Method and system for solving QUBO problems using a classical-quantum hybrid solver

Abstract

To provide a method for encoding a quadratic unconstrained binary optimized (QUBO) problem to a corresponding QUBO graph problem regarding the use of a meso-quantum (NISQ) device.SOLUTION: A method includes the steps of: encoding a coefficient of a term including both binary variables in which each binary variable of a QUBO problem corresponds to a node of a QUBO graph problem and an edge between two nodes corresponds to two nodes or receiving the QUBO graph problem of a QUBO graph representation; providing the problem as an input to a trained graph neural network on a processing system; retrieving a predictive performance metric for solving the QUBO problem using a variational quantum solver from an output of the trained graph neural network for the QUBO problem provided at the input; and based on the predictive performance metric, providing the QUBO problem to the variational quantum solver implemented in a quantum hardware or providing the QUBO problem to a classical solver implemented in a classical processing system.SELECTED DRAWING: Figure 3

Description

[Technical field]

[0001] The present invention relates to the field of quantum computing, and more particularly to finding solutions to quadratically unconstrained binary optimization problems using hybrid quantum-classical solvers. [Background technology]

[0002] Quantum computers provide a platform for controllable quantum mechanical systems whose states and interactions can be controlled to perform computations. Computation is realized through the deterministic evolution of controllable quantum mechanical systems, e.g., qubits, as the quantum analogue of classical bits, whose states can be measured to determine the computational results.

[0003] The control operations on these qubits are called quantum gates. Quantum gates can act coherently on qubits to cause a change in the state of a single qubit (so-called single-qubit gates) and on multiple qubits (so-called multi-qubit gates), e.g., to entangle the states of multiple qubits and any combination of them. For example, a single-qubit gate may cause a rotation of the spin state of an electron by a selectable value, e.g., π / 2. A multi-qubit gate may act coherently on more than one qubit, such as a coherent CNOT operation on the states of two qubits. Multiple quantum gates can be applied to the qubits of a quantum computer in parallel or sequentially to perform a computation. Finally, after applying a series of quantum gates, the states of the qubits may be repeatedly measured to determine the probability of each possible computation outcome.

[0004] To compute solutions to problems that are considered intractable for classical computers, quantum computers can exploit the special properties of quantum mechanical states, in particular the superposition and entanglement of different quantum states, to find the solution in a relatively small number of computational steps.

[0005] However, because the superposition / entangled states of quantum mechanical systems are inherently volatile (e.g., suffer from decoherence), and the control and measurement of these systems are subject to fidelity margins, state-of-the-art quantum computers are currently limited in both the number of controllable quantum mechanical systems (qubits) and the number of control operations (quantum gates) that can be successively performed.

[0006] Despite these drawbacks, variational quantum algorithms such as the quantum approximate optimization algorithm (QAOA) disclosed by Friedrich van der Weiner, "Quantum Approximation Optimization Algorithm for Quantum Computing", are promising applications for quantum processors available in the near future, i.e., noisy intermediate-scale quantum (NISQ) devices, because it has been shown that QAOA can potentially be used to provide quantum advantage for multiple problem sets. QAOA specifies a variational quantum circuit specific to a given problem, where quantum gates in the circuit have variable actions on qubits in a qubit register, which are parameterized by variational parameters. After preparing qubits in known initial states and applying the variational quantum circuit with an initial guess of the variational parameters, the variational parameters can then be modified to extremize a cost function that attributes a cost to the output of the circuit, such that the output of the variational quantum circuit approaches the optimal solution for the given problem.

[0007] Non-Patent Document 2 applies a trained recurrent classical network in a feedback loop to update the variational parameters of a quantum neuron network (QNN) based on QAOA. An LSTM recurrent neural network is used to recursively propose updates to the QNN parameters, thereby acting as a classical black-box optimizer for quantum-classical optimization. At a given iteration, the RNN receives as input the estimated cost function expectation of the previous QNN query and the variational parameters on which the QNN was evaluated. After a finite number of runs, the quantum neuron network is considered to be in an initialization state, and another optimizer is used to fine-tune the solution.

[0008] Non-Patent Document 3 applies a trained graph neural network (GNN) to warm start a QNN for solving the max-cut problem. The GNN is trained to output the probability that each node is on either side of the cut, from which candidate solutions can be derived, and the candidate solutions can be used to find the initial states and initial angles of the QNN's mixer Hamiltonian to warm start the QNN. The GNN is trained with an unsupervised learning approach, and the costs attributed to the candidate solutions are extremized so that the GNN approximates the optimal solution to the max-cut problem. [Prior art documents] [Non-patent literature]

[0009] [Non-Patent Document 1] "A Quantum Approximate Optimization Algorithm" by Farhi et al. [Non-Patent Document 2] Verdon et al., "Learning to learn with quantum neural networks via classical neural networks" [Non-Patent Document 3] Jain et al., "Graph neural network initialization of quantum approximate optimization" [Non-Patent Document 4] Kipf and Welling, Semi-Supervised Classification with Graph Convolutional Networks, 2017 Summary of the Invention [Problem to be solved by the invention]

[0010] However, the quality of the solution provided by a variational quantum circuit may depend on the problem to which it is applied and may vary significantly from problem to problem. Moreover, known warm-start approaches may still require a significant number of iterations of the variational quantum circuit.

[0011] In view of this state of the art, it is an object of the present invention to improve the efficiency associated with the use of NISQ devices, particularly with regard to solution quality and computation speed. [Means for solving the problem]

[0012] This object is solved by two methods, a computer program and a system according to the independent claims. The dependent claims relate to preferred embodiments.

[0013] According to a first aspect, the present invention relates to a computer-implemented method for selecting processing hardware based on a given quadratic unconstrained binary optimization (QUBO) problem. The method includes the steps of: encoding the QUBO problem into a corresponding QUBO graph problem, where each binary variable of the QUBO problem corresponds to a node of the QUBO graph problem, and an edge between two nodes encodes the coefficients of terms that include both binary variables corresponding to the two nodes; or receiving the QUBO graph problem in a QUBO graph representation; and providing the QUBO graph problem as an input to a trained graph neural network on a processing system. The method further includes the steps of deriving a predicted performance metric for solving the QUBO problem using a variational quantum solver from an output of the trained graph neural network for the QUBO graph problem provided at the input; and providing the QUBO problem to a variational quantum solver implemented on quantum hardware based on the predicted performance metric.

[0014] Quantum hardware is currently difficult to access and would be preferred to use only if the quality of the solution obtained by quantum hardware is good. The inventors have found that a graph neural network can be trained to predict performance metrics obtainable when actually solving a problem using a variational quantum solver, such that the trained graph network can predict whether an example QUBO graph problem would benefit from quantum advantage or whether the estimated solution quality is too low to warrant the use of a variational quantum solver over the use of a classical algorithm to solve or approximate the solution of the QUBO problem. In other words, the performance metric may provide a qualitative and / or quantitative indication related to the predicted quality of the solution when actually solving the QUBO problem using a variational quantum solver. Thus, the method may automatically select appropriate processing hardware for a given problem based on the symmetry of the problem, thereby increasing the efficiency of solving QUBO (NP-hard) problems using variational quantum networks. Graph neural networks have been found to generalize classification tasks to larger problems (graphs) and to produce outputs that correlate with the actual values ​​of performance metrics when variational quantum networks are applied to variational quantum solvers to find solutions.

[0015] In general, the performance metric may classify the quality of the best solution that can be obtained by applying a variational quantum network to solve the QUBO problem. Based on the predicted performance metric satisfying a threshold condition, for example, if the performance metric is lower or higher than a predefined threshold, the QUBO problem may be provided to a different solver, such as a classical solver implemented in a classical processing system, or to a different (variational) quantum solver.

[0016] In a preferred embodiment, a trained graph neural network is trained to estimate a quantum approximation ratio when solving a QUBO problem using a variational quantum solver, or a performance metric based on the quantum approximation ratio.

[0017] The quantum approximation ratio may be defined as the ratio of the cost associated with the best solution from a variational quantum solver to the cost of the optimal solution or the best approximate solution obtained using a classical solver, where the cost is calculated based on a cost function that attributes a quantitative quality metric to candidate solutions of the QUBO problem.

[0018] A QUBO problem is an optimization problem in which quadratic functions on binary variables are extremized such that the cost is a quadratic function or a cost based on a quadratic function that attributes a performance metric to the solution of the QUBO metric given the QUBO problem. A QUBO problem is a problem in which the objective function f Q The binary vector x that is the minimum (maximum) for * may be defined as finding f Q is a quadratic polynomial in a binary variable

number

number

number

number

number

number

[0019] And the quantum approximation ratio is the variational quantum network f Q (x variational ) to obtain the best solution objective function for the problem f Q (x optimal) divided by the objective function of the best solution that can be obtained by brute-force attack, or a function based on, mathematically equivalent to, or proportional to that ratio.

[0020] A QUBO problem may be encoded into a QUBO graph problem by specifying each binary variable as a graph node. The edge weight between two nodes may be equal to the coefficient of the term in the objective function that includes the two variables that these nodes represent. If the edge weight is 0, the edge can be removed. The node weight may be equal to the coefficient of the term that includes only the variables represented by that node. The resulting QUBO graph representation may capture the symmetry / structure of the QUBO problem and may be processed by a graph neural network. However, some QUBO problems, such as max-cut problems, may have an inherent QUBO graph representation and may be directly fed into the graph neural network so that no encoding is necessary.

[0021] A graph neural network is a neural network, typically implemented on classical processing hardware, that prescribes a set of trainable transformations for graphs, which can be viewed as a set of nodes interconnected by edges, where the nodes may be associated with feature vectors and / or the edges with respective weights. Graph neural networks may process graphs such that the output of the graph neural network captures the symmetries of the graph and can generalize to different equivalent representations of graphs with the same or similar structure, such as the same output for graphs where the nodes have simply been relabeled.

[0022] In a preferred embodiment, the trained graph neural network is a convolutional graph neural network, in particular a spatial graph neural network.

[0023] Convolutional graph networks involve generalized convolution operations on the nodes / edges of a graph, which may be classified as spectral or spatial approaches. A spectral approach to graph neural networks may involve applying a graph Fourier transform that transforms the input graph signal into a projection into a space whose basis is the eigenvectors of the normalized graph Laplacian, and the convolution may be defined by the Fourier transform. Conversely, in a spatial approach, the convolution may be similar to the convolutions defined for image operations acting on the nodes / edges of a graph, and may use a message passing paradigm.

[0024] For example, a graph neural network may include nodes that update their associated values ​​(feature vectors) using information about the values ​​of neighboring nodes in multiple graph convolutional layers.

[0025] In a preferred embodiment, the trained graph neural network includes multiple graph convolutional layers, each of which includes, for each node in the QUBO graph problem, an aggregation step in which feature vectors of adjacent nodes are aggregated into an aggregate feature vector, and a transformation step in which the original feature vector and the aggregate feature vector of the node are transformed, for each node, into an updated feature vector according to a trainable transformation. For example, the feature vector may include information about the edges connected to the node.

[0026] The aggregation step may be performed in multiple ways, for example by summing or averaging the features of adjacent nodes, by selecting the minimum or maximum of adjacent nodes, and / or based on an attention mechanism that involves the features of adjacent nodes, and the aggregated features of adjacent nodes may be combined with the features of the processed node and / or transformed to obtain an updated feature vector. Trainable transformations generally involve the application of machine learning techniques, such as by using a multi-layer perceptron to transform the aggregated features into an updated feature vector. For example, in a graph isomorphic (neural) network, GIN, the features h of each node in layer k-1 are v k-1 teeth,

number

[0027] Edge weights may be taken into account as part of the feature vector, as part of the convolution (e.g., a transformation function such as equation (2)), and / or by treating edges as nodes in the graph neural network. For example, an encoding may be used to indicate whether a particular node in the graph neural network originates from a graph node or a graph edge. The graph can then be treated as having only node features. Additionally or alternatively, the convolution step may include a transformation, e.g., for each adjacent node, based on the respective node, its adjacent nodes, and the connecting edges.

[0028] As another example, the graph convolutional network introduced by [4] is a multi-layered network that performs the propagation step.

number

number

number

[0029] The resulting update values ​​(feature vectors / activation matrices) after multiple propagation steps may then be processed by further layers of the neural network, which may include a graph-level readout, which converts the updated feature vectors / update values ​​of the nodes into output vectors and further layers, e.g., a fully connected layer, which may output depending on the graph symmetries extracted by the aggregation layer.

[0030] In a preferred embodiment, the trained graph neural network includes a pooling layer for pooling the output of at least one of the plurality of graph convolutional layers, and a fully connected layer between the pooling layer and the output of the trained graph neural network.

[0031] Similar to the aggregation step, different pooling techniques may be used in embodiments, e.g. by summing or averaging the features of the nodes after the graph convolutional layers, by selecting the minimum or maximum of the nodes, and / or based on attention mechanisms related to the features of the nodes, to obtain a feature vector derived from the updated feature vectors of all the nodes, etc.

[0032] For example, in GIN, the graph neural network may use sum pooling for global pooling / graph-level readout, i.e., all of the feature vectors may be summed, and the summed feature vector may be the graph-level readout result (pooled features). Subsequently, multiple fully connected layers of neurons may be used to transform the pooled features into the desired output. The fully connected layers may include artificial neurons with any suitable activation function, e.g., ReLu, where each artificial neuron in a layer may be connected to all artificial neurons in the previous layer. As an example, the fully connected layer may include three or four layers of fully connected artificial neurons.

[0033] In a preferred embodiment, the trained neural network has been trained by receiving a training set, the training set including a plurality of QUBO graph problems and a performance metric corresponding to each of the plurality of QUBO graph problems, the performance metric indicating a ratio of a quality metric of a candidate solution determined using a variational quantum solver for the QUBO graph problem to a similar quality metric of a classical solution to the QUBO graph problem, the quality metric being, in particular, a cost associated with the candidate solution, and training a graph convolutional network on the training set, the graph convolutional network receiving, at its input, the plurality of QUBO graph problems, and trainable parameters of the graph convolutional network being iteratively updated such that an output of the graph convolutional network for a given QUBO graph problem approaches the performance metric.

[0034] The performance metric of a solution by a variational quantum solver may be, for example, the quantum approximation ratio of the solution as defined above, or may be a performance metric indicative of the quantum approximation ratio for a given QUBO graph problem.

[0035] The training set may be constructed from previous applications of the variational quantum solver to multiple QUBO graph problems, e.g., a quality metric may be calculated for the best solutions obtained / obtainable using the variational quantum solver, and classical solutions may be determined using classical methods, e.g., optimal solutions obtained via brute force attacks or classical approximation algorithms.

[0036] In a preferred embodiment, the variational quantum solver includes a variational quantum network for determining an output indicative of a solution to the QUBO problem, and a gate configuration of the variational quantum network is selected based in particular on the QUBO problem, and the variational quantum network is preferably based on a quantum approximate optimization algorithm (QAOA), and / or the gate configuration is preferably dependent on a cost function that attributes a cost to the solution of the QUBO problem, with an optimal solution being associated with a global extremum of the cost function.

[0037] Variational quantum solvers may generally include multiple qubits, whose quantum states may be manipulated by the application of quantum gates applied sequentially or in parallel to single qubits and / or multiple qubits.

[0038] The qubits may be initialized to an initial state, such as a basis state for each qubit. In some embodiments, after initializing the qubits to their basis states, a superposition state for each qubit in the qubit register is prepared, for example, by applying a Hadamard gate.

[0039] Subsequently, multiple quantum gates may be applied to the qubits to transform their states into output states. In a variational quantum solver, the operation of at least some of the quantum gates in a variational quantum network is parameterized, so that the measured output is a function of variational parameters that parameterize the variable actions of the (variational) quantum gates. The combined operation of the (at least partially parameterized) quantum gates may be referred to as a variational quantum network, as the operating principle may be similar to that of a neural network.

[0040] In some examples, a layer of quantum gates may act on qubits to link qubits in a variational quantum network. A layer of quantum gates may include the cumulative action of multiple coherent operations on the state of qubits in a qubit register. The cumulative action of coherent operations in a layer should generally act on all qubits in a qubit register involved in a computation, in other words, a layer of quantum gates should directly affect the state of all qubits in the qubit register. Each layer should include at least one multi-qubit gate and at least one variational quantum gate (which in principle could be the same gate). Those skilled in the art will appreciate that multiple quantum gates in a layer may be applied to qubits in parallel to shorten the sequence of coherent operations on the state of qubits in the layer. Multiple layers of quantum gates may then be applied to the qubits to form a variational quantum network parameterized by a variational parameter for each layer.

[0041] A layer may include quantum gates of the same type and may be applied sequentially to the qubit register. For example, each layer may feature the same architecture of quantum gates, but different elements of variational parameters may be applied to the variational gates of a layer. In other words, the layers may feature the same quantum gate architecture, but the action of the quantum gates on the qubits in each layer may differ based on the variational parameters.

[0042] In some embodiments, the layers of quantum gates have an identical arrangement of quantum gates in each layer, and the quantum gates in each layer specifically include a plurality of multi-qubit quantum gates that operate together on all qubits of the qubit register.

[0043] After the layer of quantum gates acts on the qubits, the qubits can be measured to obtain a characteristic result of the variational quantum network with respect to a known initial state. The result of the quantum mechanical calculation can be linked to a classical solution of the problem via the computational basis states of the qubits. The computational basis states may be orthogonal basis states of the Hilbert space spanned by the tensor product of the basis states of each qubit.

[0044] The initial variational parameters of the variational quantum gates can encode an initial (random) guess, and the result of the evaluation of the variational quantum network with the variational parameters can be measured to determine the corresponding solution. Based on that solution, a cost function for the QUBO problem may be evaluated classically to attribute a cost to the solution, in other words, a measure of how good the solution is is calculated. Typically, the variational parameters are then updated (iteratively) using a feedback loop implemented in a classical processing system such that the output approaches the optimal solution, making the variational quantum solver a hybrid quantum-classical algorithm.

[0045] For example, in QAOA, the quantum gate configurations of each of p layers of quantum gates are determined based on the structure of the QUBO problem and include the variable actions associated with the quantum gates. Specifically, each layer is transformed from an initial state |Ψ0〉 to an output state

number

[0046] By varying the variables β, γ using an appropriate (classical) optimizer algorithm, the measured output quantum state can be made to approach an output that encodes a solution to the problem. However, other quantum algorithms that can be tailored to the particular underlying quantum hardware may be utilized in the embodiments.

[0047] The trained graph neural network is preferably trained based on an optimal solution obtained by a variational quantum solver, and new QUBO problems are provided to the variational quantum solver if the performance metric predicted by the graph neural network meets a threshold indicative of predicted quantum dominance. Nonetheless, in some embodiments, the variational quantum solver to which the QUBO problem is provided may also be different from the variational quantum solver on which the graph neural network was trained, e.g., the trained graph neural network may indicate to some extent whether quantum dominance can be utilized for a given QUBO problem. In other embodiments, multiple graph neural networks are trained to predict performance metrics of different variational quantum solvers, such as to select a suitable variational quantum solver for a given QUBO problem.

[0048] The results of implementing this method can provide a performance metric that can indicate whether the application of a variational quantum solver is efficient relative to the application of a classical solver, which can make it possible to quickly identify problem examples that are suitable for solving using a variational quantum solver.

[0049] Furthermore, the inventors have also found that, in particular based on the QAOA algorithm, a graph neural network can be used to directly predict optimal variational parameters of a variational quantum network, which optimal variational parameters extremize the cost function or at least provide a solution close to the optimal solution that can be obtained using a variational quantum solver when iteratively updating the variational parameters towards the best solution.

[0050] In a preferred embodiment, the method further includes providing the QUBO graph problem to a second trained graph neural network as part of providing the QUBO graph problem to a variational quantum solver implemented in quantum hardware, the second trained graph neural network being trained to predict variational parameters for optimal solutions by the variational quantum solver for a plurality of QUBO graph problems, and receiving the predicted variational parameters from the second trained graph neural network at an output of the second trained graph neural network. The method further includes providing the predicted variational parameters and the problem to the variational quantum solver to determine candidate solutions to the QUBO problem using the variational quantum solver.

[0051] Thus, the candidate solution may be determined based on the variational parameters received from the output of the second trained graph neural network. The candidate solution thus obtained may be used as the final solution, or a variational quantum solver may be used to refine the candidate solution and extremalize the cost function. The inventors have found that a graph neural network trained to predict the variational parameters based on a given QUBO graph problem may not only provide a solution close to the optimal solution that can be obtained using a variational quantum solver, but may also generalize to some extent the problem of finding optimal variational parameters (variable actions) for QUBO problems that are more complex than those in the training set. Note that both parallel graph neural networks may be provided with a QUBO problem, and if the performance metric meets a threshold, the pre-computed output of the second graph neural network may be provided to the variational quantum solver. Compared to the prior art, where neural networks are usually trained to predict the best solution, this approach may be directly generalized to graphs with more nodes / variables than the problems in the training set.

[0052] According to a second aspect, the present invention relates to a computer-implemented method for solving a given quadratic unconstrained binary optimization (QUBO) problem. The method includes the steps of: encoding the QUBO problem into a corresponding QUBO graph problem, where each binary variable of the QUBO problem corresponds to a node of the QUBO graph problem, and an edge between two nodes encodes the coefficients of terms that include both binary variables corresponding to the two nodes; or receiving a QUBO graph problem in a QUBO graph representation; and providing the QUBO graph problem as an input to a trained graph neural network on a processing system. The method further includes receiving predicted variational parameters for solving the QUBO problem using a variational quantum solver from an output of the trained graph neural network for the QUBO graph problem provided at the input; and providing the predicted variational parameters for the QUBO problem to a variational quantum solver implemented on quantum hardware.

[0053] In a preferred embodiment, the variational quantum solver includes a variational quantum network for determining an output indicative of a solution to the QUBO problem, a quantum gate configuration of the variational quantum network being specific to the QUBO problem, variational parameters defining an output of the variational quantum network specific to the QUBO problem, and the variational quantum network being based, in particular, on a quantum approximate optimization algorithm (QAOA).

[0054] According to a third aspect, the present invention relates to a system for solving a given quadratic unconstrained binary optimization (QUBO) problem. The system includes a processing system that obtains a QUBO graph problem corresponding to the QUBO problem, where each binary variable in the QUBO problem corresponds to a node in the QUBO graph problem, and an edge between two nodes encodes a coefficient of a term that includes both binary variables corresponding to the two nodes, or receives a QUBO graph problem in a QUBO graph representation and provides the QUBO graph problem as an input to a trained graph neural network on the processing system. The processing system further receives predicted performance metrics for solving the QUBO problem using a variational quantum solver from an output of the trained graph neural network for the QUBO graph problem provided at the input, and provides the QUBO problem to a variational quantum solver implemented on quantum hardware based on the predicted performance metrics.

[0055] The system may implement the method according to the first aspect, the second aspect, or any combination of embodiments thereof.

[0056] In some embodiments, the system may provide the QUBO problem to a classical solver implemented in a classical processing system based on predicted performance metrics, e.g., as an alternative or in addition to providing the QUBO problem to a variational quantum solver implemented in quantum hardware.

[0057] In a preferred embodiment, obtaining the corresponding QUBO graph problem includes encoding, by a processing system, the QUBO problem into the corresponding QUBO graph problem.

[0058] In a preferred embodiment, the system comprises, inter alia, a machine-readable model executable on a processing system, and / or the system further comprises AI processing hardware implementing the trained graph neural network, the AI ​​processing hardware including, inter alia, a GPU, a neural processing unit, analog memory-based hardware, or neuromorphic hardware.

[0059] The machine-readable model, when run on suitable processing hardware, may implement the trained graph neural network. The processing system may include a single processing unit or may include multiple processing units that may be operatively connected. The processing units may include microcontrollers, ASICs, PLAs (CPLAs), FPGAs, or other processing units including processing units operating based on software, hardware, firmware, or combinations thereof, such as the AI ​​processing hardware mentioned above. The processing units may include integrated memory and / or communicate with external memory, and may further include interfaces for connecting to sensors, devices, equipment, integrated logic circuits, other controllers, etc., and the interfaces may send and receive signals, such as electrical signals, optical signals, radio signals, acoustic signals, etc.

[0060] In a preferred embodiment, the processing system further provides the QUBO graph problem to a second trained graph neural network, the second trained graph neural network being trained to predict variational parameters of a variational quantum solver for an optimal solution by the variational quantum solver for a plurality of QUBO graph problems, receives the predicted variational parameters from the second trained graph neural network at an output of the second trained graph neural network, and provides the predicted variational parameters and the problem to the variational quantum solver to determine a candidate solution to the QUBO problem using the variational quantum solver.

[0061] According to a fourth aspect, the present invention relates to a computer program comprising machine readable instructions which, when executed by a processing system, cause the processing system to perform a method according to any embodiment of the first or second aspect and / or to implement a system according to any embodiment of the third aspect.

[0062] The features and many advantages of the method and system of the present invention will be best understood from a detailed description of the preferred embodiment when read in conjunction with the accompanying drawings. [Brief description of the drawings]

[0063] [Figure 1] FIG. 1 is a schematic diagram of an example hybrid quantum computing system. [Diagram 2] FIG. 2 is a schematic diagram of an example of a variational quantum solver based on the QAOA algorithm. [Diagram 3] FIG. 3 illustrates a flowchart of a method for dynamically selecting processing hardware based on a given QUBO problem, according to one example. [Figure 4] Figure 4 shows a schematic flow chart illustrating the working principle of an exemplary convolutional graph neural network based on the spatial approach. [Diagram 5] FIG. 5 is a flowchart of a method for dynamically selecting variational parameters for solving a QUBO problem using a variational quantum network. [Figure 6] FIG. 6 illustrates an example of a flowchart of a method for solving a given QUBO problem. [Figure 7] FIG. 7 illustrates an example flowchart of a method for solving a given max-cut optimization problem. [Figure 8] FIG. 8 illustrates a flowchart of an exemplary method for constructing training sets for training two graph neural networks. [Figure 9] FIG. 9 illustrates results for predicting performance metrics and optimal parameters of a variational quantum solver implementing the QAOA algorithm, according to some examples. [Figure 10A] FIG. 10A illustrates an example graph containing nine nodes interconnected by equal weight edges. [Figure 10B] FIG. 10B illustrates an example graph containing nine nodes interconnected by edges of equal weight. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0064] 1 illustrates generally one example of a hybrid quantum computing system 10 for implementing and operating a variational quantum network. System 10 comprises a qubit register 12 including a plurality of qubits. A plurality of quantum gates 14 may act on the qubits of qubit register 12 to perform a computation, and the variable actions of the plurality of quantum gates 14 may be parameterized by variational parameters. The computation results may be measured by a measurement sensor 16 that projects the state of the qubits onto a computational basis state of hybrid quantum computing system 10. The results may be received by a controller 18.

[0065] The controller 18 may repeatedly execute a computation sequence. The computation sequence may include, for example, initializing the qubits in the qubit register 12 to a ground state of each qubit before each computation to form an initial state of the qubits of |++...+>. The controller 18 may then apply a plurality of quantum gates 14 to the qubits in the quantum register 12 to operate a coherent expansion of the qubits. First, the controller may generate a superposition state of all qubits, for example by applying a Hadamard gate to each of the qubits, and then apply a plurality of quantum gates 14, including a variational quantum gate with a variable action. Following the coherent expansion, the state of the qubits in the qubit register 12 may be measured with a sensor 16. The measurement sensor 16 may be a plurality of single qubit state detectors for measuring the state of each qubit after expansion by the plurality of quantum gates 14. By repeating the measurements, it may be possible to determine the probability of each measurement outcome, which may be used to find a solution to the QUBO problem. Based on the measurements, the controller 18 can classically calculate the “energy” / “cost” of the solution using a cost function for the QUBO problem being solved.

[0066] For example, the cost function may be a maximum cut problem on an arbitrary graph, i.e., the problem of finding a cut through a set of connected nodes such that the total weight of the edges between two disjoint sets is as large as possible. This problem can be mapped to several equivalent problems, such as chip layout optimization in VLSI circuit design, and thus finding the correct solution is of technological importance. A maximum cut problem is a problem that requires a cost function

number

number

[0067] Controller 18 may then conventionally repeat the computation sequence with variable actions adjusted based on the results, such as to gradually improve a solution to a QUBO-type problem associated with the measurement results. For example, controller 18 may repeat the computation sequence with adjusted operating parameters of the variational quantum gates to determine gradients or energy landscapes associated with the plurality of quantum gates 14 from the measurement results, and may update the variational parameters based on the estimated gradients to gradually tune the variational quantum network toward an improved solution.

[0068] 2 shows an example of a variational quantum solver 20 based on the QAOA algorithm, which includes a variational quantum network 22 in which a number of quantum gates 14 are implemented in a quantum device 23. The variational quantum network 22 is N qubit register 12 includes a plurality of qubits having a state of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 2

[0069] The qubits may be initially initialized to a basis state, e.g., |+>. The plurality of quantum gates 14 may include a plurality of Hadamard (H) gates or another initialization sequence (not shown) that operates on each qubit in the qubit register 12 after the qubits are initialized to prepare each qubit in a superposition state. The variational quantum network 22 may then include a plurality of p layers 24a-24c of quantum gates having equal structure, where the layers 24a-24c represent a plurality of quantum operations on the qubits in the qubit register 12 that are subsequently applied.

[0070] Each layer 24a-24c includes a number of variational qubit gates whose variable actions are parameterized by variational parameters β, γ. Specifically, in QAOA, the algorithm calculates the possible computational basis states

number

[0071] Then, a series of p unitary operations, i.e., layer 24c, are applied, each characterized by angles β and γ. After these transformations, the state is

number

[0072] In the QAOA algorithm, the first unitary is usually

number

number

number

[0073] The classical computing system 30 then iteratively calculates the cost <C Z > and then determining the updated parameter values ​​γ, β in parameter updating module 34 based on some optimization function.

[0074] However, the availability of quantum hardware is currently still limited, and the efficiency of using quantum hardware to solve QUBO problems may also depend on the particular problem instance. In particular, some QUBO problems may not benefit from quantum superiority when solved by the variational quantum solver 20, or the solution cost may deviate significantly from the optimal solution. Therefore, it would be advantageous to apply quantum hardware only to problems for which the variational quantum solver 20 is expected to provide a good solution.

[0075] 3 shows a flowchart of a method for dynamically selecting processing hardware based on a given QUBO problem, according to an example. The method includes the steps of encoding the QUBO problem into a corresponding QUBO graph problem, where each binary variable of the QUBO problem corresponds to a node of the QUBO graph problem, and an edge between two nodes encodes the coefficients of the terms that include both binary variables corresponding to the two nodes, or receiving a QUBO graph problem in a QUBO graph representation (S10), and providing the QUBO graph problem as an input to a trained graph neural network on a processing system (S12). The method further includes the steps of deriving a predicted performance metric for solving the QUBO problem using a variational quantum solver 20 from the output of the trained graph neural network for the QUBO graph problem provided as input (S14), and providing the QUBO problem to the variational quantum solver 20 implemented in the quantum hardware based on the predicted performance metric (S16).

[0076] For example, the graph neural network may use the expected value of the normalized cost function associated with the QUBO problem, or C, as a performance metric for a given QUBO problem provided in the input, e.g., based on the past performance of a variational quantum solver20 for similar QUBO problems. z and theoretical maximum value C max The quantum approximation ratio r defined as the expectation value of the ratio of

number

[0077] Graph neural networks may extract features from graphs without discarding spatial information and have been found by the inventors in experiments to be well suited to the task of performance metric prediction for variational quantum solvers.

[0078] Figure 4 shows a schematic flow chart illustrating the working principle of an exemplary convolutional graph neural network based on the spatial approach.

[0079] In the first step (2), for each node in the input graph (1), the features of all its neighboring nodes are aggregated. Some common aggregation methods include summing, averaging, selecting the minimum or maximum, or applying an attention mechanism to the feature vectors associated with the neighboring nodes. In the second step (3), the feature vector on each node containing the result of the first step (2) with the aggregated features of the neighboring nodes is transformed into a new vector using a trainable transformation, which may result in an updated graph (4) that contains the structural information extracted by the previous two steps (2, 3).

[0080] Steps (2, 3) may be applied to the updated graph (4), such as to form multiple aggregation and transformation layers, where each aggregation layer and each transformation layer may be applied iteratively to aggregate features of increasingly distant neighboring nodes.

[0081] The results of the repeated application of the aggregation and transformation steps may then be read out in a pooling / readout / smoothing layer (5). The graph-level readout (5) may convert the graph, now including the feature vectors associated with each node, into vectors. Furthermore, different approaches such as max pooling, min pooling, average pooling, and sum pooling may be used in embodiments. Jump connections from previous layers may be used so that features at the graph-level readout (5) may depend on some or all of the intermediate feature vectors of previous layers, e.g., on intermediate results of previous aggregation and transformation steps. However, in some embodiments, only the final feature vectors of the last transformation and aggregation steps may be used to perform the graph-level readout (5).

[0082] The resulting vector may contain information about the graph. At this point, several fully connected layers may be applied and an output may be obtained, which may be a predicted performance metric when solving the input problem (1) using a variational quantum solver 20, for example based on the QAOA algorithm.

[0083] Additionally or alternatively, a graph neural network may be trained to predict optimal variational parameters of a variational quantum solver 20 at output based on a QUBO graph provided at input.

[0084] 5 shows a flowchart of a method for dynamically selecting variational parameters for solving a QUBO problem. The method includes the steps of encoding a QUBO problem into a corresponding QUBO graph problem, where each binary variable of the QUBO problem corresponds to a node of the QUBO graph problem, and an edge between two nodes encodes the coefficients of terms that include both binary variables corresponding to the two nodes, or receiving a QUBO graph problem in a QUBO graph representation (S20) and providing the QUBO graph problem as an input to a trained graph neural network on a processing system (S22). The method further includes the steps of receiving predicted variational parameters for solving the QUBO problem using a variational quantum solver 20 from an output of the trained graph neural network for the QUBO graph problem provided as input (S24), and providing the predicted variational parameters for the QUBO problem to the variational quantum solver 20 implemented in quantum hardware (S26).

[0085] Therefore, a graph neural network may be used to predict variational parameters of a variational quantum network, such as the variational parameters γ, β of the QAOA algorithm implemented in the variational quantum solver 20.

[0086] As before, the graph neural network may be trained using past results of applying the variational quantum solver 20 to a set of QUBO (graph) problems and recording the variational parameters γ, β for the optimal solution. The graph neural network may then be trained to predict the variational parameters γ, β for a new given QUBO problem based on the given QUBO problem. For the QAOA algorithm, this may include predicting the two parameters γ, β per layer 24a-24c of the variational quantum network implementing the QAOA algorithm, and may be independent of the size of the problem graph 26.

[0087] Thus, a graph neural network may be used to predict, based on a given QUBO problem, whether the variational quantum solver 20 is likely to generate a “good” solution and parameters that are likely to result in a “good” solution, where a “good” solution has a cost that approaches the cost of the optimal solution to the QUBO problem, for example, when brute force attacking the QUBO problem using classical hardware.

[0088] FIG. 6 illustrates an example of a flowchart of a method for solving a given QUBO problem. In a first step (1), a combinatorial optimization problem is provided to a processing system, which may be at least partially implemented in classical hardware. The processing system may encode the combinatorial optimization problem into a graph representation for processing by a first graph neural network, if the combinatorial optimization problem is not already provided in a graph representation (2). If the input problem is not a QUBO graph representation, the input problem may be optionally converted to a QUBO representation and encoded by the processing system into a graph representation processable by the first graph neural network. For example, converting the problem to QUBO form may involve quadratizing a polynomial unconstrained binary optimization problem into its corresponding quadratic unconstrained binary optimization problem. The QUBO problem may then be encoded into a graph representation.

[0089] The first graph neural network may then process the graph representation of the combinatorial optimization problem to provide a performance metric that may indicate whether the variational quantum solver 20 is likely to provide a good solution to the combinatorial optimization problem. For example, the graph neural network may provide a normalized performance metric, such as a quantum approximation ratio, that may indicate how close the cost of an optimal solution by the variational quantum solver 20 is to the cost of a truly good solution to the combinatorial optimization problem.

[0090] Based on the performance metric, a suitable algorithm for solving the combinatorial optimization problem may be selected (3). If the performance metric indicates that a certain quantum algorithm implemented in the variational quantum solver 20 is expected to obtain a good solution, the processing system may select that quantum algorithm (4).

[0091] A second graph neural network may process the graph representation of the combinatorial optimization problem to predict the variational parameters γ, β of the quantum algorithm (5). The variational parameters γ, β may be provided to a variational quantum solver 20 which may generate candidate solutions at its output (6), which may be used as a solution to the combinatorial optimization problem (9) or may be used as part of a warm start procedure, and the predicted variational parameters γ, β may then be updated to further refine the solution towards an optimal solution (9) obtainable using the variational quantum solver 20.

[0092] If the first graph neural network predicts poor performance of the variational quantum solver 20, the processing system may select a classical algorithm for solving or approximating a solution to the combinatorial optimization problem (7), and the classical algorithm may run on classical hardware to provide a solution to the combinatorial optimization problem (9).

[0093] FIG. 7 illustrates an example of a flowchart of a method for solving a given max-cut optimization problem. The method begins by receiving a max-cut optimization problem (1), which may already be in a QUBO graph representation. If the input problem is not in a QUBO graph representation, the input problem may be optionally converted to a QUBO representation (2) and encoded by a processing system into a graph representation that can be processed by a first graph neural network (3). For example, each binary variable may correspond to a graph node. An edge weight between two nodes may be equal to the coefficient of a term in the objective function that includes the two variables that the nodes represent. If the edge weight is zero, the edge may be removed. A node weight may be equal to the coefficient of a term that includes only the variables represented by that node.

[0094] In the illustrated example, a trained graph isomorphic network is used as the graph neural network to predict (4) the quantum approximation ratio that could be obtained by using a quantum algorithm implemented in a variational quantum solver 20 for a problem graph 26. Based on this prediction, an appropriate quantum or classical algorithm can be selected.

[0095] If the predicted quantum approximation ratio of the variational quantum solver 20 is good (5), then the problem may be provided to the variational quantum solver 20. In the illustrated example, the variational quantum solver 20 implements QAOA as the quantum algorithm.

[0096] The method may further include the step of predicting (6) optimal variational parameters γ,β of the quantum algorithm using a second graph neural network. For example, the graph neural network may be trained on another problem graph 26 for which the optimal variational parameters γ,β are already available. The trained graph neural network may then be used to predict the optimal variational parameters γ,β for the new problem graph.

[0097] Finally, a solution to the max-cut problem can be calculated based on the results determined using the variational quantum solver 20. If the prediction quality is not good (8) or as part of a quality control mechanism, the problem can be solved using classical methods (9). Some examples of possible classical methods for finding or approximating the optimal solution are "qbsolv" or the "Goemans-Williamson Algorithm". Finally, based on the solution determined using the variational quantum solver 20 and / or using a classical solver, an assignment of a solution to the max-cut problem may be made (10).

[0098] FIG. 8 shows a flowchart of an exemplary method for constructing training sets for training two graph neural networks that may be used in the methods shown in FIGS. 3 and 5-7.

[0099] The method includes the step of receiving (1) a (random) problem example of a given combinatorial optimization problem, such as a random max-cut problem, which may be selected from a set of combinatorial optimization problems. The problem example may be converted, if necessary, into QUBO format using a suitable algorithm implemented in classical hardware (2), and a graph representation of the problem example may be obtained, for example by encoding the problem example (4).

[0100] In parallel, the example problem may be provided to a variational quantum solver 20 (3), which may obtain the optimal variational parameters γ, β (5) and an optimal solution to the example problem. Based on the optimal solution obtained by the variational quantum solver 20 and either the truly optimal solution or a classical approximation of the solution (obtained using a classical algorithm), a quantum approximation ratio may be calculated (6) as a performance metric for the obtained optimal solution.

[0101] The graphical representation of the QUBO problem example (4) and the optimal variational parameters γ, β (5) for the optimal solution obtained by the variational quantum solver 20 may be added to a first database (7). The graphical representation of the QUBO problem example (4) and the approximation ratio for the optimal solution obtained by the variational quantum solver 20 (6) may be added to a second database (8).

[0102] Steps (1)-(8) may be repeated multiple times to construct a first training set and a second training set in the first database and the second database, respectively, which may then be used to train a graph neural network to predict the optimal variational parameters γ, β and the approximation ratio as an output of the graph neural network when a QUBO graph problem 26 is provided as an input to the graph neural network.

[0103] Figures 9(a)-9(c) show the results obtained by the inventors for predicting the performance metrics and optimal parameters of a variational quantum solver 20 implementing the QAOA algorithm with some examples. Two graph neural networks were trained on a training dataset containing graphs with 2 to 8 nodes (about 12,000 graphs) mainly composed of graphs with 8 nodes (about 11,000 graphs). The validity of the results was confirmed with a test dataset with problem graphs with 9 nodes (about 260,000 graphs). Node and edge weights were equal in all graphs.

[0104] Results are presented for three different algorithms, where "p" denotes the depth of the QAOA circuit, i.e., the number of layers 24a-24c and the associated pair of variational parameters γ, β.

[0105] Figure 9(a) shows the training statistics of the graph isomorphic networks used to predict the quality of the solution (performance metric) when solving each problem using a variational quantum solver 20 implementing the QAOA algorithm. The root mean squared quantum approximation error is plotted against the number of epochs for 1000 epochs with a learning rate of 0.0001. As can be observed, in all three cases the error decreases rapidly within the first 200 epochs and then slows down.

[0106] Figure 9(b) shows the training statistics of the graph isomorphic networks used to predict the optimal variational parameters, γ,β, of the QAOA algorithm for solving each problem. The root-mean-square error of the optimal parameters is plotted against the number of epochs for 1000 epochs with a learning rate of 0.0001.

[0107] Figure 9(c) plots the average cost delta and its standard deviation over 1000 training epochs for the same graph isomorphism network as in Figure 9(b). The cost delta is defined as the difference between the cost obtained from the predicted parameters and the cost obtained from the optimal parameters for a particular graph in QAOA. This statistic is averaged over multiple training graphs, and the area showing the standard deviation is colored over all epochs. As can be observed, both the average cost delta and its standard deviation decrease over time. This can be understood as the graph isomorphism network gradually learning to predict the variational parameters γ, β that produce good quality solutions.

[0108] 10A and 10B show two example graphs containing nine nodes interconnected by edges of equal weight. The two example graphs are random graphs selected from a test dataset to which the method illustrated in the flowchart of FIG. 7 was applied. For both graphs, the quantum approximation ratios for solving the corresponding max-cut problems were determined by simulating the QAOA-based variational quantum solver 20 on classical hardware to determine the optimal solution obtainable by the variational quantum solver 20, and by solving the problem exactly using a classical algorithm and calculating the quantum approximation ratios therefrom ("Actual Performance"). Additionally, the optimal variational parameters for the optimal solution obtainable by the variational quantum solver 20 were recorded ("Actual Optimal Parameters").

[0109] In addition, for each of the graphs, we recorded the predicted outputs of two graph neural networks trained on a training dataset containing graphs with up to eight nodes to predict the quantum approximation ratio ("Predicted Performance") and the optimal variational parameters ("Predicted Optimal Parameters").

[0110] Tables 1 and 2 show the recorded results of the simulated variational quantum solver 20 and the predicted output of the graph neural network for the graphs shown in Figures 10A and 10b, respectively, for three different circuit depths p of the QAOA algorithm, and for additional reference, the cost functions using the simulated variational quantum solver 20 output for the predicted and actual optimal parameters are provided.

[0111] [Table 1]

[0112] [Table 2]

[0113] Based on the above results, it can be seen that the graph neural network can predict the approximation ratio for each QAOA problem when solved by the variational quantum solver 20. Furthermore, the second graph neural network can also accurately predict the optimal variational parameters γ, β, so that the cost of the solution using the predicted variational parameters γ, β is close to the cost using the actual optimal variational parameters γ, β.

[0114] Note also that the graph neural network was trained on a smaller graph than those in Figures 10A and 10B, showing that graph neural networks may be used to predict performance and optimal parameters for problems that are more complex than the training set, and may be used for problems that are difficult to find classical solutions to.

[0115] In the preceding discussion, the graph neural network was trained based on an approximation ratio with an exact solution. However, one skilled in the art will appreciate that for larger graphs, determining an exact solution can be computationally expensive. Thus, the performance metric may also be based on a ratio of the quantum solution to a classical solution, which may be an approximate solution for the respective QUBO problem, or a performance metric that may be independent of the classical solution, such as the normalized cost of the solution by the variational quantum solver 20, may be used.

[0116] Furthermore, it should be noted that since many problems can be converted to QUBO problems before applying the method, the method can also be applied to other problems that can be reformulated as or approximated by a corresponding QUBO problem.

[0117] The description of the preferred embodiment and the drawings serve only to illustrate the invention and its associated beneficial effects, and should not be understood as implying any limitation, the scope of the invention being to be determined solely by the appended claims. [Explanation of symbols]

[0118] 10. System 12 qubit register 14 Multiple quantum gates 16 Measurement Sensors 18 Controller 20 Variational Quantum Solvers 22 Variational Quantum Networks 23 Quantum Devices 24a-24c Quantum gate layer 26 Problem Graph 28 Candidate solutions 30 Classical Computing Systems 32 Cost Calculation Module 34 Parameter Update Module

Claims

1. 1. A computer-implemented method for selecting processing hardware based on a given quadratic unconstrained binary optimization (QUBO) problem, the method comprising: encoding the QUBO problem into a corresponding QUBO graph problem, where each binary variable in the QUBO problem corresponds to a node in the QUBO graph problem, and an edge between two nodes encodes a coefficient of a term that includes both binary variables corresponding to the two nodes; or receiving a QUBO graph problem in a QUBO graph representation; providing the QUBO graph problem as an input to a trained graph neural network on a processing system; deriving a predicted performance metric for solving the QUBO problem using a variational quantum solver (20) from the output of the trained graph neural network for the QUBO graph problem provided at the input; and providing the QUBO problem to the variational quantum solver (20) implemented on quantum hardware (23) or to a classical solver implemented on a classical processing system based on the predicted performance metric. method.

2. The trained graph neural network is trained to estimate a quantum approximation ratio when solving the QUBO problem using the variational quantum solver (20), or a performance metric based on the quantum approximation ratio. The method of claim 1.

3. The trained graph neural network is a convolutional graph neural network, in particular a spatial graph neural network. The method of claim 1.

4. The trained graph neural network includes a plurality of graph convolution layers, each of which includes, for each node of the QUBO graph problem, an aggregation step of aggregating feature vectors of adjacent nodes into an aggregate feature vector, and a transformation step in which the original feature vectors of the nodes and the aggregate feature vectors are transformed, for each node, into an updated feature vector according to a trainable transformation. The method of claim 1.

5. The trained graph neural network includes a pooling layer for pooling the output of at least one of the plurality of graph convolution layers, and a fully connected layer between the pooling layer and an output of the trained graph neural network. The method according to claim 4.

6. The trained graph neural network comprises: receiving a training set, the training set including a plurality of QUBO graph problems and a performance metric corresponding to each of the plurality of QUBO graph problems, the performance metric indicating a ratio of a quality metric of a candidate solution (28) determined using the variational quantum solver (20) for the QUBO graph problem to a similar quality metric of a classical solution to the QUBO graph problem, the quality metric being, in particular, a cost associated with the candidate solution (28); training a graph convolutional network on the training set, where the input of the graph convolutional network receives the plurality of QUBO graph problems, and trainable parameters of the graph convolutional network are iteratively updated such that the output of the graph convolutional network for a given QUBO graph problem approaches the performance metric. The method of claim 1.

7. The variational quantum solver (20) includes a variational quantum network (22) for determining an output indicative of a solution to the QUBO problem, a gate configuration of the variational quantum network (22) being selected based on the QUBO problem, the gate configuration being dependent on a cost function that attributes a cost to a solution to the QUBO problem, and an optimal solution being associated with a global extremum of the cost function. The method of claim 1.

8. The method further comprises, as part of the step of providing the QUBO graph problem to the variational quantum solver (20) implemented in quantum hardware (23), providing the QUBO graph problem to a second trained graph neural network, the second trained graph neural network being trained to predict variational parameters for optimal solutions by the variational quantum solver (20) for a plurality of QUBO graph problems; receiving predicted variational parameters from the second trained graph neural network at an output of the second trained graph neural network; and providing the predicted variational parameters and the problem to the variational quantum solver (20) to determine candidate solutions (28) of the QUBO problem using the variational quantum solver (20). The method of claim 1.

9. 1. A computer-implemented method for solving a given quadratic unconstrained binary optimization (QUBO) problem, the method comprising: encoding the QUBO problem into a corresponding QUBO graph problem, where each binary variable in the QUBO problem corresponds to a node in the QUBO graph problem, and an edge between two nodes encodes a coefficient of a term that includes both binary variables corresponding to the two nodes; or receiving a QUBO graph problem in a QUBO graph representation; providing the QUBO graph problem as an input to a trained graph neural network on a processing system; receiving predicted variational parameters for solving the QUBO problem using a variational quantum solver (20) from an output of the trained graph neural network for the QUBO graph problem provided at the input; providing the predicted variational parameters for the QUBO problem to the variational quantum solver (20) implemented in quantum hardware (23). method.

10. The variational quantum solver (20) includes a variational quantum network (22) for determining an output indicative of a solution to the QUBO problem, a quantum gate configuration of the variational quantum network (22) being specific to the QUBO problem, the predicted variational parameters defining the output of the variational quantum network (22) specific to the QUBO problem, and the variational quantum network (22) being based, inter alia, on a quantum approximate optimization algorithm (QAOA).

10. The method of claim 9.

11. A system (10) for solving a given quadratic unconstrained binary optimization (QUBO) problem, the system (10) comprising: A QUBO graph problem corresponding to the QUBO problem is obtained, and each bibliography of the QUBO problem is obtained. a binary variable corresponds to a node of the QUBO graph problem, and an edge between two nodes encodes a coefficient of a term including both binary variables corresponding to the two nodes; or, receiving a QUBO graph problem in a QUBO graph representation; providing the QUBO graph problem as an input to a trained graph neural network on a processing system; receiving a predicted performance metric for solving the QUBO problem using a variational quantum solver (20) from an output of the trained graph neural network for the QUBO graph problem provided at the input; and a processing system that provides the QUBO problem to the variational quantum solver (20) implemented in quantum hardware (23) or to a classical solver implemented in a classical processing system based on the predicted performance metric. System (10).

12. Obtaining the corresponding QUBO graph problem includes encoding, by the processing system, the QUBO problem into the corresponding QUBO graph problem. The system (10) of claim 11.

13. The system (10) in particular comprises a machine-readable model executable on the processing system, and / or the system (10) further comprises AI processing hardware implementing the trained graph neural network, the AI ​​processing hardware including in particular a GPU, a neural processing unit, analog memory-based hardware, or neuromorphic hardware. The system (10) of claim 11.

14. The processing system further comprises: providing the QUBO graph problem to a second trained graph neural network, the second trained graph neural network being trained to predict variational parameters of the variational quantum solver (20) for an optimal solution by the variational quantum solver (20) for a plurality of QUBO graph problems; receiving predicted variational parameters from the second trained graph neural network at an output of the second trained graph neural network; The predicted variational parameters and the problem are provided to the variational quantum solver (20) to determine a candidate solution (28) of the QUBO problem using the variational quantum solver (20). The system (10) of claim 11.

15. A computer program comprising machine readable instructions, which when executed by a processing system, cause the processing system to perform the method of any one of claims 1 to 10 and / or implement a system (10) of any one of claims 11 to 14. Computer program.

Images