Method for solving a system of at least one differential equation relating to at least one real-valued function over a given domain of definition

Abstract

The present invention relates to a method for solving a system of at least one differential equation relating to at least one real-valued function over a given domain of definition, the method comprising implementing, by a hybrid quantum-classical computer (1, 2), the steps of: (a) for each function, encoding, by means of a VQC, the expression of a candidate solution of the function in an orthogonal polynomial basis, the VQC being defined by a set of quantum parameters; (b) for each function, evaluating, at each of a set of predefined points, values of both the candidate solution and at least one derivative of the candidate solution by means of the VQC of the function, the derivative of the solution being expressed in the orthogonal polynomial basis by differentiating the expression; (c) optimizing the value of the quantum parameters of the VQCs of each function so as to minimize a cost function defined by the system of differential equations and depending on the evaluations, for each function, of the candidate solution and its derivative; (d) for each function, reconstructing an optimal solution of the function as encoded by the VQC with the optimized parameters.

Description

[0001]Description Title of the invention: Method for solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition. GENERAL TECHNICAL FIELD The present invention relates to the field of quantum computing. More specifically, it concerns a method for solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition, using a hybrid quantum-classical computer. STATE OF THE ART Quantum computers hold promise for simulating the behavior of natural systems, a task that is generally difficult for "classical" computers (i.e., binary logic computers with conventional processors, typically silicon-based). The simulation of Hamiltonians is the basis of analog quantum computing.In this scheme, a relevant problem is mapped to the dynamics of a quantum system. Usually, the solutions to problems are encoded in the "ground state" of a Hamiltonian. Finding these ground states is particularly simple with quantum annealers, which utilize the adiabatic transition from a simple initial Hamiltonian to the (complex) final state—that is, the adiabatic theorem. In practice, quantum annealers using the Ising model have already been built, and modest claims of quantum advantage have begun to emerge. This model can be translated into a quadratic unconstrained binary optimization (QUBO) problem, which has numerous practical applications. QUBO is a simple problem whose optimal solution, however, is very difficult to find using classical methods.It can be adapted to many combinatorial optimization problems. Instead of tackling problems for which classical computers are inefficient, some researchers have begun developing algorithms for typical problems, such as solving differential equations (DEs). Indeed, these equations govern many physical systems, and their solution is of paramount importance in numerous fields. Moreover, solving several partial differential equations (PDEs) using analytical system techniques remains complicated and, in practice, limited to a few specific cases. In quantum computing, the heat equation... a recently solved by quantum annealing in a hybrid configuration, a fully quantum method was used to solve the wave equation (∆^^⃗ =^ ^²^ ^⃗ ^^²), and another for the Navier-Stokes equations. Nonlinear differential equations, such as these, are among the most difficult to solve using numerical methods and have been the subject of recently developed algorithms. In one case (M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Variational quantum algorithms for nonlinear problems, Phys. Rev. A 101 (January 2020), no. 1, 010301), a nonlinear Schrödinger equation was solved by a variational quantum algorithm (VQA) that included a Hadamard test as a subroutine. The approach is similar to classical finite difference methods because the function domain is discretized. The authors claim that approximately 20 corrected qubits are sufficient to compete with state-of-the-art supercomputers. However, most of these methods are limited to one type of differential equation (partial or non-partial, linear, etc.).See, for example, request EP3971793. A recent method, described in the paper O. Kyriienko, A.E. Paine, and V.E. Elfving, "Solving nonlinear differential equations with differentiable quantum circuits," which is slightly more versatile (although it does not yet allow for handling PDEs), uses quantum circuits to evaluate the function and its derivative at points, and then evaluate a cost function defined by the equation. The circuit parameters can be iteratively optimized to minimize the cost function. This method is satisfactory but requires specific quantum circuits, called differentiable quantum circuits (DQCs), to evaluate the derivatives without intrinsic numerical errors. The main drawback is that the number of n-qubit circuits required grows proportionally to nm, where m is the highest order of derivatives in the differential equation.so that this solution quickly becomes unusable. The present invention improves the situation by proposing a universal solution for solving differential equations, which requires only a limited number of qubits and circuits, and which remains very accurate. PRESENTATION OF THE INVENTION The present invention relates, in a first aspect, to a method for solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition. The method comprises the implementation, by a hybrid quantum-classical computer, of the following steps: (a) For each function, encoding, using a quantum variational circuit (QVC), the expression of a candidate solution of said function in a basis of orthogonal polynomials, said QVC being defined by a set of quantum parameters; (b) For each function, evaluation at each of a set of predefined points of said given domain of definition.(c) Values ​​of both said candidate solution and at least one derivative of said candidate solution by means of the QVC of said function, by expressing the derivative of said solution in said basis of orthogonal polynomials by differentiating said expression of the candidate solution of said function in said basis of orthogonal polynomials; (c) Optimization of the values ​​of said quantum parameters of the QVCs of each function, so as to minimize a cost function defined by said system of differential equations and function of said evaluations for each function of the candidate solution and its derivative; (d) For each function, reconstruction of an optimal solution of said function as encoded by the QVC with the optimized parameters. According to advantageous and non-limiting characteristics: Each QVC is based on a Hardware Efficient Ansatz, HEA, type structure.said quantum parameters being rotation parameters. Said orthogonal polynomials are Chebyshev polynomials. The process includes a step (a0) of initializing the quantum parameters randomly. At least one boundary condition is associated with said system of differential equations, the cost function is of the form ^(^) = ^^^^^(^) + ^. ^^^^^^^^(^), where θ is the set of quantum parameters, ^^^^^ a first term representing an error on the differential equation(s), ^, ^^^^^^^ is a second term representing an error on the boundary condition(s), and η is a predefined coefficient. The first term ^ ^^^^ is the average squared error of all the differential equations for the set of predefined points of said given domain of definition, and the second term ^ ^^^^^^^is the sum of all errors on all boundary conditions for the set of predefined points of said given domain of definition. This candidate solution fc is evaluated at a predefined point xS of said given domain of definition by the formula est the output state of the VQC, λ a scaling factor, and Oc a diagonal observable, functions of the expression of said candidate solution in the basis of orthogonal polynomials. ^ ^ ^ The q-th derivative of said candidate solution ^ ^^ ^^ …^^ ^^ is evaluated in one ^ ^ ^ é ^ (^ point p ^ ) r defined xS of said domain of definition given by the formula ^^ ^^ …^^ ^^ = xj1…xjq are q components of x, ^ ^ ^ ^a diagonal observable, functions of the q-th derivation of said expression of the candidate solution of said function in said basis of orthogonal polynomials. Said predefined points of the given domain of definition, also called collocation points, are chosen so as to be uniformly distributed in said given domain of definition. Said system of at least one differential equation models a physical system, said real-valued function expressing a quantity of said physical system. According to a second aspect, the invention relates to a quantum-classical hybrid computer, characterized in that it is configured to solve a system of at least one differential equation on at least one real-valued function on a given domain of definition, by implementing the following steps: For each function, encoding by means of a quantum variational circuit, VQC, of ​​the expression of a candidate solution of said function in a basis of polynomialsorthogonal, said QVC being defined by a set of quantum parameters;^ For each function, evaluation at each of a set of predefined points of said given domain of definition, of the values ​​of both said candidate solution and at least one derivative of said candidate solution by means of the QVC of said function, expressing the derivative of said solution in said basis of orthogonal polynomials by differentiating said expression of the candidate solution of said function in said basis of orthogonal polynomials;^ Optimization of the value of said quantum parameters of the QVCs of each function, so as to minimize a cost function defined by said system of differential equations and a function of said evaluations for each function of the candidate solution and its derivative;^ For each function, reconstruction of an optimal solution of said function as encoded by the QVC with the optimized parameters. According to a third and fourth aspect,The invention relates to a computer program product comprising code instructions for executing a process according to the first aspect of solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition; and a computer-readable storage means on which is stored a computer program product comprising code instructions for executing a process according to the first aspect of solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition. PRESENTATION OF FIGURES Other features and advantages of the present invention will become apparent from the following description of a preferred embodiment. This description will be given with reference to the accompanying drawings in which: [Fig. 1] Figure 1 is a schematic of a system for implementationof the method according to the invention; [Fig. 2] Figure 2 is a flowchart illustrating the steps of an embodiment of the method according to the invention; [Fig. 3] Figure 3 schematically represents an example of a variational quantum circuit used in an embodiment of the method according to the invention; [Fig. 4a] Figure 4a shows the results of the present method in solving a first example of a system of differential equations; [Fig. 4b] Figure 4b shows the results of the present method in solving a second example of a system of differential equations; [Fig. 4c] Figure 4c shows the results of the present method in solving a third example of a system of differential equations; [Fig. 4d] Figure 4d shows the results of the present method in solving a fourth example of a system of differential equations on a real quantum computer. DETAILED DESCRIPTION Architecture The present invention relates to a method for solving aA system of at least one differential equation involving at least one real-valued function over a given domain of definition, implemented in a quantum-classical hybrid computer 1, 2 as shown in Figure 1. A quantum-classical hybrid computer 1, 2 is understood to be a system of a classical computer 1 and a quantum computer 2 connected according to a known architecture. A terminal 10 can be connected to the classical computer 1 to act as an interface. The terminal 10, the classical computer 1, and / or the quantum computer 2 can be connected via a network 20 such as the internet, even though the classical and quantum computers 1, 2 are generally located in close proximity. It should be noted that several quantum computers 1 can be operated in parallel. The classical computer 1, i.e., a binary logic computer operating on bits (state 1 or 0), includes data processing means 11 themselvesClassical computers, i.e., a conventional processor typically based on silicon, and generally data storage means, i.e., memory, for example, a hard drive. This is typically an online platform, to which the terminal is connected via an API. The quantum computer, or quantum machine, works on qubits whose quantum state has a quantum value with several simultaneous possibilities. It classically includes means for initializing qubits, applying quantum gates, and measuring states. We denote by n the number of qubits, and in this case, we will advantageously have n ≤ 10, in particular n ≤ 7. The classical computer is configured to either process data autonomously or send instructions to the quantum computer and receive data in response. Connected classical and quantum computers of all kinds can be used, for example, quantum computers.trap ions, or even neutral atoms. As we will see, some steps of the process are in practice implemented exclusively by the classical computer 1, and others by the quantum computer 2 under the control of the classical computer 1. The terminal 10 can be any equipment such as a PC or a smartphone. The present process aims at solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition. A system of differential equations, sometimes simply called a differential system, is a set of coupled differential equations, that is, differential equations that cannot be solved separately. A differential equation is an equation whose "unknown(s)" are functions, in this case the said real-valued function(s), and it is presented in the form of a relation between these unknown functions and their successive derivatives.Let E be the set of equations and e one of the equations. Mathematically, we put all the terms of each equation so that we can write the system in the form {DEe(x)=0}, e∈E. Let F be the set of real-valued functions on which the equations of the system are based, and f ∈ F one of these functions. Each is real-valued, on a given domain of definition D, which means that it takes as input x = (x1, x2, …, xv) ∈ D ⊂ ℝv. We can have at least one boundary condition associated with this system of differential equations (also called the Cauchy condition), that is, an expected value of one of the functions or one of its derivatives (for example, f(0) = f0). Note that the number of real-valued functions on which the system's equations focus (i.e., the size of F) is generally equal to the number of equations in the system. This method, as we will see, is particularly universal insofar as the system can include any numberof equations (including a single differential equation, and therefore solving this system will amount to solving this single differential equation), the equations can be ordinary differential equations (ODEs – the unknown function(s) sought depend on only one variable, i.e., v=1) or partial differential equations (PDEs – the unknown function(s) sought may depend on several independent variables, i.e., v>1), and they can be of any order. Preferably, said system of at least one differential equation models a physical system, and said real-valued function expresses a quantity of said physical system. Differential equations are present in all fields of science, and it will not be possible either to list them or to limit the scope of the invention to a particular field. This will be seen at the end of theThis document presents examples of three systems of differential equations, two of which model a specific physical system: 1 / Simple case of a first-order system of 2 ODEs, arbitrarily taking f(0) = g(0) = 0 on a domain [0, 0.95]. 2 / Case of a damped harmonic oscillator, which is a system of a single second-order ODE modeling a physical system (the oscillator), the said quantity of said physical system expressed by the function is the position of an oscillating body as a function of time, denoted x(t): represents the undamped frequency of the oscillator and ζ the damping coefficient, respectively at 9 / 8 and 45 / 8. The domain is any interval of ℝ +For convenience, we take [0, 1]. 3 / Case of a deformation of a material in a hypoelastic regime, which is (considering a one-dimensional deformation in a three-dimensional Cartesian coordinate system of the material – typically the stretching of a metal strip) a first-order system of two ODEs modeling a physical system (the material), the two quantities of said physical system expressed by the two functions being the displacement of a point of the material and the component of the stress tensor along the axis of the deformation at this point, as a function of its position along this axis, denoted u(x) and σxx(x):^^ ^^= ^^^(^^^)^ ^^ ^^ Hypoplastic regime means that the stress-strain relationship ^^ + ^^ = 0 ^ is not linen ^^ ^ eaire but reversible, i.e ^ ^ , with b, K, ε0 and σ0 √^^ ^material parameters, and with u(0)=0 and σxx(L)=t, where L is the length of the strip (extending from coordinates x=0 to x=L) and t is the applied force. We can take b=10, ε0 =0.1, σ0 = 5, n=4, K=100, L=0.9 and t=2. The domain is an interval of ℝ+ slightly larger than [0 ; L], for example again [0,0.95]. The present method aims at solving the system, i.e., determining a solution of the function(s) on which the system is based. This is an analytical solution, meaning that for a function f ∈ F, we determine a functional expression of an optimal solution fo approximating said function f.It is understood that an exact solution could exist that could be obtained through algebraic methods, but the present method remains essentially numerical, so that the optimal solution may in some cases be said optimal solution fo, but will most often only approximate it, although the approximation can be as precise as desired. The system of equations can be entered via terminal 10, and the solution(s) retrieved via the same terminal 10. Moreover, terminal 10 can be used to define a number of parameters (number of qubits, number of circuits, depth of the circuit(s), number of iterations, choice of functions from the polynomial basis, etc. – see below). Procedure: As will be seen, the present procedure cleverly uses one or more Variational Quantum Circuits (VQCs).A quantum circuit is a model of quantum computing, similar to classical electronic circuits, in which quantum computing is expressed as a sequence of quantum gates, measurements, qubit initializations, etc. A VQC is a known type of quantum circuit, also called a "parameterized quantum circuit." Indeed, a VQC is defined by a set of quantum parameters. Note that the term "quantum" parameter means that these parameters are relative to a quantum circuit, but they remain classical, i.e., real, parameters. VQCs can be of any existing type, for example, based on a Hardware Efficient Ansatz (HEA) structure, in which case the quantum parameters are rotational parameters. This example will be described in more detail later. Preferably, there will be fewer than 50 quantum parameters, particularly fewer than 30. By analogy with machine learning models, a VQC can be trained, i.e.The values ​​of the quantum parameters can be optimized for a specific task, particularly approximating the solution function by minimizing a cost function, as proposed in the paper by O. Kyriienko, A.E. Paine, and V.E. Elfving, "Solving nonlinear differential equations with differentiable quantum circuits." Note that this paper required specific, so-called differentiable circuits, which is not the case for the present method. To optimize the values ​​of the quantum parameters, as shown in Figure 2, the method advantageously begins with a step (a0) of initializing these quantum parameters, notably in a random manner. In practice, a quantum parameter value (QPV) is taken (and initialized) for each function on which the system operates.Next, in an original step (a), the process involves, for each function, encoding, using the corresponding VQC, the expression of a candidate solution of said function in a basis of orthogonal polynomials. By basis of orthogonal polynomials, we mean an infinite sequence of polynomials with real coefficients, such that the i-th polynomial is of degree i and such that the polynomials in the sequence are pairwise orthogonal for a given inner product. Any function can be approximated by a combination of polynomials from this basis; this is the idea behind step (a). This is called the spectral decomposition of the function. To rephrase, for each function in the system, we express the function in the basis of orthogonal polynomials and encode it using the VQC. It is important to understand that we will be encoding the function itself and not its variable dependency. Indeed, in the document O. Kyriienko, AE Paine, and VEIn Elfving's work, "Solving nonlinear differential equations with differentiable quantum circuits," it was proposed to encode x and its neighboring values ​​to obtain the derivative. The present solution, as we will see, proves to be much more efficient. These orthogonal polynomials are preferably Chebyshev polynomials, and we will use this example in the rest of the description, but they could also be Legendre, Laguerre, Fourier, or other known polynomials such as NURBS or Walsh functions. Note that it is also possible to modify (translate) and / or combine existing polynomial bases. For example, in the case of functions depending on a single variable (we will see the multivariable case later), we can use the following Chebyshev polynomial basis: cos(^ arccos ^) ^^ |^| ≤ 1cosh(^ arccosh ^) ^^ ^ ≥ 1 , where x∈D ⊂. ^ cosh(^ arccosh(−^)) ^^ ^ ≤ −1ℝ and k∈ℤ+ is the order of the polynomial. Then f(x) can be expressed as ∑^^^ ^^^ ^^^h^^(^, ^), with c0, c1,…, cC-1 the Chebyshev coefficients (scalars). We can then use the n-qubit state generated by VQC to represent the candidate solution. Note that we are talking about a candidate solution of the function (denoted fc if we are looking for the function f) since the quantum parameters are not yet optimized; we will converge, using this method, towards an optimal solution. To rephrase, the solution fc as encoded with arbitrary values ​​of the quantum parameters of the QVC is called the candidate solution, and the solution fo as encoded with optimized values ​​of these parameters is called the optimal solution of the function. Denoting |^^^ is the output state of the QVC of the function f and Uθ the gates of the QVC parameterized by the parameters θ, we have |^^^ = ^^|0^^^ = According to a first embodiment, called base-state encoding, which will be considered in the remainder of this description, the candidate solution is encoded in the amplitudes of the state, preferably in the associated probabilities pi (pi = |ai|²). Each polynomial of the basis is thus associated with a basis state, and the coefficient of this polynomial is the probability of measuring this basis state. Since the probabilities are always positive, in order to also have negative coefficients and thus be able to express all functions, we can, for example, duplicate the polynomials with a negative sign: if P(k,x) is a polynomial, we consider P'(k,x) = -P(k,x). Instead of having negative ck, we have ck = 0 and ck' = -ck. To return to the example of Chebyshev polynomials, in the particular case of x in [-1; 1], Cheb(k,x) remains in [-1 ; 1] and therefore the combination of Chebyshev polynomials remains bounded.To express any function f with values ​​outside this interval, it suffices to introduce a scalar factor λ (another scalar). In the end, we advantageously obtain the following encoding, adapted to Chebyshev polynomials but potentially applicable to any decomposition in a basis of orthogonal polynomials: ^^(^) =. According to a second alternative embodiment, called Pauli-monomial encoding, the candidate solution in the observable is encoded by considering a linear combination of Pauli monomials instead of the probabilities of the basis states. This provides another general framework for computing the evaluation of the test function, with more freedom regarding the number of orthogonal functions and the choice of the Pauli monomial appearing in the decomposition, which always grows exponentially in the case of the basis states. This is particularly interesting when dealing with barren plateaus, measurement strategies, and read errors; see the paper by M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, "Cost function dependent barren plateaus in shallow parametrized quantum circuits," Nature Communications 12 (2021), no. 1, 1791.In summary, the VQC, for quantum parameters θ, models the state that encodes the decomposition of the function in the basis of orthogonal polynomials. As explained, VQCs based on a HEA-type structure are preferred. Referring to Figure 3, which shows an example of a 5-qubit architecture, an HEA comprises a succession of blocks of a single-qubit rotation gate (parameterized by a rotation parameter θi) and a 2-qubit "entanglement" gate (a controlled gate, in particular CNOT). This block is repeated d times, where d is the depth of the VQC. As d or n increases, the number of rotation parameters θ increases proportionally, resulting in a circuit with greater expressive power, but the number of parameters remains reasonable. In the embodiment of Figure 3 with d=2 and n=5 qubits, there are therefore 10 rotation parameters.These "layers" of parameterized gates make the structure of the VQC comparable to that of classical neural networks, where rotation gates play the role of neurons and angles that of synaptic weights. By adjusting the values ​​of the circuit parameters, the state of the output is modified. Therefore, as explained, the VQC can be trained by minimizing a cost (loss) function in a machine learning-type loop. In general, the rotation layer includes rotations around different axes, namely Rx, Ry, and Rz, but in the example shown, this layer is composed solely of Ry gates parameterized for each qubit. This choice is motivated by the fact that there is currently no need to generate states with complex amplitudes since only the associated probabilities are used to encode the solution function. Thus, the search space is cleverly restricted to real states, for which it suffices to use Ry gates.It is also empirically observed that the results for various cost functions already achieve satisfactory accuracy with a single rotation gate in a layer (at least in the ideal case, i.e., with perfect operations in the circuit). It should be understood, however, that we are not limited to this particular architecture, nor even to HEAs in general: it suffices to have VQCs with quantum parameters on which the function's expression can be encoded in the basis of orthogonal polynomials. Note that, generally, to function on any quantum computer 2, the circuit must conventionally undergo a transpilation process to be expressed using the native gates supported by that quantum computer 2.In this case, directly constructing the ansatz using supported gates can help avoid additional transpilation steps that typically increase circuit depth and thus the risk of gate noise propagation and decoherence during execution. Directly considering qubit connectivity in qubit selection or ansatz design also helps reduce the number of additional gates (and therefore noise) introduced during the transpilation step.The procedure then comprises a step (b), for each function, of evaluating, at each of a predefined set of points in the given domain of definition, the values ​​of both the candidate solution and at least one (often several) derivative of the candidate solution using the QQC of the function, by expressing the derivative of the solution in the basis of orthogonal polynomials by differentiating the expression of the candidate solution of the function in the basis of orthogonal polynomials. A distinction is made between the evaluation of the function, denoted (b1), and the evaluation of one of its derivatives, denoted (b2). Here, a derivative is understood to be a derivative of any order, and therefore a differentiation, once or several times depending on the order, of the expression. For example, in a second-order differential equation, it will be necessary to differentiate twice. Evaluation is naturally limited to the order of the derivatives involved in the differential equations of the system.For example, if it's a second-order ODE with no first-order terms, we only evaluate the function and its second derivative. The great strength of this method is that we can evaluate both the function and its derivatives with the same circuit. The solution is thus much lighter, while remaining universal because the derivatives can be of any order. These points in the domain of definition, denoted ^^ ∈ ^, are "examples" in which the candidate solution corresponding to the current quantum parameters will be tested. Preferably, these predefined points in the given domain of definition are chosen so that they are uniformly distributed within said domain of definition, but alternatively, they could, for example, be chosen randomly (Monte Carlo method). We will see examples later. The more predefined points we have, the longer the evaluation will be, but the more precise the result will be.For example, we could take on the order of tens or hundreds of predefined points on the interval, for example 20. Evaluating a QVC is a well-known process; it involves observing its output state and determining the probabilities. This step (b) is therefore performed on the quantum computer 2. In this case, in the first encoding implementation in the basis state, the candidate solution fc is preferentially evaluated at one of the predefined points xS of the domain of definition given by the formula |||| = |||||||, where |||| is the output state of the QVC, λ a scaling factor, and Oc a diagonal observable, each a function of the expression of the candidate solution in the basis of orthogonal polynomials. More precisely, λ and |||| are functions of the decomposition of the function, and Oc is a function of the polynomials themselves. In particular, we can express the observable as ^^(^) = ^^ ⊗. And by using the advantageous expression of fc in terms of probabilities, we have: As an example, suppose the function f(x) = 2x - 3. With Chebyshev polynomials, we have Cheb(0,x) = cos(0) = 1 and Cheb(1,x) = x, i.e., f(x) = -3*Cheb(0,x) + 2*Cheb(1,x), which can be rewritten as f(x) = 5((0 - 3 / 5)Cheb(0,x) + (2 / 5 - 0)Cheb(1,x)). We can thus easily evaluate f by taking λ = 5 (2-qubit state) Regarding derivatives, as explained, it suffices to differentiate derivative. The q-th derivative of said candidate solution fc is thus simply evaluated preferentially at one of said predefined points xS of said domain of definition given by the formula is the output state of the same VQC, λ the scale factor, and ^^^^ the diagonal observable, each a function of the q-th derivation of said expression of the candidate solution of said function in said basis of orthogonal polynomials. ^ ^ ^ ^has exactly the same structure as Oc, but with the diagonal elements replaced by the q-th derivatives of the corresponding polynomials, which can usually be expressed directly. For example, for Chebyshev polynomials: Note that this is a special summation in which the term for k=0 must be halved if it appears. Similarly, in the second implementation using Pauli demonomial encoding, the observable can now be expressed as ^^(^) = ∑ ^^^ ^^^ ^^ ∙ ^(^, ^) ∙ ^^ , with ^^ ∈ {^, ^, ^, ^} ⊗^ and ^ ^positive coefficients for i <M, avec M le nombre de monômes de Pauli considérés. Un sous-ensemble spécifique de monômes de Pauli présentant un intérêt particulier, lorsqu'on tente de définir la fonction de cout de manière plus locale,est celui des observables k-locaux (voir le document M. Cerezo, A. Sone, T.Volkoff, L. Cincio, and P. J Coles, Cost function dependent barren plateaus inshallow parametrized quantum circuits, Nature communications 12 (2021), no.1, 1791. Cité avant), définis comme des observables pour lesquels les monômesde Pauli apparaissant dans leur décomposition agissent de manière non triviale sur au plus k qubits. En d'autres termes, pour un monôme de Pauli donné Pi =Pi,1 ⊗ Pi,2 ⊗ · · · ⊗ Pi,n, on dit qu'il agit de manière non triviale sur k qubits(également appelé interaction k-corps) s'il y a exactement n − k atomes de Pauli Pi,j qui sont identiques. Le nombre maximal Mk de monômes de Pauli apparaissant dans un In summary, the present method is particularly efficient in all cases, as it only requires as many circuits as there are functions. The same circuits can be reused to evaluate the functions and their derivatives at as many points in the domain as desired. The method then includes a step (c) of optimizing the values ​​of the quantum parameters of the QVCs of each function, so as to minimize a cost function defined by the system of differential equations and a function of the evaluations for each function in the candidate solution and its derivative. To reformulate, knowing the values ​​of the functions and their derivatives at the predefined points, these values ​​can be applied to the system and an error calculated. If the error is zero or nearly zero, then fc is an excellent solution. The optimal solution fo is the "best" of the candidate solutions fc. This step can be consistent with what is described in document O.Kyriienko, AE Paine, and VE Elfving, Solving nonlinear differential equations with differentiable quantum circuits. As explained, the goal is to evaluate the cost function (which represents the error) and, if necessary, repeat steps (a) and (b) with different values ​​of the quantum parameters in order to progressively minimize this cost function and converge towards optimized parameter values, so as to "train" the quantum circuits, in the same way that neural networks would be trained. Generally, each iteration consists of a first substep (c1) for calculating the cost function, i.e., the error, followed by a substep (c2) for verifying a criterion, for example, comparison with an acceptance threshold: if the error is still above the threshold, a new iteration is implemented, modifying the parameter values.Alternatively, a predefined number of iterations can be implemented to avoid checking the criterion each time. Any known optimization algorithm can be used on the parameter values, for example, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method. Preferably, the cost function is of the form ^(^) = ^^^^^(^) +. where θ is the set of quantum parameters, ^^^^^ a first term representing an error on the differential equation(s) (called the residue),^ ^^^^^^^is a second term representing an error on the boundary condition(s), and η is a predefined coefficient. The latter controls the weight of the boundary conditions in the optimization. It can, for example, be chosen as a value greater than the maximum possible value of the left-hand term. In the absence of boundary conditions, or if they are very simple, η can be taken as 0 to speed up the process. Any classic cost function expression can be used, and for example, the first term can be the Mean Squared Error (MSE) of all the differential equations for the set of predefined points in the given domain, and / or the second term can be the sum of all the errors on all the boundary conditions for the set of predefined points in the given domain. We then have is a differential equation of the system, and He is the set of evaluations of the terms of equation e. Similairement, on a ^ ^^ (^ ^^ ))², where nBC is the number of boundary conditions, fBC and xBC the relevant values. At the end of the last occurrence of step (c), i.e., when the chosen criterion is satisfied (for example, that the error is below the acceptance threshold, or simply that enough iterations have been implemented), the optimization is considered complete, and therefore acceptable values ​​of the quantum parameters have been reached. Then, in a final step (e), for each function, the optimal solution of said function, as encoded by the VQC with the optimized parameters, is reconstructed. This simply involves expressing the optimal solution fo as the combination of orthogonal polynomials with coefficients corresponding to the optimized values ​​of the quantum parameters, as a function of the probabilities (recall that in the case of a basis-state encoding −^^^^^^^) ^(^, ^), and that fo is the last fc). Note that in practice, we could reconstruct each candidate solution in this way to evaluate it in this manner at step (b). Case of functions of several variables The present method is universal, insofar as it also allows us to solve systems including PDEs, i.e., equations in which the unknown function(s) sought may depend on v>1 independent variables. As indicated at the beginning, we always write f(x) but with v real components (x = (x1, x2, …, xv) ∈ D ⊂ ℝv). In this case, at step (b) the q-th derivative of said candidate solution ^ ^ ^ ^ ^^ ^^ …^^with j1…jq ∈ {1, …, v}q S^^ can be evaluated at a predefined point x (with v components) of said domain of definition given always by the formula where, ^^^^ a diagonal observable, each function of the q-th derivation of the said expression of the candidate solution of said function in said basis of orthogonal polynomials (again, only the diagonal observable changes). Indeed, since the variable dependence is encoded only in the observables, moving from a 1-dimensional space to a v-dimensional space does not change the algorithm; therefore, the same VQCs can always be used. In the one-dimensional case, each basis state is used to represent, in binary notation, the order of a polynomial. With the exception of the first qubit, which is preferentially used to determine the sign in front of the polynomial, all the remaining qubits of the basis states are typically used to encode the corresponding order. In the multivariate case, each variable xj can be assigned a specific number of qubits that will be used to encode the order of the corresponding polynomial.Therefore, each basis state |i^ of the (n + 1)-qubit state modeling the solution is used to encode the different indices of each variable. The binary decomposition of the basis state |i^ (in decimal notation) can be written as |i^ = |i0^ |i1i2i3 ... in^. More preferably, for each variable xj, lj qubits are allocated to encode the order of the polynomial (with l1 + l2 +… lv = n), and the binary decomposition of the order is denoted by Lj. This allows us to separate the basis state |i^ as the tensor product of the first qubit (designated) with v states, such that. Thus, the selection of the number of qubits lj for each variable xj is free, which is very advantageous when certain variables require more terms in the spectral decomposition. The only thing that might be slightly more complex to compute than in the single-variable case is the derivation of the expression for the candidate solution of the function in the basis of orthogonal polynomials. This again has no impact on the quantum part of the present method and can be greatly facilitated by cleverly choosing the basis of orthogonal polynomials. For example, Chebyshev polynomials in several variables are simply expressed as products of Chebyshev polynomials in only one variable. variable Taking Lj as the binary decomposition of the order, then the partial derivative of a Chebyshev polynomial is easily written: Furthermore, we note that the right-hand product ∏^ ^^^,^^^ ^h^^(^^, ^^)= which reduces the number of multiplications necessary, especially since all or part of these factors are already calculated to evaluate the (underived) solution at a predefined point. When we go up another order, there are two cases: - either we differentiate again with respect to the same variable xj, i.e., we go up to order 2, and then the calculation is easy: - either we differentiate with respect to a different variable, and after simplification we obtain: Thus, in the preferred case of Chebyshev polynomials in several variables To generalize any partial derivative of order q with respect to the variables xj1, …, xjq, we construct a set of h≤q distinct variables xk1, …, xkq, each appearing respectively q1, …, qh times with q1 + … + qh = q. For example, if h=1, we have a derivative q times with respect to the same variable. We can therefore rewrite de là : Note that if h > v / 2, it is more advantageous to return to the equation above in order to the number of products, and therefore: ^ ^ ^h^^ ^ ^^^ … ^^ Results The present method was implemented to solve the three examples of systems of differential equations mentioned above. 1 / Simple case of a first-order system of 2 ODEs by arbitrarily taking f(0) = g(0) = 0 on a domain [0, 0.95]. With only 4 qubits, two HEAs with a depth of 3, 150 iterations in a BFGS optimization, and a set of 20 predefined points evenly distributed over the domain, we obtain the result shown in Figure 4a (f(x) on the left and g(x) on the right), with the exact solution shown in solid lines and the optimal solution in dashed lines. The curve at the bottom represents the standard deviation as a function of x, which, as we can see, does not exceed 0.004. The final value of the cost function is only 5.12 x 10⁻¹². -5 The optimal solutions of the two functions are indistinguishable from the exact solution. ^^ 2 / Case of a damped harmonic oscillator ^^ + 2^^ ^^+ ^^^ = 0, with x(0)=2 and x'(0)=0, where ω = 9 / 8 and ζ = 45 / 8, over a domain [0, 1]. Since we move to order 2, theoretically many more parameters are required. With only 5 qubits, a HEA with a depth of 5, and 525 iterations in a BFGS optimization, we obtain the result shown in Figure 4b (single function x(t)), with the exact solution shown in solid lines and the optimal solution in dashed lines. The curve at the bottom represents the standard deviation as a function of x, which we see does not exceed 0.3. The final value of the cost function is still only 2.69 x 10 -3 3 / Case of deformation of a material in a hypoelastic regime, σxx(L)=t, b=10, ε0 =0.1, σ0 = 5, n=4, K=100, L=0.9 and t=2, in a domain [0, 0.95]. Despite the complexity of the nonlinear relationship of the system, remaining with the VQCs of the first example (4 qubits, HEA with a depth of 3), still a set of 20 predefined points arranged regularly on the domain and going up to 400 iterations in a BFGS optimization, we obtain the result visible in Figure 4c (u(x) on the left and σxx(x) on the right), with the exact solution in solid form and the optimal solution in dashed lines, respectively. The curve at the bottom represents the standard deviation as a function of x, which we see does not exceed 0.0005 for u(x) and 0.001 for σxx(x). The final value of the cost function is only 1.05 x 10⁻³. We can therefore see that in all cases the present method remains very accurate even with few parameters. In practice, 6 or 7 qubits, and a depth of 4 or 5 (i.e., 20-30 parameters) are more than sufficient to solve most of the differential equation systems encountered.and in particular those modeling a physical system. 4 / Test on a real NISQ quantum computer While the three previous tests were implemented in practice on an emulator (computer simulation of a quantum computer), the present method was also successfully tested on a real IBM Brisbane-type quantum computer with an Eagle r3 processor, on the system of a first-order ODE f'(x)-f(x)=0, with the condition f(0)=1 over a domain [0, 0.9], with 16 uniformly distributed predefined points. The method was implemented using 3 qubits, on a typical ansatz of depth 2 comprising a succession of a Ry gate and a CNOT gate, and 20 iterations in an SLSQP (sequential least squares programming) optimization. The circuit was transpiled using Qiskit 1.2.4. At each iteration, the values ​​of the candidate solution function and its derivative were evaluated against the state generated using 20,000 shots.The solution was calculated analytically using the parameter shift rule. The boundary condition was precisely imposed using floating-point boundary treatment (this method iteratively shifts the candidate solution to force it to match the boundary conditions). The result is shown in Figure 4d, with the exact solution represented by a solid line and the optimal solution by a dashed line. The curve at the bottom represents the standard deviation as a function of x, which does not exceed 0.05. An overall validation score of V = (1.57 × 10⁻², 1.46 × 10⁻⁴) is obtained, representing the maximum and mean squared distances to the exact solution (the exponential function). The final value of the cost function is only 0.052, despite the very small number of parameters (3 qubits and a depth of 2). Server According to a second aspect, the invention relates to the hybrid quantum-classical computer 1,2 for the implementation of the process according to the first aspect. This hybrid computer comprises, as explained, a classical computer 1 and a quantum computer 2, connected, and optionally a terminal 10 for data input and output. Said quantum-classical hybrid computer 1, 2 is configured to solve a system of at least one differential equation involving at least one real-valued function over a given domain of definition, by implementing the following steps:^ For each function, encoding, using a quantum variation circuit (QVC), the expression of a candidate solution of said function in a basis of orthogonal polynomials, said QVC being defined by a set of quantum parameters;^ For each function, evaluation, at each of a set of predefined points in said given domain of definition, of the values ​​of both said candidate solution and at least one derivative of said candidate solution using the QVC of said function,by expressing the derivative of said solution in said basis of orthogonal polynomials by differentiating said expression of the candidate solution of said function in said basis of orthogonal polynomials;^ Optimization of the value of said quantum parameters of the QVCs of each function, so as to minimize a cost function defined by said system of differential equations and function of said evaluations for each function of the candidate solution and its derivative;^ For each function, reconstruction of an optimal solution of said function as encoded by the QVC with the optimized parameters. Computer program product According to a fourth and a fifth aspect, the invention relates to a computer program product comprising code instructions for execution (on the quantum-classical hybrid computer 1,2) a method according to the first aspect of solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition; and a storage means (for example, the data storage means of a conventional computer) on which this computer program product is found.

Claims

CLAIMS 1. A method for solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition, the method comprising the implementation, by a quantum-classical hybrid computer (1, 2), of steps of: (a) For each function, encoding by means of a quantum variational circuit, QVC, of ​​the expression of a candidate solution of said function in a basis of orthogonal polynomials, said QVC being defined by a set of quantum parameters; (b) For each function, evaluation at each of a set of predefined points of said given domain of definition, of the values ​​of both said candidate solution and of at least one derivative of said candidate solution by means of the QVC of said function, by expressing the derivative of said solution in said basis of orthogonal polynomials by differentiation of said expression of the candidate solution of said function in said basis of orthogonal polynomials;(c) Optimization of the values ​​of said quantum parameters of the QVCs of each function, so as to minimize a cost function defined by said system of differential equations and function of said evaluations for each function of the candidate solution and its derivative; (d) For each function, reconstruction of an optimal solution of said function as encoded by the QVC with the optimized parameters.

2. A method according to claim 1, wherein each QVC is based on a Hardware Efficient Ansatz (HEA) structure, said quantum parameters being rotational parameters.

3. A method according to any one of claims 1 and 2, wherein said orthogonal polynomials are Chebyshev polynomials.

4. A method according to any one of claims 1 to 3, comprising a step (a0) of initializing the quantum parameters randomly.

5. A method according to any one of claims 1 to 4, wherein at least one boundary condition is associated with said system of differential equations, the cost function is of the form ^(^) = ^^^^^(^) + ^. ^^^^^^^^(^), where θ is the set of quantum parameters, ^ ^^^^ a first term representing an error in the differential equation(s), ^ ^^^^^^^ is a second term representing an error on the boundary condition(s), and η is a predefined coefficient.

6. Method according to claim 5, wherein the first term ^ ^^^^ is the average squared error of all the differential equations for the set of predefined points of said given domain of definition, and the second term ^ ^^^^^^^is the sum of all errors on all boundary conditions for the set of predefined points of said given domain of definition.

7. A method according to any one of claims 1 to 6, wherein said candidate solution fc is evaluated at a predefined point xS of said given domain of definition by the formula ^^(^^) = ^^^^^^^(^^)^^^^, where |^^^ is the output state of the VQC, λ a scaling factor, and Oc a diagonal observable, functions of the expression of said candidate solution in the basis of orthogonal polynomials.

8. A method according to claim 7, wherein the derivative q-^ ^ ^th of said candidate solution ^ ^ is evaluated at a predefined point xS ^^ ^ …^^ ^^ said domain of definition given by the formula (^^)^^^^, where xj1…xjq are q components of x, a diagonal observable, functions of the q-th derivation of said expression of the candidate solution of said function in said basis of orthogonal polynomials.

9. A method according to any one of claims 1 to 8, wherein said predefined points of the given domain of definition are chosen so as to be uniformly distributed in said given domain of definition.

10. A method according to any one of claims 1 to 9, wherein said system of at least one differential equation models a physical system, said real-valued function expressing a quantity of said physical system.

11. A quantum-classical hybrid computer (1, 2), characterized in that it is configured to solve a system of at least one differential equation on at least one real-valued function on a given domain of definition, by implementing steps of: For each function, encoding by means of a quantum variational circuit, VQC,of the expression of a candidate solution of said function in a basis of orthogonal polynomials, said QVC being defined by a set of quantum parameters;^ For each function, evaluation at each of a set of predefined points of said given domain of definition, of the values ​​of both said candidate solution and at least one derivative of said candidate solution by means of the QVC of said function, by expressing the derivative of said solution in said basis of orthogonal polynomials by differentiating said expression of the candidate solution of said function in said basis of orthogonal polynomials;^ Optimization of the value of said quantum parameters of the QVCs of each function, so as to minimize a cost function defined by said system of differential equations and a function of said evaluations for each function of the candidate solution and its derivative;^ For each function,Reconstruction of an optimal solution of said function as encoded by the VQC with the optimized parameters.

12. Computer program product comprising code instructions for executing a process according to one of claims 1 to 10 of solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition, when said program is executed on a computer.

13. Computer-readable storage means on which is recorded a computer program product comprising code instructions for executing a process according to one of claims 1 to 10 of solving a system of at least one differential equation involving at least one real-valued function over a given domain of definition.

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