Ugo Nzongani
email: ugo.nzongani@universite-paris-saclay.fr

Pablo Arnault
email: pablo.arnault@inria.fr

Adjustable-depth quantum circuit for position-dependent coin operators of discrete-time quantum walks

, which achieves a depth that is linear at the cost of introducing an exponential number of ancillas. In this paper, we provide an adjustable-depth quantum circuit for the exact implementation of the position-dependent coin operator. This adjustable-depth circuit consists in (i) applying in parallel, with a linear-depth circuit, only certain packs of coin operators (rather than all of them as in the original linear-depth circuit [1]), each pack contributing linearly to the depth, and in (ii) applying sequentially these packs, which contributes exponentially to the depth.

I. INTRODUCTION

Quantum walks are models of quantum transport on graphs [START_REF] Kempe | Quantum random walks: An introductory overview[END_REF][START_REF] Venegas-Andraca | Quantum walks: a comprehensive review[END_REF]. They exist both in continuous time [START_REF] Farhi | Quantum computation and decision trees[END_REF] and in discrete time [START_REF] Feynman | Quantum Mechanics and Path Integrals[END_REF][START_REF] Aharonov | Quantum random walks[END_REF][START_REF] Bialynicki-Birula | Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata[END_REF]. In terms of computer science, quantum walks are a model of computation, which has been shown to be universal, both in the continuous- [START_REF] Childs | Universal computation by quantum walk[END_REF][START_REF] Childs | Universal computation by multiparticle quantum walk[END_REF] and in the discrete-time case [START_REF] Lovett | Universal quantum computation using the discrete-time quantum walk[END_REF], that is, any quantum algorithm can be written in terms of a quantum walk; moreover, many algorithms solving a variety of tasks have been conceived with quantum walks [START_REF] Rivosh | Coins make quantum walks faster[END_REF][START_REF] Ambainis | Quantum walk algorithm for element distinctness[END_REF][START_REF] Tulsi | Faster quantum-walk algorithm for the twodimensional spatial search[END_REF][START_REF] Roget | Grover search as a naturally occurring phenomenon[END_REF][START_REF] Fredon | Quantum spatial search with electric potential: Longtime dynamics and robustness to noise[END_REF]. In terms of physics, quantum walks are particularly suited to simulate quantum partial differential equations such as the Schrödinger equation or the Dirac equation [START_REF] Strauch | Relativistic quantum walks[END_REF][START_REF] Shikano | From discrete time quantum walk to continuous time quantum walk in limit distribution[END_REF][START_REF] Arrighi | The Dirac equation as a quantum walk: higher dimensions, observational convergence[END_REF][START_REF] D'ariano | Quantum cellular automata and free quantum field theory[END_REF][START_REF] Arnault | Discrete-time quantum walks as fermions of lattice gauge theory[END_REF][START_REF] Di Molfetta | A quantum walk with both a continuous-time limit and a continuous-spacetime limit[END_REF][START_REF] Arnault | Clifford algebra from quantum automata and unitary Wilson fermions[END_REF][START_REF] Debbasch | Action principles for quantum automata and Lorentz invariance of discrete time quantum walks[END_REF][START_REF] Arnault | A single-particle framework for unitary lattice gauge theory in discrete time[END_REF] -the latter being the equation of motion for matter particles which are both quantum and relativistic -, or models of solid-state physics [START_REF] Arnault | Quantum walks in weak electric fields and Bloch oscillations[END_REF].

Discrete-time quantum walks (DQWs), in particular, are discretizations of the Dirac equation which respect both unitarity -as continuous-time quantum walks (CQWs) -and strict locality of the transport (contrary to CQWs), that is, concerning the latter point, they preserve relativistic locality on the lattice [START_REF] Debbasch | Action principles for quantum automata and Lorentz invariance of discrete time quantum walks[END_REF][START_REF] Arrighi | Discrete Lorentz covariance for quantum walks and quantum cellular automata[END_REF][START_REF] Bisio | Quantum walks, deformed relativity and Hopf algebra symmetries[END_REF][START_REF] Debbasch | Discrete geometry from quantum walks[END_REF]. These DQWs combine, as basic ingredients, shifts on the lattice which depend on the internal state of the particle, together with internal-state rotations, called coin operators, that "reshuffle the cards" regarding whether one goes in one direction or another. The Dirac equation coupled to a variety of gauge fields has been shown to be simulatable with DQWs having a coin operator which depends on the position of the walker on the graph [START_REF] Di Molfetta | Quantum walks as massless Dirac fermions in curved space[END_REF][START_REF] Di Molfetta | Quantum walks in artificial electric and gravitational fields[END_REF][START_REF] Arnault | Landau levels for discretetime quantum walks in artificial magnetic fields[END_REF][START_REF] Arnault | Quantum walks and discrete gauge theories[END_REF][START_REF] Arnault | Quantum walks and non-Abelian discrete gauge theory[END_REF][START_REF] Arnault | Quantum walks and gravitational waves[END_REF][START_REF] Arrighi | Quantum walking in curved spacetime: (3+1) dimensions, and beyond[END_REF][START_REF] Arrighi | Dirac equation as a quantum walk over the honeycomb and triangular lattices[END_REF][START_REF] Márquez-Martín | Electromagnetic lattice gauge invariance in two-dimensional discrete-time quantum walks[END_REF][START_REF] Jay | Dirac quantum walks with conserved angular momentum[END_REF]. Position-dependent coin operators also arise when considering randomly chosen coin operators, which are a model of noise in DQWs [START_REF] Kendon | Decoherence can be useful in quantum walks[END_REF][START_REF] Kendon | Decoherence in quantum walks -a review[END_REF][START_REF] Schreiber | Decoherence and disorder in quantum walks: From ballistic spread to localization[END_REF][START_REF] Ahlbrecht | Asymptotic behav-ior of quantum walks with spatio-temporal coin fluctuations[END_REF][START_REF] Di Molfetta | Discrete-time quantum walks in random artificial gauge fields[END_REF][START_REF] Arnault | Quantum simulation of quantum relativistic diffusion via quantum walks[END_REF].

In Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], we have presented different quantum cir-cuits that achieve the implementation of a DQW on the line with such a position-dependent coin operator. In this paper, we propose a family of quantum circuits with adjustable depth, parametrized by a parameter m ∈ N, the extremes of which are (i) the naive circuit of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF] for m = 0, which means that all coin operators are implemented sequentially, and (ii) the linear-depth circuit of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF] for m = n (where n is the number of qubits used to encode the position of the walker in base 2), which means that all coin operators are implemented in parallel. A higher (lower) m means that more (less) coin operators are implemented in parallel, so that one can choose m as best suited for the experimental platform, knowing that a higher m, and hence a smaller depth, requires more ancillary qubits.

In Sec. II, we recall the system on which we work, which is that of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], namely, a DQW on the cycle with N = 2 n nodes, n ∈ N. Such a DQW is made of two operations: a coin rotation C (n) , and a coin-dependent shift operation S (n) . Still in Sec. II, we recall how to implement S (n) with a quantum circuit. In Sec. III, we introduce our adjustable-depth quantum circuit for the implementation of the position-dependent coin rotation C (n) . The idea of this circuit is to (i) apply in parallel, with a linear-depth circuit such as that introduced in Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], only certain packs of coin operators (rather than all of them as in the original linear-depth circuit [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF]), each pack contributing linearly to the depth, and to (ii) apply sequentially these packs, which contributes exponentially to the depth. In Sec. IV, we implement our adjustable-depth quantum circuit on IBM's QASM, the classical simulator of IBM's quantum processors. In Sec. V, we conclude and discuss our results.

II. FRAMEWORK

A. The walk

The system we consider is the same as that of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], namely, a DQW on a cycle with N = 2 n nodes, n ∈ N. Let us briefly recall the features of this system. Each node, labeled as k = 0, . . . , N -1, is associated to a position quantum state |k . Let H pos. be the 2 n -dimensional Hilbert space spanned by the position basis {|k } k=0,...,N -1 . The quantum state of the DQW has an additional, "internal" degree of freedom, which is called the coin. Such a coin belongs to a two-dimensional Hilbert space H 0 , the basis of which is (|↑ , |↓ ) ≡ (|0 , |1 ) ≡ ((1, 0) , (0, 1) ), where denotes the transposition. The total Hilbert space to which the quantum state belongs is therefore H := H pos. ⊗ H 0 . Such a quantum state at time j ∈ N decomposes as follows on the basis of H,

|ψ j = N -1 k=0 ψ ↑ j,k |k |0 + ψ ↓ j,k |k |1 , (1)
where the complex numbers ψ ↑ j,k and ψ ↓ j,k are the coefficients of the decomposition.

The evolution of the quantum state, Eq. ( 1), is governed by the following dynamics,

|ψ j+1 = W (n) |ψ j , (2)
where the walk operator W (n) is composed of two operations,

W (n) := S (n) C (n) . (3)
The first operation is a possibly position-dependent total coin operator,

C (n) := N -1 k=0 |k k| ⊗ C k , (4)
where each C k is a coin operator, that is, here, a 2 × 2 complex matrix acting on H 0 . The second and last operation is a coin-dependent shift operator,

S (n) := N -1 k=0 |k -1 mod N k| ⊗ |0 0| + |k + 1 mod N k| ⊗ |1 1| . (5)

B. The encoding in base 2

What is the minimum number of entangled qubits, i.e., of wires, that we need in our quantum circuit in order to encode the N = 2 n nodes of the cycle: this minimum number is n, and we call these qubits position qubits.

This naturally provides a base-2 encoding of the position of the walker. Let |k 2 be the writing of |k in base 2, that is:

|k ≡ |k 2 ≡ |b n-1 ...b 0 , (6)
where b p = 0 or 1 with p = 0, ..., n -1, such that k = n-1 p=0 b p × 2 p . One can thus rewrite Eqs. ( 4) and [START_REF] Feynman | Quantum Mechanics and Path Integrals[END_REF] as

C (n) ≡ 2 n -1 k=0 |k 2 k 2 | ⊗ Ck2 , (7)
where

Ck2 := C k , (8)
and,

S (n) ≡ 2 n -1 k=0 |(k -1 mod N ) 2 k 2 | ⊗ |0 0| + |(k + 1 mod N ) 2 k 2 | ⊗ |1 1| . (9)

C. Final remarks

In order to be able to implement suck a walk W (n) with a quantum circuit, one has to be able to implement the two operators C (n) and S (n) . There are different ways of implementing the coin-dependent shift operator S (n) using a quantum circuit, as recalled in Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF]. The aim of this paper is the quantum-circuit implementation of a position-dependent total coin operator C (n) with a circuit having a depth that can be adjusted at will, which is the subject of the next section.

As mentioned in the introduction, the general idea of the adjustable-depth circuit we introduce in this paper is to (i) apply in parallel, with a linear-depth circuit such as that introduced in Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], only certain packs of coin operators Ck2 (rather than all of them as in the original linear-depth circuit [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF]), each pack contributing linearly to the depth, and to (ii) apply sequentially these packs, which contributes exponentially to the depth.

The total number of coin operators Ck2 is 2 n . The size of the packs, i.e., the number of coin operators that we apply in parallel, is the tunable parameter of our model, and we write it as a power of 2 to simplify the discussion, that is, we write it M = 2 m , m ∈ N. The number of packs is thus 2 n /2 m = 2 n-m . Let us call U (n,m) i , i = 0, ..., 2 n-m -1, the circuit that implements the ith pack of coin operators Ck2 in parallel. The total circuit, which we are going to show implements the coin operator C (n) , thus reads

U (n,m) := 2 n-m -1 L i=0 U (n,m) i , ( 10
)
where the superscript L means that the terms are multiplied in increasing index order from right to left. The number i is called the stage number.

The ancillae, and more precisions

The number of ancillae necessary to apply each U (n,m) i is: 2 m ancillary position states, and 2 m -1 ancillary coin states. We depict in Fig. 1 the different registers used to implement our circuit U (n,m) . In Fig. 2, we illustrate Eq. ( 10). As in Eq. ( 26) of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], the fact that U (n,m) does the job, i.e., implements C (n) , means that it coincides with C (n) on the Hilbert space spanned by the position qubits plus the coin qubit, provided that we have correctly initialized the ancillary qubits. We are going to detail this in the next paragraph.

Including the ancillae means extending the total Hilbert space H introduced in Sec. II into H := H ⊗ H coins ⊗ H pos. . The last two Hilbert spaces contain respectively the quantum states of the coin and position ancillary qubits. A correctly initialized quantum state |S ∈ H is a state that is arbitrary on H, but has to be equal to |s = 0 |b = 0 on H coins ⊗ H pos. , that is,

|S := 2 n -1 k=0 s0=0,1 α k,s0 |k 2 |s 0 |s = 0 |b = 0 , (11)
with the α k,s0 's being complex numbers such that

2 n -1 k=0 s0=0,1 |α k,s0 | 2 = 1.
As we said above, that U (n,m) does the job, i.e., implements C (n) , means the following,

U (n,m) |S = C (n) ⊗ I 2 (2 m -1) ⊗ I 2 (2 m ) |S . ( 12
)
Notice that in Eq. [START_REF] Rivosh | Coins make quantum walks faster[END_REF] we have chosen to represent, in that order: the state of the position qubits |k 2 , the principal coin |s 0 , the ancillary coins |s , and finally the ancillary position |b . This choice has been made for a clearer formulation of the equations. In contrast, in the diagrammatic representations of our circuits, see Figs. 1 and2, the principal coin was placed under the ancillary coins. This choice has been made for a better visual understanding of the functioning of the circuits. , as written in Eq. ( 10).

B. General structure of U (n,m) i : that of U (n) lin. of Ref. [1]
As the circuit

U (n) lin. in Ref. [1], each U (n,m) i is made of several operations; more precisely, it reads U (n,m) i := Q (n,m) 1,i † Q (n,m) 2 † Q (n,m) 0,i Q (n,m) 2 Q (n,m) 1,i . (13)
Let us first briefly recall the operating principle of U . Equation ( 13) is illustrated in Fig. 3.

Now, the central operation, Q (n,m) 0,i , is the same as Q (n) 0 in U (n)
lin. of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], except that we only apply 2 m coin operators Ck2 in parallel instead of 2 n . More precisely, Q

(n,m) 0,i reads Q (n,m) 0,i := I 2 n ⊗   2 m -1 L k=0 K b k ,s k (C i2 m +k )   , (14)
where I 2 n is applied on the position qubits, and where K a,b (C) corresponds to applying the one-qubit gate C on qubit |b while controlling it on qubit |a (we apply C only if a = 1). In Fig. 4, we illustrate the definition of Q

(n,m) 0,i in Eq. [START_REF] Roget | Grover search as a naturally occurring phenomenon[END_REF]. For m = n, we have a single operator Apart from the fact that we have less ancillae for m < n than for m = n, and that Q (n,m) 0,i only applies 2 m coin operators Ck2 in parallel, the only difference between U (n,m) i and the circuit U (n) lin. of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], is in the operation that initializes the ancillary positions, namely, Q

U n,m=n 0 = U (n) lin. .
(n,m) 1,i
. Let us explain this difference.

Previously, the operator [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF] encoded the ancillary position |b for any position |k 2 . Now, as we apply in parallel packs of 2 m coin operators only, one must only encode the ancillary position for 2 m position states only, not all of them.

Q (n) 1 of Ref.
We call current position state at stage i a position state |k

(i) 2
such that after stage i the coin operator Ck (i) 2 has been applied to the principal coin state |s 0 . There are thus 2 m current position states at each stage i, namely, those for which k (i) = i2 m , ..., (i + 1)2 m -1 (see Eq. ( 14) and Fig. 4).

In Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], the ancillary position is encoded for every position state thanks to the fact that the qubit which encodes the least significant bit of the ancillary position, namely |b 0 , is flipped using a NOT gate, before the application of the series of controlled-SWAP operations (see the operation Q 11 in Fig. 15 of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF]).

Main explanations

To encode only the current position states at each stage i, one has to flip the same qubit |b 0 but only for these position states. Let us explain how to do that. Let |k 2 = |b n-1 ...b 0 be the input position state. The current position states at any stage i have their last n -m bits in common in their binary writing, which is of the form k

(i) 2 := b (i) n-1 ...b (i) 0 .
More specifically, it turns out that these n -m bits in common actually code for the binary writing i 2 of i, that is, we have

i 2 := h n-m-1 ...h 0 (15a) := b (i) n-1 ...b (i) m . (15b)
Hence, flipping |b 0 only for the current position states at stage i can be done by controlling the NOT gate with positive (i.e., on 1) and/or negative (i.e., on 0) controls on the first n -m position qubits from top to bottom, starting from |b n-1 , such that the NOT gate is activated if and only if

b n-1 ...b m = i 2 . ( 16
)
This corresponds to applying a certain generalized (nm)-Toffoli gate, where "generalized" means with positive and/or negative controls. As a reminder, an n-Toffoli gate is a NOT gate controlled positively by n qubits.

Note that a 0-Toffoli gate thus denotes a NOT gate.

The encoding performed by Q

(n,m) 1,i is then |b = 0 → |b = (2 k (i) -i2 m ) 2
for the current position states k (i) , and |b = 0 → |b = 0 (i.e., the identity) for the other position states. In Fig. 5, we illustrate the above-mentioned generalized

(n-m)-Toffoli gates of each Q (n,m) 1,i
for n = 3 and m = 1. These generalized (n -m)-Toffoli gates replace the NOT gate at the beginning of

Q 11 in Q 1 of U (n)
lin. in Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF]. In Appendix A, we give the explicit definition of Q

(n,m) 1,i
. In Appendix B, we, for the sake of completeness, write explicitly

Q (n,m) 2
, which initializes the ancillary coins, but as mentioned the only change with respect to Q (n) 2 of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF] is the number of ancillary qubits. In appendices A and B, we have omitted certain identity tensor factors in certain equations, in order to lighten the writing.

In total, since at each stage the ancillary positions are encoded only if we have a current position state at stage i, then it means that the coin operator at a given position is applied only if that position is a current position at stage i: in other words, U (n,m) i coincides on H with (i) the coin operator at a given position if that given position is a current position at stage i, and with (ii) the identity otherwise, which achieves our goal.

D. Depth

The purpose of this adjustable-depth circuit is that, depending on the parameter m, its width and depth will be modified, one for the benefit of the other. Let w(•) and d(•) be respectively the functions that return the width and the depth before compilation of an operator. One also needs to define functions ε w (x) and ε d (x) which return respectively the number of ancillary qubits that may be needed to implement an x-Toffoli gate, and the depth after compilation of the latter. Finally, δ i,j is the Kronecker symbol.

Since one uses n position qubits, 2 m coin qubits and 2 m ancillary position qubits, the width of U (n,m) reads As for the total depth of the circuit, we show in Appendix C that it is given by

w(U (n,m) ) = n + 2 m+1 + ε w (n -m) . (17)
d(U (n,m) ) = 2 n-m (20m + 2ε d (n -m) + 8δ m,0 -5) -2 .
(18) As shown in Ref. [START_REF] Saeedi | Linear-depth quantum circuits for n-qubit toffoli gates with no ancilla[END_REF], one can implement an n-Toffoli gate linearly in n without using any ancilla. Therefore, one can consider the term ε d (n-m) to be linear in n-m, and ε w (n -m) = 0. We finally remark the exponential dependence in the number of packs n-m, namely, 2 n-m , and the linear dependence in the number m of position qubits involved per pack, namely, 20m (plus the linear dependence of ε d (n -m) in n -m).

In Fig. 6, we present the different width and depth complexities of U (n,m) for some remarkable values of m, with N = 2 n . m Depth complexity Width complexity

m = 0 O(log 2 (N )N ) O(log 2 (N )) m = n 2 O(log 2 (N ) √ N ) O( √ N ) m = n O(log 2 (N )) O(N )

IV. IMPLEMENTATION

We have implemented our adjustable-depth quantum circuit on IBM's QASM, the classical simulator of IBM's quantum processors, thanks to the software Qiskit.

In Fig. 7, we show how this quantum circuit looks like for n = 2 position qubits and an adjustable parameter m = 0, 1, 2. In Appendix E, we give the pseudo-code used in order to generate these circuits.

In Fig. 8, we show the probability distribution obtained after 100 time steps of running different circuits, with coin operators parametrized by

K(α, θ, φ, λ) := e iα cos θ 2 -e iλ sin θ 2 e iφ sin θ 2 e i(φ+λ) cos θ 2 , ( 19
)
with angles taken at random for each coin operators C i , in the intervals

α, θ ∈ [0, π[ and φ, λ ∈ [-π, π[ . ( 20
)
The values obtained are given in Table 9.

In Fig. 10, we show, as a function of m, the depth, width and size of our adjustable-depth quantum circuit, after compilation by QASM. The size is the number of one-and two-qubit gates involved in the circuit. The compilation has been done with the following set of universal gates:

R X (θ) := cos θ 2 -i sin θ 2 -i sin θ 2 cos θ 2 (21a) R Y (θ) := cos θ 2 -sin θ 2 sin θ 2 cos θ 2 (21b) R Z (λ) := e -i λ 2 0 0 e i λ
CN OT :=       1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0       . ( 21e
)
As m increases, the depth decreases but the width increases, which is expected (the decrease of the depth being the purpose of increasing m, and the increase of the width being the consequence of needing to add ancillary wires in order to decrease the depth). Moreover, it is interesting to note the following non-trivial behavior: the number of gates decreases as m increases, which may come from the fact that, as m increases, the number of multi-Toffoli gates decreases (indeed, these multi-Toffoli gates have a high cost in terms of number of one-and two-qubit gates needed to implement them).

V. CONCLUSIONS AND DISCUSSION

In this paper, we have provided a family of quantum circuits that implement a DQW on the line, with the following characteristics. This family is parametrized by m ∈ N. For m = 0, the circuit coincides with the naive circuit of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], which means that all the coin operators at each site of the line are implemented sequentially. For m = n, the number of qubits used to encode the position of the walker, the circuit coincides with the lineardepth circuit of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], which means that all the coin operators are implemented in parallel. A circuit with a given arbitrary m means that the circuit contains 2 n-m packs, each of them containing 2 m coin operators implemented in parallel. A higher (lower) m means that more (less) coin operators are implemented in parallel, so that one can choose m as best suited for the experimental platform, knowing that a higher m, and hence a smaller depth, requires more ancillary qubits. It would be interesting to characterize the specific properties of DQWs, such as strict locality, at the level of their quantum-circuit translation [START_REF] Piroli | Quantum cellular automata, tensor networks, and area laws[END_REF]. It surely would be very interesting to extend the results of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF] and of this paper to quantum cellular automata (QCAs), which are multiparticle generalizations of DQWs [START_REF] Arrighi | An overview of quantum cellular automata[END_REF][START_REF] Farrelly | A review of quantum cellular automata[END_REF]. Given that there are already some QCAs which simulate quantum electrodynamics in 1 + 1, 1 + 2 and 1 + 3 dimensions [START_REF] Arrighi | A quantum cellular automaton for one-dimensional QED[END_REF][START_REF] Sellapillay | A discrete relativistic spacetime formalism for 1 + 1-QED with continuum limits[END_REF][START_REF] Eon | A relativistic discrete spacetime formulation of 3+1 QED[END_REF], this would mean having a quantum circuit which simulates some quantum field theory while implementing the strict locality of the transport. If more properties or symmetries of the continuum model are preserved by the QCA that simulates it, such as Lorentz symmetry [START_REF] Debbasch | Action principles for quantum automata and Lorentz invariance of discrete time quantum walks[END_REF][START_REF] Arrighi | Discrete Lorentz covariance for quantum walks and quantum cellular automata[END_REF][START_REF] Bisio | Quantum walks, deformed relativity and Hopf algebra symmetries[END_REF][START_REF] Debbasch | Discrete geometry from quantum walks[END_REF] or gauge invariance [START_REF] Arrighi | A quantum cellular automaton for one-dimensional QED[END_REF][START_REF] Sellapillay | A discrete relativistic spacetime formalism for 1 + 1-QED with continuum limits[END_REF][START_REF] Eon | A relativistic discrete spacetime formulation of 3+1 QED[END_REF], such a quantum-Figure 8: Final probability distribution obtained by running 10000 times a given circuit with n = 3 position qubits up to 100 time steps. In the left figure, we show the histograms for the naive circuit and the linear-depth circuit of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], for our adjustable-depth circuit with m = 2, and for a direct simulation of the walk (i.e., without mapping it onto a quantum circuit), which we have called "Matrix simulation" in Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF] instead of "Direct simulation"; the angles used for the different coin operators are given in Table 9. In the right figure, we show the histograms for different choices of the adjustable parameter m = 0, 1, 2, 3. circuit translation program would provide quantum circuits for quantum field theories, which respect many of the symmetries of the continuum model, which is not only a guarantee of numerical accurateness, but also provides an alternate, lattice definition of these theories, endowed with all the symmetries required by physical principles, thus creating a new, lattice paradigm for such physical theories, phrased in terms of quantum circuits and thus directly implementable on most-used quantum hardware. Finally, a question which is interesting is the following. Imagine that a certain algorithm is conceived with quantum walks, say, DQWs, and we want to run it with a quantum circuit. If we use the known algorithms that translate a DQW into a quantum circuit, do we obtain an algorithm that is as efficient as the original one made with a DQW, or do we have to modify it to reach the original efficiency?

α θ φ λ C0 2

STATEMENT OF ABSENCE OF CONFLICT OF INTEREST

On behalf of all authors, the corresponding authors state that there is no conflict of interest. . So, what we can do is simply applying a NOT gate before applying a positive control in order to obtain a negative control, and then we just wait until the application of Q (n,m) 11,i † to undo these operations on the n -m last position qubits. One thus has to flip the qubit, i.e., to (i) apply a NOT gate on |b j , with j = m, . . . , n -1, whenever the bit h j-m of i 2 is equal to 0, and then to (ii) apply a positive control, which delivers in total a negative control on |b j .

Thus, we replace T (n,m) i by

T (n,m) i := K multi α,b 0 (X) (n -m)-Toffoli       n-1 k=m g bk i,k-m bit flips ⊗I 2 m ⊗ I 2 (2 m ) ⊗ I 2 (2 m )       , ( A7
) which we call almost generalized multi-Toffoli gate. In this equation, the function g b k i,j indicates when to place the NOT gates on the control position qubit |b k : (A9)

g b k i,j := X if i 2 j mod 2 = 0 I 2 otherwise . ( A8
In Appendix D, we present an alternative method for reducing the amount of NOT gates used in the generalized multi-Toffoli gate.

b. The controlled SWAPs

We can write

B j :=   2 j -1 k=1 E s j+k-1 b k ,b k +2 j   E bj b 0 ,b 2 j , ( A10
)
where E a b,c denotes the controlled-SWAP operation that swaps |b and |c , controlling this by |a .

:= I 2 n ⊗ I 2 (2 m ) ⊗   m-2 L j=0   2 m-j-1 -2 k=0 K b 2 m -1-2 j+1 ( 1 2 +k) ,b 2 m -1-k2 j+1 (X)     , (B2) while the explicit definition of Q (n,m) 21 reads Q (n,m) 21 := I 2 n ⊗   m-1 L j=0 C m,j   2 j -1 k=0 E b (k+1)2 m-j -1 s k2 m-j ,s 2 m-j ( 1 2 +k)     , (B3)
where the CNOTs following the controlled SWAPs are given by Let us notice that h b k i,j implements the NOT gates as in the naive circuit of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF]. In Fig. 14

C m,j :=    2 j+1 -2 l=0 K b 2 m -1-2 m-j-1 ( 1 2 +l) ,b 2 m -1-l2 m-j-1 (X) if j < m -1 I 2 (2 m ) ⊗ I 2 (2 m ) if j = m -1 (B4)

Figure 1 :

 1 Figure 1: Registers necessary for the implementation of U (n,m) .

Figure 2 :

 2 Figure 2: Decomposition of U (n,m) in 2 n-m packs U (n,m) i

;

  we then swap the state of the principal coin |s 0 onto the ancillary coins via Q (n,m) 2 ; we then apply the running, pack i of coin operators in parallel, via Q (n,m) 0,i ; finally, the ancillary coin states are reset via Q (n,m) 2 † , and the ancillary position states are reset via Q (n,m) 1,i †

Figure 3 :

 3 Figure 3: Decomposition of each U (n,m) i

Figure 4 :

 4 Figure 4: Quantum circuits implementing Q (n=3,m=1) 0,i for i = 0, 1, 2, 3, that is, we implement, at each stage, 2 m = 2 coin operators (out of 2 n = 8) in parallel.

Figure 5 :

 5 Figure 5: We choose n = 3 and m = 1. On the far left, there is a circuit with only a NOT gate on |b 0 , which is what would be used at the beginning of the controlled-SWAP operations of each Q (n,m) 1,i , if there was no difference

Figure 6 :

 6 Figure 6: Depth and width complexity of our adjustable-depth quantum circuit U (n,m) , implementing the position-dependent coin operator, for remarkable values of m, where we recall that N = 2 n .

2

 2

Figure 7 :

 7 Figure 7: Our adjustable-depth quantum circuit for n = 2 position qubits, and packs of size 2 m , m = 0 (top circuit), 1 (middle circuit), and 2 (bottom circuit), which means that 2 m coin operators are executed in parallel at each of the 2 n-m stages in the quantum circuit.

Figure 9 :

 9 Figure 9: Values of the angles of the coin operators, chosen for our implementation of the different circuits in Fig. 8. The parametrization is given in Eq. (19).

Figure 10 :

 10 Figure 10: Depth, width, and size of the adjustable-depth quantum circuit, after compilation, for n = 4 (left plots) and n = 8 (right plots), as a function of the adjustable parameter m.

Figure 11 :

 11 Figure 11: Quantum circuits implementing Q (n=3,m=2) 1,i for i = 0, 1.

  ) Moreover, K multi α,b 0 (C) is the multiply controlled operation that applies gate C on qubit |b 0 whenever all qubits are 1 in the set of n -m control qubits α := n-1 k=m {|b k } .

Figure 12 :

 12 Figure 12: Quantum circuits implementing Q (n=3,m) 2, for m = 0, 1, 2, 3.

Figure 13 :

 13 Figure 13: A way of expressing the SWAP operation, with 3 CNOT gates.

  , we show how part of the NOT gates of the almost generalized multi-Toffoli gates ( T (n,m) i ) † and T (n,m) i+1cancel between each other, and which NOT gates remain, that we encode via h b k i,j .

Figure 14 :

 14 Figure 14: In the top figure, we show U (n=3,m=0) . In the middle figure, we replace the generalized multi-Toffoli gates by the T (n,m) i

DATA AVAILABILITY

Data will be made available upon reasonable request.

of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF] is, apart from the number of ancillary qubits, the fact that the first NOT gate of

11 is replaced by a generalized (n -m)-Toffoli gate in Q (n,m) 11,i (that we are going to define). To describe Q (n,m) 1,i explicitly, we thus follow exactly the construction of Q (n) 1 in Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF]Appendix C3]. Thus, the explicit definition of operator Q (n,m) 1,i can be written

where Q

(n,m) 10

makes the copies of the values of the position qubits on the ancillary coins, in order to be able to perform the controlled SWAPs in parallel via Q (n,m) 11,i , and then one undoes the copies via

. The amount of copies and SWAPs is no longer quantified by n as in Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF], but by the parameter m. On Fig. 11, we have depicted the quantum circuits implementing Q We have

where ∀j ≥ 0, q (n,m) 10

with

where we have defined

and where we remind the reader that K a,b (C) corresponds to applying the one-qubit gate C on qubit |b while controlling it on qubit |a (we apply C only if a = 1). , and of the controlled-SWAP operations. We can write it as

where we are going to define T (n,m) i and the B j 's below.

a. The generalized multi-Toffoli gate

What we call generalized multi-Toffoli gate is a multi-Toffoli gate for which the controls can be positive (i.e., on 1) or negative (i.e., on 0). The question here is how to express a negative control in terms of a positive control.

A first, naive idea is to express a negative control by the sequence of a NOT gate, then, a positive control, and finally another NOT gate. But, have in mind that we do not apply Q In this appendix, we give an explicit definition of

, which is the same as that of Q (n) 2 of Ref. [START_REF] Nzongani | Quantum circuits for discrete-time quantum walks with position-dependent coin operator[END_REF] except for the number of ancillary wires. In Fig. 12, we show the quantum circuits implementing Q (n,m) 2

for n = 3 and In this appendix, we prove the result for the depth of the circuit, given in Eq. ( 18). First, we recall that the depth of a SWAP operation counts for 3, as shown in Fig. 13. Now, the depth of U (n,m) , defined in Eqs. ( 10) and ( 13), is

Since we apply 2 m coin operators in parallel with Q

, defined in Eq. ( 14) and illustrated in Fig. 4, we have that

The depth of the operator

, defined in Eqs. (B1), (B2), (B3) and (B4), and illustrated in Fig. 12, is

If m = 0, we get a negative depth for d(Q

), and the minimum value of the depth must be 0. To solve this problem we perform the following modification,

Moreover, we have that

) .

(C5)

Lastly, the depth of the operator Q

, defined in Eqs. (A1), (A2) and (A6), and illustrated in Fig. 11, is,

where

and

If m = 0 we obtain a negative depth for d(Q

(n,m) 10,i ); we therefore perform the following modification,

As shown in Eq. (A6), the operator Q (n,m) 11,i can be separated into two operations: the (n -m)-Toffoli gate on |b 0 , and the series of controlled SWAPs. Let us first treat the (n -m)-Toffoli gate. As mentionned in Sec. III C, one has to flip some of the position qubits before applying the (n -m)-Toffoli gate; the only pack i for which no flip is needed is when i = 2 n-m -1, i.e., the last pack; therefore, we get a contribution 1

11,i ), where we recall that ε d (n -m) denotes the depth of the (n -m)-Toffoli gate. Let us now treat the controlled SWAPs. The depth of the non-parallelized controlled SWAPs is 3 m-1 k=0 2 k = 3(2 m -1); however, as one parallelizes this step in the circuit, the depth becomes only 3m. Therefore, the depth of Q (n,m) 11,i finally reads

Inserting Eqs. (C7), (C9) and (C10) into Eq. (C6), one gets,

Moreover, we have that

Inserting now Eqs. (C5) and (C12), and then (C2), (C4) and (C11d), into the expression of d(U

) given by Eq. (C1b), we obtain

The only term of d(U (n,m) i

) which depends on i is -2δ i,2 n-m -1 , which is equal to -2 when i = 2 n-m -1 and 0 for the rest; thus, inserting Eq. (C13d) into Eq. (C1a), we get

which is the result announced in Eq. ( 18). The NOT gates applied in order to realize the almost generalized multi-Toffoli gate T (n,m)

Appendix E: Pseudo-code

The pseudo-code used to code, with Qiskit, the adjustable-depth quantum circuit, is given below:


Yaëlle Bellahsen-Harrar

Mélanie Lubrano

Charles Lépine

Aurélie Beaufrère

Claire Bocciarelli

Anaïs Brunet

Franck Elise Decroix

Franck Neil El-Sissy

Bettina Fabiani

Aurélien Morini

Cyprien Tilmant

Thomas Walter

Cécile Badoual

AI-Augmented Pathology for Head and Neck Squamous Lesions Improves Non-HN Pathologist Agreement to Expert Level

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Head and Neck Squamous cell carcinomas (HNSCC) are a significant global health concern, ranking sixth worldwide in both incidence and mortality rates [START_REF] Bray | Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries[END_REF] . These cancers are notoriously associated with poor prognosis and high morbidity in the laryngeal and pharyngeal regions [START_REF] Johnson | Head and neck squamous cell carcinoma[END_REF][START_REF] Liao | Health-related quality of life and utility in head and neck cancer survivors[END_REF] . One reason for these figures is the late diagnosis of invasive lesions and their dysplastic counterparts. Early detection of dysplasia is essential in preventing invasive carcinomas [START_REF] Mehta | Premalignant Conditions of Larynx[END_REF] , and accurate grading is decisive as the grade remains the most important prognostic factor for the biological behaviour of disease, guiding the physicians in their care strategy [START_REF] Zidar | Update from the 5th Edition of the World Health Organization Classification of Head and Neck Tumors: Hypopharynx, Larynx, Trachea and Parapharyngeal Space[END_REF] . Pathological examination is the gold standard diagnostic method [START_REF] Gale | Current Views and Perspectives on Classification of Squamous Intraepithelial Lesions of the Head and Neck[END_REF] but poses many challenges. The small size of the samples impairs their optimal embedding orientation, often resulting in difficult-toanalyse tangent cuts. Changes in epithelium thickness between anatomical locations [START_REF] Eckel | European Laryngological Society position paper on laryngeal dysplasia Part II: diagnosis, treatment, and follow-up[END_REF] and within a lesion itself 8 can make it challenging to differentiate reactive epithelial changes such as basal hyperplasia, from true dysplastic lesions. Moreover, dysplasia grading is a complex task that requires simultaneous consideration of multiple cytological and architectural features [START_REF] Idar | Laryngeal and hypopharyngeal epithelial dysplasia[END_REF][START_REF] Thompson | Laryngeal Dysplasia, Squamous Cell Carcinoma, and Variants[END_REF] . Unlike most anatomical locations such as the uterine cervix or the digestive tract, the grading of head and neck dysplasia lacks immunohistochemical markers for guidance, thereby exclusively relying on Haematoxylin and Eosin (HE) staining and morphological assessment [START_REF] Idar | Laryngeal and hypopharyngeal epithelial dysplasia[END_REF][START_REF]Organisation mondiale de la santé[END_REF][START_REF] Hellquist | Developing Classifications of Laryngeal Dysplasia: The Historical Basis[END_REF] . The complexity of grading is evident from the multiple classifications proposed since the 1960s each with its own terminology (squamous intraepithelial neoplasia (SIN) by Friedmann and Osborn in 1976, intraepithelial neoplasia of the larynx by Crissman and Fu in 1986, laryngeal intraepithelial neoplasia (LIN) by Friedmann and Ferlito in 1988, squamous intraepithelial lesion (SIL) by Gale et al in 2014) and number of dysplasia categories [START_REF] Hellquist | Developing Classifications of Laryngeal Dysplasia: The Historical Basis[END_REF][START_REF] Gale | Evaluation of a new grading system for laryngeal squamous intraepithelial lesions-a proposed unified classification[END_REF] . This multitude of options has led to ambiguity and dissonance without a single approach standing out as superior. Additionally, dysplastic lesions of the oral cavity are graded using a different system without a solid explanation for the rationale behind this distinction.

Numerous studies have highlighted the mediocre inter-rater agreement among pathologists, reflecting the significant challenges of pathological examination [START_REF] Sarioglu | Inter-observer Agreement in Laryngeal Preneoplastic Lesions[END_REF][START_REF] Fleskens | Interobserver variability of laryngeal mucosal premalignant lesions: a histopathological evaluation[END_REF] . In an attempt to address this issue, the WHO proposed in 2017 to simplify dysplasia grading by combining moderate and severe dysplastic lesions into a larger "high grade dysplasia" category [START_REF]Organisation mondiale de la santé[END_REF][START_REF] Gale | Evaluation of a new grading system for laryngeal squamous intraepithelial lesions-a proposed unified classification[END_REF] . However, despite this simplification, reproducibility between pathologists remained unsatisfactory [START_REF] Gale | Laryngeal Dysplasia: Persisting Dilemmas, Disagreements and Unsolved Problems-A Short Review[END_REF] . Notably, the latest study on this topic by Mehlum et al [START_REF] Mehlum | Laryngeal precursor lesions: Interrater and intrarater reliability of histopathological assessment: Assessment of Laryngeal Precursor Lesions[END_REF] compared all reproducibility studies on head and neck squamous lesions in the literature, reported mediocre inter-rater agreement, demonstrating difficulties in providing reliable diagnosis even with the simplified binary system. Moreover, the scarcity of head and neck pathology specialists exacerbates the difficulties in getting optimal patient care. Therefore, the development of new tools to assist pathologists in their diagnoses is critical.

Recently, several studies have shown the benefits of AI models for improved diagnostic accuracy and reproducibility among pathologists, leading to what could be called an "augmented pathology". However, most of these works have focused on classification between different cancer subtypes or carcinoma gradings [START_REF] Raciti | Novel artificial intelligence system increases the detection of prostate cancer in whole slide images of core needle biopsies[END_REF][START_REF] Eloy | Artificial intelligence-assisted cancer diagnosis improves the efficiency of pathologists in prostatic biopsies[END_REF][START_REF] Bulten | Artificial intelligence assistance significantly improves Gleason grading of prostate biopsies by pathologists[END_REF][START_REF] Zheng | A deep learning model and human-machine fusion for prediction of EBV-associated gastric cancer from histopathology[END_REF][START_REF] Ba | Assessment of deep learning assistance for the pathological diagnosis of gastric cancer[END_REF] . In a previous work [START_REF] Lubrano | Diagnosis with Confidence: Deep Learning for Reliable Classification of Squamous Lesions of the Upper Aerodigestive Tract[END_REF] , we proposed a deep learning model for classifying head and neck squamous lesions with an indication of the model's confidence. However, we didn't assess its effectiveness in a real-life setting.

The objective of the present study was to assess whether AI-Assistance could increase accuracy and reproducibility among pathologists, ultimately leading to more effective and efficient management of patients, in the challenging task of detecting and grading laryngeal and pharyngeal (oropharynx and nasopharynx excluded) squamous dysplastic and invasive lesions.

Materials and Methods

Deep Learning Model Implementation and Evaluation

Based on the widely used Attention-MIL architecture [START_REF] Ilse | Attention-based Deep Multiple Instance Learning[END_REF] , we developed, trained and validated a model for automatic grading of head and neck squamous lesions [START_REF] Lubrano | Diagnosis with Confidence: Deep Learning for Reliable Classification of Squamous Lesions of the Upper Aerodigestive Tract[END_REF] . For each slide, it generates two outputs: the predicted lesion (ranging from non-dysplastic, low grade dysplasia, high grade dysplasia, to carcinoma) and an associated confidence score. The confidence score is specifically designed to measure the model's level of confidence for lesions on the same spectrum: it measures the extent to which the model hesitated with the second most probable (adjacent) class, as described in a previous work [START_REF] Lubrano | Simple and Efficient Confidence Score for Grading Whole Slide Images[END_REF] . The confidence threshold is optimised to reach an overall AUC > 0.9 on the validation set (thus settled at 0.5). The model was trained using a dataset of 1949 digitised Haematoxylin, Eosin and Saffronstained slides obtained from 456 patients who underwent either biopsies or surgical resection at Hôpital Européen Georges Pompidou (AP-HP, Paris, France). Each slide was associated with one class based on the most severe lesion present in the sample. The slides were digitised at 20X magnification using a Hamamatsu NanoZoomer® s360 scanner, resulting in a pixel resolution of 0.45 μm. To properly evaluate the model's performance, an independent subsample of 115 biopsies was used as the test set. The classes for these slides were determined using a dual-blind review by two pathologists with expertise in head and neck squamous lesions, followed by a consensus meeting to thoroughly discuss any slides on which they disagreed. This dataset was used to evaluate the performance of the AI model and to assess the effectiveness of AI-assisted reviews in terms of reproducibility between pathologists and their diagnostic performances. Finally, the model was validated on an external dataset from another centre (Hôpital Tenon, AP-HP, Paris, France) including 87 slides from 67 patients [START_REF] Lubrano | Diagnosis with Confidence: Deep Learning for Reliable Classification of Squamous Lesions of the Upper Aerodigestive Tract[END_REF] . Details about the datasets are shown in Supplementary Table A.

Randomized Protocol and Pathologists Panel

The panel consisted of eight pathologists with varying experience levels and practice backgrounds: two residents in their last year of residency, three pathologists specialised in head and neck pathology, and three pathologists with no routine practice in head and neck pathology (details in Supplementary Figure A). The two expert pathologists who labelled the 115 slides of the reference test set were not included in the panel. All panel members were tasked with reviewing the slides from the reference standard test set with and without AI-Assistance. Residents and non-specialized pathologists were provided with the references of the latest WHO grading system beforehand to update their knowledge. The study was designed as a randomised crossover trial, where each participant was randomly assigned to start with either the AI-assisted review or the non-assisted review, following protocols used in other studies [START_REF] Bulten | Artificial intelligence assistance significantly improves Gleason grading of prostate biopsies by pathologists[END_REF][START_REF] Nasir-Moin | Evaluation of an Artificial Intelligence-Augmented Digital System for Histologic Classification of Colorectal Polyps[END_REF][START_REF] Kiani | Impact of a deep learning assistant on the histopathologic classification of liver cancer[END_REF] . The pathologists independently reviewed the 115 slides and assigned a diagnosis to each of them without external input, in one uninterrupted session. A mandatory washout period of at least two weeks was required between the two reviews to avoid potential carryover effects.

Digital Platform and Review Process

The reviews were conducted using the EyeDo© digital platform (Tribun Health), a web-based viewer allowing for simultaneous visualisation of the slides, the model's prediction, and the confidence score. A user guide was provided to the participants. Both the assisted and nonassisted reviews were performed on the same platform, but the slide names were changed between the two reviews to ensure blinding. During the non-assisted review, the pathologists had access only to the slides and were blind to any other information related to the case. For the assisted reviews, they were provided with the model's prediction, the confidence score expressed as a percentage, a categorization of the confidence (high or low, following the threshold established beforehand) and a heatmap that could be toggled on and off, highlighting regions of the slide that contributed to the prediction, as shown in Figure 1 and Supplementary Figure B. For each slide, the reviewers were asked to fill in a table with their diagnosis, with the slide names pre-filled in the order of appearance on the platform.

Statistical Analysis

After all the panel members completed the assisted and unassisted reviews, their diagnoses were compared to the reference internal test set. Cohen's kappa with linear weights was used as the primary metric to measure the reproducibility, to compare it with the other studies published in the literature. Other standard classification metrics (accuracy, sensibility, specificity, negative and positive predictive values) were computed per class in a one-versusrest manner. Confidence intervals of the AI algorithm model were computed using 10000 bootstraps. Statistical differences of the metrics between the AI-Assisted and the non-assisted reviews were assessed with a paired t-test testing the null hypothesis that the two reviews by the panel have identical average. To account for possible bias in the reference standard of the internal test set, the pairwise agreement between all panel members individually were computed for the two reviews. All statistical analyses were performed using python (v3.6.9), pandas (v1.1.5), scikit learn (v1.2.0) and scipy (v1.6.0).

Results

Standalone Model's Performances

On the internal test set, the standalone AI model achieved an average AUC of 0.878 (95% CI: [0.801-0.937]) across the four classes, with an AUC > 0.9 for the detection of carcinoma, and an average linear kappa of 0.675 (95%CI: [0.575-0.760]). For the correct predictions, the confidence score had an average value of 0.846 +/-0.153, compared to 0.288 +/-0.150 for incorrect predictions, showing a good correlation between high model's confidence and correct predictions. The overall AUC improved by 5.3% (0.931 [0.888-0.968]) when removing slides with low confidence. Conversely, overall AUC computed on the uncertain slides was 0.694 [0.580-0.797]. The linear kappa was 0.833 [0.737-0.905] on high confidence slides and 0.275 [0.077-0.449] on low confidence slides (+55.8%). Similar performances were observed on the external test set, where the model achieved an average AUC of 0.886 (95% CI: [0.813-0.947]). All metrics are detailed in Supplementary Table B.

Agreement of the Pathologists' Panel with and without AI-Assistance

Agreement comparisons are presented in Figure 2. The results demonstrate that AI-Assistance significantly improved inter-rater agreement, as indicated by the reduced range of kappa range values (as shown in Fig 2A ). Furthermore, the combined analysis of pathologist and AI showed greater efficiency compared to the standalone AI model (0.73 vs 0.675, p < 0.001). When considering pairwise agreement within the panel without taking the reference standard labels into account, the median agreement was 0.619 in the non-assisted review and 0.734 in the assisted review. These results show that AI-Assistance led to increased consistency in grading among the pathologists (as shown in Fig 2B ).

Pathologists' Performances Improvement with AI-Assistance depending on their Experience

The overall performances of pathologists depending on their category (resident, non-HN specialist, HN specialist) are presented in Figure 3. The results show that the assisted residents and non-HN specialists outperformed the standalone AI model and demonstrated significant improvement across all metrics. These findings highlight the powerful impact of the AI-Assistance for non-HN specialists and young pathologists, suggesting its promising benefits in routine practice. However, it is worth noting that the HN specialists, who already achieved better performances than the standalone model, did not benefit from much improvement from the AI-Assistance. Detailed results are shown in Supplementary Table C.

Pathologists' Performances depending on the Confidence Score

When considering the model's confidence scores, the results indicate that there was no improvement in pathologist's agreement on low confidence predictions, and a significant improvement when the confidence score was high (as shown in Figure 4). There was no difference in the kappa values between assisted and non-assisted reviews when the confidence score was below the threshold, suggesting that the pathologists did not take the model's predictions into account in these cases. Conversely, on high confidence predictions, reproducibility was significantly higher and with a reduced distribution of kappa values (high confidence slides: linear kappa 0.8 assisted vs 0.73 non-assisted, p < 0.001). This indicates that pathologists were more inclined to consider the predictions and that it facilitated higher diagnostic performances.

Metrics Improvement with AI-Assistance depending on the Type of Lesion

Pathologists' performances per diagnostic class are shown in Table 1. Globally, the pathologists were more performant with the AI-Assistance. Specificity improved significantly for low grade dysplasia, indicating better discrimination of this subtle lesion (non-assisted pathologists: 0.814 vs assisted pathologists: 0.855, p = 0.021). Moreover, performances showed drastic improvements for high grade dysplasia and carcinoma, the pathologists becoming more efficient than the standalone model (for instance, accuracy for high grade dysplasia: 0.774 for the standalone model vs 0.806 for the Augmented Pathologist, p = 0.003). These results show that the AI tool assisted pathologists in identifying these harmful lesions with the highest therapeutic impact, and the combination of the pathologist's expertise and the AI analysis proved to be complementary and more powerful when used together.

Discussion

To the best of our knowledge, the benefit of an AI-Assistance in the difficult task of detecting and grading dysplasia in the laryngeal and pharyngeal regions wasn't assessed before. Due to their limited number, studies on dysplasia detection and classification by AI models have not assessed whether these models enhance the performance of pathologists. Furthermore, almost all of them didn't follow any official classification system. For instance, Tomita et al [START_REF] Tomita | Attention-Based Deep Neural Networks for Detection of Cancerous and Precancerous Esophagus Tissue on Histopathological Slides[END_REF] combined low and high grade dysplasias as well as true invasive adenocarcinoma and high grade suggestive of invasion due to their dataset's limited size. Other works on Barrett's esophagus [START_REF] Guleria | Deep learning systems detect dysplasia with human-like accuracy using histopathology and probe-based confocal laser endomicroscopy[END_REF][START_REF] Faghani | Development of a deep learning model for the histologic diagnosis of dysplasia in Barrett's esophagus[END_REF] didn't grade dysplasia either. One major limitation of these studies is the scarcity of the datasets, which results in inadequate training sets to learn such nuanced pathological aspects. Benefiting from a substantial training set of nearly 2000 slides, our model was subsequently validated on two different test sets. To our knowledge, only one previous work on gastric dysplasia successfully differentiated between epithelial regeneration change and dysplasia and graded the latter in a large cohort [START_REF] Shi | Deep learning for automatic diagnosis of gastric dysplasia using whole-slide histopathology images in endoscopic specimens[END_REF] . However, this study did also not assess if the AI tool could improve pathologist performance in a real-world setting. Indeed, it is imperative to remember that even if an efficient standalone AI model is developed, the use of such a tool without medical control is currently impossible due to possible errors, with potential harmful patient outcomes. Thus, pathologist validation of the AI predictions is mandatory. Our findings demonstrate that pathologists supplemented with AI were more efficient than the standalone deep learning model. Notably, AI-Assistance significantly improved the panel's accuracy and reproducibility, especially for non-HN specialists and less experienced pathologists. These results show that our solution combining an efficient model with a confidence score is a helpful tool for the diagnosis of laryngeal and pharyngeal squamous lesions for non-specialists and younger pathologists. Our AI-assisted method enabled our pathologist panel to surpass the performances reported in earlier studies [START_REF] Mehlum | Laryngeal precursor lesions: Interrater and intrarater reliability of histopathological assessment: Assessment of Laryngeal Precursor Lesions[END_REF] and to perform equivalently to the expert panel of Gale et al [START_REF] Gale | Evaluation of a new grading system for laryngeal squamous intraepithelial lesions-a proposed unified classification[END_REF] which achieved a weighted kappa of 0.80 among ten globally recognized expert pathologists. The agreement was even greater on high confidence predictions, showing that the confidence score is efficient in guiding pathologists in their diagnosis, emphasising the need to integrate model confidence indicators in AI-assisted workflows.

The implications of our study are significant for clinical practice, since the use of our AI model by pathologists across varied backgrounds could lead to more precise and uniformed diagnoses, and improved patient care management. Additionally, the tool's explanatory features, such as reliability scores and heatmap regions that influenced the prediction, provide the opportunity for pathologists to improve their own knowledge in the field. The educational potential of our confidence-based model could be explored in future work.

Limitations

This study does have several limitations. Primarily, the model was trained using a single slide per sample, which was selected by a pathologist as the best representative of the lesion. In routine practice, a pathologist would examine multiple slides for the same sample, with a variation of the difficulty depending on the cut level. This selection of the best slide could have helped in the training of the model. However, all cuts on the selected slide were scanned and incorporated into the training and validation processes. Another limitation is the absence of clinical information, which is provided to the pathologist in a real-life setting. Finally, the relatively small size of the pathologist panel could have potentially restrained the statistical power of our tests. Despite these limitations, our results still highlight the great potential of AIassisted pathology in the diagnosis of laryngeal and pharyngeal squamous lesions.

Conclusions

In this study, we demonstrate the feasibility and effectiveness of authentic Augmented Pathology in the challenging task of diagnosing laryngeal and pharyngeal squamous lesions. For the first time, we describe a methodology that truly improves inter-rater agreement such as no classification system alone achieved before. By providing a reliable diagnosis and an efficient confidence score, we believe that our AI model has the potential for broad acceptance in clinical practice, thereby greatly improving patient care and management.

Table 1 .

 1 Classification Metrics per Type of Lesion One sided paired t tests. Results with statistical significance are in bold.

	Type of Lesion

*

Funding ML was supported by a CIFRE PhD fellowship founded by Keen Eye, Paris, France and ANRT (CIFRE 2019/1905). Furthermore, this work was supported by the French government under management of Agence Nationale de la Recherche as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute).

Data Availability Statement

The WSI dataset described in the manuscript were subject to hospital regulations and could not be publicly released. Data sharing with other research teams is possible under formal agreement with Assistance Publique -Hôpitaux de Paris (contact first and last authors for more information).

Conflict of interest

The authors declare no competing interests.

Ethics Approval and Consent to Participate

Our study was approved by the ethics committee of Assistance Publique -Hôpitaux de Paris Centre (CERAPHP. Centre -Institutional Review Board registration #00011928). All the patients were informed by a notification letter of the study and the possibility to refuse the use of their medical data, in line with current legislation. The study was performed in accordance with the Declaration of Helsinki.

Author contributions

Concept and design: CB, SB, TW. Ethical approvals processes: CB, SB. Creation of the clinical and pathology database, slides selection and labelling: YBH. Reference standard test set review: CB, YBH. Data management and processing, software implementation and statistical analyses: ML. Pathologists panel: CL, AB, CB, AB, ED, FEL, AM, CT. Results analysis: YBH and ML. Discussion of results: YBH, ML, CB, TW. Manuscript writing and review: all authors. All authors read and approved the final paper.


G Laisné

J-M Salotti

N Rezzoug

Genetic algorithms for force polytopes prediction

Introduction

Knowledge of human's force capacities enables the design of physical Human-Robot Interaction (pHRI) workspaces. As measuring force capacities for all postures is time consuming, predicting force capacities from a subset of measurements performed in a limited number of postures is crucial. The force capacities can be described as a convex polytope by means of a personalized musculoskeletal (MSK) model [START_REF] Skuric | On-Line Feasible Wrench Polytope Evaluation Based on Human Musculoskeletal Models: An Iterative Convex Hull Method[END_REF]. However, the tuning of a MSK model is difficult due to the high number of parameters. Thanks to its constraint-free nature on the optimization function, a genetic algorithm is implemented to find a MSK model parameter set, which fits and predicts force polytopes [START_REF] Van Soest | The Merits of a Parallel Genetic Algorithm in Solving Hard Optimization Problems[END_REF][START_REF] Lo | A practical EMG-driven musculoskeletal model for dynamic torque estimation of knee joint[END_REF].

Methods

Force polytope

The upper-limb is considered as a n degree-offreedoms kinematic chain actuated by d muscles. In isometric conditions, the (convex) force polytope localized at the center of the hand is defined as:

𝑃 = { 𝐟 ∈ 𝐑 𝟑 | 𝐽 𝑇 𝐟 = -𝐿 𝑇 𝐭, 𝐭 ∈ [𝐭 𝐦𝐢𝐧 , 𝐭 𝐦𝐚𝐱 ] } (1)
where 𝐽 𝑇 ∈ 𝐑 𝑛×3 and -𝐿 𝑇 ∈ 𝑹 𝑛×𝑑 map respectively the forces at the end-effector 𝐟 ∈ 𝐑 3 and the muscle tensions 𝐭 ∈ [𝐭 𝐦𝐢𝐧 , 𝐭 𝐦𝐚𝐱 ] onto the torque space. These mappings and the muscle tensions depend on the joint configuration 𝐪 ∈ 𝐑 𝑛 . The Iterative Convex Hull method with a tolerance of 1N is used to approximate the polytope vertices [START_REF] Skuric | On-Line Feasible Wrench Polytope Evaluation Based on Human Musculoskeletal Models: An Iterative Convex Hull Method[END_REF]. For an arbitrary 𝑘 ∈ 𝐍, the discretization of a polytope is a scaled-down representation in 2𝑘 vertices. It ensures easier comparisons. The polytope is intersected with lines 𝐿 1 , 𝐿 2 , … , 𝐿 𝑘 passing through the origin and 𝑘 points evenly distributed on the upper sphere, with 𝐿 1 passing through (1,0,0). Each line produces two vertices noted 𝑣 𝐿 𝑖 + and 𝑣 𝐿 𝑖 -. The discretized polytope is defined as:

𝑃 𝐷 = (𝑣 𝐿 1 + , 𝑣 𝐿 1 -, … , 𝑣 𝐿 𝑘 + , 𝑣 𝐿 𝑘 -) ∈ 𝐑 2𝑘 (2)

Musculoskeletal model

A biorbd-compatible version of Stanford's upper-limb model [START_REF] Holzbaur | A Model of the Upper Extremity for Simulating Musculoskeletal Surgery and Analyzing Neuromuscular Control[END_REF] is used and consists of 50 muscles and 7 degrees-of-freedom. The muscle tensions are computed using Thelen's muscle model.

Optimization problem

Our goal is to find a MSK model parameter set 𝐌 * which generates discretized forces polytopes close to discretized polytopes computed from an unknown model parameter set in p postures:

𝐌 * = argmin 𝐌 𝑓 𝑝 (𝐌) (3) with 𝑓 𝑝 (𝐌) = 1 𝑝 ∑ MSE (𝑃 ̂𝑖𝐷 , 𝑃 𝑖 𝐷 (𝐌)) 𝑝 𝑖=1 (4)
with 𝐌 a model parameter set, 𝑝 the number of postures considered, 𝑃 ̂𝑖𝐷 the discretized polytope to estimate at posture 𝑖, and 𝑃 𝑖 𝐷 (𝐌) the computed discretized polytope at posture 𝑖.

Genetic algorithm

Genetic algorithms are based on the biological concept of evolution. They are particularly efficient compared to classic optimization methods to find solutions for hard optimization problems, with conditions such as unsteadiness, non-derivability, or noise [START_REF] Van Soest | The Merits of a Parallel Genetic Algorithm in Solving Hard Optimization Problems[END_REF]. A genetic algorithm begins with randomly or manually initialized solutions called the initial generation. The solutions are then assessed according to a cost function. The best solutions, called parents, define a new generation by means of small changes called mutations. The process is iterated until a terminating criterion is met.

Simulations

A genetic algorithm is used to minimize (4). A solution is a set of 150 parameters defining Stanford's MSK model. This set includes 3 parameters per muscle (the maximum isometric force, the optimal muscle length at which the muscle creates its maximal isometric force and the tendon slack length). These parameters have direct impact on muscle tensions computed using the force-length relationship. Each generation has 128 solutions. An initial generation is created using uniform random perturbations up to 30% (arbitrarily chosen due to computation considerations) of the Stanford's model parameter set. Small perturbations, up to 5%, are also applied on each muscle origin and insertion points. Five initial generations 𝐺 1 , 𝐺 2 , 𝐺 3 , 𝐺 4 and 𝐺 5 are defined. To assess the impact of the number of postures in the optimization process, two sets of postures 𝑆 4 and 𝑆 6 are considered: 𝑆 4 includes 4 common upper-limb postures (𝐐 1 , 𝐐 2 , 𝐐 3 , and 𝐐 4 ), while 𝑆 6 includes 2 more (𝐐 5 and 𝐐 𝟔 ). The model is defined in the global reference frame following the ISB recommendations: the 𝑥-axis is normal to the coronal plane, the 𝑦-axis normal to the transverse plane and the 𝑧-axis normal to the sagittal plane. The postures are described by 7 Euler angles in degrees using a 𝑦-𝑧′-𝑦′′ sequence for the shoulder, a 𝑧-𝑦′ sequence for the elbow and a 𝑥-𝑧′ sequence for the wrist: 𝐐 1 = (31°, 12°, -34°, 74°, 17°, 15°, 15°), 𝐐 2 = (79°, 54°,-73°,53°,90°,0°,0), 𝐐 3 = (124°, 56°, -75°, 32°, 88°, 0°, 0°), 𝐐 4 = (27°, 66°, -59°, 34°, 88°, 0°, 0), 𝐐 5 = (-41°, 79°, -6°, 104°, -46°, 6°, 0), and 𝐐 6 = (108°, 104°, 7°, 31°, -46°, 6°, 0°). A cost function is defined per posture set and are respectively 𝑓 4 and 𝑓 6 , using (4). The 5 best solutions which minimize the cost function define the parents. An elitist strategy is used to keep the parents in the next generation. For each parent, the mutation process generates 5 new parameter sets randomly selected in the neighbourhood of the parent parameter set. The neighbourhood size varies depending on the number of times a solution has been selected as a parent: below 4 times, the neighbourhood includes up to 10% around the parent parameters. Between 5 and 9 times, up to 1%. Between 10 and 29, up to 0.5% and above 30 times, up to 0.1%. Each generated parameter set also includes up to 1% random perturbations for the muscle origin and insertion points. They induce small perturbations in the generated force polytopes, allowing us to verify the genetic algorithms robustness.

Results and discussion

A genetic algorithm was used to tune a MSK model using force polytopes discretized in 26 vertices. A simplified version of the Stanford's MSK model (Holzbaur et al., 2015) with 7 degree-of-freedoms and 50 muscles was used as the model parameter set to estimate. For the five different initial generations considered using each posture sets, the algorithm converged to a solution (Fig 1). Predictions show better results using 6 postures, due to overfitting when using 4 postures only (Table 1).

Figure 1 Force polytope fitting for posture 𝐐 1 from loop 1 to loop 500 using the posture set 𝑆 6 .

Posture number should be optimized to get better results. Also, found solutions were expectedly distant from the model parameter set to estimate, due to redundancy in joint torques generation. 1 Mean±SD of discretized force polytopes RMSEs (N) between Stanford's and solutions found for 𝐺 1 to 𝐺 5 using 𝑆 4 and 𝑆 6 in fitting postures (white) and in 10 new prediction postures (grey).

𝐺 1 𝐺 2 𝐺 3 𝐺 4 𝐺 5 𝑆 4 4±1 3±1 4±1 5±1 5±2 𝑆 4 14±7 16±13 32±45 31±39 31±53 𝑆 6 5±2 6±1 6±1 6±2 9±3 𝑆 6 20±14 16±13 11±9 11±8 14±8 Table

Conclusions

Simulation results attest the performance of finding a model parameter set which fits and predicts force polytopes for different postures (Fig 2). The cost functions should be adapted to consider the polytopes geometry. The method was tested in silico and will be validated experimentally using a similar protocol for force polytopes measurements as in [START_REF] Rezzoug | Comparison between model-based and measured force polytopes: towards isometric force capacity evaluation[END_REF].

Figure 2 Example of force polytope predictions for 3 new postures.

Acknowledgements

The simulations were carried out using the PlaFRIM experimental testbed (Inria, CNRS, LABRI, IMB, Université de Bordeaux, Bordeaux INP and Conseil Régional d'Aquitaine) (see https://www.plafrim.fr).


Thomas Bonte
email: thomas.bonte@minesparis.psl.eu

Maxence Philbert

Emeline Coleno

Edouard Bertrand

Arthur Imbert

Thomas Walter
email: thomas.walter@minesparis.psl.eu

Learning with minimal effort: leveraging in silico labeling for cell and nucleus segmentation

Keywords: Segmentation, Transfer learning, Pretext task, In Silico Labeling, Fluorescence microscopy

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Detection and segmentation of cells and nuclei, among other cell structures, are essential steps for microscopy image analysis. Deep Learning has provided us with very powerful methods to perform these segmentation tasks. In particular, recently published neural networks, such as NucleAIzer [START_REF] Hollandi | nucleAIzer: A Parameter-free Deep Learning Framework for Nucleus Segmentation Using Image Style Transfer[END_REF], Cellpose [START_REF] Stringer | Cellpose: a generalist algorithm for cellular segmentation[END_REF] or StarDist [START_REF] Schmidt | Cell Detection with Star-Convex Polygons[END_REF], trained on hundreds of images of different modalities, give excellent results, outperforming by far traditional methods for image segmentation. However, the main drawback of state-of-the-art networks is the need for large amounts of fully annotated ground truth images, which can take a significant amount of time to create. Here, we present an alternative strategy, where we pretrain our segmentation models using In Silico Labeling (ISL) before fine-tuning them on a very small data set to perform nucleus and cell segmentation.

ISL was first introduced by [START_REF] Christiansen | In Silico Labeling: Predicting Fluorescent Labels in Unlabeled Images[END_REF], aiming to predict fluorescent labels from bright-field inputs. Fluorescence microscopy is the major technique employed in cellular image-based assays, as the use of fluorescence labels allows to highlight particular structures or phenotypic cell states. However, the number of fluorescent labels is limited (typically up to 4). In addition, phototoxicity and photobleaching can also represent serious drawbacks.

To tackle these limitations, several variants have been proposed. In [START_REF] Ounkomol | Label-free prediction of three-dimensional fluorescence images from transmitted-light microscopy[END_REF], ISL is applied to predict fluorescent labels from transmitted-light images (DIC), or immunofluorescence from electron micrographs. Besides, Generative Adversarial Networks (GAN) are used in [START_REF] Rivenson | Virtual histological staining of unlabelled tissueautofluorescence images via deep learning[END_REF] to predict different stains: H&E, Jones Silver or Masson's trichrome. They underlie staining standardization as an advantage of ISL. In another paper [START_REF] Rivenson | PhaseStain: the digital staining of label-free quantitative phase microscopy images using deep learning[END_REF] GANs are also used on different transmitted light images: quantitative phase images (QPI). Moreover, in [START_REF] Li | Deep Learning for Virtual Histological Staining of Bright-Field Microscopic Images of Unlabeled Carotid Artery Tissue[END_REF] conditional GANs (cGAN) generate H&E, PSR and Orcein stained images from unstained brightfield inputs. In [START_REF] Liu | Global Pixel Transformers for Virtual Staining of Microscopy Images[END_REF], using the same data set and same tasks as [START_REF] Christiansen | In Silico Labeling: Predicting Fluorescent Labels in Unlabeled Images[END_REF], the authors add attention blocks to capture more information than usual convolutions. Finally, stained images of human sperm cells are generated in [START_REF] Yoav | Holographic virtual staining of individual biological cells[END_REF], from quantitative phase images. They use these virtually stained images to recognize normal from abnormal cells. The principle of ISL has also been proposed for experimental ground truth generation for training cell classifiers for the recognition of dead cells [START_REF] Boyd | Experimentally-Generated Ground Truth for Detecting Cell Types in an Image-Based Immunotherapy Screen[END_REF][START_REF] Hu | Live-dead assay on unlabeled cells using phase imaging with computational specificity[END_REF], tumour cells [START_REF] Jingfang | Automatic Colorectal Cancer Screening Using Deep Learning in Spatial Light Interference Microscopy Data[END_REF] embryo polarization [START_REF] Shen | Stain-free detection of embryo polarization using deep learning[END_REF] or the cell cycle phase [START_REF] Yuchen | Cell Cycle Stage Classification Using Phase Imaging with Computational Specificity[END_REF].

In this paper we show that models trained to generate fluorescence microscopy images with nuclear or cytoplasmic markers can be used efficiently to pretrain segmentation networks for nuclear and cell segmentation, respectively. To the best of our knowledge, no previous work has used ISL as a pretext task for segmentation of cell structures. This provides us with a powerful strategy to minimize the annotation burden for a given application, and to train models on large data sets, requiring only minimal effort in terms of manual annotation.

Materials and Methods

Image Acquisition

We work on two different data sets. The first dataset has been generated by the Opera Phenix™ Plus High-Content Screening System (Perkin Elmer). It contains 960 images of dimension (2160, 2160). For each position, we acquired brightfield images and DAPI, both at 4 different focal planes. DAPI is a very common fluorescent stain binding to AT-rich regions of the DNA, which can thus be used to locate the nucleus in eukaryotic cells. Additionally we have a phase contrast image, computationally created from the 4 bright-field images by a proprietary algorithm of the Opera system. Images contain on average 15.6±19.6 cells.

Our second data set contains 100 images of dimension (1024, 1024). We used Differential Interference Contrast (DIC) as label-free microscopy technique, and we marked the cytoplasmic membrane with the CellMask™ marker (Life Technologies). Images contain on average 52.4±15.1 cells.

Nucleus Segmentation

Nucleus segmentation is one of the most important segmentation tasks in biology, as nuclear morphologies are indicative of cellular states, and because they are visually very different from the cytoplasm. Segmentation of the nucleus is usually a comparatively simple segmentation task, and for this reason we assumed that this might be a good first segmentation problem to investigate our ISL-based pretraining.

DAPI prediction as pretraining task

The first step of our strategy for nucleus segmentation is the prediction of DAPI images from bright-field inputs.

We used a data set of 421 images of dimension (2160, 2160), divided into 384 images for training and 37 images for testing. 5 images of dimension (512, 512) were randomly cropped from each initial image (see Fig. 1). Note that we only included images containing at least one nucleus.

Inspired by the work of [START_REF] Christiansen | In Silico Labeling: Predicting Fluorescent Labels in Unlabeled Images[END_REF], the model is a U-net-shape model [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] with a densenet121 architecture [START_REF] Yakubovskiy | Segmentation Models Pytorch[END_REF]. It has been previously trained on ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF], hence it is referred to as 'on steroids' in the following. As input we used 3 channels, 2 being bright-field images of the same field-of-view with different focal planes, and the third the corresponding phase-contrast image. As output we used only one channel, the maximum intensity projection of our DAPI images (z-stack, 4 focal planes) that we have for each field-of-view.

We did not use any data augmentation. All training details are reported in Supplementary Table Transfer Learning for Nucleus Segmentation. In a first step, we aimed at investigating how pretraining on fluorescent markers impacts semantic segmentation. For this, we turned to nucleus segmentation.

In order to generate the ground truth, we applied Cellpose [START_REF] Stringer | Cellpose: a generalist algorithm for cellular segmentation[END_REF], a widely used segmentation technique in bioimaging, based on a U-net-shaped network, trained on massive amounts of heterogeneous data. We applied Cellpose to the DAPI channel and corrected the segmentation results manually. As segmentation of nuclei from DAPI images with high resolution is a fairly simple task, as expected the results were overall excellent.

Next, we used training sets with different sizes N ∈ {1, 10, 50, 100, 200, 500}, composed of images of dimension (2160, 2160) and evaluated the accuracy for each N . Testing is always performed on the same withheld 190 images. 5 images of dimension (512, 512) were randomly cropped from each initial image.

To investigate whether our pretraining scheme is useful for segmentation, we compare two different models. The first model is composed of the U-net 'on steroids' followed by a sigmoid activation function in order to output, for each pixel, its probability of belonging to a nucleus (Fig. 2a). The second model has the same U-net architecture but is pretrained on DAPI images, and has an activation function displayed in equation ( 1) that takes a different range into account (Fig. 2b). The reason for this choice is that the model pretrained on DAPI images is likely to output values between 0 and 1, so we centered the following activation function around 0.5.

f (x) = 1 1 + exp(-(x -0.5)) (1)
We did not use any data augmentation. All training details are reported in Supplementary Table 1.

Cell Segmentation

We next turned to the application of our pretraining scheme to cell segmentation, a more difficult multiple instance segmentation scenario.

CellMask™ Prediction as Pretraining Task In our pretraining strategy, the first step of cell segmentation is the prediction of CellMask™ (Fig. 3b) images from DIC microscopy as inputs (Fig. 3a).

We used a data set of 100 images of dimension (1024, 1024), divided into 90 images for training and 10 images for testing. 5 images of dimension (512, 512) were randomly cropped for each initial image.

For comparison, we again used the U-net 'on steroids'. We did not use any data augmentation. All training details are reported in Supplementary Table 2.

Transfer Learning for Cell Segmentation. Segmentation of cells is usually more difficult than nuclear segmentation, because cells tend to touch each other, and the precise detection of the contact line can be challenging. Indeed, we need to turn to multiple instance segmentation, where object properties are predicted together with pixel labels. Again, we used Cellpose [START_REF] Stringer | Cellpose: a generalist algorithm for cellular segmentation[END_REF] with manual correction to generate this instance segmentation ground truth images from associated CellMask™ images (Fig. 4a, Fig. 4b).

As for nuclear segmentation, we used training sets of different sizes N ∈ {1, 10, 50, 80} of dimension (1024, 1024) and evaluated the accuracy for each of them. Testing is always performed on the same 17 images. 5 images of dimension (512, 512) were randomly cropped from each initial image.

To tackle the issue of instance segmentation, we implemented a model predicting both a cell semantic segmentation image (Fig. 4c) and a distance map, i.e. an image where pixels values get higher as they are closer to the center of the cell, the background remaining black (Fig. 4d), as proposed in [START_REF] Naylor | Nuclei Segmentation in Histopathology Images Using Deep Neural Networks[END_REF], [START_REF] Naylor | Segmentation of Nuclei in Histopathology Images by Deep Regression of the Distance Map[END_REF]. Like in the previous section we compare two models to investigate whether transfer learning from an ISL model can significantly improve the accuracy of our segmentation. The first model is the U-net 'on steroids', outputting 2 channels (Fig. 5a). The second model has the same U-net architecture but is pretrained on CellMask™ images, thus outputting only 1 channel. Hence we add two Conv2d layers at the end to upscale to 2 channels (Fig. 5b).

We did not use any data augmentation. All training details are reported in Supplementary Table 2. Both models use CombinedLoss α , presented in equation ( 2), as loss function. MSELoss stands for the usual Mean Square Error, while BCEWithLogitsLoss combines a sigmoid layer with the Binary Cross Entropy loss. y represents the output of our model, with the two channels y d and y s standing for the distance and semantic segmentation image, respectively. The factor α is used to balance the weights of the different losses during training. It has been set as α = 2000, 2000 being the initial ratio between MSELoss and BCEWithLogitsLoss. This has been inspired by the loss function used in Cellpose [START_REF] Stringer | Cellpose: a generalist algorithm for cellular segmentation[END_REF], which also uses a loss function computed as the sum of two loss functions, one for each output channel. Finally, we apply a post-processing step to get the final results. For this, we apply the h-maxima transformation of the predicted distance map, with h = 10. The h-maxima transformation is defined as the reconstruction by dilation of f -h under f : HM AX h (f ) = R δ f (f -h), and removes insignificant local maxima. f stands for the initial image, which is in our case the reconstructed distance map displayed in Fig. 6. h stands for the minimum local contrast for a local maximum to be kept; otherwise it will be removed. Each local maximum represents a cell.

The local maxima of HM AX then serve as seed for the watershed algorithm, which splits semantic segmentation result into individual regions, one for each maximum of HM AX. This leads to an instance segmentation image, such as the one presented in Fig. 6. Fig. 6: Pipeline to get instance segmentation image from both distance map and semantic segmentation image. H-maxima transform followed by watershed algorithm enable to segment cells instance-wise.

Results

Evaluation metrics

The metric used to evaluate DAPI and CellMask™ prediction performance is the Pearson Correlation Coefficient (PCC, equation ( 3)). PCC is defined as the covariance of two variables divided by the product of their standard deviations. In the equation, x and y are two images to compare, and x is the average of x.

PCC(x, y) = n i=0 (x i -x)(y i -ȳ) n i=0 (x i -x) 2 n i=0 (y i -ȳ) 2 (3)
To evaluate nucleus semantic segmentation, we use the Jaccard index (equation ( 4)). The Jaccard index, or Intersection Over Union (IoU), is a very popular metric in segmentation, as it equally penalizes both False Positive and False Negative pixels. A perfect segmentation would lead to IoU of 1, while IoU corresponds to an entirely missed object (no intersection).

IoU(x, y) = 1 if x ∪ y = 0 x∩y x∪y otherwise (4)
While the IoU is perfectly suitable to make pixel-wise comparisons for semantic segmentation, the performance of instance segmentation needs to incorporate an object-wise comparison that does not only penalize wrong pixel decisions, but also fused or split objects. For this, we choose to use the Mean Average Precision (mAP), which is a popular metric for instance segmentation evaluation. For this, a connected component from the ground truth is matched with a connected component from the segmentation result, if the IoU of the two components is above a given threshold. In this case, the object is considered as a TP. Unmatched connected components from the ground truth and the segmentation result are considered as FN and FP, respectively. Thus, given an IoU threshold one can compute the precision as defined in equation ( 5).

Precision = TP TP + FP + FN

(5)

Precision is computed for all 10 IoU thresholds in {0.5 [START_REF] Liu | Global Pixel Transformers for Virtual Staining of Microscopy Images[END_REF]]}. The final result is the mean of these 10 values, hence called mean AP, or mAP.

+ i × 0.05, i ∈ [[0,

Nucleus Segmentation

DAPI prediction yields very good results, with a PCC of 0.95±0.08.

Using the Jaccard index (or IoU) as metric, the U-net 'on steroids' gives 0.64±0.2 after training on 1 single image. In comparison, the model pretrained on DAPI reaches 0.84±0.1, improving the previous score by 31.3% (Fig. 7a). This improvement decreases as the size of the training set increases, being 4.8% (respectively 1.1%, 0.0%, 0.0%, -1.1%) after training on 10 (respectively 50, 100, 200, 500) (Fig. 7b).

Results from both models trained on 1 single image are displayed in Fig. 8.

Cell Segmentation

CellMask™ prediction also yields very good results, with a PCC of 0.97±0.02. Using mAP as metric, the U-net 'on steroids' gives 0.17±0.1 after training on 1 single image. In comparison, the model pretrained on CellMask™ reaches 0.33±0.09, improving the previous score by 94.1% (Fig. 9a). As in the previous section this improvement decreases as the size of the training set increases, being 18.5% (respectively -3.0%, -2.9%) after training on 10 (respectively 50, 80) (Fig. 9b).

Results from both models trained on 1 single image are displayed in Fig. 10.

Discussion

The results presented in the previous sections show that pretraining with In Silico Labeling as pretext task significantly improves the performance of a segmentation model trained on a very small data set. Indeed, the accuracy raises by 31.3% and 94.1% for nucleus semantic segmentation and cell instance segmentation, respectively, after training on 1 single image, using a model pretrained in an ISL setting.

The fact that pretraining on DAPI images helps to generate a nucleus semantic segmentation was actually expected since the two outputs (DAPI and binary segmentation maps) are very close to each other. On the other hand, cell instance segmentation is a much more complex problem, and our results clearly indicate that also in this situation, pretraining with fluorescent marker prediction as a pretext task significantly improves segmentation accuracy for small datasets. We also observe that transfer learning is useful if we work on a very small data set (1 to 10 images), but that for both nucleus and cytoplasmic segmentation, the accuracy difference disappears if the models are trained on more than 10 images. This being said, if one has access to fluorescent images, it makes sense to use our proposed method to pretrain the network.

From a practical point of view, this idea provides an interesting alternative to manual annotation, in particular in the context of High Content Screening, where it is fairly easy to generate large amounts of data that contain both label-free and fluorescently labeled microscopy images. In this case, we can train efficient models for fluorescence prediction, and use these models in a pre-training scheme to reduce the manual annotation burden. Finally, we showed here that this pretraining scheme is effective for segmentation of nuclei and cells, but we also believe that this could be effective for any other type of cell structures as soon as you can get the associated fluorescent images available. Furthermore, it will be interesting to investigate to which extent the pre-training scheme provides good starting points for generalist networks, applicable to a wide variety of modalities.

Conclusion

In this paper, we demonstrated that pretraining on the prediction of relevant fluorescent markers can be very useful to segment nuclei or cells. We showed that a model trained to predict some fluorescent structures from label-free microscopy can learn to segment these structures from a very small data set, down to 1 single image. We believe that this can be of great help for applications where fluorescent data are easily available, if one wants to avoid tedious manual annotation to build large ground truth datasets for the training of neural networks. With only a few images, it is possible to fine-tune a pretrained model achieving performances matching those obtained by ImageNet-pretrained state-of-the-art networks finetuned on a much larger set of images. Our pre-training scheme can thus help biologists to save time and money without sacrificing any accuracy. IoU Table 1: Training details for DAPI and nucleus segmentation models. Segmentation models were supposed to be trained on 20 000 training samples, yet for non ISL-pretrained models the training was stopped earlier as the model was not learning anything after some point. All models have the same U-net densenet121 architecture. They were trained with an ADAM optimizer on GPU hardware. BF stands for bright-field and PC for phase contrast.

1

 1

  .

Fig. 1 :

 1 Fig. 1: Images from the same field-of-view, for a given focal plane. (a) Brightfield image. (b) Phase contrast image, computationally generated by the Opera system. (c) Fluorescent DAPI image.

Fig. 2 :

 2 Fig. 2: Compared models to predict nucleus semantic segmentation. (a) U-net 'on steroids' which has not been trained on DAPI images. (b) U-net 'on steroids' pretrained on DAPI images. Note the difference in the activation functions.

Fig. 3 :

 3 Fig. 3: Images from the same field-of-view, for a given focal plane. (a) DIC image. (b) Fluorescent CellMask™ image.

Fig. 4 :

 4 Fig. 4: (a) Fluorescent CellMask™ image. (b) Corresponding cell instance segmentation image generated by Cellpose. (c) Cell semantic segmentation image generated from Cellpose output. (d) Distance map generated from Cellpose output.

2 )Fig. 5 :

 25 Fig. 5: Models compared to predict cell instance segmentation: (a) U-net 'on steroids' which has not been trained on CellMask™ images. (b) U-net model pretrained on CellMask™ images.

Fig. 7 :

 7 Fig. 7: Nucleus segmentation results. (a) Intersection Over Union (IoU) score for non ISL-pretrained and ISL-pretrained models, after training on 1 image. (b) Evolution of IoU average score for both models for different training set sizes.

Fig. 8 :

 8 Fig. 8: Input bright-field images, DAPI images, DAPI predictions generated by U-net 'on steroids', ground truth instance segmentation generated by Cellpose, non ISL-pretrained U-net 'on steroids' segmentation prediction, ISL-pretrained U-net 'on steroids' segmentation prediction. Segmentation is performed after training on 1 image for both models.

Fig. 9 :

 9 Fig. 9: Cell segmentation results. (a) mAP score for non ISL-pretrained and ISLpretrained models, after training on 1 image. (b) Evolution of mAP average score for both models for different training set sizes.

Fig. 10 :

 10 Fig. 10: Input DIC images, CellMask™ images, CellMask™ predictions generated by U-net 'on steroids', ground truth instance segmentation generated by Cellpose, non ISL-pretrained U-net 'on steroids' segmentation prediction, ISLpretrained U-net 'on steroids' segmentation prediction. Segmentation is performed after training on 1 image for both models.

Table 2 :

 2 Training details for CellMask™ and cell segmentation models. All models have the same U-net densenet121 architecture. They were trained with an ADAM optimizer on GPU hardware.

Acknowledgments This work has been supported by the French government under management of Agence Nationale de la Recherche (ANR) as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute), the Q-Life funded project CYTODEEP (ANR-17-CONV-0005) and the ANR project TRANSFACT (ANR-19-CE12-0007). Furthermore, we also acknowledge support by France-BioImaging (ANR-10-INBS-04).

Code availability Code (pre-processing, training and testing, post-processing pipelines), is available at https://github.com/15bonte/isl segmentation.

Supplementary material