Device and processing method for solving linear systems

Abstract

Device and processing method for solving linear systems Processing method in a computer (1) comprising an electronic processing device (2) and an optimization module (3) Ising or QUBO according to which to solve an initial linear system Ax + B=0, x = [xj] j = 0 to N-1, A =[aij] i,j = 0 to N-1 and B =[bj] j = 0 to N-1 , comprising: transposition of the linear system comprising, in each row i, the replacement of , and by their binary representation of resolution r, thus giving, for each row i of the initial linear system, r equations each corresponding to a respective bit weight from 0 to r-1, for each row i, additional binary variables being further added to take into account the binary carries resulting from the operations performed in the equations at the different bit weights and the transfer of said carries to higher bit weights;determination of a QUBO type or Ising Hamiltonian representation of the transposed linear system and resolution by optimization calculation of the optimization module (3) Figure for the abstract: Fig. 1;

Description

Title of the invention: Device and processing method for solving linear systems. Technical field

[0001] The invention lies in the field of solving linear systems, in particular using a quantum computer. Previous technique

[0002] Quantum computers operate with quantum bits, or qubits, which, unlike classical computer bits having a value of either 0 or 1, can be a superposition of the states 0 and 1. This characteristic means that quantum computers could be much faster than classical computers for many tasks and could also be used to solve certain problems that a classical computer cannot solve. Qubits can be fabricated from different hardware platforms or building blocks, such as superconducting qubits, elementary particles, or trapped ions. Other emerging methods include photonic quantum processors that use light. Electronic processing modules, including optical blocks such as lasers, lenses, and mirrors, control the qubits.

[0003] The imperfect nature of current hardware gives rise to quantum errors that can prevent the final result of the calculation from being achieved. The main cause of these errors is qubit decoherence, which destroys the quantum character of the qubits and reverts them to the state of classical bits. Decoherence is caused by the interaction of the qubits with their environment.

[0004] Quantum error correction (QEC) techniques, which involve using a large number of qubits to create a "logic qubit" that is much less prone to errors, can be used with the aim of approaching FTQC ("Fault Tolerant Quantum Computer") type operation, i.e., "fault-tolerant quantum computing".

[0005] Moreover, the resolution of linear systems is of major interest, both scientifically and industrially and societally: it is involved in multiple fields, ranging from various aspects of numerical simulation, such as plasma physics, electromagnetism, fluid mechanics, and naturally extends to aspects of analysis of natural disaster scenarios (meteorological, seismic, tsunami, ...) but also climate forecasting scenarios, planetology, astrophysics, particle physics, etc.

[0006] In this particular field, a landmark article, "Aram W. and Hassidim, Avinatan and Lloyd, Seth, Quantum Algorithm for Linear Systems of Equations, 2009," corresponds to the publication of the algorithm known as "HHL," named after the initials of the three authors, which demonstrates the existence of a possible exponential advantage of this type of calculation on a quantum computer. This advantage is conditioned by several particularities of the system of linear equations and the use of the solution:

[0007] - the system matrix must be hollow,

[0008] - the vector result must be used in the form of a product scalar (it is not possible to obtain the solution vector explicitly without abandoning all hope of exponential advantage).

[0009] Despite these limitations, the HHL algorithm is one of the most anticipated algorithms for the use of quantum machines. However, this algorithm assumes the use of a perfect quantum computer, known as a FTQC. Currently, no FTQC computers exist. Current quantum computers and those expected in the coming years have relatively high noise characteristics (>10⁴), and the number of physical qubits (the units for storing and using quantum information) remains relatively limited, between a few dozen and a few hundred. Within this framework, algorithms capable of providing exponential acceleration, such as HHL, prove infeasible even at modest scales.

[0010] There are other types of quantum computers, the so-called analog quantum computers, which probably do not have the characteristics to provide exponential acceleration except on potentially physical simulations (so-called "many-body" problems) but which can offer a possibility of polynomial improvement of algorithms and / or a possibility of having reduced energy consumption compared to "classical" computers: cf. the D-Wave® quantum computers which are based on a superconducting qubit technology, and those of Pasqal® which manipulate neutral atoms in optical matrices.

[0011] Quantum computers exist that are suitable for performing a quantum annealing procedure (also called adiabatic annealing) on ​​a two-dimensional Ising-type Hamiltonian basis. This type of approach to quantum computing is particularly well suited to optimization calculations, as it is easily shown that they are formally equivalent to specialized solvers for unconstrained optimization problems in binary variables, often referred to by their English acronym QUBO ("Quadratic Unconstrained Binary Optimization"). In general, QUBO problems are NP-complete, but allocating a non-trivial-scale NP-complete problem (at least several hundred variables) on these computers is not necessarily a simple problem in itself. Moreover, there are no results regarding any computational advantage of this calculation technique compared to a conventional computer.

[0012] Below is briefly recalled the operating principle of a quantum computer using an adiabatic calculation on a two-dimensional Ising Hamiltonian basis.

[0013] Quantum computer using adiabatic computation on a two-dimensional Ising Hamiltonian basis.

[0014] Quantum annealing is a specific quantum computing technology with performance levels intermediate between those of traditional supercomputers and universal quantum computers. The general principle of the adiabatic / quantum annealing computer consists of establishing connections between qubits using weights, then making the system find an equilibrium point by modifying these weights to identify an energy minimum for the system, which corresponds to the desired solution to the problem. The process is called adiabatic because there is no energy transfer between the computer's chipset and its environment: the same energy level is maintained throughout the adiabatic annealing process, and therefore excited states are eliminated exponentially (conditions of the so-called "adiabatic theorem" of quantum mechanics).

[0015] The basic principle of the adiabatic quantum computer consists of preparing a “Hamiltonian”, i.e., a quantum system with several interconnected qubits. This Hamiltonian is initialized in a state that is the solution to an easy problem. The computer will then evolve this Hamiltonian adiabatically towards the Hamiltonian of the problem posed, respecting prior constraints and corresponding to a minimum energy, and thus provide a solution to the difficult problem that one seeks to solve.

[0016] The principle of adiabatic quantum computing is as follows: first, a complex Hamiltonian is determined whose ground state describes a solution to the problem under study (the final Hamiltonian is derived from the cost function whose lowest-energy solution is sought, and which will correspond to the ground state (lowest energy) of this Hamiltonian). Next, a system possessing a simpler Hamiltonian is prepared, which is initialized in its ground state. This Hamiltonian is then adiabatically evolved towards the complex Hamiltonian that has been determined; according to the adiabatic theorem, the system remains in the ground state, and its final state describes a solution to the problem under consideration.

[0017] Consider an initial Hamiltonian and a final Hamiltonian, corresponding to the following forms: (Equation 1)

[0018] where: ^4; ; j ; are the usual Pauli operators acting on spins, i.e. the complex 2*2 matrices that represent the spin of the following particles: - THE / 4 are biased magnetic fields and - the matrix f → → is an inter-spin coupling matrix.

[0019] A time parameter θ, called the quantum annealing time, is defined, along with two monotonic functions x and θ, respectively decreasing and increasing, with support on θ, such that 4(θt) and 4, 44; 4, so that the internal quantum system of the computer quantum is subject to the following time-dependent Hamiltonian during the annealing time:

[0020] (Equation 2)

[0021] If the spins of the quantum computer are prepared in the ground state of the Hamiltonian and if the quantum annealing time ~ 5 is sufficiently long, then the quantum computing system will be at time y in the ground state of the Hamiltonian % y (adiabatic theorem of quantum mechanics).

[0022] The fundamental state of the Hamiltonian W? allows us to trivially deduce the minimum cost solution (or a solution) of the QUBO (Quadratic Unconstrained Binary Optimization) type problem directly deduced from this Hamiltonian.

[0023] A classical optimization problem called QUBO is, in the general case, NP-hard. It is characterized by a cost function C(x):

[0024] y .......T y -j (Equation 3)

[0025]

[0026]

[0027] jxj is a matrix usually written in triangular form superior, but which can be any matrix with real coefficients and where x is a binary vector of dimension n, x' :u Now, the QUBO problem is equivalent, by simple change of variables, to the well-studied quantum problem of Ising Hamiltonians. In the latter case, instead of studying binary variables (0 or 1), one studies spin orientations (value $ equal to +1 or -1 on a selected axis which is often - by convention - the axis , vertical). With the following variable substitution: , , with ... < s ,-,, the .....Mr. • r) Hamiltonian is written:

[0028] (Equation 4)

[0029]

[0030] with (Equation 5)

[0031]

[0032] As mentioned earlier, analog quantum computers based on the Ising Hamiltonian are well-suited for physical spin-glass simulations and optimization problems thanks to the Ising-QUBO equivalence. To solve systems of linear equations, the initial representation of the linear problem must be transformed into a quadratic problem. Furthermore, the goal is to ensure the best possible solveability of the Hamiltonian resulting from this transformation. Starting from the system of linear equations expressed in vector form using a matrix of constant coefficients * $ MA^i^e the set of square matrices of dimension N x N with coefficients (^1 = OàN-letj = OàNl) in and of a vector r æ , (vector with component b,, i = 0 to N-1 ) we seek to solve the following equation: (Equation 6) .Ax 4- 23 0

[0034] where A, B are known and v ... is the solution that is sought (vector of A. components xi5 i = 0 to Nl).

[0035] To enable the solution of the equation, it is necessary to transpose it into a Hamiltonian formulation that respects the following principles:

[0036] - the minimum energy for the chosen Ising Hamiltonian must correspond to the solution to equation #6,

[0037] - the spectral gap of the defined Hamiltonian should be as large as possible in order to to increase the probabilities of resolution through adiabatic quantum computing (by virtue of the adiabatic theorem),

[0038] - the scale of value of the relative couplings defined below is the lowest possible in order to avoid the intrinsic resolution and noise problems of the quantum computer, with:

[0039] (Equation 7)

[0040] To limit the problems of resolution and inherent noise, the ideal is to have < $ $ and with current technologies it is recommended to remain in all cases with < Us).

[0041] The article Kyungtaek Jun, "QUBO formulations for a System of linear equations, 2024") studies the problem of solving equation no. 6 by the least squares method, on a D-Wave Advantage machine and illustrates the state of the art in the field.

[0042] It uses a binary representation of each element of the solution vector X. And a fixed-point resolution or (equivalently) an integer representation in which a common factor yA is introduced and implied in the rest of the document (i.e. all the coefficients have been implicitly multiplied by y- to get back to an integer equation).

[0043] For the binary representation, it is necessary, according to the method described in this article, to choose a resolution > 4 1 and each element x, of the solution vector x of the equation can be written:

[0044] _ y*" ! < - y (Equation 8) ~*ir~

[0045] where each x'^ , j = 0 to r, is a binary variable. Each x. , i = 0 to Nl, is therefore an integer between and -1: consequently, is the intrinsic integer resolution of the resolution method.

[0046] The least squares method considered in the aforementioned article consists of constructing a cost function for a QUBO problem whose minimum corresponds to the solution of the system of equations we are seeking. For each row of equation no. 6, we have:

[0047] The i-th line (i = 0 to Nl) of equation 6 is written as follows: yAM _ 0, or still like this: 100481 + z > 0 (Equation 9)

[0049] Since the partial cost function is classically defined by / \ \2, it AI / = A partial cost function is therefore defined here such that:

[0050]

[0051] It turns out that w ... ? (it's a square) and that a sufficient condition, For this partial cost function to be zero, æ must be a solution to equation #6. We can define a global cost function:

[0052] v .. (Equation 11) C(x) ss ' / 4X)

[0053] which is positive or zero; therefore the condition for it to vanish is that x is a solution of equation no. 6. If the equation has no integer solution, then there is no solution at 0 of the cost function but the minimum of the configuration in binary variables does indeed correspond to the integer vector closest to the exact real-valued solution.

[0054] It is theoretically possible to exceed the intrinsic resolution by using a quadratic interpolation of the value to 0 between the lowest-cost results revealed by the search for the minimum. In practice, this will potentially be limited by the noise of the QUBO or Ising solving system.

[0055] Note that in QUBO formulation, the minimum cost will not be 0, but a constant corresponding to the sum of the squares of the , due to the formulation matrix which implies that the cost in QUBO formulation is 0 for a zero vector.

[0056] Starting from the cost function defined in equations no. 10 and no. 11, we observe an exponential explosion problem in solving the coefficients of the QUBO matrix: the coefficient (Cr) multiplying the term in 2- in this formula will lead to growth as follows:

[0057] j (Equation 12) G a

[0058] The fact that we are in QUBO or Ising representation changes little the fact that, since the implementation of the coefficients of the Ising or QUBO matrix cannot have infinite precision, and since current technologies limit it even around p = 10⁻¹, the capacity of a machine will be limited to r (in practice, it is *...... G even less so, because of intrinsic qubit noise and interaction noise between qubits (crosstalks).

[0059] To obtain a correct precision of 32 bits, for example, it would be necessary to have a resolution quality on the coefficients that is better than $ j ce which is completely unrealistic: it is therefore necessary to find solutions which behave at worst linearly with the increase in resolution s and dimension A of the equation that we are trying to solve. Summary of the invention

[0060] To this end, according to a first aspect, the present invention describes a processing method in a computer comprising an electronic processing device and an optimization module adapted to perform optimization calculations of a specific type among Ising and QUBO and any type reducible to an Ising or QUBO optimization, according to which said method comprises the following steps implemented by the electronic processing device following the receipt, by the electronic processing device, of a command to solve an initial linear system Ax + B=0, where x is the solution vector to be determined x = [Xj] j=0 to N4, A =[a;j] ij=0 to N-i and B =[bj] j -0 to N-i with in line i of the initial linear system, i =0 to Nl, the equation: jxj —

[0061] a / transposition of the linear system comprising, in each line i, the replacement of each variable xj by a binary representation of xj based on r binary variables xjk, j = 0 to Nl and k = 0 to r-1, where r indicates the resolution of the binary representation;

[0062] b / determination of a representation in a predefined format of QUBO type or of Ising Hamiltonian type of the linear system thus transposed with variables to be determined at least the said binary variables xjk, said representation being a function of at least the coefficients of said transposed linear system;

[0063] c / provision of said determined representation of the linear system to the optimization module,

[0064] d / The method further comprises determining, based on said provided representation, the values ​​of the binary variables xjk satisfying said system transposed by optimization calculation of said determined type implemented by said optimization module

[0065] said process being characterized in that:

[0066] at step a:

[0067] - the transposition further includes the replacement of aij and bt by their binary representation based on, respectively, aijk and k = 0 to r-1, thus giving, for each line i of the initial linear system, r equations each corresponding to a respective bit weight from 0 to r-1, the transposed linear system thus comprising N*r equations and

[0068] - for each line i, additional binary variables are further added to take into account the binary carries resulting from the operations performed in the equations at different bit weights and the transfer of said carries to higher bit weights;

[0069] in step b: a representation is determined in the predefined format of the linear system thus transposed with variables to be determined at least the said binary variables xjk and the additional binary variables, said representation being determined as a function of at least the coefficients of said transposed linear system;

[0070] in step d, said optimization calculation further determines values ​​of additional binary variables satisfying said transposed system.

[0071] The present invention therefore offers a solution for solving linear systems that is particularly advantageous in terms of complexity for determining the solution of the system.

[0072] In embodiments, such a method will further comprise at least one of the following features:

[0073] - said optimization module includes an analog quantum computer adiabatic quantum annealing,

[0074] and in step d, said determination is carried out by quantum annealed adiabatic analog quantum computing implemented by said quantum annealed adiabatic analog quantum computing;

[0075] - at step a, for each line i: the r lines associated with line i are processed successively by increasing bit weight, carrying over the carries from the lower bit weights and introducing at each of the r lines the at most \ „ , 5 s. 10 necessary deductions to guarantee the equality of line i to 0;

[0076] - in step a, for each row i, a tree summation method is applied binary according to which the partial sums of successive terms to be summed in the row are further considered, and then iteratively, at a given level of the tree, the partial sums of the partial sums of the previous level are considered; and

[0077] in step b, the additional binary variables further include the binary representations of the partial sums of the different levels of the tree;

[0078] - in step b, the spectral gap of said representation is increased while maintaining the table of truth.

[0079] According to another aspect, the invention describes a computer comprising an electronic processing device and an optimization module adapted to perform optimization calculations of a specific type among Ising and QUBO and any type reducible to an Ising or QUBO optimization, wherein said electronic processing device is adapted to receive a command to solve an initial linear system Ax + B=0, where x is the solution vector to be determined x = [Xj] j = OàN-b A =[aÿ] ij = oàN-i and B =[bj] j-oàN- with in line i of the initial linear system, i =0 to Nl, the equation: yv* , said device being further adapted to perform the The following treatments:

[0080] a / transposition of the linear system comprising, in each line i, the replacement of each variable xj by a binary representation of xj based on r binary variables xjk, j = 0 to Nl and k = 0 to r-1, where r indicates the resolution of the binary representation;

[0081] b / determination of a representation in a predefined format of QUBO type or of Ising Hamiltonian type of the linear system thus transposed with variables to be determined at least the said binary variables xjk, said representation being a function of at least the coefficients of said transposed linear system;

[0082] c / provision of said determined representation of the linear system to the optimization module,

[0083] said optimization module being adapted to determine, based on said provided representation, values ​​of the binary variables xjk satisfying said system transposed by optimization calculation of said determined type implemented by

[0084] said computer being characterized in that:

[0085] at step a:

[0086] - the transposition further includes the replacement of aij and b{ by their binary representation based on, respectively, aijk and , k = 0 to r-1, thus giving, for each row i of the initial linear system, r equations each corresponding to a respective bit weight from 0 to r-1, the transposed linear system thus comprising Nxr equations and

[0087] - for each line i, additional binary variables are further added to take into account the binary carries resulting from the operations performed in the equations at different bit weights and the transfer of said carries to higher bit weights;

[0088] in step b: a representation is determined in the predefined format of the linear system thus transposed with variables to be determined at least the said binary variables xjk and the additional binary variables, said representation being determined as a function of at least the coefficients of said transposed linear system;

[0089] in step d, said optimization calculation further determines values ​​of additional binary variables satisfying said transposed system.

[0090] In embodiments, such a device will further comprise at least one of the following features:

[0091] - said optimization module includes an analog quantum computer adiabatic quantum annealing,

[0092] and in step d, said determination is carried out by quantum annealed adiabatic analog quantum computing implemented by said quantum annealed adiabatic analog quantum computing;

[0093] - at step a, for each line i: the r lines associated with line i are processed successively by increasing bit weight, carrying over the carries from the lower bit weights and introducing at each of the r lines the at most y .... deductions necessary to ensure that line i is equal to 0;

[0094] - in step a, for each row i, a tree summation method is applied binary according to which the partial sums of successive terms to be summed in the row are further considered, and then iteratively, at a given level of the tree, the partial sums of the partial sums of the previous level are considered; and

[0095] in step b, the additional binary variables further include the binary representations of the partial sums of the different levels of the tree.

[0096] According to another aspect, the invention describes a computer program product intended to be stored in the memory of an electronic processing device and further comprising a microcomputer, said computer program comprising instructions which, when executed on the microcomputer, put into implements the steps incumbent upon the processing device of a process according to the first aspect of the invention.

[0097] The invention also describes a computer-readable recording medium for storing such a computer program. Such recording media may include a storage means, such as a ROM, for example a CD-ROM or a microelectronic circuit ROM, or a magnetic recording means, for example a USB flash drive or a hard drive. Such recording media may be transmissible, such as an electrical or optical signal, which can be transmitted via an electrical or optical cable, by radio, or by other means, so that the computer program it contains is executable remotely. The programs according to the invention can, in particular, be downloaded onto a network, for example, the Internet.Such recording media may include an integrated circuit in which the program is incorporated, the circuit being adapted to execute or to be used in the execution of the aforementioned display control method. Brief description of the drawings

[0098] The invention will be better understood and other features, details and advantages will become clearer from the following description, given by way of non-limiting reason, and from the accompanying figures, given by way of example.

[0099] [Fig-1] Fig-1 is an illustration of a computer comprising a device processing electronics and an optimization module in one embodiment of the invention;

[0100] [Fig.2] The [Fig.2] describes the steps of a treatment process in one embodiment of the invention;

[0101] [Fig.3] The [Fig.3] illustrates the consideration of binary carries in an example of an embodiment of the invention known as simple summation by bit line;

[0102] [Fig.4] The [Fig.4] illustrates the consideration of binary carries in the embodiment of the invention of the [Fig.3];

[0103] [Fig.5] The [Fig.5] illustrates the use of a register and a multiplexing selection in one embodiment of the invention;

[0104] [Fig.6] Fig.6 illustrates the use of QUBO constraint adders in an embodiment of the invention with a 2-to-2 binary summation tree.

[0105] Identical references may be used in different figures when they refer to identical or comparable elements. Description of the implementation methods

[0106] [Fig.1] illustrates an electronic processing device 1 comprising an electronic processing block 2 and an optimization module 3 in one embodiment of the invention.

[0107] In the embodiment considered, the electronic processing block 2 comprises a hardware unit of the FPGA or ASIC type, or a memory comprising software instructions and a "classical" processor adapted to execute these software instructions (here, "classical technology" means a non-quantum processor).

[0108] In one embodiment, the optimization module 3 includes a quantum accelerator, adapted to receive as input the Ising or QUBO representations of linear systems to be solved and to determine the solutions of such systems for example by implementing a procedure of the type of adiabatic quantum annealing on a basis of two-dimensional Ising Hamiltonian, for example D-Wave® or Pasqal®.

[0109] In one embodiment, the optimization module 3 comprises, instead of a quantum accelerator, a QPU (Quantum Processing Unit). This invention can also be implemented with a QUBO or Ising computing accelerator (or more generally adapted to determine an extremum: for example, in the case of QUBO, a minimum of the cost function and in the case of Ising, a minimum of the Hamiltonian energy) that is simply analog, instead of a quantum computer.

[0110] Figure 2 shows the steps of a treatment process 100 implemented according to the invention. The treatment device 1 is adapted to implement this process in one embodiment of the invention.

[0111] In a step 101, the matrix A and the vector B as previously introduced, defining the linear system to be solved as indicated in equation 6, are provided as input to the processing block 2, which determines their respective fixed-point binary representation:

[0112] «... ex• • J ; and

[0113]

[0114] ... v _ A Here, the numbers will be coded in binary according to The two's complement method (thus, with 4 bits, we have 1000b for -8, 1111b for -1, and 0111b for 7). By convention, and as is common practice in quantum computing, binary numbers are written here in "little-endian binary," i.e., with the least significant bits on the left. However, another convention may be chosen for implementing the invention.

[0115] In a step 102, further considering the chosen binary representation (two's complement) of the scalar component xj5, i.e.:

[0116] v _ (vrvi

[0117] the linear system of equation 6 is transposed by the processing block 2 using these binary representations of xj5 a;j and b;.

[0118] The linear system to be solved is thus transposed using on the one hand these binary coefficients and on the other hand these binary variables whose value is to be determined and furthermore by introducing additional variables corresponding to the binary carries, which makes it possible to avoid the exponential explosion of the coefficients encountered when using the previously mentioned standard least squares method.

[0119] Matrices are defined corresponding to the products from equation no. 6 with the use in binary notation of the constants ay, bj and the scalar values ​​Xj with j between 0 and N-1, where A is the number of equations in the system.

[0120] Indeed, the product . x : can be expressed as a set of simple products, shifts and additions, which allows us to obtain a binary matrix corresponding to the product ;X ; in the form of a matrix-vector product 4^^ : thus (we limit ourselves here to a resolution of r; however, we may want a (hidden) resolution of 2r, but if this solution is possible, it is advantageous to replace it with a normalization of the rows to a fixed i: this will drastically limit the spurious cases):

[0121]

[0122] (Equation 13)

[0123] As can be seen illustrated in [Fig.5], each of the coefficients of Ay is simple to obtain from the binary representation of using a register and a multiplexing selection: the output bit (“out”) corresponds to the bit of the register aik (RGSTR) selected by the sum 1+c (1 and c being the indices of, respectively, row and column of the multiplication matrix Aik that we are looking for).

[0124] According to the invention, in equation 9 (an equation 9 for each i = 0 to Nl), each term ayx / of the polynomial is then replaced by the representation of AyX j from equation 13 without taking into account the multiplicative factors # of this equation 9 due to the binary representation used here, and b; is also replaced by its binary components [3ik, k=0 to r-1], which gives, for each i, r corresponding lines of equations, each of these r lines of equations corresponding to a respective bit weight k, from k = 0 (lowest bit weight) to r-1 (highest bit weight). And line by line of bit weights, starting with k = 0, additional binary variables are added to account for the binary carries resulting from the sums performed in the line of equation corresponding to the current bit weight; these carries are then carried over to higher bit weights, and the carries from lower bit weights are carried over to the current line of weight. Taking these carry variables into account ensures that the current sum is strictly equal to zero (and not equal to 0 modulo powers of 2).

[0125] Two different methods (hereinafter referred to as method I, method II respectively) for example implemented in step 102 to take into account this principle of taking into account the carries, for bit by bit additions, after transposition of the system are detailed below (other methods may be used): the first (method I) performs all the additions at once, adding all the carries necessary for this addition and the correct transfer of the carries to the following orders in bit weights; the second (method II) performs the addition in a binary tree with systematic output of a carry and an overflow bit, and the optional input of a carry.

[0126] Method I (simple summation): This method is simpler and minimizes the number of auxiliary variables (= retained variables) since they are only introduced as needed. Its drawback is that it can potentially generate slightly higher coefficients and does not easily detect potential overflows of the integer encoding capacity. However, it will be easy to implement a mitigation / rectification of this type of configuration using processing block 2.

[0127] With a complete binary representation of the equation, one can now seek to solve it bit by bit, i.e. for each fixed i, i = 0 to Nl, by considering the r binary equations, one after the other, i.e. with each fixed k (k in the interval 0, r-1) from the equation corresponding to the bit weight k = 0 to that corresponding to the bit weight r-1).

[0128] For example, the first line of bits will give the following equation:

[0129] £ E {0.1, ' - , r......1}, 52 +

[0130]

[0131]

[0132]

[0133]

[0134] (Equation 14) It should be noted that in one embodiment, to go further in taking into account the carryovers, the indices can go up to 2r. To obtain equality instead of a modulo operation, processing block 2 introduces a set of binary summation carries, which are further carried over to the next line (corresponding to the higher bit weight). Since there is at most j jV 4 1 binary terms per line, it is intrinsically added to the most — nwd rM 4 1) I carries for the row (here the relation f.. s denotes the smallest integer part, that is, the part closest to 0 for positive values). In general, the actual number of carries is lower, because the maximum number of carries can only be obtained for uniform values ​​of -1 for the elements of the matrix. Moreover, for the following rows (i.e., corresponding to the higher rank weights), there are fewer coefficients "yi" resulting from the product, as can be seen in equation 13. For the first row (i.e., for i = 0), equation 14 becomes, taking into account the carries:

[0135]

[0136] (Equation 15) where nc, k is the number of carries (Nc) and the binary variables c <w sont les retenues en notation binaires. Il faut reporter à chaque étape les retenues à la ligne de résolution aux lignes suivantes de résolution, donc on ajoute la variable cOoi à la ligne de résolution suivant au rang ?■ i, la variable c002 au rang r ™ 2 et ainsi de suite. La prise en compte de ces nouvelles variables à chaque ligne peut ajouter de nouvelles retenues par cascade, mais l’évolution du nombre de retenues est logarithmique en fonction du nombre de variables et de retenues à un rang de résolution donné, la progression en nombre de variable est donc bornée et rapidement contenue.

[0137] With the carry-overs taken into account, equality to 0 can be found, as long as the resolution is sufficient to encode the set of numbers needed (including intermediate sum values) (from these equations incorporating the carry-overs, block 2 can then define a QUBO cost function and / or its Ising equivalent).

[0138] The basic principles are stated as follows: - Baselines (i.e., without carry-over) are written using binary variables and binary constants And - basic carries (so-called "basic" carries are those resulting from the current equation under consideration without taking into account carries of lower rank (weights)) are added according to the number of binary terms from the previous step (i.e. in the baseline mentioned in the previous paragraph) and the number of carries carried over from the lower bit weight lines,

[0139] - the deductions are carried over to the following lines;

[0140] - if (it is known that) all the values ​​are positive, then in the last line (that of the most significant bit), only the carries necessary to ensure the nullity of the most significant line (which is also the sign bit) are carried over (if this is not the case, it is necessary to carry over to the line following the most significant bit to ensure the nullity of the carry-over);

[0141] - otherwise, we can limit ourselves to this method by performing post-processing to do reset the results to 0 for the sum if it is a multiple of 2N (by performing additions modulo 2N on the "spurious" result vector), or by adding constraints on the carries so that the sign bit overflow also results in 0. The principle is the same as for the other bit lines, but there are no input variables, just the carries and the carries generated to ensure 0; - alternatively, spurious results can be eliminated by taking the unknowns into account. x, which do not appear in the cost function (columns with zeros, which is easy to verify), and either determine their values ​​a posteriori (since there are generally not many, this is not necessarily a problem, see above), or add lines to account for carries, but truncated to r bits. At most, there will be 2r lines, and therefore as many carry lines to add. It is not difficult to limit the number of lines to consider to ensure that all binary variables are indeed taken into account in the binary equations. What can be lost by adding The more variables there are to retain, the lower the probability of finding an optimal solution to the cost function.

[0142] Method II (2-by-2 binary tree summation method): The binary tree method consists of performing 2-by-2 summations of elements in each row. It requires more resources and more auxiliary variables since these include the intermediate sums (the result bits of 2-by-2 summation) in addition to potential carries (the carries are at most 1 per bit of the partial sum). However, this can sometimes be compensated for by the fact that the coefficients are generally always the same and range between -2 and 1. The summation can be performed using an inverted binary tree, which allows the number of summation levels to be limited proportionally to the base-2 logarithm of the number of terms in a row. The principle of multiplication by constants is the same as for the previous method I, however the bit summation is done in pairs with carry-over, and in a binary tree of width JV b I and depth $4 is similar to what can be achieved in a wired multiplier. The goal here is to create intermediate calculations using pairwise addition: thus, the first two terms of the first row of the equation (i.e., i = 0 among the i = 1 to N) are summed by processing block 2: (after their transposition into binary product according to equation 13) and on the same line the following two, «02*2 + ^03^3 = ^2, etc. up to the last two ao.2V-l^-l + ~ ^Vl (v°ir plus l°'n an example described with reference to [Fig.7]).

[0143] The translation into a mini cost function will be carried out by moving the terms to the other side of the equation and squaring each equation. As in Method I, processing block 2 introduces (after binary product transposition according to Equation 13 of the terms of each pairwise sum), for each weight level, the carries, which are carried over to the next weighted bit level. The procedure is continued with the partial sums S'j by combining them pairwise: Sq + S? — , in the same way ^4 + ^- S until' + y .yj = 5' . The first summation line has , (rounded to the nearest integer 2 upper term, at the second level) terms; at the second level, there are only jy 4.1, and thus in -4 4. 4 lines, the total sum is determined.

[0144] This requires the set of additional variables y, S'k, ... ^w) with r resolution bits each, plus the carry variables that must be introduced from the Similar to Method I, the problem is solved using equations that are the final result of the complete sum of the row, including all variables and intermediate sums. Each of these sums is then converted to binary notation, and binary carries are taken into account as before. However, there are fewer carries per partial cost function because there are fewer terms.

[0145] In a step 103, the processing block 2 determines a representation of the transposed linear system obtained in step 102 comprising these binary coefficients and these binary variables whose value is to be determined in a predetermined format. In the case considered, the processing block 2 is adapted to generate a partial Ising Hamiltonian and / or the partial QUBO matrix for each binary equation obtained in step 102 (r equations for each i, i = 0 to Nl), and then to generate a global Hamiltonian or global QUBO matrix as a function of the partial Ising Hamiltonians or the partial QUBO matrices.

[0146] The reference cost considered (-R0) is determined simultaneously with the QUBO matrix and / or the Ising Hamiltonian. As is known, the QUBO cost function is the sum of the squares of each row of the transposed system (which must be equal to 0). The Ising Hamiltonian is derived using formula 5.

[0147] Since the verification of the solution (step 106) is simpler to perform in QUBO form, the processing block 2 generates for example the QUBO matrix at the same time as the Ising Hamiltonian.

[0148] In an optional step 104, the processing block 2 performs an optimization (here of the Ising Hamiltonian) to increase the spectral gap.

[0149] Indeed, the spectral gap is an important quantity in the context of the resolution by a quantum accelerator based on quantum annealing and the adiabatic theorem of quantum mechanics, of the Ising problem as generated in step 103: typically, the quantum annealing time on a quantum annealer must be much greater than a time inversely proportional to the spectral gap (or even to a power of the spectral gap).

[0150] Consequently, the larger the spectral gap, the greater the probability of solving the problem by a machine based on quantum annealing, and this increase is very significant. It is therefore useful in this case for the relative spectral gap to be as large as possible. It should be noted that the same is true in an embodiment using an algorithm that imitates the adiabatic process, such as QAOA or Quantum Approximate Optimization Algorithm, which can be used on a gated quantum machine.

[0151] The proposed solution and its implementation by hardwired and microprogrammed function differs from a classical Boolean calculation since it provides the coefficients of a Boolean satisfiability problem expressed in QUBO or Ising form whose solution (or at least one solution, the others being easily reducible to the solution sought) corresponds to the solution of the system of linear equations.

[0152] Another difference with the least squares method is that processing block 2 does not seek to determine the coefficients which necessarily correspond to the result of the application of this method, but uses combinations of coefficients which are likely to bring a greater probability of success in solving the system of equations (increase of the spectral gap on the target Hamiltonian in the context of Ising models, therefore increases the chances of also having a higher minimum spectral gap in an adiabatic evolution of the Hamiltonian, which by the adiabatic theorem makes it possible to validate the conditions of the quantum mechanics theorem).

[0153] Obtaining the QUBO matrix or the Ising matrix is ​​of the order of complexity for a dense matrix (less for a sparse matrix) and therefore, if The process of assigning coefficients to the quantum system for solving the Ising problem is integrated, and the probability of solving the problem by the quantum computer is important, then this provides a polynomial advantage to quantum systems of Ising Hamiltonian calculations for linear algebra calculations — the best known complexity for the classical solution of a system of linear equations being y. Note that this remains true for solving sparse systems of linear equations.

[0154] By way of illustration, in the case of 3 binary variables (or equivalently three spin variables), processing block 2 calculates the partial QUBO matrix (or equivalently, the partial Ising coupling matrix and the partial bias vector) as follows:

[0155] - before spectral gap correction with respectively least significant bits of a unknown x, of an unknown y and first carry-over of the addition, in the case where: (x00 + x10 + l-2c0)2-

[0156] - the associated QUBO matrix, is therefore equal to:

[0157]

[0158] but it corresponds to a relative QUBO gap of only 0.11 (the / 3 3 -4\ |0 3 —4 j \0 0 0 / (the spectral gap considered being the minimum energy difference between the ground energy level and the first "excited" level of the Hamiltonian), whereas with the choice of the following improved matrix, Q instead:

[0159] 9 (Equation 16) | ü i -a \0 0 -4 /

[0160] processing block 2 results in the same truth table (including the -ordered- ranking in energy or cost of the spin or binary configurations) for the partial cost function, but with a relative spectral gap of -0.42.

[0161] Such partial optimization is performed by processing block 2 for different configurations (i.e., according to the number of bits to be considered; for example, the matrix Q\n is the 2-bit configuration plus a constant of 1; for each possible configuration, a different matrix is ​​required, which can be pre-calculated) in order to significantly increase the relative spectral gap of each QUBO (equivalently, of each partial Ising). One way to do this is to have a bank of partial QUBOs or Isings depending on the number of input variables and the number of carries, and depending on the values ​​of the constant (|3k). Note that it may be useful to use additional (auxiliary) qubits to increase this gap for certain configurations (for example, when the constant |3k=0).

[0162] In Ising notation, the matrix , Q' is written:

[0163] (Equation 17)

[0164] In Ising, the relative spectral gap is 0.21 (minimum at -6.75, next level at -2.75, maximum at 12.25), and 0.11 for the relative gap of the original matrix (minimum at -2.5, gap at -1.5, and maximum at 6.5). By systematically doubling the spectral gap, we can expect to obtain an additional order of magnitude in the probabilities of success.

[0165] The method for choosing substitution matrices may depend on the platform. It is not necessarily useful to compile a list of such matrices, as offline solvers can be used to search for partial Hamiltonians that provide the largest relative spectral gap.

[0166] Following step 103 (if no step 104) or otherwise following step 104, in a step 105, the quantum accelerator 3 is adapted to, based on the input data received in the predetermined format defining the linear system to be solved (either directly or through an automatic mapping (embedding) system as is known), implement an optimization procedure to determine the values ​​of the binary variables that are solutions of the linear system, the procedure optimization being in the example considered an adiabatic annealing (more generally, depending on the methods and technologies used for the implementation of the invention, an extremum (generally a minimum) is sought in this step: for example, of the cost function in the QUBO case, of the energy of the Hamiltonian for the Ising case ...).

[0167] In a step 106, if applicable, at least one determined solution is provided by the quantum accelerator 3 to the processing block 2.

[0168] In the event that no solution could be determined by the quantum accelerator 3, it provides, in one embodiment, several lower-cost "solutions" discovered, and the processing block then searches for a solution by interpolating these solutions. One possibility for this is to take at least 3 (potentially more if the costs are close to ensure slightly greater accuracy) lower-cost solutions found by the quantum accelerator 3, and then calculate a local gradient of the cost function. The approximate solution should lie at the intersection of the two tangents constructed on the gradient. This is only valid under the assumption that the cost function is locally convex and that we are indeed close to the solution.This can be partially verified by the case of 3 reference tangents: if the 3 combinations of 2 tangents arrive more or less at the same value of vector x, the solution obtained is associated with a good level of confidence.

[0169] If the two tangents are on the same plane or approximately on the same plane, it may be more advantageous for the processing block 2 to perform an extrapolation on the passing to 0 of the derivative (or a quadratic extrapolation which amounts to the same thing).

[0170] Processing block 2 in one embodiment further tests the validity of this solution by calculating the QUBO costs of the binary variable values ​​of the solution. And if the reference cost considered (determined simultaneously with the QUBO matrix in step 103) is reached, or if a satisfactory tangential interpolation is obtained, the validity of the solutions is then confirmed, benefiting from a polynomial advantage (therefore a reduction in complexity: decrease in processing time and energy required to perform the calculation) compared to classical algorithms for solving linear systems.

[0171] Otherwise, the search is closed with a finding of failure.

[0172] The invention thus proposes in particular a solver of linear algebra problems for quantum computer based on 2D Ising model.

[0173] In one embodiment, a final preprocessing of matrix A by block 2 is further performed before solving the system in order to drastically reduce potential problems with spurious results and the need to take into account more carry-overs: if a row of matrix A is divisible by 2, it is necessary to divide This row is divided by 2 (which implies dividing the value b_i by two as well). This operation must be repeated as long as there is a row divisible by 2. The loss in resolution on the vector B is part of the limitations of fixed-point resolution anyway. We can keep the cost function with the original matrix A if we want to interpolate to a fractional solution for the vector X.

[0174] In the case of using method I: the obtaining of the partial QUBO matrices and the summations of the cost function is detailed

[0175] Once the rx N lines in binary equality have been completed with the carries at the end of step 102, the partial values ​​of the QUBO matrix are obtained by determining the coefficients (unit or multipliers of the variables) obtained by raising the row (of the matrix representing the system considered containing the coefficients of aijk and cijk and [3kj ) to the square as is known (as these are coefficients exact powers of 2, the calculation of the coefficients does not require a full multiplier, but simply shift registers and simple logic to take into account the sign).

[0176] The principle implemented by processing block 2 can be broken down as follows:

[0177] - for each non-zero coefficient vall of each row i of this matrix representative of the system under consideration:

[0178] - for each coefficient val2 in column j:

[0179] add the value vall^vaU to the QUBO matrix in (i,j) if i>j and (j,i) otherwise (this saves on the memory required, although it is not mandatory);

[0180] update the matrix J by adding vall^va^H in (ij) or (j,i) with the same conditions as for the QUBO matrix but only if i^j;

[0181] update the vector h with the value vall^vaUH in position i and j (only once if i=j)

[0182] update the diagonal of Q (QUBO mode, i.e., add to the diagonal) and / or the vector h (Ising notation), with twice the constant (we are in binary, so twice the constant beta_ik is either 0 or 2 (or the conversion to Ising) - update the value of the optimal result S; by adding the value 2xva / 7 (the value of (Lj is initialized to 0 at the beginning of the process).

[0183] At the end of this process, the QUBO matrix and its optimal value for selecting the solution in the samples from the Ising or QUBO calculation process are obtained, and an Ising Hamiltonian in the form of its Jet matrix of its bias vector h.

[0184] However, for an Ising Hamiltonian optimized with respect to its spectral gap, it will be necessary to use for each row another method using a Hamiltonian whose spectral gap will be pre-optimized.

[0185] In one embodiment, the solution sought is a sufficiently high probability of being discovered by the annealing processor (or the quantum processor) that in one or two dozen samples, there is a very high probability of having at least one solution.

[0186] To further illustrate the implementation of the solution according to the invention and using the notations introduced previously, let us now consider the following initial linear system (corresponding to AX + B), with N = 2, a00 = 1, aOi = 1, ai0 = -1, an = 1, b0 = l, b1 = 3:

[0187] [ x0 + Aj + 1 = 0 (Equation 18) [ -Xo + X( + 3 = 0

[0188] Considering that r = 3 (3-bit resolution), during step 101, for each row of this initial system, a binary equation is obtained for each bit of resolution, before taking into account the carries, as shown in [Fig. 3], Jk representing the integers that will subsequently be derived as binary carries for the binary equation relating to the bit weight k, resulting from the transposition of the first row of the linear system above, with k = 0 to rl(=2) and J'k representing the integers that will subsequently be derived as binary carries for the binary equation relating to the bit weight k, resulting from the transposition of the second row of the linear system above. The two rows of equation 18 become 2x3 rows in binary notation; the constants '1' and '3' have been converted into three constants '1'; The multiplication by '-1' (i.e. '111' in two's complement binary notation) is visible from the triangular shape of the submatrix.

[0189] Figure 4 illustrates step 102 of taking into account binary carries according to method I (ci and c'i, 1=0 to 2) to ensure that each line is equal to 0. The carries are carried over to the lines corresponding to the highest bit weight. The first line has two binary variables and a constant, so |_log?3 J = 1 carry is sufficient to satisfy the equality to 0. For the last line, there are four binary variables and a carry carried over from the previous line; therefore, 2 carries are necessary to satisfy the equality.

[0190] In step 103, the QUBO matrix corresponding to these 2 x 3 (= N xr) equations translating the initial linear system at each bit weight with additional binary variables corresponding to the carries is constructed and the associated cost function is determined.

[0191]

[0192]

[0193]

[0194]

[0195]

[0196]

[0197] For the equation xOi + Xn + c0 -2ci = 0, where xOi + Xn are the variables (1 bit each) to be determined and c0 (carry received at input, coming from a lower bit weight), Ci (carry supplied at output, for a higher bit weight) are the additional carry-type variables, the cost function corresponds to the polynomial of degree 2, (xOi + Xn + c0 -2ci)2, and we determine QP, the partial matrix QUBO corresponding to this equation: (xqi + Xn + c0 -2ci)2 = 0 The complete matrix is ​​composed of these successively determined partial matrices. Example when using method II: We consider a case where step 102 implements Method II (binary tree summation method, 2-by-2 summation), using two-number adders As and Bs as basic blocks. Each basic block is encoded using QUBO constraints, and the variables resulting from the addition are propagated to the next summation level in the tree. Figure 6 illustrates such an implementation: Ss is the result of the partial summation of the two binary elements As and Bs, here with s = 0 to 3 (the input and output carry bits are, respectively, carry-in and carry-out and "overflow"). Recall that overflow is a situation detected in the most significant bit row when summing two numbers in binary: the most significant bit should not carry if the sign bits are 0, and conversely, it should carry if the sign bits are 1.The overflow, or "overflow indicator," is generally generated by an exclusive OR operation of the internal carry inside and outside the sign bit. Since the sign bit is the same as the most significant bit of a number considered unsigned, the overflow indicator is "meaningless" and normally ignored when adding or subtracting unsigned numbers. Each of the four adders considered here is a basic 4-bit adder (we are considering the case of 4-bit resolution). These are full adders (2-bit adders with carry-in and carry-out). They are not logical adders but QUBO constraint adders: that is, adders whose minimum cost is verified if and only if the bit configuration conforms to the truth table of a binary adder. For each addition step, a partial QUBO cost function is calculated: (this refers to the bitwise summation of Aset Bs. The carry-ins and the intermediate sum result are binary values ​​that are carried bit by bit from the least significant bit (without carry-in) to the most significant bit (also called the sign bit)).

[0198] By performing the additions two elements at a time for a given line, we arrive at .yj summation outputs for this first level of summation 9 partial (rounded up to the nearest whole number). Repeating the process for whole numbers s, , s >., Si, etc. the complete summation (composed of summations of 2 elements each time) of the first line of the equation, is thus finally obtained with 1 partial cost function by summation elements, and as many intermediate variables.

[0199] Figure 7 illustrates the partial sum operations illustrated by considering for example the equation So there will be 3 intermediate results on r bits and the carries that go with them, all integrated into a sum of partial QUBO cost function.

[0200] In the embodiment considered, N is any integer greater than or equal to 2; in one embodiment, the resolution r is any integer greater than or equal to 2, respectively to 3, respectively to 4.

Claims

1. Demands A processing method in a computer (1) comprising an electronic processing device (2) and an optimization module (3) adapted to perform optimization calculations of a specific type among Ising and QUBO and any type reducible to an Ising or QUBO optimization, wherein said method comprises the following steps implemented by the electronic processing device (2) following the receipt by the electronic processing device of a command to solve an initial linear system Ax + B=0, where x is the solution vector to be determined x = [xj]j=0 to N-i, A = [a;j] îj = 0 to n1 and B = [bj]j-0 to N-i with line i of the initial linear system, i = 0 at Nl, the equation: L^ijXj = 0' a / transposition of the linear system comprising, in each line i, the replacement of each variable xj by a binary representation of xj based on r binary variables xjk, j = 0 to N-1 and k = 0 to r-1, where r indicates the resolution of the binary representation; b / determination of a representation in a predefined format of QUBO type or of Ising Hamiltonian type of the linear system thus transposed with variables to be determined at least the said binary variables xjk, said representation being a function of at least the coefficients of said transposed linear system; c / provision of said determined representation of the linear system to the optimization module, d / the process further comprises a determination, based on said provided representation, of the values ​​of the binary variables xjk satisfying said system transposed by optimization calculation of said determined type implemented by said optimization module (3) said process being characterized in that: at step a: - the transposition further includes replacing aij and b with their binary representations based on, respectively, aijk and k = 0 to r-1, thus giving, for each line i of the initial linear system, r equations each corresponding to a bit weight from 0 to r-1 respectively, the transposed linear system thus comprising Nxr equations and - for each row i, additional binary variables are further added to take into account the binary carries resulting from the operations performed in the equations at the different bit weights and the transfer of said carries to higher bit weights; in step b: a representation is determined in the predefined format of the linear system thus transposed with variables to be determined at least the said binary variables xjk and the additional binary variables, said representation being determined as a function of at least the coefficients of said transposed linear system; in step d, said optimization calculation further determines values ​​of the additional binary variables satisfying said transposed system.

2. Processing method according to claim 1, wherein said optimization module (3) comprises an adiabatic quantum analog annealer, and in step d, said determination is performed by adiabatic quantum analog annealer implemented by said adiabatic quantum analog annealer.

3. Processing method according to claim 1 or 2, wherein at step a, for each line i: the r lines associated with line i are processed successively by increasing bit weight, carrying over the carries from the lower bit weights and introducing at each of the r lines the at most jV. ik®rM 4 1 ) 1 carries necessary to guarantee the equality of line i to 0.

4. A processing method according to claim 1 or 2, wherein in step a, for each row i, a binary tree summation method is applied, further considering the pairwise partial sums of successive terms to be summed in the row, and then iteratively, at a given level of the tree, the pairwise partial sums of the partial sums of the previous level are considered; and in step b, the additional binary variables further include the binary representations of the partial sums of the different levels of the tree.

5. A processing method in a computer (1) according to any one of the preceding claims, wherein, in step b, the gap spectral of said representation is augmented while maintaining the truth table.

6. Computer program, intended to be stored in the memory of an electronic processing device (2) and further comprising a microcomputer, said computer program comprising instructions which, when executed on the microcomputer, implement the processing device steps of a process according to any one of the preceding claims.

7. A computer (1) comprising an electronic processing device (2) and an optimization module (3) adapted to perform optimization calculations of a determined type from Ising and QUBO and any type reducible to an Ising or QUBO optimization, wherein said electronic processing device is adapted to receive a command to solve an initial linear system Ax + B=0, where x is the solution vector to be determined x = [xj]j = 0 to N i, A = [a;j]ij=0 to N-i and B = [bj]j = 0 to N-i with in line i of the initial linear system, i =0 to Nl, the equation: VV! , _ said device (2) being further j=üaijxj + — u adapted to perform the following processing: a / transposition of the linear system comprising, in each line i, the replacement of each variable xj by a binary representation of xj based on r binary variables xjk, j = 0 to N-1 and k = 0 to r-1, where r indicates the resolution of the binary representation;b / determination of a representation in a predefined format of QUBO type or of Ising Hamiltonian type of the linear system thus transposed with variables to be determined at least the said binary variables xjk, said representation being a function of at least the coefficients of said transposed linear system; c / provision of said determined representation of the linear system to the optimization module, said optimization module (3) being adapted to determine, as a function of said provided representation, values ​​of the binary variables xjk satisfying said transposed system by optimization calculation of said determined type implemented by said computer (1) being characterized in that: at step a: - the transposition further includes the replacement of aij and bi by their binary representation based on, respectively, aijk; and Pk = 0 to r-1, thus giving, for each row i of the initial linear system, r equations each corresponding to a respective bit weight from 0 to r-1, the transposed linear system thus comprising Nxr equations and - for each row i, additional binary variables are further added to take into account the binary carries resulting from the operations performed in the equations at the different bit weights and the transfer of said carries to higher bit weights; at step b: a representation is determined in the predefined format of the linear system thus transposed with as variables to be determined at least the said binary variables xjk and the additional binary variables, said representation being determined as a function of at least the coefficients of said transposed linear system; at step d, said optimization calculation further determines values ​​of the additional binary variables satisfying said transposed system.

8. Computer (1) according to claim 7, wherein said optimization module (3) comprises an analog adiabatic quantum computer with quantum annealing, and in step d, said determination is performed by analog adiabatic quantum computing with quantum annealing implemented by said analog adiabatic quantum computer with quantum annealing.

9. Computer (1) according to claim 7 or 8, wherein at step a, for each line i: the r lines associated with line i are processed successively by increasing bit weight, carrying over the carries from lower bit weights and introducing at each of the r lines the at most y ... < Fd carries necessary to guarantee the equality of line i to 0.

10. Computer (1) according to claim 7 or 8, wherein in step a, for each row i, a binary tree summation method is applied whereby the pairwise partial sums of successive terms to be summed in the row are further considered, and then iteratively, at a given level of the tree, the pairwise partial sums of the partial sums of the previous level are considered; and in step b, the additional binary variables further include the binary representations of the partial sums of the different levels of the tree.

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