FR3140190A1 - Method for determining a solution to an optimization problem - Google Patents

Abstract

Method for determining a solution to an optimization problem The present invention relates to a method for determining a solution to an optimization problem, the method comprising the following steps: - obtaining a number P of data experimental data comprising arrangement data and state data, the arrangement data relating to a spatial arrangement of N atoms forming a network of qubits, the state data relating to the measured state of each of the N atoms of the spatial arrangement following the application of a sequence of laser pulses to the N atoms, - obtaining, for each experimental data, an optimal ordering for the atoms of the spatial arrangement among a set of possible scheduling, - determining a set of P potential solutions to the optimization problem based on the P experimental data and the optimal scheduling obtained, and - determining the best solution to the optimization problem among the P potential solutions . Figure for abstract: Figure 3

Description

Method for determining a solution to an optimization problem TECHNICAL FIELD OF THE INVENTION

The present invention relates to a method for determining a solution to an optimization problem. The present invention also relates to a related computer program product.

BACKGROUND OF THE INVENTION

A wide variety of industrial problems can be expressed as combinatorial optimization problems. Quantum computing is a growing technology that, among other applications, aims to solve these problems by leveraging information stored in quantum states. To access this information, however, it is necessary to produce classical information by measuring the quantum system. In particular, neutral atom platforms make it possible to obtain quantum states by controlling Rydberg atoms in optical traps. The atoms are arranged to define the nodes of a graph, the edges of which are represented by the dipole-dipole interactions between the atoms in their Rydberg states. This makes it possible, for example, to use the Ising Hamiltonian to solve optimization problems.

In the case of quadratic unconstrained binary optimization (QUBO), many problem instances are in the form of matrices that cannot be reproduced by the Ising Hamiltonian. Also, one approach consists of integrating the problem information into a graph, using the atoms whose corresponding interactions correspond as best as possible to the problem matrix, called the QUBO matrix. As these interactions are in , r being the interatomic distance, only a few families of matrices can be constructed faithfully.

To generate these graphs, several steps of atom rearrangement involving optical traps are implemented. However, as the number of atoms in the graph increases, the total duration of the steps lengthens, while efficiency decreases. Thus, constructing a desired pattern of atoms for each repetition of a measurement requires significant additional time after loading the atoms into the traps. Furthermore, in hybrid quantum-classical variational algorithms, each iteration generally requires a high number of repetitions of the quantum processing cycle resulting in a measurement (also called a "shot"). This allows sufficient statistics to be accumulated to estimate the properties of the quantum state produced.

Multiple learning techniques can be used to produce optimized positions of the atoms, where the interactions better mimic the given QUBO problem. However, the supposed gain from this optimization does not always compensate for the resources necessary for its implementation. Although rapid rearrangement algorithms have been developed to meet the need to quickly produce initial registers, this step remains the longest of all four steps performed, among atom loading, atom rearrangement, sending the sequence of pulses to the atoms and measuring the states of the atoms. In particular, the rearrangement alone lasts several hundred milliseconds, for an overall duration of the four steps of between 500 milliseconds and 1 second.

Thus, these rearrangement steps and the accumulation of measurements constitute important bottlenecks for the implementation of variational optimization schemes by quantum processors with neutral atoms.

There is therefore a need for a method of solving optimization problems that is quicker to implement.

1. l’obtention d’un nombre P de données expérimentales acquises au cours de P expériences distinctes, chaque donnée expérimentale comprenant une donnée d’agencement et une donnée d’états, la donnée d’agencement étant relative à un agencement spatial de N atomes formant un réseau de qubits, N étant un nombre entier, la donnée d’états étant relative à l’état mesuré de chacun des N atomes de l’agencement spatial à la suite de l’application d’une séquence d’impulsions laser aux N atomes,
2. le traitement des P données expérimentales pour obtenir, pour chaque donnée expérimentale, un ordonnancement optimal pour les atomes de l’agencement spatial parmi un ensemble d’ordonnancements possibles, chaque ordonnancement possible correspondant à une matrice d’interactions carrée de taille N quantifiant les interactions entre les atomes de l’agencement en fonction de l’ordonnancement desdits atomes, l’ordonnancement optimal correspondant à la matrice d’interactions la plus proche de la matrice du problème selon un critère de proximité,
3. la détermination d’un ensemble de P solutions potentielles au problème d’optimisation en fonction des P données expérimentales et de l’ordonnancement optimal obtenu pour chaque donnée expérimentale, et
4. la détermination de la meilleure solution au problème d’optimisation parmi les P solutions potentielles en fonction d’un critère d’optimisation.

To this end, the present description relates to a method for determining a solution to an optimization problem, the optimization problem being characterized by a square matrix of size N, called the problem matrix, the method being implemented work by computer and comprising the following steps:

1. obtaining a number P of experimental data acquired during P distinct experiments, each experimental data comprising arrangement data and state data, the arrangement data relating to a spatial arrangement of N atoms forming a network of qubits, N being an integer, the state data being relative to the measured state of each of the N atoms of the spatial arrangement following the application of a sequence of laser pulses to the N atoms,
2. processing the P experimental data to obtain, for each experimental data, an optimal ordering for the atoms of the spatial arrangement among a set of possible orderings, each possible ordering corresponding to a square interaction matrix of size N quantifying the interactions between the atoms of the arrangement according to the ordering of said atoms, the optimal ordering corresponding to the interaction matrix closest to the problem matrix according to a proximity criterion,
3. determining a set of P potential solutions to the optimization problem as a function of the P experimental data and the optimal scheduling obtained for each experimental data, and
4. determining the best solution to the optimization problem among the P potential solutions according to an optimization criterion.

According to particular embodiments, the method comprises one or more of the following characteristics, taken in isolation or in all technically possible combinations:

1. le chargement aléatoire d’atomes dans des sites de piégeage d’atomes de sorte à obtenir un agencement spatial quelconque de N atomes piégés formant un réseau de qubits, l’agencement spatial quelconque obtenu permettant d’obtenir la donnée d’agencement,
2. l’application d’une séquence d’impulsions laser aux N atomes de l’agencement spatial, et
3. la mesure de l’état de chaque atome à la fin de la séquence d’impulsions, les états mesurés permettant d’obtenir la donnée d’état ;

- each of the P experimental data is obtained during an experiment implementing the following sub-steps:

1. the random loading of atoms into atom trapping sites so as to obtain any spatial arrangement of N trapped atoms forming a network of qubits, the arbitrary spatial arrangement obtained making it possible to obtain the arrangement data,
2. applying a sequence of laser pulses to the N atoms of the spatial arrangement, and
3. measuring the state of each atom at the end of the pulse sequence, the measured states making it possible to obtain the state data;

- each off-diagonal coefficient of each interaction matrix quantifies the interaction between two distinct atoms of the spatial arrangement according to the corresponding ordering, the interaction depending on the spatial distance between the two atoms considered;

- for each possible ordering, a difference is determined between the off-diagonal coefficients of the corresponding interaction matrix and the off-diagonal coefficients of the problem matrix, the proximity criterion stipulating that the optimal ordering corresponds to the matrix of interactions with the smallest difference;

- the deviations for possible scheduling are determined according to the following formula:

Or :

- expresses a difference between the interaction matrix M considered and the problem matrix Q for an ordering of atoms corresponding to a permutation of length N,

- = Or is a coefficient depending on a Rydberg level chosen as the excited state for the atoms, denotes the position of atom i, and A is a constant proportionality coefficient, and

- designates the standard of .

- each of the P potential solutions is a vector whose coefficients correspond to the state data of one of the P experimental data according to the optimal ordering obtained for the experimental data;

- the optimization problem aims to find the antecedent of a function minimizing said function, the function depending on the matrix of the problem, the optimization criterion stipulating that the best solution is the solution minimizing the function among all of the P potential solutions;

- the optimization problem is a quadratic binary optimization problem without constraints, called the QUBO problem;

- the sequence of pulses sent to the N atoms of each spatial arrangement to obtain the state data is a sequence of Ramsey type pulses;

- the method comprises a step of implementing the P experiments by a quantum device to obtain the P experimental data;

The present description also relates to a computer program product on which a computer program comprising program instructions is stored, the computer program being loaded onto a data processing unit and causing the implementation of a determination method as described above when the computer program is implemented on the data processing unit.

The present description also relates to a system configured to implement a method as previously described, the system comprising a quantum device configured at least to implement the P experiments making it possible to obtain the P experimental data.

The present description also relates to a readable information medium on which a computer program product comprising program instructions is stored, the computer program being loaded onto a data processing unit and causing the implementation of a determination method as described above when the computer program is implemented on the data processing unit.

1. l’obtention d’un nombre P de données expérimentales acquises au cours de P expériences distinctes, chaque donnée expérimentale comprenant une donnée d’agencement et une donnée d’états, la donnée d’agencement étant relative à un agencement spatial de N atomes formant un réseau de qubits, N étant un nombre entier, la donnée d’états étant relative à l’état mesuré de chacun des N atomes de l’agencement spatial à la suite de l’application d’une séquence d’impulsions laser aux N atomes,
2. le traitement des P données expérimentales pour obtenir, pour chaque donnée expérimentale, un ordonnancement optimal pour les atomes de l’agencement spatial parmi un ensemble d’ordonnancements possibles, chaque ordonnancement possible correspondant à une matrice d’interactions carrée de taille N quantifiant les interactions entre les atomes de l’agencement en fonction de l’ordonnancement desdits atomes, l’ordonnancement optimal correspondant à la matrice d’interactions la plus proche de la matrice du problème selon un critère de proximité,
3. la détermination d’un ensemble de P solutions potentielles au problème d’optimisation en fonction des P données expérimentales et de l’ordonnancement optimal obtenu pour chaque donnée expérimentale, et
4. la détermination de la meilleure solution au problème d’optimisation parmi les P solutions potentielles en fonction d’un critère d’optimisation.

The present description also relates to a device for determining a solution to an optimization problem, the optimization problem being characterized by a square matrix of size N, called the problem matrix, the device being configured to implement the following steps:

1. obtaining a number P of experimental data acquired during P distinct experiments, each experimental data comprising arrangement data and state data, the arrangement data relating to a spatial arrangement of N atoms forming a network of qubits, N being an integer, the state data being relative to the measured state of each of the N atoms of the spatial arrangement following the application of a sequence of laser pulses to the N atoms,
2. processing the P experimental data to obtain, for each experimental data, an optimal ordering for the atoms of the spatial arrangement among a set of possible orderings, each possible ordering corresponding to a square interaction matrix of size N quantifying the interactions between the atoms of the arrangement according to the ordering of said atoms, the optimal ordering corresponding to the interaction matrix closest to the problem matrix according to a proximity criterion,
3. determining a set of P potential solutions to the optimization problem as a function of the P experimental data and the optimal scheduling obtained for each experimental data, and
4. determining the best solution to the optimization problem among the P potential solutions according to an optimization criterion.

Other characteristics and advantages of the invention will appear on reading the following description of embodiments of the invention, given by way of example only and with reference to the drawings which are:

, , a schematic example of a (classical) calculator interacting with a computer program product, the calculator receiving data from a quantum calculator,

, , a flowchart of an example of implementation of a method for determining a solution to an optimization problem, and

, , an illustrative example of a spatial arrangement of atoms and two possible orderings for the atoms, each corresponding to a distinct interaction matrix, the top ordering having been obtained randomly, while the bottom ordering was chosen so that the corresponding interaction matrix is closer to the matrix of the problem to be solved.

DETAILED DESCRIPTION OF EMBODIMENTS

The method of determining a solution to an optimization problem which will be described in the remainder of the description is implemented by a calculator 10, that is to say is implemented by computer, this calculator being in interaction with a computer program product 12. The interaction of the calculator 10 and the computer program product 12 is represented in .

The calculator 10 is in this example a classical calculator (that is to say a calculator different from a quantum calculator) which in this case comprises a processor comprising a data processing unit, memories and possibly a media reader information, a man-machine interface, such as a screen, and a display.

The calculator 10 is capable of receiving data from a quantum device, such as a quantum calculator 14, illustrated in . The data are the result of a number P of experiments carried out by the quantum computer 14, as will be described in the remainder of the description. The data is received directly from the quantum computer or indirectly via a memory or a server on which the data has been previously stored.

The quantum computer 14 includes a quantum processor. The quantum computer includes in particular the equivalent of the data processing unit, memories and possible information support readers, man-machine interface, and display, of a conventional processor.

A quantum processor is a network of qubits (also called a qubit register), as well as hardware allowing these qubits to be manipulated. A quantum processor is suitable for performing quantum operations on qubits.

A quantum processor uses the quantum properties of matter, such as superposition and entanglement, to perform operations on data. Unlike a classical computer based on transistors working on binary data (coded on bits, 0 or 1), the quantum processor works on qubits whose quantum state can take a number of values, no longer discrete but continuous.

In particular, a qubit refers to a two-level quantum mechanical system. For example, a qubit includes two basic quantum states I0> and I1> representing the possible quantum states of the qubit. According to the superposition principle of quantum mechanics, any superposition of the form aI0> + bI1> (a and b being complex numbers and aa*+bb*=1) is a possible quantum state of the qubit.

The quantum processor considered is, for example, based on neutral atom technology. In this case, the quantum processor includes neutral atoms capable of being manipulated to form qubits. The manipulations are carried out by light beams, such as lasers.

A neutral atom is an electrically neutral atom. Such neutral atoms are, for example, excited towards Rydberg levels when using the quantum processor. Neutral atoms are, for example, Rubidium atoms.

The components of such a quantum processor with neutral atoms are, for example, detailed in paragraph 2 of the article Loïc Henriet, Lucas Beguin, Adrien Signoles, Thierry Lahaye, Antoine Browaeys, Georges-Olivier Reymond, and Christophe Jurczak. Quantum computing with neutral atoms. Quantum , 4:327, September 2020. ISSN 2521-327X. doi:10.22331/q-2020-09-21-327.

Alternatively, the computer 10 is itself a quantum computer. In this case, the computer 10 is confused with the quantum computer 14.

The computer program product 12 includes an information medium. The information medium is a medium readable by the computer, usually by the data processing unit. The readable information medium is a medium suitable for storing electronic instructions and capable of being coupled to a bus of a computer system. The computer program comprising program instructions is stored on the information carrier.

The computer program is loadable on the data processing unit and is adapted to cause the implementation of the method of determining a solution to an optimization problem when the computer program is implemented on the the computer processing unit.

An example of a method for determining a solution to an optimization problem will now be described with reference to the flowchart of the and to the illustrating some of the steps of the process.

The determination method is implemented by the calculator 10 in interaction with the computer program product, that is to say it is implemented by computer.

The optimization problem is typically a problem aimed at finding the antecedent of a function minimizing said function. The optimization problem is characterized by a problem matrix Q whose coefficients are specific to the problem. The problem matrix Q is a square matrix of size N, N being a positive integer. Preferably, the problem matrix Q is a symmetric matrix with positive off-diagonal coefficients.

Preferably, the optimization problem is an unconstrained quadratic binary optimization problem, called QUBO problem (in English “Quadratic Unconstrained Binary Optimization (QUBO) problems”). A QUBO problem involves finding the bit string , that is to say an N-dimensional vector of 0 and 1, minimizing the cost function , with Q a symmetric real matrix. Q is in this case the matrix of the QUBO problem.

The determination method comprises a step 100 of obtaining P experimental data D ₁ , …, D _P.

The experimental data D ₁ , …, D _P were acquired during P separate experiments, P being an integer. In practice, P can be greater than 100, although other choices are nevertheless possible. Each experimental data D _i includes arrangement data A and state data E.

The arrangement data A relates to a spatial arrangement of N atoms forming a network of qubits, N being an integer. In particular, the arrangement data A indicates the relative position of the N atoms with respect to each other.

The state data E relates to the measured state of each of the N atoms of the spatial arrangement following the application of a sequence of laser pulses to the N atoms. In particular, the measured state of each atom is transcribed by a binary value, “zero” indicating a first state and “one” indicating a second state. In practice, by convention, the “zero” state indicates that the atom is in a de-excited state and the “one” state indicates that the atom is in an excited state. However, other alternative conventions are possible.

Preferably, the sequence of laser pulses sent to the N atoms of each spatial arrangement is the same for each of the P experiments.

Preferably, the sequence of laser pulses sent to the N atoms of each spatial arrangement to obtain the E state data is a sequence of square pulses separated by periods of free interaction. To be more precise, a square pulse is applied for a time t_a with a constant amplitude Ω and a period of free interaction lasts a time t_b during which the amplitude Ω is left at 0. A sequence is therefore characterized by its total number of square pulses, denoted p, as well as by the time of each of these pulses, t_a^{i=1,…,p}, the time of each of the periods of free interactions (number of p-1), t_b^{i=1,…,p-1} as well as the value of Ω chosen. Preferably, the sequence obtained with p = 2, Ω/2π =2 MHz, t_a^1 = t_a^2 = π/2Ω and t_b^1 = 0.2 us is chosen. This sequence is relatively short and easy to implement. It is called the Ramsey sequence. In particular, an isolated atom will be excited by the effective π-momentum, adding little energy to the system. On the contrary, atoms in strong interactions with their neighbors will see their probability of excitation decrease, preventing them from adding too much energy to the system.

More general sequences can be used, notably with p>2, t_a^1 = … = t_a^p = π/pΩ and with specific choices for t_b^i. Likewise, the value of Ω can be modified, particularly in relation to the spacing of the spatial arrangement of the atoms.

Other pulse sequences can be used alternatively.

In an example of implementation, the obtaining step 100 consists of loading, into the computer 10, the experimental data D ₁ , ..., D _P which have been obtained beforehand. The experimental data D ₁ , …, D _P have for example been stored in a memory or a server while waiting for their loading.

In another example of implementation, the obtaining step 100 includes carrying out the P experiments in order to obtain each experimental data D _i . In this case, the P experiments are carried out using a quantum computer (for example the quantum computer 14 described previously in the description). When the quantum computer 14 is the computer 10 implementing the method, the experimental data D ₁ , ..., D _P are directly stored in a memory of said computer. When the quantum computer 14 is different from the computer 10 implementing the method, the experimental data D ₁ , ..., D _P are obtained by the quantum computer 14 before being loaded into the computer 10 implementing the method.

In this other example of implementation, each of the P experimental data D _i is preferably obtained during an experiment implementing the following sub-steps:

- the random loading of atoms into atom trapping sites so as to obtain any spatial arrangement of N trapped atoms forming a network of qubits. Any spatial arrangement obtained makes it possible to obtain the arrangement data A. Preferably, the arrangement of the trapping sites has been defined beforehand to obtain a configuration of trapping sites adapted to the combinatorial problem. Such a configuration determines to some extent the level of entanglement resulting from the quantum state. The configuration is, for example, a rectangular, triangular or even Kagome configuration.

- the application of a sequence of laser pulses to the N atoms of the spatial arrangement. As noted previously, the parameters that determine the sequence of laser pulses can be chosen to vary the final quantum state.

- measuring the state of each atom at the end of the laser pulse sequence. The measured states make it possible to obtain the state data E.

In particular, during each of the P experiments, the N atoms trapped in the trapping sites form a random spatial arrangement. These atoms are not rearranged in the trapping sites to satisfy a predefined spatial arrangement. Thus, the spatial arrangement of N atoms is potentially different from one experiment to another.

The determination method comprises a step 200 of processing the P experimental data D ₁ , ..., D _P at the end of which it is obtained, for each experimental data D _i , an optimal ordering O for the atoms of the spatial arrangement among a set of possible orderings.

An ordering is a way of ordering the N atoms of the spatial arrangement considered. For example, an ordering consists of assigning N increasing numbers to each atom so as to define an order for the atoms. For example, the illustrates two examples of ordering 10 atoms for which the integers between 0 and 9 were assigned differently to the 10 atoms.

The possible orderings correspond for example to all the possibilities of ordering atoms, that is to say N! (N factorial). Alternatively, the possible schedules are only a predetermined number of schedules among all the schedule possibilities.

The possible schedules were, for example, obtained by discrete optimization methods. According to one example, the possible orderings are obtained by successive permutations of the order assigned to the atoms starting from a first ordering obtained randomly. Alternatively, the possible orderings can be obtained differently.

Alternatively, the possible orderings result from an algorithm carrying out a predetermined number of random drawings during which a random order is assigned to the N atoms.

Each possible ordering corresponds to an interaction matrix M square of size N quantifying the interactions between the atoms of the arrangement as a function of the ordering of said atoms.

Preferably, each interaction matrix M is a symmetric matrix having positive off-diagonal coefficients.

Preferably, each off-diagonal coefficient M _ij of each interaction matrix M quantifies the interaction between two distinct atoms of the spatial arrangement according to the corresponding ordering. The interaction depends on the spatial distance between the two atoms considered.

More precisely, each off-diagonal coefficient M _ij quantifying the interaction between an atom i and an atom j, is of the form Or is a coefficient depending on a Rydberg level chosen as the excited state for the atoms, designates the position of atom i, and A is a constant proportionality coefficient (for example equal to 1).

For example, if the coefficient M _ij of the interaction matrix corresponds to the interaction between atom i and atom j and the atoms are numbered from 1 to N, by permuting the numbers 2 and 6, we understand that the coefficients involving atoms 2 and 6 (M _{2 , j} , M _{i , 2} , M _{6 , j} and M _{i , 6} ) will be modified.

The optimal scheduling O corresponding to the interaction matrix M closest to the problem matrix Q according to a proximity criterion.

Preferably, for each possible ordering, a difference is determined between the off-diagonal coefficients of the corresponding interaction matrix M and the off-diagonal coefficients of the problem matrix Q, the proximity criterion stipulating that the optimal ordering O corresponds to the interaction matrix M having the smallest deviation.

Preferably, the deviations for the possible schedules are determined according to the following formula:

Or :

- expresses a difference between the interaction matrix M considered and the problem matrix Q for an ordering of atoms corresponding to a permutation of length N,

- = Or is a coefficient depending on a Rydberg level chosen as the excited state for the atoms, denotes the position of atom i, and A is a constant proportionality coefficient, and

- designates the standard of . The choice of standard is not imposed. For example, the standard is the absolute value.

For example, the illustrates the interaction matrices obtained for the same spatial arrangement of atoms but different orderings. In particular, the optimal scheduling O at the bottom of the makes it possible to obtain an interaction matrix M closer to the problem matrix Q than the ordering at the top of the .

The determination method comprises a step 300 of determining a set of P potential solutions S ₁ , …, S _P to the optimization problem as a function of the P experimental data D ₁ , …, D _P and the optimal scheduling O obtained for each experimental data D _i . In particular, for each of the P experimental data D _i , a solution S _i is obtained.

In an implementation example, each of the P potential solutions S _i is a vector whose coefficients correspond to the state data E of one of the P experimental data D _i according to the optimal scheduling O obtained for the experimental data D _i . The vectors considered are thus binary vectors (sequence of 0 and 1).

For example, atoms are numbered from 1 to N and each coefficient of the vector corresponds to an atom of increasing number. Thus, the state of the atom 1 is assigned to the coefficient 1 of the vector, etc. We understand that if the numbers 2 and 6 had been permuted during step 200, the coefficient 2 of the vector would have been permuted with the coefficient 6 of the vector.

The determination method comprises a step 400 of determining the best solution to the optimization problem among the P potential solutions S ₁ , …, S _P according to an optimization criterion.

As seen previously, the optimization problem aims to find the antecedent of a function minimizing said function, the function depending on the matrix of the problem Q. The optimization criterion states that the best solution is the solution minimizing the function among the set of P potential solutions S ₁ , …, S _P.

Preferably, in the case of a QUBO problem, the best solution among the potential solutions S₁, …, S_P is the solution minimizing the cost function , with Q the matrix of the QUBO problem.

Thus, the present method makes it possible to avoid the time-consuming steps of rearrangement of the atoms of a qubit register which are conventionally carried out before sending sequences of laser pulses to the atoms. Instead of such rearrangement steps, the present method offers post-processing of the data measured on atoms, which is much faster and very suitable for solving optimization problems.

In particular, the present method is applicable for solving different types of problems, such as QUBO problems, graph combinatorial problems (such as the unit disk maximal independent set problem), quantum simulation applications where a function of the target Hamiltonian must be quickly estimated, and in general, for any problem requiring finding "low energy" binary strings (or a function thereof) to propose a solution to a binary vector search or to guide a search for parameters in iterative protocols.

EXPERIMENTAL RESULTS

The inventors carried out a study during which several applications of the process described, based on a reordering of atoms in post-processing, were tested and compared to three other alternatives.

The first, simplest alternative is a classic random draw without rearrangement of the atoms, carried out on the 2N possible bit chains.

The second alternative is a variational algorithm of the QAOA type (Quantum Approximate Optimization Algorithm) with three parameters to optimize (two pulse durations and the duration of the interaction period), 10 iterations (classic iterations of the optimizer with a certain number of sequences, each with its budget) and 100 prints (shots) per iteration for a total of 1000 prints made. As is usual for this type of algorithm, the atom register is fixed during the 10 iterations and therefore requires the implementation of an atom rearrangement. Therefore, a shot of this method lasts longer than a shot of the random graph sampling method.

In these examples, each “shot” designates an operation comprising the implementation of the steps of loading the atoms, rearrangement of the atoms, sending the sequence of pulses to the atoms and measuring the states of the atoms.
Finally, the third alternative consists of random sampling (abbreviated as RGS from English “random graph sampling”) of the graph without optimized scheduling.

The method of the invention (RGS with reordering) is evaluated to find low energy states (or even the ground state) with a fixed number of 1000 shots for the instance of the QUBO problem described in the (N = 10) and for a family of 10 dense QUBO instances of size N = 15.

The results of this study show that rescheduling for RGS accelerates convergence by a factor of 2 to 5, making it more efficient than traditional scheduling on average.

In addition, the RGS process with reordering converges more quickly than the other 3 alternatives, finding with the allocated budget (the maximum number of shots authorized) the 3.4 best bit strings while the QAOA variational algorithm only finds on average the 7 best.

Additionally, the RGS method with reordering samples the sampling space more efficiently, exploring it further and avoiding getting stuck in certain areas with local minima. In the experiments carried out for a size N=15, the firing budget is better used by the RGS than by the QAOA variational algorithm. Thus, on average, the sampled bit strings have an energy below the 10% threshold.

Furthermore, in the experiments carried out, on average, the sample obtained using the RGS presents a likelihood distribution more concentrated in the low energies. Reordering further improves performance.

Finally, the performance of the RGS method with reordering was evaluated on a given set of QUBO problems. In particular, the solution time was calculated, where "solution" means a given higher percentage of the ordered solution (this can only be known with an exhaustive search, which is possible for N ≤ 15 qubits ). The calculations focused on TT10, the average time at 10% and TT1, the average time at 1% as benchmarks for sampling performance. The time TT10 is for example defined as being the duration necessary to implement the average number of shots at the end of which we obtain a solution to the QUBO problem which is part of the 10% of the best possible solutions, the number of which here is order of 2 ^N. The time TT1 is for example defined in an analogous way, for the 1% of the best possible solutions. It appears that these figures improve as the size N increases. In particular, for the size N=15, a bit string which is part of the family of 10% of the best solutions is obtained almost every time, even in the presence of read errors.

Claims (13)

Method for determining a solution to an optimization problem, the optimization problem being characterized by a square matrix of size N, called the problem matrix (Q), the method being implemented by computer and comprising the following steps :

1. obtaining a number P of experimental data (D ₁ , …, D _P ) acquired during P distinct experiments, each experimental data (D _i ) comprising arrangement data (A) and state data (E), the arrangement data (A) being relative to a spatial arrangement of N atoms forming a network of qubits, N being an integer, the state data (E) being relative to the measured state of each of the N atoms of the spatial arrangement following the application of a sequence of laser pulses to the N atoms,
2. the processing of P experimental data (D ₁ , …, D _P ) to obtain, for each experimental data (D _i ), an optimal ordering (O) for the atoms of the spatial arrangement among a set of possible orderings, each possible ordering corresponding to a square interaction matrix (M) of size N quantifying the interactions between the atoms of the arrangement as a function of the ordering of said atoms, the optimal ordering (O) corresponding to the interaction matrix ( M) closest to the problem matrix (Q) according to a proximity criterion,
3. determining a set of P potential solutions (S ₁ , …, S _P ) to the optimization problem as a function of the P experimental data (D ₁ , …, D _P ) and the optimal scheduling (O) obtained for each experimental data (D _i ), and
4. determining the best solution to the optimization problem among the P potential solutions (S ₁ , …, S _P ) according to an optimization criterion.

Method according to claim 1, in which each of the P experimental data (D_i) is obtained during an experiment implementing the following sub-steps:

1. the random loading of atoms into atom trapping sites so as to obtain any spatial arrangement of N trapped atoms forming a network of qubits, the arbitrary spatial arrangement obtained making it possible to obtain the arrangement data (A) ,
2. applying a sequence of laser pulses to the N atoms of the spatial arrangement, and
3. measuring the state of each atom at the end of the pulse sequence, the measured states making it possible to obtain the state data (E).

Method according to claim 1 or 2, in which each off-diagonal coefficient (M _ij ) of each interaction matrix (M) quantifies the interaction between two distinct atoms of the spatial arrangement as a function of the corresponding ordering, the interaction depending on the spatial distance between the two atoms considered. Method according to any one of claims 1 to 3, in which for each possible ordering, a difference is determined between the off-diagonal coefficients of the corresponding interaction matrix (M) and the off-diagonal coefficients of the problem matrix ( Q), the proximity criterion stipulating that the optimal scheduling (O) corresponds to the interaction matrix (M) having the smallest deviation. Method according to claim 4, in which the deviations for the possible schedules are determined according to the following formula:

Or :
- expresses a difference between the interaction matrix M considered and the matrix of the problem Q for an ordering of atoms corresponding to a permutation of length N,
- = Or is a coefficient depending on a Rydberg level chosen as the excited state for the atoms, denotes the position of atom i, and A is a constant proportionality coefficient, and
- designates the standard of . Method according to any one of claims 1 to 5, in which each of the P potential solutions (S _i ) is a vector whose coefficients correspond to the state data (E) of one of the P experimental data (D _i ) according to the optimal scheduling (O) obtained for the experimental data (D _i ). Method according to any one of claims 1 to 6, in which the optimization problem aims to find the antecedent of a function minimizing said function, the function depending on the matrix of the problem (Q), the optimization criterion stipulating that the best solution is the solution minimizing the function among the set of P potential solutions (S ₁ , …, S _P ). Method according to any one of claims 1 to 7, in which the optimization problem is an unconstrained quadratic binary optimization problem, called QUBO problem. Method according to any one of claims 1 to 8, in which the sequence of pulses sent to the N atoms of each spatial arrangement for obtaining the state data (E) is a sequence of Ramsey type pulses. Method according to any one of claims 1 to 9, in which the method comprises a step of implementing the P experiments by a quantum device to obtain the P experimental data (D ₁ , …, D _P ). System configured to implement a method according to claim 10, the system comprising a quantum device configured at least to implement the P experiments making it possible to obtain the P experimental data (D ₁ , …, D _P ). Computer program product, on which a computer program comprising program instructions is stored, the computer program being loaded onto a data processing unit and causing the implementation of a determination method according to the any one of claims 1 to 9 when the computer program is implemented on the data processing unit. Device for determining a solution to an optimization problem, the optimization problem being characterized by a square matrix of size N, called the problem matrix (Q), the device being configured to implement the following steps:

1. obtaining a number P of experimental data (D ₁ , …, D _P ) acquired during P distinct experiments, each experimental data (D _i ) comprising arrangement data (A) and state data (E), the arrangement data (A) being relative to a spatial arrangement of N atoms forming a network of qubits, N being an integer, the state data (E) being relative to the measured state of each of the N atoms of the spatial arrangement following the application of a sequence of laser pulses to the N atoms,
2. the processing of P experimental data (D ₁ , …, D _P ) to obtain, for each experimental data (D _i ), an optimal ordering (O) for the atoms of the spatial arrangement among a set of possible orderings, each possible ordering corresponding to a square interaction matrix (M) of size N quantifying the interactions between the atoms of the arrangement as a function of the ordering of said atoms, the optimal ordering (O) corresponding to the interaction matrix ( M) closest to the problem matrix (Q) according to a proximity criterion,
3. determining a set of P potential solutions (S ₁ , …, S _P ) to the optimization problem as a function of the P experimental data (D ₁ , …, D _P ) and the optimal scheduling (O) obtained for each experimental data (D _i ), and
4. determining the best solution to the optimization problem among the P potential solutions (S ₁ , …, S _P ) according to an optimization criterion.