domain: math phys

text:
Amir R Arab
email: amir.arab@phystech.edu

On states of quantum theory

Keywords: States on C * -algebras, GNS construction, Covariant cloning, Gelfand-Pettis integral

In this paper the generalized quantum states, i.e. positive and normalized linear functionals on C * -algebras, are studied. Firstly, we study normal states, i.e. states which are represented by density operators, and singular states, i.e. states can not be represented by density operators. It is given an approach to the resolution of bounded linear functionals into quantum states by applying the GNS construction, i.e. the fundamental result of Gelfand, Neumark and Segal on the representation theory of C * -algebras, and theory of projections. Secondly, it is given an application in quantum information theory. We study covariant cloners, i.e. quantum channels in the Heisenberg and the Schrödinger pictures which are covariant by shifting, and it is shown that the optimal cloners can not have a singular component. Finally, we discuss on the representation of pure states in the sense of the Gelfand-Pettis integral. We also give physical interpretations and examples in different sections of the present work.

Introduction

A generalized quantum state is a state on a C * -algebra that generalizes the notion of a density matrix in quantum mechanics. A density matrix represents a quantum state, both a mixed state and a pure state. A density matrix in turn generalizes the notion of a state vector, which only represents a pure state. In the C * -algebraic formulation of quantum mechanics, states in the previous sense correspond to physical states, i.e. mappings from physical observables to their expected measurement outcome.

In 1943, Gelfand and Neumark defined what is now called a C * -algebra and proved the basic theorem that every C * -algebra is isomorphic to the norm-closed * -algebra of operators on a Hilbert space. Their paper [START_REF] Gelfand | On the imbedding of normed rings into the ring of operators in Hilbert space[END_REF] also contained the rudiments of what is now called the GNS construction, connecting states to representations. In its present form, this construction is due to Segal [START_REF] Segal | Postulates for general quantum mechanics[END_REF], a great admirer of von Neumann, who generalized von Neumann's idea of a state as a positive normalized linear functional from B(H) to arbitrary C * -algebras. Moreover, Segal returned to von Neumann's motivation of relating operator algebras to quantum mechanics.

Starting with Segal in 1947, C * -algebras have become an important tool in mathematical physics, where traditionally most applications have been to quantum systems with infinitely many degrees of freedom, such as quantum statistical mechanics in infinite volume and quantum field theory. Many important examples of states in quantum physics (for instance, the Gibbs states for free Bose gas) are normal states, i.e. they are of the form ρ(a) = tr(Λa) with a certain density matrix Λ.

In 1966, Dixmier proved that there exist states on B(H), the bounded linear operators on a separable complex Hilbert space, which are not normal [START_REF] Dixmier | Existence de traces non normales[END_REF]. Dixmier states have the further property to be "singular", i.e. they vanish on the finite rank operators. Quantum states have been classified in a wide range of studies, for instance see [START_REF] Kadison | Extensions of Pure States[END_REF][START_REF] Takesaki | Theory of operator algebras[END_REF][START_REF] Landsman | Foundation of Quantum Theory[END_REF][START_REF] Schmüdgen | On trace representation of linear functionals on unbounded operator algebras[END_REF][START_REF] Kadison | States and Representations[END_REF]. We are motivated by two papers [START_REF] Amosov | Geometrical properties of systems of vector states and representing of states in the form of Pettis integrals[END_REF][START_REF] Amosov | On Analogs of Spectral Decomposition of a Quantum State[END_REF] which studied states on B(H) in the classical case by considering finitely additive measures and ultrafilters, and provided some results on representation of them. In the present paper, states on B(H) are generalized to states on a C * -algebra.

This paper is devoted to the description of generalization of quantum states, applying this treatment in quantum mechanics, quantum field theory, quantum dynamics and quantum information theory. This paper is organized as follows. In Section 2, we provide basic definitions and theorems on weak topologies on B(H), C * -algebras, von Neumann algebras, and representation of C * -algebras. In Section 3, we introduce different types of states and formulate a fundamental theorem on normal states. We also address the formulation of quantum mechanics in terms of worlds that is verified by realization of singular states. In Section 4, we introduce the GNS construction and universal representation of a C * -algebra. We also provide an approach to quantum field theory. In Section 5, we prove some results on extension of bounded linear functionals, topological properties with respect to weak topologies on B(H), relation between a representation of a C * -algebra and its universal representation, and classification of normal and singular states by a central projection. We also deal with generalized states in quantum dynamics. In Section 6, we introduce phase space, Weyl operators and covariant cloning, and by applying the notion of fidelity it is shown that the optimal cloner can not have a singular component. And finally in Section 7, we have a short discussion on the representation of pure states in the sense of the Gelfand-Pettis integral. We also discuss on the process of measurement in the formalism of quantum mechanics in terms of worlds.

Preliminaries

Let B(H) be the set of bounded linear operators on a complex Hilbert space H. The induced topology on B(H) by the family of semi-norms, (i)

p x (T ) = T (x) is called the strong-operator topology (SOT); (ii) p x,y (T ) = | T (x), y |, is called the weak-operator topology (WOT); (iii) p {xn,yn} (T ) = | n T (x n ), y n |, where n ( x n 2 + y n 2 ) < ∞ is called the ultraweak topology.
The SOT and the ultraweak topology are stronger than the WOT. On every norm-bounded set the WOT and the ultraweak topology are the same, particularly the unit ball is compact in both topologies.

We assume that A is an associative algebra over the field C with the unit element 1 A . Let . be a norm on A, as a vector space on C, and * : A-→A, a -→ a * , be an antilinear map. Then (A, . , * ) is called a C * -algebra, if (A, . ) is complete and for each a, b in A, we have:

1. a * * = a (involution), 2. (ab) * = b * a * (anti-homomorphism), 3. ab ≤ a b (submultiplicativity), 4. a * = a (isometry), 5. a * a = a 2 (C * -property). B(H) with the usual operator norm is a C * -algebra. An algebra A is called self-adjoint if for each a ∈ A, a * ∈ A is so. A von Neumann algebra is a self-adjoint algebra A ⊆ B(H)
that is closed with respect to the WOT. Every von Neumann algebra is a C * -algebra, but the converse is not true, for instance the C * -algebra K(H) of compact operators on an infinite dimensional Hilbert space H is not a von Neumann algebra. Let A -be the strong-(equivalently weak-)operator closure of a C * -algebra A ⊆ B(H). According to the Kaplansky density theorem, (A) 1 , the unit ball of A, is strong-operator dense in (A -) 1 , the unit ball of A -. The predual of a von Neumann algebra A is the linear space of linear functionals on A that are weak-operator continuous on (A) 1 and we denote it by (A) * .

A representation of a C * -algebra A on H is a * -homomorphism π : A -→ B(H) (i.e. homomorphism with this property π(a * ) = π(a) * for each a in A). If, further, π is one-to-one, it is called * -isomorphism or faithful representation. Every * -homomorphism is norm decreasing and then is continuous. The image of a C * -algebra under a * -homomorphism is a C * -algebra, specially π(A) is a C * -subalgebra of B(H). If there is a vector x in H for which the linear subspace

π(A)x = {π(a)x : a ∈ A},
is dense in H, π is called a cyclic representation, and x is termed as a cyclic vector for π. Two representations π 1 : A -→ B(H 1 ) and π 2 : A -→ B(H 2 ) are equivalent, if there is a unitary operator U : H 1 -→ H 2 such that U π 1 (a) = π 2 (a)U where a ∈ A.

States

A state on a C * -algebra A is a linear functional ρ : A -→ C such that ρ(a * a) ≥ 0 for each a in A (positivity) and ρ(1 A ) = 1 (normalization). For example, when A ⊆ B(H) then every unit vector x ∈ H defines a state ω x (T ) := T (x), x that is called a vector state. A state ρ on A is called faithful if ρ(a * a) > 0 unless a = 0. The set of states on A is called the state space of A and we denote it by S(A). The state space is a compact (with respect to the weak- * topology) convex set. For example the state space of the algebra of 2 × 2 complex matrices is the Bloch sphere (the unit ball in R 3 ). A state ρ on A is called pure (extreme) if it is not a convex combination of other states, i.e. if ρ = tρ 1 + (1 -t)ρ 2 for some 0 < t < 1, then ρ = ρ 1 = ρ 2 . In fact pure states are elements of the boundary of S(A). A state that is not pure is called mixed. A state ρ on a C * -algebra A is called normal, if a i a then ρ(a i ) ρ(a), i.e. if for every increasing net {a i } i∈I of positive elements of A (i.e. elements may be written in the form b * b), we have ρ(sup i∈I a i )=sup i∈I ρ(a i ). For example with {e i } n i=1 an orthonormal basis of C n , ρ(T ) = n i=1 p i T (e i ), e i , where p i ≥ 0 (1 ≤ i ≤ n) and n i=1 p i = 1, is a normal state on the algebra C n×n . For a unit vector e, ρ(T ) = T (e), e is a normal pure state.

Theorem 1. ([6])

The following conditions for a state ρ on a von Neumann algebra A are equivalent, 1. ρ is normal; 2. ρ(T ) = tr(ΛT ) for some density operator Λ on H (Λ ≥ 0, tr(Λ) = 1); 3. ρ is ultraweakly continuous; 4. ρ is weak-operator continuous on (A) 1 ; 5. ρ is strong-operator continuous on (A) 1 ; 6. ρ = ∞ n=1 ω xn where ∞ n=1 x n 2 = 1; 7. ρ(∨ i∈I e i ) = i∈I ρ(e i ) for every orthogonal set of projections {e i } i∈I (i.e. e 2 i = e i = e * i and e i e j = δ ij e i ).

Physical interpretation and example

We do not measure vectors in a Hilbert space. What we measure are expectation values. This is precisely the meaning of a state ρ as defined above. A state assigns an expectation value to each physical observable. That is what a physical state is: a list of expectation values for all our observables satisfying reasonable postulates. It means that if a is self-adjoint, then ρ(a) is the expectation value of a and ρ(a 2 ) -(ρ(a)) 2 is the variance. Therefore, the definition of state includes not only expectation values of observables, but also their moments.

Formulation of quantum mechanics in terms of worlds

Let Q be a quantum system with a separable Hilbert space H. A world of Q is an orthonormal basis of H [START_REF] Chen | Mathematical formalism of many-worlds quantum mechanics[END_REF]. Since there are many orthonormal bases for a Hilbert space H, then there are many worlds for Q. For two worlds W = {e n } and W = {e n }, W is identified with W whenever for each n, we have e n = c n e n where c n is a complex number with |c n | = 1. For a given world W = {e n } and unitary operator U , W = {U e n } is also a world. Furthermore, any operator mapping an orthonormal basis into another one is a unitary operator. It results that the evolution of worlds is described by U t+s W = U t U s W (t, s ≥ 0), where W 0 = W and U t 's are unitary operators. Consequently, the evolution satisfies the following Schrödinger equation:

dW t dt = iBW t (t > 0),
with W 0 = W , where B is a self-adjoint operator. For a certain world W = {e n } of Q, an observable is a self-adjoint operator on H which is diagonal in the basis {e n }. The set of all observables relative to W , is denoted by O(W ).

Let Q = Q 1 × Q 2 be a bipartite system composed of two subsystems Q 1 and Q 2 , with Hilbert space H 1 ⊗ H 2 where H 1 and H 2 are Hilbert spaces of Q 1 and Q 2 , respectively. For two given observables O 1 and O 2 in the worlds

W 1 = {e 1 j } (of Q 1 ) and W 2 = {e 2 k } (of Q 2 ), respectively, O 1 ⊗ O 2 is an observable in the world W = {e 1 j ⊗ e 2 k } of Q. Conversely, for a given observable O in W = {e 1 j ⊗ e 2 k }, there are observables O 1 and O 2 in the worlds W 1 = {e 1 j } and W 2 = {e 2 k } such that O = O 1 ⊗ O 2 .
Similarly, we can study observables in a world of multipartite systems.

In the formalism of quantum mechanics in terms of worlds, observables are considered relative to a certain world of the system. This relativity goes back to the fundamental works of Dirac and 't Hooft. Dirac in his principles of quantum mechanics [START_REF] Dirac | The Principles of Quantum Mechanics[END_REF] introduced the notion of orthonormal representation as an orthonormal basis and 't Hooft in his deterministic quantum mechanics [START_REF] Hooft | The cellular automaton interpretation of quantum mechanics[END_REF] introduced the notion of ontological basis as an orthonormal basis.

Let M(W ) be the von Neumann subalgebra of B(H) generated by all bounded operators in O(W ). A state in W is defined to be a state on M(W ). For a world W = {e n }, w n (T ) =

T (e n ), e n defines a pure state corresponding to the pure state |e n in the conventional formulation of quantum mechanics.

We will continue our discussion on this formalism and realization of singular states (which will be defined in Section 5) in Section 7.1.

The GNS construction

The basic idea of the GNS construction is that given an algebra A of observables and a state ρ on this algebra, we can construct the Hilbert space on which the algebra of observables acts on. From the mathematical point of view, the algebra of observables is a C * -algebra.

The Gelfand-Neumark-Segal (GNS) theorem constructs a representation of a C * -algebra A starting from a state ρ : A -→ C. The main idea is to define an inner product via a, b := ρ(b * a). If ρ is not faithful, one has to define the elements of the sought Hilbert space in terms of equivalent classes and consider the set A/N with N = {a ∈ A : ρ(a * a) = 0}. By completing A/N we get a Hilbert space H ρ and define π ρ (a) as the left multiplication by a on H ρ , i.e. π ρ (a)(b) = ab. We can summarize this discussion in the following theorem,

Theorem 2. ([1]) Given a state ρ : A -→ C, there is a cyclic representation π ρ : A -→ B(H ρ ) with a unit cyclic vector x ρ ∈ H ρ , such that ρ = ω xρ • π ρ ; that is ρ(a) = π ρ (a)x ρ , x ρ where a ∈ A. If π : A -→ B(K)
is another cyclic representation with unit cyclic vector x such that ρ(a) = π(a)x, x , then π ρ and π are equivalent. [START_REF] Balachandran | Entanglement, Particle Identity and the GNS Construction: A Unifying Approach[END_REF]. One of the important results of the GNS theorem is that every C * -algebra has a faithful representation (Gelfand-Neumark theorem). In other words, for every C * -algebra A, there is a Hilbert space H such that A is a * -isomorphic to a C * -subalgebra of B(H). Suppose that A is a C * -algebra with the state space S(A). The representation Π = ρ∈S(A) π ρ of A is faithful and is called the universal representation of A.

For example with

A = C 2×2 , H ρ ∼ = C 4 or C 2 [

Physical interpretation and example

The GNS construction is closely related to constructing a Hilbert space from a vacuum state by acting on it with creation operators and thereby adding particles. If we think a and a * as annihilation and creation operators, respectively, the meaning of a, a := ρ(a * a) is the expectation value of the particle number for the state ρ. When a representation is irreducible, every non-zero vector is cyclic-by using annihilation operators, one can get to the ground state. One should take this comment with a grain of salt due to domain and distribution issues. A finite-dimensional and completely rigorous example occurs in the theory of spin by use of ladder operators [START_REF] Parzygnat | From Observables and States to Hilbert Space and Back: A 2-Categorical Adjunction[END_REF].

Approach to quantum field theory

Given an n-dimensional manifold M, the cotangent bundle of M is defined to be the 2ndimensional manifold comprised by all points x of M together with all cotangent vectors at x. The phase space P is defined to be the cotangent bundle of an n-dimensional configuration manifold M such that the symplectic form Ω ab , i.e. nondegenerate closed two-form, on P is given by

Ω ab = n µ=1 2(∇p µ ) [a (∇q µ ) b] ,
where (q 1 , ..., q n ; p 1 , ..., p n ) are canonical coordinates on P.

In classical mechanics, an observable is a smooth map f : P -→ R. The set of classical observables O c is an infinite dimensional vector space. The inverse symplectic form Ω ab gives rise to an algebraic structure on O c , called the Poisson bracket, which is defined by

{f, g} = Ω ab ∇ a f ∇ b g.
In quantum mechanics, an observable is a self-adjoint operator on an infinite-dimensional separable Hilbert space F. The set of quantum observables is denoted by O q . Letˆ: O c -→ O q be the map taking classical observables to quantum observables such that for any pair of classical observables f, g we have

[ f , ĝ] = i{ f, g}, where = 1.
Let P be a symplectic vector space, i.e. a vector space such that the symplectic form Ω ab has constant components in a globally parallel basis. Therefore, Ω may be considered as an antisymmetric bilinear map Ω : P × P -→ R rather than a tensor field. In this notation, the Poisson bracket of the fundamental observables, i.e. linear functions, is given by

{Ω(ψ 1 , .), Ω(ψ 2 , .)} = -Ω(ψ 1 , ψ 2 ),
and consequently

[ Ω(ψ 1 , .), Ω(ψ 2 , .)] = -iΩ(ψ 1 , ψ 2 )I, (1)
for all ψ 1 , ψ 2 ∈ P, where I is the identity operator. It is convenient to work with the exponential version of (1). Let W (ψ) = e iΩ(ψ,.) and seek a correspondence map " " for which Ŵ (ψ) is unitary, varies continuously with ψ in the SOT and satisfies the following relations:

Ŵ (ψ 1 ) Ŵ (ψ 2 ) = e iΩ(ψ 1 ,ψ 2 ) 2 Ŵ (ψ 1 + ψ 2 ), Ŵ * (ψ) = Ŵ (-ψ).
These relations known as the Weyl relations which uniquely determine (F, Ŵ (ψ)).

In the framework of general relativity, spacetime structure is defined by a four-dimensional manifold M , on which there is a Lorentz metric g ab . Furthermore, g ab is related to the matter distribution by Einstein's equation. For the spacetime structure (M, g ab ), where g ab is smooth, we study the formulation of the quantum theory of a Klein-Gordon scalar field φ. The curved spacetime version of the Klein-Gordon equation is

∇ a ∇ a φ -m 2 φ = 0, ( 2
)
where ∇ a is the derivative operator compatible with g ab . Let Σ ⊂ M be any closed set which is achronal, i.e. no pair of points P, Q ∈ Σ can be joined by a timelike curve. The domain of dependence of Σ is the set of all points P of M such that every (past and future) inextendible causal curve [START_REF] Wald | General Relativity[END_REF] through P intersects Σ and it is denoted by D(Σ). If D(Σ) = M , then Σ is said to be a Cauchy surface for the spacetime (M, g ab ). A spacetime is called globally hyperbolic if it admits a Cauchy surface. Given a globally hyperbolic spacetime (M, g ab ). Equation ( 2) has a well posed initial value formulation, with the initial data (φ, τ ) on a Cauchy surface Σ, where τ = n a ∇ a φ, with n a the unit normal to Σ. Let the classical phase space P of the Klein-Gordon theory be the set of pairs (φ, τ ) of smooth and of compact support functions on Σ. Let Ξ be the space of solutions to (2) which arise from initial data in P. Ξ is independent of the choice of Σ and each (φ, τ ) ∈ P, gives rise to a unique element of Ξ [START_REF] Wald | Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics[END_REF]. The symplectic structure Ω on P is given by

Ω[(φ 1 , τ 1 ), (φ 2 , τ 2 )] = (τ 1 φ 2 -τ 2 φ 1 )d 3 x. (3)
Ω is conserved for solutions, so (3) gives rise to a bilinear map Ω : Ξ × Ξ -→ R.

We construct a quantum field theory for the Klein-Gordon field as follows: Let ν : Ξ × Ξ -→ R be a bilinear map such that for all ψ 1 ∈ Ξ,

ν(ψ 1 , ψ 1 ) = 1 4 sup ψ 2 =0 [Ω(ψ 1 , ψ 2 )] 2 ν(ψ 2 , ψ 2 ) . (4)
There is a wide class of ν's satisfying (4) [START_REF] Wald | Quantum Field Theory in Curved Spacetime[END_REF]. Given a bilinear map ν satisfying (4), we complete Ξ in the inner product 2ν. By using Ω, we convert the completion of Ξ into a complex Hilbert space H [START_REF] Wald | Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics[END_REF]. Consequently, a projection map L : Ξ -→ H gives rise. To define the quantum field theory, we choose the Hilbert space F to be the symmetric Fock space F s (H) = +∞ n=0 (H s n ), where H s n is the subspace of the n-fold tensor product H n consisting of the maps which are totally symmetric in the n-variables. Finally, we define the observables Ω(ψ, .) as self-adjoint operators on F by Ω(ψ, .) = ia(Lψ) -ia * (Lψ), where a, a * are the annihilation and creation operators, respectively. However, the construction has the potentially unsatisfactory feature that it appears to depend on our choice of ν. Let H 1 and H 2 be the Hilbert spaces associated with bilinear maps ν 1 and ν 2 satisfying (4), respectively. In the case when dim P < ∞, it follows from the Stone-von Neumann theorem [START_REF] Reed | Methods of Modern Mathematical Physics[END_REF] that {F s (H 1 ); Ω1 (ψ, .)} and {F s (H 2 ); Ω2 (ψ, .)} are unitarily equivalent, i.e. there is a unitary operator U such that U -1 Ω2 (ψ, .)U = Ω1 (ψ, .) for all ψ. But in the case when dim P = ∞ (general quantum field theory in curved spacetime), unitarily inequivalent field theory constructions give rise.

In the usual formulation of a quantum theory, one first constructs states in a Hilbert space F and then defines observables as operators on F. But in our approach, observables are constructed as elements of an abstract algebra and then states are defined as objects which act upon observables. The key observation that makes our approach applicable is that although unitarily inequivalent field theory constructions exist, the algebraic structure of the field operators in the unitarily inequivalent constructions are the same.

Let L(F) be the C * -algebra of all bounded linear maps on F with the *-operation corresponding to taking adjoints. Let W = Span{ Ŵ (ψ) : ψ ∈ Ξ}, where Ŵ (ψ) = e i Ω(ψ,.) and the closure is taken in the norm provided by L(F). W is a C * -subalgebra of L(F) which is called the Weyl algebra. In fact, although two bilinear maps ν 1 and ν 2 satisfying (4) may define unitarily inequivalent quantum field theory constructions, the associated C * -algebras W 1 and W 2 which give rise are isomorphic [START_REF] Slawny | On factor representations and the C * -algebra of canonical commutation relations[END_REF]. According to this fact, we can define the fundamental observables for quantum field theory in a curved spacetime to be elements of the Weyl algebra W. Then we define a state w of the quantum field to be a linear map w : W -→ C satisfying w(A * A) ≥ 0 for all A ∈ W and w(I) = 1. A natural question that gives rise is: What is the relationship between the notion of states defined here and states of the Hilbert space F = F s (H)? We take into account that given any density matrix ρ on F, which carries a representation π : W -→ L(F), we obtain a state w : W -→ C by w(A) = tr[ρπ(A)]. Therefore, all states arising in all quantum field theory constructions give rise to states defined here. The converse of this result is given by Theorem 2.

The crucial advantage of our approach is that it allows us to treat all states, in particular states arising in unitarily inequivalent quantum field theory constructions, on an equal footing. Thus, one can define the theory without the need to select a preferred construction.

Resolution of bounded linear functionals into normal and singular states

Let π and Π be a representation and the universal representation of a C * -algebra A on a Hilbert space H, respectively, also B = Π(A) and B π = π(A). Firstly, we need to prove the following two lemmas on extension of the bounded linear mappings.

Lemma 1. There is an isometric isomorphism from B * onto (B -) * .

Proof. According to Theorem 2, for every state ξ on A, there is a representation π ξ with a unit vector x such that ξ = ω x • π ξ . Hence, ξ = ω u • Π, where u = ρ∈S(A) u ρ ∈ ρ∈S(A) H ρ , defined by u ξ = x, u ρ = 0 for ρ = ξ. The mapping ξ -→ ξ • Π -1 sends the state space of A onto the state space of B. Thereby, every state on B is a vector state. It is well-known that every bounded linear functional on B can be written as a linear combination of at most four states. Every vector state ω x on B extends to ω x on B -, therefore f extends to the corresponding linear combination of vector states on B -. It shows that f extends to a weakoperator continuous linear functional f on B -. This extension is unique by continuity. With Lemma 2. Let A be a C * -subalgebra of B(H), J be a Hilbert space, and α : A -→ B(J) be an ultraweakly continuous * -homomorphism. Then α extends uniquely to an ultraweakly continuous * -homomorphism α : A --→ B(J) with α(A -) = α(A) -.

Proof. Without loss of generality, we may assume that α = 1. For each positive real number r, α maps the ball (A) r (i.e. ball with radius r) into the ball (B(J)) r . Since (A) - r = (A -) r and (B(J)) r is ultraweakly compact then α| (A)r extends uniquely to an ultraweakly continuous mapping α r : (A -) r -→ (B(J)) r . With 0 < r < s, we have α r (T ) = α s (T ) for each T ∈ (A -) r . Hence, we can define α : A --→ B(J) by α(T ) = α r (T ) where T ∈ (A -) r . Given S, T ∈ A -, there are nets {S i }, {T j } in A, converging ultraweakly to S, T , respectively, such that S i ≤ S , T j ≤ T . Given scalars s, t, let r = max{ T , S , |s| S +|t| T }, therefore, S i , T j , sS i +tT j all lie in (A -) r , and α(sS i +tT j ) = s α(S i )+t α(T j ), by the linearity of α = α| A . When S i → S and then T j → T , we have α(sS + tT ) = s α(S) + t α(T ), by the ultraweak continuity of α| (A -)r . Hence, α is a linear mapping. Similarly, we can prove that α(T * ) = α(T ) * , α(ST ) = α(S) α(T ) S, T ∈ A -, by the ultraweak continuity of α and the adjoint operation, and the separate ultraweak continuity of multiplication. Hence, α is a * -homomorphism. It is clear that α extends α and is norm decreasing, then α = α = 1. If f is an ultraweakly continuous linear functional on B(J), the composition f • α is ultraweakly continuous on (A -) r for each r > 0. Then by Theorem 1, f • α is ultraweakly continuous on A -. It results that α is ultraweakly continuous on A -, and then α = α| A is ultraweakly continuous on A.

With M = α(A), we have (M) 1 = α((A) 1 ). Since α is ultraweakly continuous and (A -) 1 is ultraweakly compact, α((A -) 1 ) is an ultraweakly compact set containing (M) 1 and then containing its closure (M -) 1 . Thus M -⊆ α(A -) and therefore α(A -) = M -= α(A) -, by the ultraweak continuity of α.

In the proof of the following theorem, we apply the fact that the weak-operator closed left ideals in a von Neumann algebra R are left principal ideals of the form RP for a projection P in R.

Theorem 3. There is a projection E in the center of B -, and a * -isomorphism ϕ :

B -E -→ B - π , such that ϕ(Π(T )E) = π(T ) for each T in A. Proof. Since Π is a faithful representation, µ = π • Π -1 : B -→ B π is a * -homomorphism
and then is bounded. If f is an ultraweakly continuous linear functional on B π , the linear functional f • µ on B is bounded and thus, by applying the argument of Lemma 1, is ultraweakly continuous. It results that µ is ultraweakly continuous (see Remark 1). Therefore, µ extends to an ultraweakly continuous * -homomorphism µ : B --→ B - π , by Lemma 2. The kernel of µ has the form B -P for some projection in the center of B -. Let E = I -P , then for each T in B -,

µ(T ) = µ(T I) = µ(T E + T P ) = µ(T E), (5)
We define ϕ = µ| B -E that according to [START_REF] Takesaki | Theory of operator algebras[END_REF], has the same range as µ. Since B -E∩B -P = {0}, therefore ϕ is one-to-one, and thus is a * -isomorphism from B -E onto B - π . For each T in A,

ϕ(Π(T )E) = µ(Π(T )E) = µ(Π(T )E + Π(T )P ) = µ(Π(T )) = µ(Π(T )) = π(T ).
Remark 1. Given x, y ∈ H, let f (T ) = T (x), y , where T ∈ B - π . By extending f • µ to an ultraweakly continuous linear functional g on B -, or equivalently weak-operator continuous on (B -) 1 , it results that µ is ultraweakly continuous.

Let f be a bounded linear functional on A. Then g = f •Π -1 is an ultraweakly continuous linear functional on B. According to Lemma 1, we can extend g to g such that

g(Π(T )) = f (T ), T ∈ A. ( 6
)
Let E be the projection in Theorem 3 when π is the inclusion mapping and define λ E on A * in the following manner, λ E (f )(T ) := g(Π(T )E), T ∈ A. Proof. Suppose that f is ultraweakly continuous. Then f extends to an ultraweakly continuous linear functional f on A -. According to Theorem 3, we have

T = π(T ) = ϕ(Π(T )E), T ∈ A. (7)
It follows from ( 6), [START_REF] Schmüdgen | On trace representation of linear functionals on unbounded operator algebras[END_REF] that

g(Π(T )) = f (T ) = f (T ) = f (ϕ(Π(T )E)), T ∈ A.
Accordingly,

g(S) = f (ϕ(SE)), S ∈ B. (8)
Because of the ultraweak continuity of g, f , and ϕ, the equality ( 8) holds for each S ∈ B -. When S is replaced by SE, the right-hand side of ( 8) is unchanged, therefore

g(SE) = g(S), S ∈ B -.
Particularly,

f (T ) = g(Π(T )) = g(Π(T )E) = λ E (f )(T ), T ∈ A.
Now, suppose that λ E (f ) = f holds. It follows from ( 7) that f = g •ϕ -1 | A and the ultraweak continuity of g and ϕ -1 results that f is ultraweakly continuous.

The mapping f -→ λ E (f ) is a norm-decreasing projection from A * onto a closed subspace A N consisting of ultraweakly continuous linear functionals or, by Theorem 1, normal states on A. Non-zero elements of the complementary closed subspace

A S = {f ∈ A * : λ E (f ) = 0},
are described as singular states on A. In fact, this projection maps the dual space onto the space of normal states parallel to the space of singular states.

Corollary 1. ([5]

) Each f in A * can be decomposed, uniquely, in the form

f = f N + f S , with f N in A N and f S in A S . Proof. Let f N = λ E (f ), f S = (I -λ E )(f ).
As a result of Corollary 1 , every pure state is a normal or a singular state. According to Theorem 1, singular states annihilate all one-dimensional projections, and therefore all compact operators. Let H be an infinite dimensional separable Hilbert space, T ∈ B(H) a positive compact operator, and {η(k, T )} ∞ k=1 an eigenvalue sequence of T counted with multiplicities and arranged in a non-increasing order. Suppose that

sup n≥1 1 log(1 + n) n k=1 η(k, T ) < ∞.
If ρ is a state on the space l ∞ of bounded sequences such that the following conditions hold, (i) ρ vanishes on finitely supported sequences, i.e. S = {(x 1 , x 2 , ...) : ∃N s.t.

x n = 0 ∀n ≥ N }, (ii) ρ(x 1 , x 2 , ...) = ρ(x 1 , x 1 , x 2 , x 2 , ...) for each x ∈ l ∞ , then Tr ρ (T ) := ρ 1 log(1 + n) n k=1 η(k, T ) ,
is a non-trivial state that vanishes all compact operators and thereby is a singular state [START_REF] Connes | The local index formula in noncommutative geometry[END_REF].

Physical interpretation and example

We can think of a non-relativistic particle localized at a sharp point, as witnessed by the expectations of all continuous functions of position. Extending from this algebra to all bounded operators, we get a singular state with sharp position, but "infinite momentum", i.e. the probability assigned to finding the momentum in any given finite interval is zero [START_REF] Werner | Physical uniformities on the state space of non-relativistic quantum mechanics[END_REF]. This shows that the probability measure on the momentum space induced by such a state is only finitely additive, but not σ-additive. This is typical for singular states.

Arising in quantum dynamics

Consider the evolution of the quantum many-body system of Bose particles with the following Hamiltonian:

H = ψ * (x)hψ(x)dx + 1 2 υ(x -y)ψ * (x)ψ * (y)ψ(x)ψ(y)dxdy,
where (i) ψ and ψ * are the annihilation and creation operators, respectively, satisfying the canonical commutation relations (CCR, see Section 6.1), as follows:

[ψ(x), ψ * (y)] = δ(x -y), [ψ(x), ψ(y)] = [ψ * (x), ψ * (y)] = 0.
(ii) h := -+ V (x) acting on the variable x, defined on the symmetric (or Bosonic) Fock space

F := +∞ n=0 (L 2 (R d , C)) s n
, where -and V (x) are the Laplacian and the external potential, respectively. Under some conditions on υ and V , it follows that H is a self-adjoint operator [START_REF] Bach | The time-dependent Hartree-Fock-Bogoliubov equations for Bosons[END_REF].

Here states, as positive linear functionals w on the Weyl CCR algebra, correspond either to nonzero densities and temperatures, or to a finite number of particles as in the case of Bose-Einstein condensate experiments [START_REF]Universal Themes of Bose-Einstein Condensation[END_REF], in traps. In the latter case, states are represented by density operators, i.e. there is a density operator D on F, such that for all observables A we have w(A) = tr(DA).

As an example for singular states, we can refer to the equilibrium state of a free Bose gas in infinite space at finite density and temperature. This state is singular (with respect to Fock space) because the probability for finding only a finite number of particles in it, is zero.

An application in quantum information theory

The no-cloning theorem says that there is no unitary operator that makes a perfect copy of an unknown pure state [START_REF] Wooters | A single quantum cannot be cloned[END_REF]. However perfect quantum cloning is impossible, it is possible to perform imperfect cloning. Accordingly, several studies on approximate duplication of quantum states have been done, e.g. see these impressive works [START_REF] Werner | Optimal cloning of pure states[END_REF][START_REF] Cerf | Non-Gaussian Cloning of Quantum Coherent States is Optimal[END_REF][START_REF] Gisin | Optimal Quantum Cloning Machines[END_REF].

Phase space and Weyl operators

A phase space consists of a real vector space V of dimension 2n (n is the number of modes) and an antisymmetric bilinear form β : V ×V -→ R that defines the scalar product β(x, y) = A quantum system is described by the canonical commutation relations (CCR) between the fields operators R k (k = 1, 2, ..., 2n) as [R k , R l ] = iβ k,l I where I is the identity operator. In quantum mechanics, fields operators are written in the vector form as

- → R = (Q 1 , P 1 , Q 2 , P 2 , ..., Q n , P n ) or - → R = (Q 1 , Q 2 , .
.., Q n , P 1 , P 2 , ..., P n ) where Q i , P i are position and momentum operators of each mode, respectively (i = 1, 2, ..., n).

Weyl operators as a family of bounded and unitary operators are defined in the following manner W x = e ix t B -→ R where x ∈ V , B = [β(e k , e l )] and W 0 = I [START_REF] Cerf | Non-Gaussian Cloning of Quantum Coherent States is Optimal[END_REF]. For x = (q 1 , p 1 , q 2 , p 2 , ..., q n , p n ) the standard form of the Weyl operators in quantum mechanics is written as W x = exp(i n k=1 (q k P k -p k Q k )). It follows from the CCR that the Weyl operators satisfy the following relations:

W x W y = e -iβ(x,y) 2 W x+y , W x W y = e -iβ(x,y) W y W x , W * x = W -x .
In our study, the Weyl operators have the irreducibility property, i.e. if for every x in V , [W x , A] = 0 then A ∝ I.

Covariant cloning

Let (R 2 , β in ) be the phase space of the input system and (R 2n , β out ) be the phase space of the output system including n subsystems (n modes) where

β out (x, y) = n j=1 β in (x j , y j ), x = (x 1 , x 2 , ..., x n ), y = (y 1 , y 2 , ..., y n ).
Let A in and A out be the CCR algebras of observables on the input and output systems, respectively. Accordingly, the state spaces are S(A in ) = B * (H in ), S(A out ) = B * (H out ) where

H in = L 2 (R 2 ) and H out = H n in L 2 (R 2n ).
A 1 to n cloner is a quantum channel, i.e. completely positive and trace-preserving map, that in the Heisenberg picture, maps A out onto A in , Φ : A out -→ A in and in the Schrödinger picture, maps S(A in ) onto S(A out ), Φ * : S(A in ) -→ S(A out ).

To compare the input and output states of the cloner we need a functional that determines how well they coincide. For two states ρ 1 , ρ 2 , this functional is defined as

F (ρ 1 , ρ 2 ) = tr (ρ 1 2 1 ρ 2 ρ 1 2 1 ) 1 2 2 ,
and is called the fidelity [START_REF] Holevo | Quantum Systems, Channels, Information A Mathematical Introduction[END_REF]. In our study, at least one of the states is pure and this definition is reduced to F (ρ 1 , ρ 2 ) = tr[ρ 1 ρ 2 ]. Accordingly, for an input coherent state ρ = |α α|, the (joint) fidelity between the output state of the cloner, Φ * (ρ) and ρ n is defined as the expectation value of the operator ρ n in the output state of the cloner as follows F (Φ, ρ) = tr Φ * (ρ)ρ n . Let C = {|α α| : α ∈ R 2 }, we choose F (Φ) = inf ρ∈C F (Φ, ρ) as the worst-case overlap between the input and output states.

For x ∈ R 2 , we define the shift cloner Φ x * by

Φ x * (ρ) = W n x * Φ * (W x ρW * x )W n x .
A cloner Φ * is called covariant if Φ x * (ρ) = Φ * (ρ) for every x, ρ. )W x and it means that Φ is also covariant.

Let Φ * be a general cloner in the form Φ * (ρ) = (Φ N * Φ S * )(ρ), where Φ N * is a normal cloner, i.e. the respective output state is a positive trace-class operator, and Φ S * is a singular cloner, i.e. the respective output state is singular. Consequently, a natural question gives rise: What is the structure of the optimal covariant cloner? This question will be answered below in case n = 2, for 1 to 2 covariant cloners.

We define K ⊂ B(H out ) as the subalgebra generated by the identity operator and operators of the form I K and K I, where K is a compact operator and I is the identity operator. A general element A of K is represented by the notation A = M ⊆{1,2} K M I M c in which K M is the compact component that denotes tensor product of compact operators in the tensor factors indexed by M and I M c denotes tensor product of the rest. For a state ω on the output states, ω can be decomposed into parts which are labeled by a set of tensor factors N on which the state is normal and hence described by a positive trace-class operator Λ N and on N c is singular, as follows:

ω(A) = ω   M ⊆{1,2} K M I M c   = N ⊆{1,2} M ⊆N tr Λ N K M I (N -M ) , (9)
we take into account that the expectation value of compact operators in a singular output state is zero, hence in the above sum, we consider only subsets M of N . Since ω is a state then the normalization condition gives rise to

N ⊆{1,2} tr[Λ N ] = 1. (10)
Let ω = Φ * (ρ)| K and define Φ N * : ρ → Λ N that maps ρ to the unique positive trace-class operator Λ N from [START_REF] Amosov | Geometrical properties of systems of vector states and representing of states in the form of Pettis integrals[END_REF]. Since Φ * is covariant, then Φ N * is so. Although Φ * (ρ) is normalized, but Φ N * (ρ) is not normalized. Let Φ N (I) = X N , then it follows from Remark 2 and (10) that for every ρ, tr[ρX N ] = tr[Φ N * (ρ)] ≤ 1, and since Φ N * is covariant, then X N commutes with all Weyl operators. Hence, it follows from the irreducibility property that X N = c N I for some constant 0 < c N ≤ 1, which does not depend on the input state ρ. We define Ψ

N * (ρ) = Φ N = Λ N c N , which is a normal cloner (since Λ N
c N is a density operator). Now we calculate the fidelity: Since Φ * is covariant, by applying Remark 2 and the equality

W α |0 = |α , F (Φ) = inf ρ∈C F (Φ, ρ) = F (Φ, |0 0|) = Φ * (|0 0|)(|α α| 2 ).
For N = {1, 2} by applying (9) and compactness of |α α| 2 ,

F (Φ) = tr[Λ N |α α| 2 ] = c N tr Ψ N * (|0 0|)|α α| 2 = c N F (Ψ N ).
To achieve the optimal cloner, we need to increase the fidelity, hence c N = 1 and by [START_REF] Amosov | On Analogs of Spectral Decomposition of a Quantum State[END_REF], for N {1, 2}, c N = 0. It shows that Φ * = Ψ N * , where N = {1, 2} and it means that the optimal cloner can not have a singular component and it is purely normal.

Discussion

Let H be a separable infinite dimensional Hilbert space. We say that a pure state ρ on B(H) is diagonalizable, if ρ(T ) = lim U T (e n ), e n for some orthonormal basis {e n } of H and some ultrafilter U in N. An atomic masa (maximal abelian self-adjoint subalgebra) is the set of all operators on H that are diagonalized with respect to some orthonormal basis. According to the Kadison-Singer conjecture which was proved in 2013 [START_REF] Marcus | Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem[END_REF] every pure state on an atomic masa of B(H) has a unique extension to a pure state on B(H). Nevertheless, assuming the continuum hypothesis, there is a pure state on B(H) whose restriction to any masa is not pure [START_REF] Akemann | B(H) has a pure state that is not multiplicative on any masa[END_REF]. It follows from this discussion that: There is a (singular) pure state on B(H) which can not be represented in the sense of the Gelfand-Pettis integral by an orthonormal basis.

Formulation of quantum mechanics in terms of worlds (continue)

From the Kadison-Singer conjecture it follows that in addition to vector states, singular states also have unique extensions to B(H). It results that unlike observables which are considered relative to a certain world, states can be (uniquely) extended to the whole system and so of absolute meaning. Although singular states do not appear in the conventional formalism of quantum mechanics, but they are defined, based on the notion of world by the Stone-Čech compactification and are (mathematically) constructed by the notion of Banach limit [START_REF] Chen | Mathematical formalism of many-worlds quantum mechanics[END_REF]. Thus, realization of such states will verify the necessity of our conceptual approach to worlds of a quantum system.

Measurement

Measurement, i.e. making observation, in a certain world is done by applying a quantum state as an external observation. Let w be a pure state in a world W = {e j }. Every observable O relative to W takes a definite value w(O), particularly w j (O) is the eigenvalue of O corresponding to eigenvector e j (w j is the vector state corresponding to e j ). For an observable O relative to another world W = {e k }, since w is a state in W , the information represented by w is incomplete for W . By the Kadison-Singer conjecture, w has a unique extension to B(H) which is also denoted by w. Therefore, the expectation value of the new measurement is O w := w(O ). Particularly, if w = w j then A natural question gives rise: What occurs in the measurement process? Let w be a state in a world W = {e j } and the observer wants to observe an observable O relative to another world W = {e k }. We take into account that O relative to W has a definite value, then we can assign a state w representing the information obtained by the observer in the world W . Similarly, w has a unique extension to B(H) which is also denoted by w . The next measurement will only depend on w and it does not depend on w.

It follows that in the measurement process we have only change of information and in contrary of conventional quantum mechanics no collapse of wave packets occurs [START_REF] Dirac | The Principles of Quantum Mechanics[END_REF].

Conclusion

It is shown the resolution of bounded linear functionals into normal and singular states. In fact, there is a projection in the von Neumann algebra generated by applying the GNS construction which maps the dual space onto the space of normal states parallel to the space of singular states. Singular states are mathematically constructed and like vector states, have unique extensions to the whole system. Realization of singular states verifies the necessity of approaching to worlds of a quantum system. Quantum field theory in curved spacetime is constructed based on a Hilbert space F which is associated with a bilinear map ν. Our treatment allows us to formulate quantum field theory in a manner which does not require the specification of even a preferred unitary equivalence class of bilinear maps. Consequently, one can define quantum theory without the need to select a preferred construction. States correspond to a finite number of particles as in the case of BEC (Bose-Einstein condensate) experiments, in traps are normal states. The equilibrium state of a free Bose gas in infinite space at finite density and temperature is a singular state. As an application in quantum information theory it results that, in the class of general covariant cloners, the optimal cloner can not have a singular component and it is purely normal. There is a singular state which can not be represented in the sense of the Gelfand-Pettis integral by an orthonormal basis. In the formalism of quantum worlds, the process of measurement only changes the information and no collapse of wave packets occurs.

  T ∈ (B) 1 , we have | f (T )| = |f (T )| ≤ f , then for each T ∈ (B -) 1 , by the Kaplansky density theorem, | f (T )| ≤ f . It results that f ≤ f and hence f = f . With g ∈ (B -) * , let f = g| B ∈ B * . Therefore, g = f on B and are ultraweakly continuous on B -, by Theorem 1, and thus g = f on B -.

Theorem 4 .

 4 Let E be the projection in the center of B -when π : A -→ B(H) is the inclusion mapping in Theorem 3. A bounded linear functional f on A is ultraweakly continuous if and only if λ E (f ) = f .

Remark 2 .

 2 For a covariant cloner Φ * and given observable O and state ρ, we havetr [ρ Φ(O)] = tr [Φ * (ρ) O] = tr W n x * Φ * (W x ρW * x )W n x O = tr ρW * x Φ(W n x OW n x * )W x ,hence, it results that Φ(O) = W * x

Theorem 5 .

 5 ([9, 10]) For every pure state ρ on B(H), there is a basis {e n } of H and measure µ such thatρ(T ) = N T (e n ), e n dµ.By applying the following notation ρ(T ) = ρ, T , and ρ n (T ) = T (e n ), e n , the result of Theorem 5, gives rise to:ρ, T = N ρ n , T dµ.It is called the representation of ρ in the sense of the Gelfand-Pettis integral.

  w j (O ) = e j |O |e j = e j | k λ k |e k e k |e j = k λ k | e j |e k | 2 .Hence, the probability that O takes the value λ k is | e j |e k | 2 . It is the same as the usual Born rule.

2n k,l=1 x t k β k,l y l ( t is transposition) where β k,l = β(e k , e l ) and {e k } is an orthonormal basis for V .

Acknowledgement . I would like to thank Professors Grigori G. Amosov, Vsevolod Zh. Sakbaev, and Oleg G. Smolyanov for motivating to write this paper.


Timothée Goubault De Brugière

Marc Baboulin

Benoît Valiron

Simon Martiel

Cyril Allouche

Quantum CNOT circuits synthesis for NISQ architectures using the syndrome decoding problem

Keywords: Quantum Circuit Synthesis, CNOT Circuits, Syndrome Decoding, Reversible Computation, Noisy Intermediate Scaled Quantum Computers (NISQ)

Current proposals for quantum compilers involve the synthesis and optimization of linear reversible circuits and among them CNOT circuits. This class of circuits represents a significant part of the cost of running an entire quantum circuit and therefore we aim at reducing the size of CNOT circuits. In this paper we present a new algorithm for the synthesis of CNOT circuits based on the solution of the syndrome decoding problem. Our method addresses the case of ideal hardware with an all-to-all qubit connectivity and the case of near-term quantum devices with restricted connectivity. Benchmarks show that our algorithm outperforms existing algorithms in both cases of partial and full connectivity.

Introduction

Quantum compilers transform a quantum algorithm into an optimized sequence of instructions (elementary gates) directly executable by the hardware. The most common universal set of gates for this task is the Clifford+T gate set, used in many quantum architectures [START_REF] Campbell | Roads towards fault-tolerant universal quantum computation[END_REF]. With this setup two resources have to be optimized in priority: the T gate and the CNOT gate. The T gate is considered to be the most costly gate to implement and many efforts have been made to reduce their number in quantum circuits [START_REF] Amy | Polynomial-time T-depth optimization of Clif-ford+T circuits via matroid partitioning[END_REF][START_REF] Heyfron | An efficient quantum compiler that reduces T count[END_REF][START_REF] Kissinger | Reducing T-count with the ZX-calculus[END_REF]]. Yet, when implementing complex quantum algorithms, e.g, reversible functions, it is estimated that the total number of CNOT gates increases much more rapidly with the number of qubits than the number of T gates, and it is likely that the CNOT cost will not be negligible on medium sized registers [START_REF] Heyfron | An efficient quantum compiler that reduces T count[END_REF][START_REF] Maslov | Optimal and asymptotically optimal NCT reversible circuits by the gate types[END_REF].

Circuits consisting solely of CNOT gates, also called linear reversible circuits, represent a class of quantum circuits playing a fundamental role in quantum compilation. They are part of the so-called Clifford circuits and the CNOT+T circuits, two classes of circuits that have shown crucial utility in the design of efficient quantum compilers [START_REF] Amy | Polynomial-time T-depth optimization of Clif-ford+T circuits via matroid partitioning[END_REF][START_REF] Heyfron | An efficient quantum compiler that reduces T count[END_REF] and error correcting codes [START_REF] Campbell | Magic-state distillation in all prime dimensions using quantum Reed-Muller codes[END_REF][START_REF] Gottesman | Stabilizer Codes and Quantum Error Correction[END_REF]. For instance the Tpar optimizer [START_REF] Amy | Polynomial-time T-depth optimization of Clif-ford+T circuits via matroid partitioning[END_REF] takes a Clifford+T circuit as input and decomposes it into a series of CNOT+T circuits separated by Hadamard gates. Then each CNOT+T circuit is optimized and re-synthesized by successive syntheses of CNOT circuits and applications of T gates.

Hence the synthesis of CNOT circuits naturally occurs in general quantum compilers and giving efficient algorithms for optimizing CNOT circuits will then be of uttermost importance.

With the current near term quantum devices, also called Noisy Intermediate Scaled Quantum Computers (NISQ) [START_REF] Preskill | Quantum computing in the NISQ era and beyond[END_REF], the synthesis of circuits is subject to constraints on the elementary operations available. In this situation, a physical qubit on the hardware can only interact with its neighbors, restricting the 2-qubit gates -such as CNOT-one can apply. Taking into account these constraints is a crucial and difficult task for the design of quantum algorithms and the optimization of the corresponding quantum circuits. In particular, in the literature several works present post-processing techniques to convert with minimum overhead a circuit designed for an ideal hardware to a circuit designed for a specific architecture [START_REF] Childs | Circuit transformations for quantum architectures[END_REF].

Contribution and outline of the paper. In this paper we focus on the size optimization of linear reversible circuits. We present a new method for the synthesis of CNOT circuits relying on solving a well-known cryptographic problem: the syndrome decoding problem. Our algorithm transforms the synthesis problem into a series of syndrome decoding problems and we propose several methods to solve this particular subproblem. This method, initially designed for a full qubit connectivity, is robust enough to be extended to partial connectivity.

The outline of the paper is the following: in Section 2 we present the basic notions and the state of the art in the synthesis of linear reversible circuits. We first present our algorithm in the case of an all-to-all connectivity in Section 3. Then we extend it to the case of restricted connectivity in Section 4. Benchmarks are given at the end of Sections 3 and 4.

Background and state of the art

Synthesis of a linear reversible function. Let F 2 be the Galois field of two elements. A linear reversible function f on n qubits applies a linear Boolean function on the inputs to each qubit. Given x ∈ F n 2 as inputs, the output of qubit i is

f i (x) = α i • x = α i 1 x 1 ⊕ α i 2 x 2 ⊕ ... ⊕ α i n
x n where ⊕ is the bitwise XOR operation and the α i 's are Boolean vectors also called parities. The action of f can be represented as an n × n binary matrix A with A[i, :] = α i (using Matlab notation for row selection) and f (x) = Ax. In other words each row of A corresponds to the parity held by the corresponding qubit after application of A. By reversibility of f , A is also invertible in F 2 . The application of two successive operators A and B is equivalent to the application of the operator product BA.

We are interested in synthesizing general linear reversible Boolean functions into reversible circuits i.e series of elementary reversible gates that can be executed on a suitable hardware. To that end we use the CNOT gate, it performs the following 2-qubit operation:

CNOT(x 1 , x 2 ) = (x 1 , x 1 ⊕ x 2 ).
where x 1 , resp. x 2 , is the parity held by the control qubit, resp. the target qubit. If applied after an operator A, the total operator (A + CNOT) is given from A by adding the row of the control qubit to the row of the target qubit. Such row operations are enough to reduce any invertible Boolean matrix to the identity matrix, so the CNOT gate can be solely used to implement any linear reversible operator. Overall, a CNOT-based circuit can be simulated polynomially: starting from A = I the identity operator, we read sequentially the gates in the circuit and apply the corresponding row operation to A.

We use the size of the circuit, i.e, the number of CNOT gates in it, to evaluate the quality of our synthesis. The size of the circuit gives the total number of instructions the hardware has to perform during its execution. Due to the presence of noise when executing every logical gate, it is of interest to have the shortest circuit possible.

Connectivity constraints. At the current time, for superconducting technologies, full connectivity between the qubits cannot be achieved. The connections between the qubits are given by a connectivity graph, i.e, an undirected, unweighted graph where 2-qubit operations, such as the CNOT gate, can be performed only between neighbors in the graph. Examples of connectivity graphs from current physical architectures are given on Fig. 1.

LU decomposition. Given the matrix representation A of a generic linear reversible operator, we can always perform an LU decomposition [START_REF] Golub | Matrix Computations[END_REF] such that there exists an upper (resp. lower) triangular matrix U (resp. L) and a permutation matrix P such that A = P LU . The invertibility of A ensures that the diagonal elements of L and U are all equal to 1. In the remainder of this paper, the term "triangular operator" stands for an operator whose corresponding matrix is either upper or lower triangular. The LU decomposition is at the core of our synthesis of general linear reversible Boolean operators: synthesizing U , L, P and concatenating the circuits gives an implementation of A.

State of the art. In the unrestricted case the best algorithm reaching an asymptotic optimum is [23, Algo. 1] and produces circuits of size O(n 2 / log 2 (n)). This algorithm is for instance used in the Tpar and Gray-Synth algorithms [START_REF] Amy | Polynomial-time T-depth optimization of Clif-ford+T circuits via matroid partitioning[END_REF][START_REF] Amy | On the controlled-NOT complexity of controlled-NOT-phase circuits[END_REF] so any improvement over [START_REF] Patel | Optimal synthesis of linear reversible circuits[END_REF]Algo. 1] will also improve any quantum compiler that relies on it. In the restricted case the first proposed approach has been to transform the circuits given by an unrestricted algorithm with swap insertion algorithms to match the connectivity constraints [START_REF] Li | Tackling the qubit mapping problem for NISQ-era quantum devices[END_REF][START_REF] Pedram | Layout optimization for quantum circuits with linear nearest neighbor architectures[END_REF][START_REF] Wille | Look-ahead schemes for nearest neighbor optimization of 1D and 2D quantum circuits[END_REF]. To produce more efficient circuits, two concomitant papers proposed a modification of the Gaussian elimination algorithm [START_REF] Kissinger | CNOT circuit extraction for topologicallyconstrained quantum memories[END_REF][START_REF] Nash | Quantum circuit optimizations for NISQ architectures[END_REF]. They synthesize the operator column by column similarly to the Gaussian Elimination algorithm but they use Steiner trees to compute the shortest sequence of CNOT gates for the synthesis of one column. In [START_REF] Kissinger | CNOT circuit extraction for topologicallyconstrained quantum memories[END_REF] the authors compare their method based on Steiner trees against two compilers: Rigetti Computing's QuilC and Cambridge Quantum Computing's t|ket that both produced state of the art results on benchmarks published by IBM [START_REF] Cowtan | On the qubit routing problem[END_REF]. The benchmarks show a consequent savings in the total number of CNOT gates in favor of the Steiner tree method, so we consider that the work in [START_REF] Kissinger | CNOT circuit extraction for topologicallyconstrained quantum memories[END_REF] is state-of-the-art and we will compare solely to their algorithm.

Algorithm for an all-to-all connectivity

In this section we present our algorithm in the case of a complete connectivity between the qubits. We focus on the synthesis of a lower triangular operator L ∈ F n×n

2

. What follows can be straightforwardly extended to the case of upper triangular operators and to general operators using the LU decomposition. With an all-to-all connectivity one can avoid to apply the permutation P by doing a post-processing of the circuit that would transfer the permutation operation directly at the end of the total circuit. This can be done without any overhead in the number of gates.

A circuit implementing L can solely consist of "oriented" CNOTs, whose controlled qubit i and target qubit j satisfy i < j. The circuit given by the Gaussian elimination algorithm is an example. For this particular kind of circuits, a CNOT applied to a qubit k does not have any influence on the operations performed on the first k -1 qubits: removing such a CNOT will not modify the result of the synthesis of the first k -1 parities. We use this property to design a new algorithm where we synthesize L parity by parity and where we reuse all the information acquired during the synthesis of the first k parities to synthesize parity k + 1.

Given

L n-1 = L[1:n -1, 1:n -1]
(again using Matlab notation), a circuit C implementing the operator Ln-1 0 0 1 and considering that we want to synthesize the operator L = Ln-1 0 s 1 the core of our algorithm consists in adding a sequence of CNOTs to C such that we also synthesize the parity s of the n-th qubit. During the execution of C, applying a CNOT i → n will add the parity currently held by qubit i to the parity of qubit n without impacting the synthesis of the first n -1 parities. In other words, if we store in memory all the parities that appeared on all n-1 qubits during the execution of the circuit C, we want to find the smallest subset of parities such that their sum is equal to s. Then when a parity belonging to this subset appears during the execution of C, on qubit i for instance, we insert in C a CNOT i → n. We ultimately have a new circuit C that implements L.

The problem of finding the smallest subset of parities whose sum equals s can be recast as a classical cryptographic problem. Assuming that H ∈ F n-1×m 2 is a Boolean matrix whose columns correspond to the m available parities, any Boolean vector x satisfying Hx = s T gives a solution to our problem and the Hamming weight of x, wt(x), gives the number of parities to add, i.e, the number of CNOTs to add to C. We are therefore interested in an optimal solution of the problem minimize

x∈F m 2 wt(x)
such that Hx = s T .

(

) 1
Problem 1 is an instance of the syndrome decoding problem, a well-known problem in cryptography. The link between CNOT circuit synthesis and the syndrome decoding problem has already been established in [START_REF] Amy | On the controlled-NOT complexity of controlled-NOT-phase circuits[END_REF], yet it was used in a different problem for proving complexity results (under the name of Maximum Likelihood Decoding problem) and the authors did not pursue the optimization. The syndrome decoding problem is presented in more details in Section 3.1.

To summarize, we propose the following algorithm to synthesize a triangular operator L. Starting from an empty circuit C, for i from 1 to n perform the three following steps:

1. scan circuit C to compute all the parities available on a single matrix H, 2. solve the syndrome decoding problem Hx = s with s the parity of qubit i, 3. add the relevant CNOT gates to C depending on the solution obtained.

Provided that the size of C remains polynomial in n, which will be the case, then steps 1 and 3 can be performed in polynomial time and in practice in a very short amount of time. The core of the algorithm, both in terms of computational complexity and final circuit complexity, lies in Step 2.

Syndrome decoding problem

In its general form, the syndrome decoding problem is known to be NP-Hard [START_REF] Berlekamp | On the inherent intractability of certain coding problems[END_REF] and cannot be approximated by a constant factor [START_REF] Arora | The hardness of approximate optima in lattices, codes, and systems of linear equations[END_REF]. A good overview of how difficult the problem is can be found in [START_REF] Vardy | Algorithmic complexity in coding theory and the minimum distance problem[END_REF].

We give two methods for solving the syndrome decoding problem. The first one is an optimal one and uses integer programming solvers. The second one is a greedy heuristic for providing sub-optimal results in a short amount of time.

Integer programming formulation. The equality Hx = s is a Boolean equality of n lines. For instance the first line corresponds to

H 1,1 x 1 ⊕ H 1,2 x 2 ⊕ . . . ⊕ H 1,m x m = s 1 .
We transform it into an "integer-like" equality constraint. A standard way to do it is to add an integer variable t and to create the constraint

H 1,1 x 1 + H 1,2 x 2 + . . . + H 1,m x m -2t = s 1 .
If we write c = (1, ..., 1, 0, ..., 0) T ∈ N m+n and A = [H| -2I n ] then the syndrome decoding problem is equivalent to the integer linear programming problem

min x∈F m 2 ,t∈N n c T • [x; t] such that A[x; t] = s. (2)
A cost minimization heuristic. Although the integer programming approach gives optimal results, it is very unlikely that it will scale up to a large number of qubits. Moreover, to our knowledge the other existing algorithms proposed in the literature give exact results, they are complex to implement and their time complexity remains exponential with the size of the problem. We therefore have to consider heuristics to compute an approximate solution in a much shorter amount of time.

We use a simple cost minimization approach: starting with the parity s we choose at each iteration the parity v in H that minimizes the Hamming weight of v ⊕ s and we pursue the algorithm with the new parity v ⊕ s. The presence of the canonical vectors in H (as we start with the identity operator) is essential because they ensure that this method will ultimately converge to a solution.

A simple way to improve our heuristic is to mimic path finding algorithms like Real-Time A* [START_REF] Korf | Real-time heuristic search[END_REF]. Instead of directly choosing the parity that minimizes the Hamming weight, we look up to a certain horizon and we make one step in the direction of the most promising path. To control the combinatorial explosion of the number of paths to browse, we only expand the most promising parities at each level. We set the maximum width to m and the depth to k so that it represents at most m k paths to explore. With suitable values of m and k we can control the total complexity of the algorithm. A limitation of such a simple approach is that we can store the same path but with different parities order: we decided to ignore this limitation in order to keep a simple implementation.

Lastly, we introduce some randomness by change of basis and we solve the problem P Hx = P s for several change of basis matrices P . Repeating this several times for one syndrome decoding problem increases the chance to find an efficient solution. This technique has been proven to be efficient for a class of cryptographic algorithms called Information Set Decoding [START_REF] Prange | The use of information sets in decoding cyclic codes[END_REF], even though the complexity of these algorithms remains exponential.

Benchmarks

All the code is written in Julia and executed on the QLM (Quantum Learning Machine) located at ATOS/BULL. We generate random operators by generating random circuits with randomly placed CNOT gates. When the number of input gates is sufficiently large we empirically note that the operators generated represent the worst case scenario.

We first generate an average complexity for different problem sizes: for n = 1..200 we generated 20 random operators on n qubits with more than n 2 gates to reach with high probability the worst cases. We run our algorithms on this set of operators in the following cases:

with the integer programming solver (Coin-or branch and cut solver), with the cost minimization heuristic with unlimited width and depth 1, with the cost minimization heuristic with width 60 and depth 2, with the cost minimization heuristic with width 15 and depth 3, with the cost minimization heuristic (width=Inf, depth=1) and 50 random changes of basis, the "ISD" case.

In the case of the ISD experiment, due to its probabilistic nature, one can hope that repeating the complete synthesis several times and keeping the shortest circuit would improve the results. Yet the experiments show that it has a minor influence on the final result.

The results are given on Fig. 2. For clarity, instead of plotting the size of the circuits we plot the ratio between the size of the circuits given by our algorithms and the state of the art algorithm [START_REF] Patel | Optimal synthesis of linear reversible circuits[END_REF]Algo. 1]. We stopped the calculations when the running time was too large for producing benchmarks in several hours.

Overall, for the considered range of qubits and for all versions of our algorithm we outperform [START_REF] Patel | Optimal synthesis of linear reversible circuits[END_REF]Algo. 1]. The integer programming solver gives the best results with with a maximum gain of more than 40% but its scalability is limited: beyond 50 qubits it requires too much computational time. Using commercial softwares for reaching larger problem sizes would be interesting to confirm the tendency toward an increasing gain.

Concerning the cost minimization heuristic, it seems better to increase the depth of search than the width. With depth 3 and width 15 we have the best results for the range 70-125 with 30% of gain. Surprisingly the ISD based method with 50 random changes of basis works well until 60/70 qubits with more than 35% of gain. Then it seems that the number of random changes is not enough to search efficiently an optimal solution and ultimately after 150 qubits the random changes have no effect at all compared to the simpler heuristic with one try. It is possible to increase this number of random changes but this comes at the price of a longer computational time and the ISD method cannot compete with the other versions of the cost minimization heuristic.

As the number of qubits increases our method performs worse. We ran a few computations for much larger problems and the results are that [23, Algo. 1] produces shorter circuits whenever n goes approximately beyond 400. This raises the question of whether it is due to the method in itself or to the solution of the syndrome decoding that becomes less and less optimal as the problem size increases. We leave this question as future work.

We now look at the performance of the algorithms on a specific number of qubits, here n = 60, but for different input circuit sizes. This experiment reveals how close to optimal our algorithm is when we synthesize an operator for which we expect a small output circuit. The results are given on Fig. 3. As the ISD method produces the best results for this size of problem we only plot the results for this method. We also plot the line y = x that shows how far we still are from the optimal solution. Again we outperform the best algorithm in the literature even for small input circuits with more than 50% of savings when the input circuit is of size 100-300 gates, with a maximum saving of 60% for 200 gates.

Extension to an arbitrary connectivity

In this section we extend the algorithm to the case where the connectivity is not complete. First we present how to adapt our algorithm based on syndrome decoding for the synthesis of triangular operators, then we extend our method to the synthesis of any general operator.

Synthesis of a triangular operator

Let G be a qubit connectivity graph and L the lower triangular operator to synthesize. We require an ordering on the nodes of G such that the subgraphs containing only the first k nodes, for k = 1..n, are connected. As we need to synthesize both L and U we need in fact this property to be true for an ordering of the qubits and the reverse ordering. An Hamiltonian path in G is enough to have this property so for simplicity we assume that the ordering follows an Hamiltonian path in G.

Even though the native CNOTs in the hardware are CNOTs between neighbor qubits in the connectivity graph, it is possible to perform an arbitrary CNOT gate but this requires more local CNOT gates. Given a target qubit q t and a qubit control q c and assuming we have a path (q c , q 1 , ..., q k , q t ) in the graph connecting the two nodes (such path always exists with the assumption we made above), it is possible to perform the CNOT q c → q t with max(1, 4k) CNOTs. An example for 4 qubits (with k = 2) is given (3) Problem 3 can be recasted again as an integer linear programming problem: we only have to change the value of c. We also propose a greedy heuristic for solving quickly and approximately the problem: we define the "basis cost" of implementing s as the sum of the costs of each canonical vector whose component in s is nonzero. Let bc(s) be this cost. Our greedy approach consists in finding among the parities of H the parity v (column i of H) that minimizes the cost

c[i] + bc(s ⊕ v).
This approach gives a good trade-off between zeroing the most costly components of s and applying parities at a very high cost. Again we can repeat the algorithm with random changes of basis to find a better solution. Especially we focused on computing bases for which the canonical vectors have the lowest possible costs.

Nonetheless, compared to the all-to-all case, solving the weighted syndrome decoding problem is not the only computational core for controlling both the quality of the solution and the computational time. Another key task lies in the enumeration of the available parities. As we will see, it is possible to generate more parities for one syndrome decoding problem instance and this increases the chances to get a low-cost solution.

Listing the parities available. Until now we set the weighted syndrome decoding instances by computing the parities appearing during the synthesis and by using the template in Fig. 4 to estimate their costs. This is in fact inefficient because it ignores some specificities of the problem:

-It is possible to add multiple parities in one shot using the template in Fig. 4. -There is not necessarily one unique path in G between the control qubit and the target qubit.

= |x t ⊕ x c ⊕ x 1 ⊕ x 2 ⊕ x 3 ⊕ x 4
Fig. 5: Fan in CNOT in LNN architecture More precisely, the template shown in Fig. 4 is the best to our knowledge, in terms of size, to apply solely the parity on qubit q c to qubit q t . However it is possible to apply any parity q t ← q t ⊕ q c ⊕ k i=1 α i q i with α i ∈ {0, 1} using less CNOTs than required for applying only q c . In fact the less costly linear combination of parities is the complete combination q c ⊕ q 1 ⊕ ... ⊕ q k , showing that 2k + 1 CNOTs are enough. Removing any parity from this combination requires 2 additional CNOTs per parity except for the qubit q k that needs only one extra CNOT. An explanatory template on 6 qubits (k = 4) is given on Fig. 5. For any parity at a distance k of the target qubit, there is at most 2 k-1 different linear combinations possible and just as many new parities to consider. Moreover the path between the control qubit and the target qubit matters as a different path will result in different linear combinations of parities. A slight modification of the A* algorithm is enough to compute all the shortest paths between two nodes in a graph.

Even for a small number of qubits the number of parities becomes quickly intractable. The number of linear combinations along a path increases exponentially with the length of the path as the number of paths for most of the architectures -a grid for instance. In practice we control the total number of parities by favoring paths over the choices in the linear combinations. This option is empirically justified but a more detailed analysis could be made. For one path we only consider the less costly linear combination, i.e, the one that adds all the parities on the way. On the other hand if possible we go through all the shortest paths between one control qubit and one target qubit. We introduce a parameter P max equal to the maximum number of shortest paths we consider between two qubits.

Synthesis of a general operator

The extension of the synthesis from triangular to general operator is not as straightforward as in the all-to-all connectivity case. We cannot simply write A = P LU and concatenate the circuits synthesizing L and U and ultimately permuting the qubits. If we want to use this algorithm as a sub-task of a global circuit optimizer for NISQ architectures we cannot afford to swap the qubits because it could break the optimizations done in the rest of the circuit.

To avoid the permutation of the qubits we have to transform the matrix A by applying a pre-circuit C such that CA = LU . Then the concatenation of C -1 and the circuits synthesizing L and U gives a valid implementation of A.

Computation of C. If

A is invertible, which is always the case, then it admits an LU factorization if and only if all its leading principal minors are nonzero. We propose an algorithm for computing C exploiting this property while trying to optimize the final size of C. We successively transform A such that every submatrix A[1:i, 1:i] is invertible. By construction when trying to make A[1 : k, 1 : k] invertible for some k we have

A[1:k-1, 1:k-1] invertible. If A[1 : k, 1 : k]
is invertible then we do nothing, otherwise we look in the parities A[k + 1:n,1:k] those who, added to A[k, 1 : k], make A[1 : k, 1 : k] invertible. By assumption A is invertible so there is at least one such row that verifies this property. Then among the valid parities we choose the closest one to qubit k in G. We can add all the parities along the path because by assumption they belong to the span of the first k -1 rows of A[1 : k, 1 : k] so it has no effect on the rank of A[1 : k, 1 : k].

Choice of the qubit ordering. A last optimization can be performed by changing the qubits ordering. The algorithm we have presented for synthesizing a triangular operator is still valid up to row and column permutations. Thus, given a permutation P of the qubits, one can synthesize P -1 LP by applying our algorithm with the order given by P . Then, instead of computing a circuit C such that CA = LU we search for a circuit C satisfying P -1 CAP = LU and

CA = P LP -1 P U P -1 = L U
where L and U can be synthesized using our algorithm. Searching for such C can be done using our algorithm on A[P, P ] (in Matlab notation, i.e. the reordering of A along the vector P ).

This means that we can choose P such that the synthesis of L and U will yield shorter circuits. Empirically we noticed that when synthesizing the k th parity of L it is preferable to have access to the parities appearing on qubits k -1, k -2 etc. in priority for two reasons: first because they can modify more bits on the k th parity and secondly because it is likely that there will be much more parities available, increasing the chance to have an inexpensive solution to the weighted syndrome decoding problem. Intuitively we want the ordering of the qubits to follow at least an Hamiltonian path in G = (V, E) which would match the previous restriction on the ordering we formulated at the beginning of the section. We formulate the best ordering π : V → [ [1, n]] as a solution of the Minimum Linear Arrangement problem minimize

π (u,v)∈E w uv |π(u) -π(v)| (4)
where w uv is the weight of the edge connecting u and v in the graph. Here we want to give priority to neighbors in the hardware: the nodes must be as close as possible in the hardware if their "numbers" are also close. A way to do so is to solve the MinLA problem, not in the hardware graph, but in the complete graph with suitable weights. Namely w ij must be large when i, j are neighbors in the hardware and w ij must be smaller if i, j are at distance 2 etc. The MinLA problem has already been used for qubit routing [START_REF] Pedram | Layout optimization for quantum circuits with linear nearest neighbor architectures[END_REF] and the problem is in general NP-Hard [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NP-Completeness[END_REF]. In our case we did not use any heuristic for solving this problem: all the architectures chosen to test our method have Hamiltonian paths and we simply chose manually a suitable ordering of the qubits. Fig. 6 features the choices we made for some architectures: nodes are labeled with their position in the linear arrangement. We leave as a future work the inclusion of the solution of the MinLA problem in our algorithm.

Benchmarks

We compare our method against the best algorithm in the literature [START_REF] Kissinger | CNOT circuit extraction for topologicallyconstrained quantum memories[END_REF] whose source code is available on the PyZX Github repository [START_REF] Kissinger | PyZX[END_REF]. For each architecture considered in their implementation we generate a set of 100 random operators and perform the synthesis using the Steiner trees. Their algorithm provides an optimization using genetic algorithms but this implements the circuit up to a permutation of the qubits. As we focus on implementing exactly the operator we considered their algorithm without this extra optimization. Our own algorithm is implemented in Julia. The experiments have been carried out on one node of the QLM (Quantum Learning Machine) located at ATOS/BULL. We set a time limit of 10 minutes for the synthesis of an operator. We recall that P max is the maximum number of shortest paths considered between two qubits. We also set N iter_syndrome to be the number of iterations for the solution of a decoding syndrome and N iter the number of times that the synthesis has been repeated. The values of the parameters for the different problem sizes are the following:

n < 36, P max = Inf, N iter = 100 and N iter_syndrome = 1, The results are summarized in Table 1. Columns 3 and 4 give the average size of the generated circuits for the method using Steiner trees in [START_REF] Kissinger | CNOT circuit extraction for topologicallyconstrained quantum memories[END_REF] and our algorithm based on syndrome decoding. The next columns detail the savings: the mean saving, the minimum saving (negative saving means that our algorithm performs worse), the maximum saving and the proportion of operators for which our circuit is actually shorter than the one provided by the state-of-theart method. The last two columns give the average time required to perform the synthesis of one operator (all iterations included for our algorithm).

We can expect our algorithm to behave better if there are more connections between the qubits. When the connectivity is as limited as possible, for instance with an LNN architecture, our algorithm does not outperform the algorithm based on Steiner trees. Except for 6% of the operators where we have a slight gain (less than 8%) we provide circuits with more gates, up to 19%. For other architectures the results are more promising. In the case of the 9-qubit square there is a lot of variance in the results: depending on the operator we can have a gain of 40% or a loss of 25%. Overall we still manage to produce a shorter circuit 66% of the time. For larger square architectures, we outperform the state-of-theart algorithm consistently with increasing savings, between 10% for the 25 qubits square to 18% for the 100 qubits square. When adding diagonal connections in the square architectures the results are even better. This shows that improving just slightly the connectivity can improve consistently the results of our algorithm compared to the state of the art method. Finally on specific architectures we also provide better results. The results for Rigetti's chip are not as good as for IBM's chips essentially because the connectivity is still close to a straight line, otherwise we manage to have a saving of around 20% for both IBM-QX5 and IBM-Tokyo chips.

Conclusion

We have presented a new framework for the synthesis of linear reversible circuits. We exploit the specific structure of triangular operators to transform the synthesis into a series of syndrome decoding problems, which are well-known problems in cryptography. Using an LU decomposition we can synthesize any quantum operator in the case of an all-to-all connectivity. Benchmarks show that we outperform a state-of-the-art algorithm for intermediate sized problems (n < 400). Our heuristics for solving the syndrome decoding problem are efficient but could be still improved, both in circuit size and computational time. For instance, some quantum algorithms have been proposed for solving the syndrome decoding problem via the Information Set Decoding algorithm [START_REF] Bernstein | International Workshop on Post-Quantum Cryptography[END_REF][START_REF] Kachigar | Quantum information set decoding algorithms[END_REF][START_REF] Kirshanova | Improved quantum information set decoding[END_REF], which gives the possibility of designing a hybrid quantum/classical compiler for this particular synthesis problem.

Then we have highlighted the robustness of our framework by extending it to an arbitrary connectivity graph having a Hamiltonian path. With a suitable pre-processing of the matrix we transform the problem into a series of weighted syndrome decoding problems. Except for the LNN architecture whose connectivity is too sparse, we consistently outperform existing algorithms by a percentage that increases with the number of qubits. As a future work, we can study how to extend our method to the case where the connectivity graph does not have a Hamiltonian path, similarly to [START_REF] Kissinger | CNOT circuit extraction for topologicallyconstrained quantum memories[END_REF].

Fig. 1 :

 1 Fig. 1: Example of qubit connectivity graphs from existing architectures

Fig. 2 :Fig. 3 :

 23 Fig. 2: Average performance of the Syndrome Decoding based algorithms versus the state of the art [23, Algo. 1].

  Fig 4.

Fig. 4 :m 2 c

 42 Fig. 4: CNOT in LNN architecture

-

  n = 36, P max = Inf, N iter = 20 and N iter_syndrome = 1, n = 49, P max = 10, N iter = 100 and N iter_syndrome = 1, n > 49, P max = 1, N iter = 100 and N iter_syndrome = 1.

Table 1 :

 1 Performance of our Syndrome Decoding based algorithm vs Steiner trees algorithm[START_REF] Kissinger | CNOT circuit extraction for topologicallyconstrained quantum memories[END_REF] for several architectures

	Architecture	#	Steiner Syndrome [17]	Saving Mean Min. Max. Positive	tSt (s) tSy (s)
	9q Square	9	60	56	6% -25.5% 40.6% 66%	0.01 0.16
	Rigetti 16q	16 272	245	10% -6% 23.1% 97% 0.022 1.6
	IBM QX 5	16 245	195	20.2% 10% 28.7% 100% 0.019	2
	16q Square	16 205	183	10.7% -7.1% 33%	93%	0.02	1.6
	19q Line	19 455	470	-6.7% -19.4% 7.9%	6%	0.045 4.5
	IBM Q20 Tokyo 20 292	239	18.1% 8.7% 26.9% 100% 0.025 1.8
	25q Square	25 512	458	10.6% 3.5% 19.1% 100% 0.04	19
	25q Sq. + diag. 25 410	324	21% 10.7% 28.3% 100% 0.035	8
	36q Square	36 1067	891	16.5% 11% 22.4% 100%	0.1	67
	36q Sq. + diag. 36 861	667	22.5% 18.2% 26.6% 100% 0.09	22
	49q Square	49 1981	1662	16% 11.9% 20.7% 100%	0.2	420
	49q Sq. + diag. 49 1607	1246	22.4% 19% 25.2% 100% 0.19 114
	64q Square	64 3374	2812	16.6% 13.9% 18.9% 100% 0.54	79
	81q Square	81 5363	4447	17% 14.4% 19.2% 100% 1.04 192
	100q Square 100 8148	6666	18.2% 16.8% 19.8% 100%	2.1	449

Acknowledgment

This work was supported in part by the French National Research Agency (ANR) under the research project SoftQPRO ANR-17-CE25-0009-02, and by the DGE of the French Ministry of Industry under the research project PIA-GDN/QuantEx P163746-484124. We thank Bertrand Marchand for comments on the manuscript.


Charles Jr

Tovignan

Relation entre le caractère stay green et stabilité du rendement grain de 250 lignées de sorgho cultivées dans divers environnements. [Memoire]

RESULTS & DISCUSSION

Haiti's diverse soils and microclimates impact sorghum performance across environments, complicating genotype selection. Therefore, choosing stable genotypes and the physiological determinism associated with them is crucial for breeding programs. Conventional phenotyping for stability is time-consuming and costly. Genomic prediction offers a cost-effective approach, aiming to shorten selection cycles and enhance genetic gains. Limited studies exist on genomic prediction and other selection methods, specifically phenomic selection on sorghum. This paper aims to optimize genomic prediction models for predicting sorghum genotype breeding values in new environments and, alternatively, to test the potential of phenomic selection. In this context, optimization involves selecting the best models and selection indexes.

Genomic prediction and characterization of grain yield and yield stability in sorghum

This study is made possible with the support of the American People provided to the Feed the Future Innovation Lab for Crop Improvement. The contents are the sole responsibility of the authors and do not necessarily reflect the views of USAID or the United States Government Support was also provided for our trials and the first author

GXE Phenomic Physiologic

Genomic prediction models, training strategies, and objectives

Genetic materials, experimental design, and phenotypic data y = X b + Z u + e with: y, vector of phenotypic data; X, the design matrix for fixed genetic effects of SNPs; b, vector of SNPs fixed effects; Z, the design matrix for the random genetic effects of SNPs; u, the vector of random genetic effects of SNPs; and e, the vector of errors..

Fig. 3: Comparison of genomic prediction to phenomic prediction

The inheritable wavelengths were insufficient to achieve high levels of prediction 45 days after sowing (Fig. 3).

Brix has an influence on all three traits; however, stay green does not exert any influence on the Kang index and superiority index (Fig. 2).

The superiority index and the average yield are the two best predictors of yield across environments. The GLUP model performs better for these two indices(Fig. 1).

▪Fig. 2 :

 2 Fig.2: Multiple regression for the superiority and Kang index based on the brix and stay-green traits

scholarship by the Haitian Ministry of Agriculture, Natural Resources, and Rural Development, and the Inter American Development Bank


Giorgio Maria Di

Pascal Elbaz
email: elbaz.pascale@isitparis.eu

Rodolfo Maslias
email: rodolfos.maslias@europarl.europa.eu

Federica Vezzani
email: federica.vezzani@unipd.it

YourTerm: A Collaborative & FAIR European Project

Theme of the Panel

The Terminology Coordination Unit (TermCoord) [START_REF] Vezzani | Methodology for the standardization of terminological resources: Design of TriMED database to support multi-register medical communication[END_REF] is a supporting service to the translation units of the Directorate-General for Translation (DG TRAD) [START_REF] Callegari | Management of diastratic variation in medical settings: On the collaboration within the YourTerm MED project[END_REF] of the European Parliament. The main role of the TermCoord is to assist translators and facilitate terminology research and terminology management in the translation units, as well as to increase the European Parliament contribution to the EU terminology database IATE3 (Interactive Terminology for Europe), the largest multilingual terminological database openly available.

Among other activities, TermCoord launched the "Terminology without Borders" projectabbreviated with YourTerm -and deployed an official separate website of the European Parliament. [START_REF] Elbaz | Quelles Ressources Terminologiques et Quelles Méthodes pour la Formation des Apprenant.es Traducteur.rices Professionnel.les en 2021 ? Analyse d'un Projet Académique Collaboratif[END_REF] This project reflects and supports the goal of DG TRAD to communicate with citizens in a clear and comprehensible language. The main aim of YourTerm is to enhance communication in different domains by means of several multilingual sub-projects (10 in total), for example for the medical domain (YourTerm MED), [START_REF] Wilkinson | The FAIR guiding principles for scientific data management and stewardship[END_REF] or the environment field (YourTerm ENVI).6 Each one of these multilingual sub-projects produces high-quality terminological resources thanks to the collaboration and coordination of several users from different universities and research institutes. Beyond the objective to collect terminology that is inserted in IATE after revision by external experts and validation by the terminologists of the EP, it provides to the terminology and linguistic departments of the collaborating universities a tool to practice terminology related to the EU legislative activity.

In 2021, the University of Padova (UNIPD) together with the Aristotle University of Thessaloniki (AUTH), the Institute of Intercultural Management and Communication (ISIT) of Paris, and TermCoord have worked together for the design, implementation, and testing of a collaborative multilingual terminological management system (TMS), named FAIRterm,7 to support the work of the terminologists in each sub-project [1, 2, 3].

In particular, Masters' students from AUTH and ISIT had the opportunity to work collaboratively, in three languages (English, French, Greek), in the domain of Natural Language Processing. During this innovative project they had to build ad hoc corpora, extract terms, find definitions or contexts, consult an expert of the domain and finally store all terminological entries in the new collaborative TMS [4].

FAIRterm is the first terminological application designed in order to meet the FAIR principles (Findability, Accessibility, Interoperability, and Reusability) [5] and implemented according to the de iure ISO TC/37 SC/3 standards8 currently in force for terminology management. The theoretical premises behind its design are based on the paradigm of the "FAIR terminology" whose goal is to provide a model for the optimal and efficient organization of terminological data. In May 2022 all participating Universities attended a workshop on the use of FAIRterm that became the compulsory tool to be used for the projects to ensure uniformity and to increase similar joint projects using the collaborative features of the application.

In this panel, we want to discuss the challenges of this project from technical and pedagogical perspectives, such as:

• How to integrate the FAIRterm resource in CAT tools?

• Would it be possible to integrate an on-the-fly search of terminological records into a Computer Assisted Interpreting tool?

• Would it be possible to search for relevant terms and automatically update the translation memory files of the translators at EU?

• Could a terminological management system be used to autocomplete/suggest queries during the translation process?

• How reliable are the terms produced within a collaborative platform and how to avoid/manage duplicate records?

• How can we assure a correct validation of the terms if more users are working on the same record?

• What are the factors to consider when coordinating a collaborative project in an academic environment?

• How students reacted to the new tool in a UX design perspective, and did it facilitate their collaboration?

• What are the intercultural and technical challenges for the coordinators of such projects?

• How to enhance students critical thinking ability toward their terminological work as a whole, and toward the terminological management tool in particular?

• How to help students have a good view of the needs of the European users of the terminological resources they build and adapt to these needs.

• When shall a tool oriented collaborative terminological project be proposed to students in a global terminological and translation training?

https://termcoord.eu/

https://www.europarl.europa.eu/translation/

https://iate.europa.eu/

https://yourterm.eu/

https://yourterm.eu/yourterm-med/

https://yourterm.eu/yourterm-envi/

https://shiny.dei.unipd.it/fairterm/

https://www.iso.org/committee/48136.html

2 Panelist and Their Roles Chair of the Panel: Rodolfo Maslias. Panelist: Giorgio Maria Di Nunzio. Panelist: Federica Vezzani. Panelist: Pascale Elbaz. Panelist: Elpida Loupaki.