |e⟩) (|q b = -β⟩ + |q b = +β⟩) |ψ⟩ a ] (4.16) = |g⟩ + |e⟩)(|q b = -β⟩ e -i √ πra + |q b = +β⟩ e +i √ πra ) |ψ⟩ a .

Figure 4 . 3 :

 43 Figure 4.3: Target state preparation with homodyne versus TLS-based readout of an ancilla grid state with 2 peaks in the q b -distribution. We consider the evolution of an infinite-energy, periodic target state (similar numerical simulation techniques as in chapter 2, Sec. 1), in absence of TLS and oscillator errors. The target oscillator, initially in vacuum, is prepared in a grid state by multiple correction cycles. After the quadrature gate exp(i √ π

Figure 4 . 4 :

 44 Figure 4.4: Schematic of target envelope correction mechanism, (similar to schematic in Fig. 2.5). For simplicity the target peaks are assumed to have much larger squeezing than the ancilla peaks. (a) Composite-state-representation before the interaction. Target q a -state distribution with a shift error and ancilla p b -state distribution perfectly centered. The TLS is initially prepared in the state |+x⟩ (b) Subsequent to the entangling gate e i √ πβ qaq b , from the perspective of the target state, the ancilla state is displaced by along the p b quadrature conditioned on the q a value. By applying a conditional displacement gate e iηp b σz between the ancilla and the TLs, a Bloch vector (copy) rotates in the σ x -σ z plane conditioned on the value of p b (correlated with the q a value of the target oscillator). (c),(d) A projective measurement of the TLS, |+z⟩ / |-z⟩ collapses (highlighted by the shaded half-discs) the ancilla distribution -and consequently the target distribution.

  cycle. In figure 4.5, we plot this quantity, averaged over all possible records, as a function of the number of TLS-e i2ηp b rounds. A few rounds, 4 -10, suffice to successfully dis-entangle the two modes and retrieve the information of the target envelope encoded in the the ancilla p b distribution .

Figure 4 . 5 :

 45 Figure 4.5: After performing of the entangling gate e i √ π β raq b , the target information encoded in the squeezed cat ancilla state is extracted by a single BSB+ p measurement and multiple TLSe i2ηp b measurements. The residual entanglement between the target and the ancilla is estimated by the purity Tr(Tr b (ρ ab ) 2 ) of the density matrix ρ ab after tracing out the ancilla mode b. The purity Tr(Tr b (ρ ab ) 2 ) is plotted versus the number of e i2ηp b rounds for different values of η = 0.025, 0.05, 0, 075, 0.10, 0.125.

  p sequence in presence of TLS errors, as it entails only a single conditional displacement by β. Therefore, for the target peak shift detection, the SB+ p sequence (e iβp b σ Z e -i ηp 2 q b σ Y ) is chosen. The conditional displacement e -i ηp 2 q b σ Y realizes a partial measurement of the q b envelope and e iβp b σ Z serves as a coherent feedback. The subsequent TLS detection along σ y accesses the stabilizer value ⟨Im(S p b )⟩. In this manner, both the ancilla and target shift error can be extracted and corrected without expanding the ancilla envelope along q b . For the detection of the target envelope position, we propose to perform multiple rounds of the BS+ q sequence (e -i2ηqp b σ Y e i π 2β q b σ Z ). A TLS-measurement along σ z contains information on p b mod π ηq , which allows to recover the target mode envelope-shift error syndrome as detailed in the previous section. In this manner, the target peak and envelope shift information encoded in the p b quadrature, is probed via TLS-S ∆ r b (SB+ r -and BSB+ r -type sequences) measurements of the ancilla oscillator, as illustrated in figure 4.6. A detailed discussion on the choice of the control parameter η r and the feedback strengths for a specific setting, is presented in the last part of this chapter.

Figure 4 . 6 :

 46 Figure 4.6: Measurement circuits of the SB+ p sequence (left) and the BS+ q sequence (right) to extract the ancilla/target peak and envelope information. Based on the TLS detection records, feedback displacements on both the ancilla and the target oscillator are applied.

Figure 4 . 7 :

 47 Figure 4.7: Circuit representation of a single q a -error-correction cycle C qa . The error syndromes of the target oscillator a are encoded in the ancilla oscillator b via the quadraturegate exp(i √ π

Figure 4 . 8 :

 48 Figure 4.8: Schematic of a principle setup in the framework of super-conducting circuits: Two 3D microwave post-cavities hosting the target (a) and ancilla (b) mode are coupled by a non-linear element (NL, possibly a pumped SNAIL device[START_REF] Arne | Quantum error correction with the gottesmankitaev-preskill code[END_REF]) that mediates the quadrature-quadrature interaction. An additional TLS (transmon + readout resonator) is coupled uniquely to the ancilla b mode for the ancilla state preparation and readout. The Wigner function depicted in the a/b mode highlights, the choice of a symmetric/asymmetric GKP state (squeezed cat state).

  squeezed cat ancilla state displaced by D(i π 4 ). Furthermore, η S p agrees with the value for which the BSB p sequence realizes a phase swap between an infinitely squeezed cat state and a TLS (see chapter 2, Sec 2.8). A single SB+ p round gives a close to unity readout of the ancilla phase information. However, an SB+ p round entails a conditional displacement along the ancilla q b -quadrature by β, that maps the 2-peak ancilla state (support at {|q b = -β⟩ , |q b = β⟩}) to a three-peak ancilla state (support at {|q b = -2β⟩ , |q b = 0⟩ , |q b = +2β⟩}) in a deterministic manner.

Figure 4 . 9 :

 49 Figure 4.9: Readout of the ancilla phase information ⟨S p b ⟩ for a squeezed cat state with β = 2 √ π, that is prepared by the SBS r sequence. (Left) Probability to detect the TLS in ⟨σ y ⟩ = ±1 (corresponding to the Kraus operators J1 ± SPp) is visualized versus a displacement D(iε) along p b for a squeezing of the ancilla q b -peaks that is set by η q = 0.02 in the state-preparation. The optimal value of η S p = π 2β is found by a sweep of η S p (Right) for the ancilla state displaced by D(i π 4 ) and for different values of η q in the state-preparation.

  √ πr ⊥a /e ±ik π β q b by a multiple (m, k) of the stabilizer periods, on both the target and the ancilla oscillator is applied. In figure4.10,In order to quantify the propagating errors, regarding the expectation values of the stabilizer operators, is not sufficient. Any single scalar quantity can not assess a 'proper' ancilla state in this specific setting. We recall that for a single quadrature gate e i √ π β raq b , shift errors along q b beyond the modular interval [-β/2, β/2] directly translate in displacements of the target state exceeding [-

Figure 4 . 11 :

 411 Figure 4.11: Analysis of TLS-induced errors during the ancilla preparation R r b rounds.The q/pancilla probability distributions P (q)/P (p) are illustrated for an initial ancilla state, prepared in absence of TLS errors (orange). The family of curves show the ancilla distributions after n = 0, 10, 20, 30 BS+ q rounds subsequent to the TLS-S ∆ p b readout. The suppression of the according first A k moments as a function of the round number n are depicted below.

Figure 4

 4 Figure 4.12: Analysis on the impact of additional SB+ q rounds on the steady state envelope size. Numerical simulations of the error correction scheme in figure4.7 for M SB+ q rounds, where the parameters of the sequence are chosen to be δ q = η q = 0.06 followed by a fixed number of 6 BSB q rounds with η q = 0.06. The average photon number in steady state is plotted versus the number of SB+ q rounds M.

Figure 4 . 13 :

 413 Figure 4.13: Numerical simulations on the optimized ancilla error suppression for a fixed number of N = 12/16 (left/right column) R q rounds for a TLS with T 1 = 200µs, T ϕ = 300µs. The R q rounds are partitioned in M SB+ q and N -M = 6 BS+ q rounds (separated by the vertical dashed line). The top row shows the optimized feedback values with respect to the moments A 0 and A 4 for each round n. For clarity, the control parameter η q = 0.12 during the BS+ q rounds is kept constant. The lower row depicts the behaviour of the first A k moments over the rounds n, for the two different feedback strategies A 0 /A 4 , that are distinguished in dotted/dashed linestyles.

  ρ ′ b = p l ρ b + (1 -p l )ρ err b .We then consider such an event happening when the target and ancilla state are entangled and that the composite state evolves asρ ′ a,b = p l ρ a,b + (1 -p l ) Tr b (ρ a,b ) ⊗ ρ err b = p l ρ a,b + Tr b (ρ a,b ) ⊗ (ρ ′ b -p l ρ b ). (4.25)where we take Tr a (ρ a,b ) as an input for ρ b and ρ ′ b is computed in numerical simulations for the respective R b round that is used. This is under the approximation of small residual entanglement shared (at the end of the cycle C r ) between the two modes to justify the partial trace over the mode a. For more rapid simulations, we further approximate Tr b (ρ a,b ) by the initial ancilla input state at the start of the two-mode simulations, such that ρ err b can be pre-computed. The specific error rates in these simulations were: T int = 5µs, T CD = 1µs, T osc = 2ms, T 1 = 100µs, T ϕ = 200µs and a Fock space truncation value of N = 120.

Figure 4 . 14 :

 414 Figure 4.14: Numerical simulation results on the logical qubit lifetime of the target oscillator as a function of the feedback strategies that are optimized on the respective A k values. (Left)The explicit values of the feedback displacements for M = 10 SB+ q rounds δ q (m) and for the L = N -M = 6 SB+ q rounds η q (l) are plotted on the left. The dashed vertical line separates the two batches. (Right) Extracted logical qubit lifetime is plotted versus the different feedback strategies that have been optimized on the respective A k values.

1 .- 2 σ 2 +s 2 δ. 9 )-+s 2 - 2 ∆ 2 +d 2 )(A. 10 ) 1 ∆2 (ϵ-ηϵϵ 0 ) 2 k e - 1 σ2 2 (A. 11 ) 2 σ 2 +s 2 ,η ϵ = ∆ 2 (1+γϵ)+d 2 ∆ 2

 12292221012121122222 Gaussian comb formalism for error correction with homodyne detection readoutP (ϕ) = dϕ 0 P (ϕ 0 ) δ(ϕ -ϕ 0 ) (A.8) P (ϵ) = dϵ 0 P (ϵ 0 ) δ(ϵ -ϵ 0 )A single peak in ϕ and ϵ, transforms according to equation A.7 asδ(ϕ -ϕ 0 ) δ(ϵ -ϵ 0 ) → e (ϕ -φ -2kα σ 2 σ 2 +s 2 ) δ(ϵ -ε). (ASubsequent to the quadrature readout, a corrective feedback proportional to the outcome r, to re-center the peak γ ϕ r and envelope γ ϵ r position is appliede 1+γ ϕ )σ 2 +ϕs 2 σ 2 2kα σ 2 σ 2 +s 2 ) δ(ϵ -r(1+γϵ)∆ 2 +ϵdFurther, the expression is integrated over all possible outcomes r, resulting in e -(ϕ-η ϕ ϕ 0 -2kαω ϕ )with ∆2 = ∆ 4 (1+γϵ) 2 ∆ 2 +d 2 , σ2 = σ 4 (1+γσ) 2 σ 2 +s 2 ,η ϕ = σ 2 (1+γσ)+s +d 2

2 | Error correction with an an- cillary two-level system

  Phase-flips of the TLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Contents
	1	Feedback strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2	Error correction for finite-energy code states . . . . . . . . . . . . . . . . . . . . . . . . . . . .
		2.1	Phase-swap sequence for an infinitely squeezed cat state . . . . . . . . . . . . . . . . .
	3	Error propagation from the TLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
		3.1	Bit-flips of the TLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
		3.2

  e -iαp cos(ϵq) e -iαp |g⟩ ⟨g| + e iαp cos(ϵq) e iαp |e⟩ ⟨e|(2.17)

-e iαp sin(ϵq) e -iαp |e⟩ ⟨g| + e -iαp sin(ϵq) e iαp |g⟩ ⟨e| , realizes a phase-swap sequence between an infinitely squeezed cat state and a TLS. We recall the Binomial GKP states from section 4.2 in chapter 1

In the numerical simulations, we consider infinite-energy states and compute the evolution in terms of the probability distributions on a periodic interval. More details on the numerical methods are presented in chapter 3.

This can prove to be advantageous for platforms where the autonomous reset is faster and/or less noisy than measurement-based reset.

Intuitively, this unravelling corresponds to the case where an observer in the environment detects the emission or absorption of a photon with a high-bandwidth photo-counter.

This scaling is achieved by keeping the repetition rate of correction rounds constant.

and the Fourier coefficients of D n are computed with

where * n denotes the n-fold discrete convolution product defined as (u * v) (k) = N l=-N u (k-l) v (l) . In simulations, we truncate this sum in order to maintain a 2n F + 1 structure for the Fourier coefficient vectors.

The distribution is then convolved with a Gaussian kernel G a modeling the effect of quadrature noise during the C q cycle (left equation in (3.29)), then convolved with the Π q b distribution to model the backaction of the quadrature gate in the following C pa cycle (right equation in (3.27), replacing P a → Q a ), and again convolved with a Gaussian kernel Ga modeling the effect of quadrature noise during the C pa cycle (right equation in (3.29) replacing P a → Q a ). In Fourier domain, it reads

with σ 2 tot = 2κT cycle + θ 2 3 κT quad .

Combining Eq. (3.43) and Eq. (3.45), the evolution through the two cycles can be expressed under a matrix form

M kl Π l qa,j-1 .

(3.46)

Note that M is real when F is odd, which is the case in the following.

GKP qubit decoherence rate and convergence rate to the code manifold by spectral analysis of the evolution matrix

The evolution matrix M is the Fourier transform of a stochastic matrix. As such, it shares the same eigenspectrum {λ i } where we arrange the eigenvalues in decreasing magnitude order. In particular λ 0 = 1, and

In the regime where the logical flip probability per cycle is small, we find that the spectrum is gapped with

Qualitatively, this gap indicates a fast convergence of the system to a 2D-manifold of probability vectors (distributions), at a rate

We interpret this fast dynamics as a convergence of the target oscillator state to a meta-stable state in the vicinity the GKP code manifold. It is followed by a slow relaxation, within this manifold, to the steady-state of the system-the probability distribution Π 0 obtained by inverse Fourier transform of the eigenvector attached to λ 0 -at a rate

In this expression, we have used that, since M is real and λ 1 does not have a conjugate eigenvalue, λ 1 is real. We interpret this slow dynamics as the relaxation of the GKP qubit towards the mixed logical state.

We confirm this intuition by representing the probability distributions Π 0 and Π 1 corresponding to λ 0 and λ 1 in Fig. 3 and Eq. (3.41) is truncated at a sufficiently high order n T (not shown). In practice, we found that n F = n T = 30 was sufficient for all numerical simulations presented in this thesis, except to estimate the smallest decay rates of Fig. Furthermore, for a given feedback parameter set, the method allows us to compute the gradient of λ 1 with respect to the cycle continuous parameters (length e j of the ancilla feedback displacements, Fourier coefficients F (k) of the feedback function and ancilla rectangularity R). To this end, we first take the derivative of the evolution rules for the target and the ancilla probability distributions (see Sec. 4 ), respectively p b -distribution, an infinite sequence of such measurements yields the equivalent information as a homodyne detection (found for instance in [START_REF] Howard | Quantum measurement and control[END_REF]). However, the accuracy of this effective homodyne detection is limited for a finite measurement round number.

In detail, a single TLS-e iηp b round gives a single classical bit of information about the ancilla/target envelope position. By multiple measurement rounds, corresponding to the Kraus operators M + = cos(ηp b + π 4 ) and M -= sin(ηp b + π 4 ), an estimate of the ancilla-and in turn of the target-envelope position can be computed based on the TLS measurement records {m j }. For the ancilla being in a momentum state at p 0 << 1 η , the probabilities of each measurement outcome m during a round read

. Following [START_REF] Clerk | Introduction to quantum noise, measurement, and amplification[END_REF], one can compute an effective homodyne measurement rate as

Therefore, using small values of η to perform an effective homodyne detection of the p b , results in a small measurement rate.

We resolve this conundrum by measuring the ancilla state in two steps. First, we use a single BSB+ p readout as described in the previous section to measure p b mod π β . This allows to access the target peak-shift error syndrome, i.e. information on the phase of the infinite-energy stabilizer S ra (during a C r cycle). At this stage, the ancilla no longer encodes information on the target modular position, but still about on the target envelope position. Second, we perform a sequence of TLS-e iηp b rounds with small value of η to perform a measurement of p b mod 2π η ∼ p b and to retrieve this envelope shift error syndrome. The corresponding measurment rate is small, so that over a finite sequence of rounds, it models a effective weak homodyne measurement.

Nevertheless, as illustrated in figure 4.4, one does not need to measure p b very accurately to recover the envelope-shift error syndrome.

In order to estimate the number of rounds needed to accurately retrieve the envelopeshift error, we compute in numerical simulations the quantum trajectories corresponding to all possible measurement records of a cycle, where the ancilla detection is performed by this a 'two-step' measurement scheme. The target and ancilla states are assumed to be pure before the cycle begins, so that for a given record m, the purity of the target mode density matrix after tracing out the ancilla mode Tr(Tr b (ρ ab ) 2 ) informs about the residual entanglement between the two modes at the end of the the average photon number of the target and ancilla in steady state is depicted as a function of the value η q for different round numbers.

Figure 4.10: Average photon number of the ancilla and target oscillator in steady state versus the values η q in the BS + q (η q ) sequence for 4 and 6 rounds. For small values of η < 0.12 envelope is not stable for target and ancilla state for only 4 BS + q (η q ) rounds.

The ancilla properties and therefore the value of η q are chosen according to an optimal average photon number of the target oscillator for its given bare lifetime.

Suppression of TLS-induced errors

In the next part, we focus on the optimization of the optimal number of TLS-S ∆ q b rounds in the presence of TLS errors. First, we consider the ancilla re-preparation, by performing Lindblad simulations on the subsystem of the ancilla oscillator + TLS only.

In figure 4.11, numerical simulations results are presented for the specific SB+ p and BS+ q sequences that have been proposed in this section. The q b /p b -ancilla probability distributions P (q b )/P (p b ) are illustrated for two different TLS decay T 1 = 200µs/20µs and de-phasing times T ϕ = 300µs/30µs. The ancilla state is assumed to be prepared initially in absence of any errors (orange curve). After the S ∆ p b readout, a family of curves depict the re-preparation of the ancilla q b -quadrature for n BS+ q rounds with a fixed value of η q = 0.12. At least 4-6 BS+ q rounds are necessary to extract sufficient information to stabilize the target envelope size, and additional rounds are performed to suppress errors along the q b quadrature that have been generated by Terrors during the S ∆ p b readout.

Our proposal is tailored to minimal hardware requirements and control techniques for near-term experiments. Except for the tuneable quadrature-quadrature gate, it only requires the conditional displacement gates that have already been demonstrated in the framework of trapped ions [START_REF] Brennan De Neeve | Error correction of a logical grid state qubit by dissipative pumping[END_REF] and superconducting circuits [START_REF] Campagne-Ibarcq | Quantum error correction of a qubit encoded in grid states of an oscillator[END_REF], [START_REF] Vv Sivak | Real-time quantum error correction beyond break-even[END_REF]. As proposed in [START_REF] Terhal | Encoding a qubit into a cavity mode in circuit QED using phase estimation[END_REF] the quadrature-quadrature gate can be engineered through single-mode squeezing and Beam-splitter operations, which have been individually demonstrated.

Moreover, this gate is a matter of active experimental research and are central to implement logical gates in the GKP code.

A | Appendix to chapter 4 1 Gaussian comb formalism for error correction with homodyne detection readout

A general Gaussian comb distribution

is characterized by 5 quantities, the peak/envelope variance σ 2 /∆ 2 of the wavefunction, the peak/envelope shift error ϕ/ϵ and the lattice spacing α. In this representation the two logical target states are identified by ϵ = 0, ϕ = 0 (|+Z L ⟩) and

The product of two Gaussian combs, with equal lattice spacing 2α, can be expressed as a sum of Gaussian combs

In order to describe the dynamics of a single error-correction cycle in terms of the target probability distribution, P ′ (q a ) = M 2 r P (q a ), the operator M r

.

is represented in the Gaussian comb representation by rewriting it in terms of the period 2α

By using equation A.2, the expression for M 2 r can be derived

This expression can be rewritten in a compact form as the sum over four Gaussian

We consider the dynamics of one q-error correction cycle, the product of the qprobability distribution of a general target state and the expression of M 2 r in equation A. [START_REF] Flühmann | Encoding a qubit in a trapped-ion mechanical oscillator[END_REF]. For clarity, the multiplication with a single Gaussian comb in the M 2 r expression indexed by λ = {0, 1, 2, 3} is computed below

This formalism allows to describe the dynamics by solely updating the variances ∆ 2 and σ 2 and the distributions of ϵ and ϕ.

General initial distributions for the peak and envelope shift errors ϵ and ϕ are considered with s 2 2 /d 2 2 being the ancilla q-peak/envelope variance.

ABSTRACT

This thesis focuses on the GKP code that allows for generic error-correction of a qubit encoded in a harmonic oscillator. Recent experiments have demonstrated the stabilization of the code manifold based on Rabi interactions with an ancillary two-level system. However, these schemes suffer from uncorrectable logical flips triggered by ancilla relaxation errors during the interaction. This thesis, proposes a protocol to stabilize the GKP code in a target mode by mapping its error syndromes to an ancillary GKP mode via a quadrature-quadrature interaction. In contrast to previously proposed schemes, coupling to solely one ancilla quadrature allows tailoring the ancilla state and its preparation accordingly to ensure a strong suppression of back-propagating errors to the target mode. The error-syndrome information is retrieved and the ancilla efficiently reinitialized using similar techniques demonstrated in the recent GKP experiments. For realistic system parameters, numerical simulations confirm the robust suppression of ancilla induced logical errors and show an enhancement of the logical qubit lifetime by an order of magnitude beyond the break-even point.