A method for sampling quantum states distributed according to a distribution in a hilbert space

Abstract

A method for sampling quantum states distributed according to a distribution in a Hilbert space The present invention relates to a method for sampling quantum states distributed according to a distribution in a Hilbert space, the method being implemented by a computing system (10) comprising a quantum processor (12) having an array of qubits, the method comprising the sampling of the quantum states of the Hilbert space using a Markov chain algorithm, the first sample being the initial quantum state of the array of qubits, each other sample being obtained one by one, by preparing the array of qubits so that its quantum state corresponds to the previous sample and by performing a quantum evolution of the quantum state of the prepared array of qubits, each quantum evolution being performed by applying a Many-Body Localization unitary evolution operator to the array of qubits.

Description

[0001]A method for sampling quantum states distributed according to a distribution in a Hilbert space TECHNICAL FIELD OF THE INVENTION The present invention concerns a method for sampling quantum states distributed according to a distribution in a Hilbert space. The present invention also concerns an associated computing system. BACKGROUND OF THE INVENTION Markov chains are ubiquitous in numerical simulations of physical systems. They are used whenever it is necessary to extract samples from a probability distribution. Obtaining samples of quantum states of a Hilbert space according to the distribution of the quantum states would be of particular interest. Such samples could be used for example for the following applications: • simulate spin systems beyond the actual constraints of the hardware (meaning arbitrary interactions, connectivity, dimensionality, …), • solving combinatorial optimization problems that are not native to the hardware (for example for solving any graph problem and not be restricted to unit-disk graphs), • preparation for arbitrary states, and • generative modelling. However, the current algorithms are either based on classical objects, or do not target the sampling of a Hilbert space. Such algorithms are therefore not adapted for sampling a distribution of quantum states over a Hilbert space. SUMMARY OF THE INVENTION Hence, there exists a need for a method enabling to sample quantum states distributed according to a distribution in a Hilbert space. To this end, the invention relates to a method for sampling quantum states distributed according to a distribution in a Hilbert space, the method being implemented by a computing system comprising a quantum processor having an array of qubits, the method comprising the sampling of the quantum states of the Hilbert space using a Markov chain algorithm, the first sample being the initial quantum state of the array of qubits, each other sample being obtained one by one, by preparing the array of qubits so that its quantum state corresponds to the previous sample and by performing a quantum evolution of the quantum state of the prepared array of qubits, each quantum evolution being performed by applying a Many- Body Localization unitary evolution operator to the array of qubits. The method according to the invention may comprise one or more of the following features considered alone or in any combination that is technically possible: - the preparation of the array of qubits comprises, starting from the array of qubits in its initial quantum state, the application of all previous Many-Body Localization unitary evolution operators to such array of qubits, enabling to obtain the array of qubit whose quantum state corresponds to the previous sample; - the sampling of the quantum states comprises the following steps: - preparing the array of qubits in the initial quantum state, - performing a quantum evolution of the quantum state of the prepared array of qubits by applying a Many-Body Localization unitary evolution operator so as to obtain an evolved quantum state, - determining whether the quantum evolution is acceptable or not depending on a probability of acceptance, the probability of acceptance being a Metropolis acceptance probability which is the ratio between the value of the distribution for the evolved quantum state and the value of the distribution for the initial quantum state, - determining a new sample of quantum state depending on the acceptance of the quantum evolution, the new sample of quantum state being the evolved quantum state if the quantum evolution is accepted and being the initial quantum state otherwise, the preceding steps being repeated N times by replacing each time the initial quantum state with the last sample of quantum state so as to obtain a Markov chain of samples; - the determination whether or not the quantum evolution is acceptable comprises the comparison of the probability of acceptance to a random number drawn at random between zero and one for each quantum evolution, the quantum evolution being accepted when the probability of acceptance is above the random number and being rejected otherwise; - the evolved quantum state is measured by a detecting device of the quantum processor, the detecting device being for example a camera, such as an Electron- Multiplying Charge-Coupled Device camera; - the applied quantum evolutions fulfill an irreducibility criterion, the irreducibility criterion stating that the chain of Many-Body Localization unitary evolution operators applied for each evolution is equivalent to a Circular Unitary Ensemble unitary evolution operator; - the applied quantum evolutions fulfill a reversibility criterion, the reversibility criterion stating that the probability of performing a quantum evolution from any quantum state to any other quantum state is the same as the probability of performing the inverse quantum evolution; - the quantum array of the quantum processor is formed of qubits which are chosen among the following elements: neutral atoms, superconducting qubits, Nitrogen-Vacancy centers, photons, ions, electrons, molecules and quantum dots; - for each quantum evolution, the Many-Body Localization unitary evolution operator is carried out with Floquet engineering which consists in controlling quantum systems using time periodic external fields; - the obtained samples are used to solve a combinatorial problem, such as a Quadratic Unconstrained Binary Optimization, said QUBO, problem; - the solving of a QUBO problem is performed by choosing as a distribution aBoltzmann distribution of the type ^^(|^^^) = where D is the Boltzmann distribution,^^ is a tunable parameter and ^^ is a QUBO cost function.The invention also relates to a computing system configured for implementing a method as previously described, the computing system comprising a quantum processor having an array of qubits. BRIEF DESCRIPTION OF THE DRAWINGS The invention will be easier to understand in view of the following description, provided solely as an example and with reference to the appended drawings in which: - Figure 1 is a schematic view of an example of a computing system configured for implementing a method for sampling quantum states distributed according to a distribution in a Hilbert space, - Figure 2 is a schematic view of an example of an organigram of a method for sampling quantum states distributed according to a distribution in a Hilbert space, - Figure 3 is a schematic view of an example of the best Maximum Independent Set (MIS) solution on a 5-node graph, - Figure 4 is a schematic view of an example of the best Maximum Cut (MAXCUT) solution on the same 5-node graph than figure 3, -Figure 5 is a schematic representation of an example of a distribution ^^^^^^^^=^^−^^^^^^^^^^ ,- Figure 6 is a flowchart of an example of a Markov chain algorithm to solve MIS and MAXCUT problem, - Figure 7 is a graphic of an example of convergence plots for MIS on Erdos-Renyi graphs with edge probability 0.4, - Figure 8 is a graphic of an example of convergence plots for MIS on Erdos-Renyi graphs with edge probability 0.7, - Figure 9 is a graphic of an example of convergence plots for MAXCUT on Erdos- Renyi graphs with edge probability 0.4, - Figure 10 is a graphic of an example of convergence plots for MAXCUT on Erdos- Renyi graphs with edge probability 0.7, - Figure 11 is a plot of an example of the performance metric for MIS on a Erdos- Renyi graph with edge probability 0.4, the specific graph instance being shown on the plot, - Figure 12 is a plot of an example of the performance metric for MIS on a Erdos- Renyi graph with edge probability 0.7, the specific graph instance being shown on the plot, - Figure 13 is a plot of an example of the performance metric for MAXCUT on a Erdos-Renyi graph with edge probability 0.4, the specific graph instance being shown on the plot, and - Figure 14 is a plot of an example of the performance metric for MAXCUT on a Erdos-Renyi graph with edge probability 0.4, the specific graph instance being shown on the plot. DETAILED DESCRIPTION OF SOME EMBODIMENTS A computing system 10 is illustrated on figure 1. The computing system 10 is configured to implement a method for sampling quantum states distributed according to a distribution D in a Hilbert space H. To sample from a distribution means to obtain a collection of samples, the occurrence of each sample depending on the probability of obtaining the sample according to the distribution (for example, there will be more samples corresponding to a higher probability of the distribution, and less samples corresponding to a lower probability of the distribution). If the variables belong to a space Χ, a distribution is a function (^^) that to each object ^^ ∈ Χ assigns a real number. This is denoted ^^: Χ → ℝ. The computing system 10 comprises a quantum processor 12 (quantum processing unit or QPU) and a calculator 14. In an example, the quantum processor 12 comprises a source of particles, a generator of trapping sites for particles, a detecting device and hardware elements. In the case of neutral atoms, the source of particles comprises a vacuum chamber in which the particles (for example atoms) are located. In particular, during use, a vacuum is created in the vacuum chamber and a dilute atomic vapor is formed in the vacuum chamber. Other embodiments are nonetheless possible. The generator of trapping sites is able to generate a laser beam which when focused generates an array of trapping sites for particles (e.g., optical tweezers). In an example of implementation, the generator of trapping sites comprises a laser source, or equivalent, able to generate a laser beam and a beam shaper able to shape the laser beam (e.g., to impart a specific intensity pattern to the laser beam) so as to obtain an array of trapping sites for particles when the laser beam is focused. The beam shaper is, for example, a spatial light modulator (SLM), such as an optical phase modulator, preferably a liquid crystal optical modulator (LCOS-SLM). In another example, the beam shaper comprises one or more acousto-optical deflector, or a device with metasurfaces. The detecting device is configured to detect the presence of particle(s) in the qubit register. The detecting device is, for example, a camera such as a charge-coupled device (CCD) camera, or an Electron-Multiplying CCD (EMCCD) camera, or any suitable imaging technology. For example, in the case of atoms, the camera is suitable to detect the fluorescence emitted from the atoms. The hardware elements enable to manipulate the particles trapped in the trapping sites and forming the qubits, so as to perform quantum operations on the qubits. Examples of quantum processors based on neutral atoms are described in 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, Septembre 2020. ISSN 2521-327X. doi:10.22331 / q-2020-09-21-327. Typically, the quantum processor 12 uses the quantum properties of matter, such as superposition and entanglement, to perform operations on the qubits. In particular, a qubit refers to a two-level quantum system. In a non-limiting example, a qubit may comprise 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 particles meant to be trapped in the trapping sites are polarizable particles sensitive to alternative current (AC) Stark shift from lasers. Preferably, the particles are atoms, and in particular electrically neutral atoms. Neutral atoms are, for example, alkali atoms (such as Rubidium or Cesium atoms) or alkaline-earth atoms (such as Strontium or Ytterbium atoms). The particles have a fundamental state (no excitation) and at least one excited state. The excited state is, for example, a Rydberg state. In a variant, the quantum array of the quantum processor 12 is formed of qubits which are chosen among the following elements : neutral atoms, superconducting qubits, NV centers (nitrogen-vacancy centers), photons, ions, electrons, molecules and quantum dots. The calculator 14 is, for example, a classical computer. The calculator 14 comprises typically a processor comprising a data processing unit, memories and a reader for information media. Eventually, the calculator 14 comprises a human machine interface, such as a keyboard, and a display. In an example of implementation, the calculator 14 interacts with a computer program product. The computer program product comprises an information medium. The information medium is a medium readable by the calculator 14, 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 computer system bus. On the information medium is stored the computer program comprising program instructions. The computer program is loadable on the data processing unit and is adapted to entail some steps of the method that will be described later in the description, when the computer program is loaded on the processing unit of the calculator. In a variant, the calculator 14 is under the form of an electronic circuit comprising microcontrollers or integrated circuits. A method for sampling quantum states |^^^ distributed according to a distribution D in a Hilbert space H, will now be described with reference to the organigram of figure 2. The method is implemented by the computing system 10. A quantum state |^^^ corresponds to the collective state of the qubits forming the array of qubits. The quantum state can be decomposed in the computational basis, each element of the computational basis being a state where each qubit typically corresponds to an integer number which is either zero (fundamental state) or one (excited state). The method comprises the sampling of the quantum states|^^^of the Hilbert space H using a Markov chain algorithm. A Markov chain is defined as a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. The first sample is the initial quantum state of the array of qubits. The firstsample is typically either the state |00 … 0^ where all qubits are in the fundamentalstate, or it is generated at random. Each other sample obtained one by one, by preparing the array of qubits so that its quantum state |^^^ corresponds to the previous sample and by performing a quantum evolution of the quantum state|^^^of the prepared array of qubits. Each quantum evolution is performed by applying a Many-Body Localization unitary evolution operator to the array of qubits. In an example, the preparation of the array of qubits comprises, starting from the array of qubits in its initial quantum state |^^0^, the application of all previous Many-Body Localization unitary evolution operators (the MBL used to obtain the previous samples) to such array of qubits, enabling to obtain the array of qubit whose quantum state|^^^corresponds to the previous sample In a variant, the preparation of the array of qubits comprises applying a state preparation algorithm to the array of qubits in any quantum state so as to obtain the array of qubit whose quantum state |^^^ corresponds to the previous sample The state preparation algorithm is for example as described in the article Grover, Lov K.. “Rapid sampling though quantum computing.” Symposium on the Theory of Computing (2000). Each quantum evolution is performed by applying a Many-Body Localization unitary evolution operator to the array of qubits. The applied Many-Body Localization unitary evolution operators can be different for the different quantum evolutions. In a variant, the applied Many-Body Localization unitary evolution operators are identical for the different quantum evolutions. The concept of Many-Body Localization (MBL) is described in the article Abanin, D., Altman, E., Bloch, I., & Serbyn, M. (2019). Colloquium: Many-body localization, thermalization, and entanglement. Rev. Mod. Phys., 91, 021001. In a nutshell, MBL is a phenomenon where a quantum system retains at all times a “memory” of its initial conditions. In a sense, we never stay too far from the initial state during the whole evolution. Hence, applying a MBL evolution operator might be considered an equivalent of a small move in a Hilbert space H. On the opposite end, if the system completely loses information of its initial conditions, we will say that the system is thermalized, and this would be the equivalent of a very long move in the Hilbert space H. In an example of embodiment, the sampling of the quantum states|^^^comprises the following steps. The first step 90 consists in preparing the array of qubits in the initial quantum state The first step 90 is, for example, carried out by the quantum processor 12 of the computing system 10. The second step 100 consists in performing a quantum evolution of the quantum state of the prepared array of qubits by applying a Many-Body Localization unitary evolution operator so as to obtain an evolved quantum state |^^′^. The second step 100 is, for example, carried out by the quantum processor 12 of the computing system 10. For example, the application of each Many-Body Localization unitary evolution operator corresponds to specific laser pulses applied on the array of qubits. In an example of implementation, for each quantum evolution, the Many-Body Localization unitary evolution operator is carried out with Floquet engineering. Floquet engineering consists in controlling quantum systems using time periodic external fields. In particular, the use of Floquet engineering is of particular interest when using a quantum processor based on neutral atoms. Several references describe how to do so, both theoretically and experimentally, with neutral atoms: - Thanasilp, S., Tangpanitanon, J., Lemonde, M.A., Dangniam, N., & Angelakis, D. (2021). Quantum supremacy and quantum phase transitions. Phys. Rev. B, 103, 165132 - Jae-yoon Choi et al. ,Exploring the many-body localization transition in two dimensions. Science 352, 1547-1552 (2016). DOI:10.1126 / science.aaf8834 - Bordia, P., Lüschen, H., Schneider, U. et al. Periodically driving a many-body localized quantum system. Nature Phys 13, 460–464 (2017). https: / / doi.org / 10.1038 / nphys4020 Floquet engineering is also relevant for other types of quantum processors, for example based on superconductors or trapped ions architectures. Preferably, the applied quantum evolutions fulfill an irreducibility criterion. The irreducibility criterion states that the chain of Many-Body Localization unitary evolution operators applied for each evolution is equivalent to a Circular Unitary Ensemble (CUE) unitary evolution operator. More precisely, irreducibility means that it is possible to explore every corner of the Hilbert space H (not necessarily in a single move). Chaining together many MBL unitaries results in what’s called a CUE unitary. This means that the Hilbert space exploration is extremely good, and we can reach a good portion of it. In the present invention, the irreducibility is ensured by the fact that the unitary evolution operators are MBL unitary evolution operators. Preferably, the applied quantum evolutions fulfill a reversibility criterion. The reversibility criterion states that the probability of performing a quantum evolution from any quantum state|^^^^^ to any other quantum state the same as the probability of performing the inverse quantum evolution. The use of Many-Body Localization unitary evolution operators carried out with Floquet engineering enables to fulfill such a reversibility criterion. In a variant, the applied quantum evolutions fulfill a pseudo-reversibility criterion, which is an approximation of the reversibility criterion. In an example of embodiment, the evolved quantum state |^^′^ is measured by a detecting device (camera) of the quantum processor 12. The third step 110 is the determination whether or not the quantum evolution is acceptable depending on a probability of acceptance Pacc. The probability of acceptance Paccis the Metropolis acceptance probability which is the ratio between the value D(|^^′^) of the distribution D for the evolved quantum state|^^′^(with the evolved quantum state |^^′^ being for example measured by the detection device) and the value D(|^^0^) of the distribution D for the initial quantum state |^^0^. The third step 110 is, for example, carried out by the calculator 14 of the computing system 10. In an example, the determination whether or not the quantum evolution is acceptable comprises the comparison of the probability of acceptance Pacc to a random number drawn at random between zero and one for each quantum evolution. The quantum evolution is accepted when the probability of acceptance Paccis above the random number and is rejected otherwise. In another example, the determination whether or not the quantum evolution is acceptable comprises the comparison of the probability of acceptance Paccto a predetermined number comprised between zero and one. The quantum evolution is accepted when the probability of acceptance Paccis above the predetermined number and is rejected otherwise. The fourth step 120 is the determination of a new sample of quantum state depending on the acceptance of the quantum evolution. The new sample of quantum state is the evolved quantum state |^^′^ if the quantum evolution is accepted and is the initial quantum state|^^0^otherwise. The fourth step 120 is, for example, carried out by the calculator 14 of the computing system 10. The preceding steps are repeated N times by replacing each time the initial quantum state|^^0^ with the last sample|^^^^^ of quantum state so as to obtain a Markov chain ofsamples In this case, the preparation step 90 is for example carriedout as previously described (applying all the previous MBL on the initial quantum state, or using a state preparation algorithm). In an example of implementation, due to the use of Markov chains, only the samples comprised between the M-th sample and the last sample are distributed according to the distribution D of quantum state in the Hilbert space H (uniformly distributed). In the following, we give an example explaining how to choose the number M, i.e. how many samples have to be discarded at the beginning of a Markov chain. The idea is that Markov chains are initialized with a random state, and the first few samples will basically be a random walk around this initial state, which is not necessarily representative of the distribution we want to sample. Therefore we need to discard these few iterations. The number of iterations that are necessary for the Markov chain to exit this initial phase of random walking around the initial state is called thermalization time, or mixing time. The number M is here estimated in a post-processing phase. Denote with x the states of the chain, with X the space where the states belong, and D the distribution, then: 1. Initialize two Markov chains from two different initial states ^^0(1)and ^^0(2). Let’s call the two chains “Markov chain 1” and “Markov chain 2”. 2. Collect a certain number of samples from both Markov chains. 3. Compute, iteration after iteration, the value of a certain function f : X → ℝ. For example, it can be the value of the distribution itself D(x) for every x that was sampled. 4. Plot the value of the function for both Markov chains, and note the iteration number where the two Markov chains start to overlap and oscillate around each other. Take twice that number, and that’s the number of iterations you need to discard in this example. Take the example where the space ^^ = ℝ and the distribution is a Gaussian ^^(^^) =^^2^^−2^^2. If we follow the above steps, initializing “Markov chain 1” from ^^(1)0 = 1000 and“Markov chain 2” from ^^(2)0 = −1000. The two Markov chains will initially perform a randomwalk around the initial state, but gradually they will move towards the correct distribution. The thermalization time can be estimated by taking twice the point where the two chains start oscillating around a common value. In an example, the obtained samples are used to solve acombinatorial problem, such as a Quadratic Unconstrained Binary Optimization (QUBO) problem. In an example, the solving of a QUBO problem is performed by choosing as adistribution D a Boltzmann distribution of the type ^^(|^^^) = where D is theBoltzmann distribution, ^^ is a tunable parameter and ^^ is the QUBO cost function. In other examples, the obtained samples |^^^^^) are used for simulatingspin systems beyond the actual constraints of the hardware, for preparing for arbitrary states or for generative modelling. More precise examples of applications of the method for sampling quantum states are describes in the paragraph “Practical applications” hereinafter. Hence, the above method enables to sample quantum states distributed according to a distribution D in a Hilbert space H. The use of many-body localization is of particular interest because it allows an extensive exploration of the Hilbert space H in a controlled way. The person skilled in the art will understand that the embodiments and variants described above in the description can all be combined provided that they are technically compatible. PRACTICAL APPLICATIONS The purpose of this section is to present how the invention can be used concretely on a neutral atom QPU to solve graph problems of practical interest. The study presented here is just an example and the capabilities of the invention go much further than this in terms of which problems can be solved. However, the problems on which we focus highlight how the invention can bypass some limitations of the QPU. Problem selection The problems we want to tackle here are Maximum Independent Set (MIS) and Maximum Cut (MAXCUT). These are well-known graph problems that are of practical interest in industry. MIS consists in finding the largest subset of nodes in a graph such that no two nodes in the subset are connected by an edge. A visual representation of the MIS problem on a 5-node graph is shown in figure 3. In this example, nodes 2 and 5, colored in white, are an optimal solution to the MIS problem because they are not connected to each other, and there are no subsets with more than 2 nodes that do not share connections. MAXCUT consists in partitioning the nodes of a graph into two disjoint subsets, such that the number of edges connecting nodes belonging to a different subset are maximized. A visual representation of the MAXCUT problem on a 5-node graph is shown in figure 4. In this example, the partition {3, 4} ∪ {1, 2, 5} is an optimal solution to the MAXCUT problem because six different edges connect white nodes with black nodes: (1,3), (2,3), (3,5), (1,4), (2,4), (4,5). There is no other partition such that more than six edges connect nodes colored differently. Neutral atom QPUs can solve natively only a few graph problems (including MIS and MAXCUT) on a restricted class of graphs called Unit Disk (UD) graphs. Here natively means that 1 node in the graph corresponds to 1 qubit in the QPU, and that a solution to the problem can be obtained from the natural quantum evolution of the system of qubits without the need for extra resources or techniques. Techniques to overcome this restriction often scale poorly in the number of extra qubits required to implement them. As this will be described in what follows, the invention instead represents a fixed prescription for solving any graph problem on any graph, without requiring extra qubits. Correspondence between solving graph problems and sampling quantum states Before explaining how the method of the invention is used to solve MIS and MAXCUT problems, we specify here the connection between the invention (Markov chain with MBL) and solving graph problems. This is a particular case of the solving of a QUBO problem bychoosing as a distribution D a Boltzmann distribution of the type ^^(|^^^) = whereD is the Boltzmann distribution, ^^ is a tunable parameter and ^^ is the QUBO cost function. In the following we will be referring to the MIS problem, but the same applies to MAXCUT or any other problem. The invention is a method to sample quantum states |ψ^ according to a distribution D in a Hilbert space H. So somehow the distribution of quantum states needs to encode the solution to the graph problem. In MIS we are partitioning the nodes of the graph into two subsets S and T. Any such partition represents a candidate solution to the problem, and we are looking for the best one. On a QPU this partitioning is represented by some qubits being in the excited state |1^ and some others being in the ground state |0^. So that for example a graph with 3 nodes can be represented in the QPU as a system of 3 qubits, and any basis state like for instance |101^ represents a possible solution to the problem. The value of the cost function CMIS(|101^) tells us how good the solution |101^ is. The lower the cost, the better the solution. A generic quantum state |ψ^ for a system of 3 qubits can be written as a superposition of all possible basis states: |ψ^ = a1 |000^ + a2 |001^ + a3 |010^ + a4 |011^ + a5 |100^ + a6 |101^ + a7 |110^ + a8 |111^ or, for brevity: The cost function can be extended to generic states in the following way: So to find the best solution to the MIS problem, we want to find the quantum state |ψ^ with the smallest cost CMIS(|ψ^). To do that we build a probability distribution of quantum states DMIS from the costfunction CMIS in the following way: where β > 0 is a free parameter and Z is an unimportant normalization factor. For large positive values of β, this probability distribution is very peaked around quantum states that have a small cost CMIS, i.e. quantum states that encode good solutions to the MIS problem. The situation is depicted schematically in figure 5. In particular, in this figure, the x- axis represents in a symbolic way the whole Hilbert space of quantum states. The distribution is peaked around those quantum states that represent good solutions to the MIS problem. The rest of the Hilbert space, which is the part representing bad MIS solutions, is associated with a low probability. Sampling from this distribution is then equivalent to finding good solutions to the MIS problem. The effect of increasing β is to tighten the peak around the good solutions, and therefore the very optimum is found by sampling from DMIS with a large β. We have thus established a correspondence: Solving MIS ←→ Sampling quantum states from DMIS. Methodology For each of the two problems, MIS and MAXCUT, we generated 20 random Erdos- Renyi graphs: 10 of them with edge probability 0.4, and 10 of them with edge probability 0.7, for a total of 40 different combinations of graph / problem. Each graph is composed of 9 nodes. We then set up an MBL Markov chain as described in the invention in order to sample from DMIS and DMAXCUT, in virtue of the correspondence between solving graph problems and sampling quantum states explained in the previous paragraph. Our implementation uses neutral atoms and Floquet engineering. We performed a total of 10000 Markov chain iterations for each of the 40 instances. A schematic representation of the algorithm is shown in figure 6. This algorithm applies to solve both MIS and MAXCUT. We chose Erdos-Renyi graphs with 9 nodes and edge probability 0.4 or 0.7. This graph is represented in a neutral atom QPU as a one- dimensional chain of 9 neutral atoms initialized in the ground state |00..0^. To this initial state we apply an MBL unitary generated with Floquet engineering. The new state obtained is used to calculate the Metropolis accept / reject step (where the cost of the new state is compared to the cost of the old state). The whole procedure is repeated iteratively until a stopping criterion is met (in our case we stopped at 10000 iterations). In order to check the convergence of the algorithm, we stored the value of the cost function at each iteration. As a metric of success of the algorithm, we take the last quantum state of the chain (i.e. the quantum state obtained after the last Markov chain iteration) and measure the probability of observing each possible value of the cost function. Results Here we present the two metrics mentioned in the methodology section. The first metric is about convergence. As is often the case in Markov chain applications, the first state of the chain, let’s call it |ψ1^, is not at all a high-probability state of the distribution we are targeting. Therefore, the value of the cost function calculated on the first state is very high (i.e. bad). The Markov chain then starts to thermalize, meaning that it quickly starts producing states which are closer and closer to the peak of the target distribution. Eventually, the chain reaches a regime where the cost function is fluctuating around some equilibrium value. At this point we can consider the Markov chain to have converged. This behaviour is quite universal and it can be seen by plotting the cost as afunction of the iteration number, i.e.:^^^^^^^^(|^^^^^) for k = 1, 2, ..., Nwhere in our case N = 10000 (and similarly for MAXCUT). We show this metric in figures 7 to 10. The convergence plot for all 10 graphs related to the same problem and graph class are collected in the same figure for convenience. The second metric is aimed at assessing the performance of the algorithm. Ultimately, we want to find a quantum state which is largely composed of bitstrings with the lowest cost. Given a graph with n nodes and a cost function C related to some optimization problem (C can be CMIS or CMAXCUT in our case), there are only a finite number of possible values of the cost function. Call these values c1, ... , cm, with m ≤ 2n. We then define the probability of observing the cost value c in a quantum state |^^^ =∑^^ ^^^^|^^^ as: in other words, P(c) is the probability of sampling from the quantum state any basis state |j^ that has cost c. Since we are targeting the distribution we expect a probability peaked at the lowest (i.e. best) cost, and exponentially decaying from that. We show this metric for one graph, in figures 11 to 14. In particular, figures 11 and 12 show the performance metric for MIS on an Erdos-Renyi graph with edge probability 0.4 (figure 11) and 0.7 (figure 12). Figures 13 and 14 show the performance metric for MAXCUT on an Erdos-Renyi graph with edge probability 0.4 (figure 13) and 0.7 (figure 14).

Claims

CLAIMS 1.- A method for sampling quantum states (|^^^) distributed according to a distribution (D) in a Hilbert space (H), the method being implemented by a computing system (10) comprising a quantum processor (12) having an array of qubits, the method comprising the sampling of the quantum states (|^^^) of the Hilbert space (H) using a Markov chain algorithm, the first sample (|^^0^) being the initial quantum state (|^^0^) of the array of qubits, each other sample (|^^^^^) being obtained one by one, by preparing the array of qubits so that its quantum state (|^^^) corresponds to the previous sampleand by performing a quantum evolution of the quantum state (|^^^) of the prepared array of qubits, each quantum evolution being performed by applying a Many-Body Localization unitary evolution operator to the array of qubits. 2.- A method according to claim 1, wherein the preparation of the array of qubits comprises, starting from the array of qubits in its initial quantum statethe application of all previous Many-Body Localization unitary evolution operators to such array of qubits, enabling to obtain the array of qubit whose quantum state (|^^^) corresponds to the previous sample 3.- A method according to claim 1 or 2, wherein the sampling of the quantum states (|^^^) comprises the following steps: - preparing the array of qubits in the initial quantum state (|^^0^), - performing a quantum evolution of the quantum state of the prepared array of qubits by applying a Many-Body Localization unitary evolution operator so as to obtain an evolved quantum state (|^^′^), - determining whether the quantum evolution is acceptable or not depending on a probability of acceptance (Pacc), the probability of acceptance (Pacc) being a Metropolis acceptance probability which is the ratio between the value (D(|^^′^)) of the distribution (D) for the evolved quantum state (|^^′^) and the value (D(|^^0^)) of the distribution (D) for the initial quantum state- determining a new sample (|^^1^) of quantum state depending on the acceptance of the quantum evolution, the new sample of quantum state being the evolved quantum state (|^^′^) if the quantum evolution is accepted and being the initial quantum state (|^^0^) otherwise,the preceding steps being repeated N times by replacing each time the initial quantum state (|^^0^) with the last sample (|^^^^^) of quantum state so as to obtain a Markov chain ofsamples4.- A method according to claim 3, wherein the determination whether or not the quantum evolution is acceptable comprises the comparison of the probability of acceptance (Pacc) to a random number drawn at random between zero and one for each quantum evolution, the quantum evolution being accepted when the probability of acceptance (Pacc) is above the random number and being rejected otherwise. 5.- A method according to claim 3 or 4, wherein the evolved quantum state (|^^′^) is measured by a detecting device of the quantum processor (12), the detecting device being for example a camera, such as an Electron-Multiplying Charge-Coupled Device camera. 6.- A method according to any one of claims 1 to 5, wherein the applied quantum evolutions fulfill an irreducibility criterion, the irreducibility criterion stating that the chain of Many-Body Localization unitary evolution operators applied for each evolution is equivalent to a Circular Unitary Ensemble unitary evolution operator. 7.- A method according to any one of claims 1 to 6, wherein the applied quantum evolutions fulfill a reversibility criterion, the reversibility criterion stating that the probability of performing a quantum evolution from any quantum state (|^^^^^) to any other quantum state is the same as the probability of performing the inverse quantum evolution. 8.- A method according to any one of claims 1 to 7, wherein the quantum array of the quantum processor (12) is formed of qubits which are chosen among the following elements: neutral atoms, superconducting qubits, Nitrogen-Vacancy centers, photons, ions, electrons, molecules and quantum dots. 9.- A method according to any one of claims 1 to 8, wherein for each quantum evolution, the Many-Body Localization unitary evolution operator is carried out with Floquet engineering which consists in controlling quantum systems using time periodic external fields.10.- A method according to any one of claims 1 to 9, wherein the obtainedsamplesare used to solve a combinatorial problem, such as aQuadratic Unconstrained Binary Optimization, said QUBO, problem. 11.- A method according to claim 10, wherein the solving of a QUBO problem is performed by choosing as a distribution (D) a Boltzmann distribution of thetype ^^(|^^^) =where D is the Boltzmann distribution, ^^ is a tunable parameterand ^^ is a QUBO cost function. 12.- A computing system (10) configured for implementing a method according to any one of claims 1 to 11, the computing system (10) comprising a quantum processor (12) having an array of qubits.

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