Hybrid quantum computing architecture for solving quadratic unconstrained binary optimization problems

Abstract

The present disclosure relates to a method of driving a quantum computing network for determining an extremum of a cost function of a solution to a quadratic unconstrained binary optimization problem, the method comprising: initializing qubits; sequentially applying quantum gate layers to the qubits, wherein each layer comprises a multi-qubit quantum gate acting on a plurality of qubits and a plurality of variational quantum gates having two eigenvalues and a variable action on the qubits, wherein the variable actions of the variational quantum gates of the quantum gate layers form a set of variational parameters; and determining an output state of the quantum computing network to obtain a solution associated with the set of variational parameters, wherein each binary variable of the quadratic unconstrained binary optimization problem is associated with a computational ground state of the register qubits and the solution is obtained by evaluating probabilities of measuring the computational ground states corresponding to the binary variables, and wherein the solution is iteratively improved by determining an output state of altered variational parameters to evaluate partial derivatives with respect to a subset of the variational parameters, thereby determining a gradient of the cost function based on the output state of the altered variational parameters; and updating the variational parameters based on an update function of a moving average over the gradient of the cost function and a moving average over the square gradient of the cost function.

Description

TECHNICAL FIELD

[0001] The present invention is in the field of quantum computing. More precisely, the present invention relates to a quantum computing architecture for finding solutions to discrete optimization problems. BACKGROUND

[0002] Quantum computers provide a platform of controllable quantum mechanical systems whose states and interactions can be controlled to perform computations. The computation is realized through the deterministic evolution of the controllable quantum mechanical systems, and the result of the computation can be determined by measuring the state of the quantum mechanical systems.

[0003] Quantum computers typically encode information in so-called qubits, which act as quantum mechanical equivalents of classical bits. A qubit is a physical system whose quantum mechanical state can be (coherently) controlled and (substantially) maintained between two ground states, denoted |0> and |1>, during the computation time. As an example, a qubit can be implemented by encoding information in the spin state of an electron, e.g. in the “up” or “down” state of the electron, but can also be encoded in the polarization state of a photon, the state of a (superconducting) oscillator, the energy level of an atom, etc.

[0004] Control operations on these qubits are referred to as quantum gates. Quantum gates can coherently act on qubits to induce changes in the state of individual qubits (so-called single-qubit gates) and on multiple qubits (so-called multi-qubit gates) (e.g. entangling the states of multiple qubits) and any combination thereof. For example, a single-qubit gate can induce a rotation of the spin state of an electron by a selectable value (e.g. p / 2). Multi-qubit gates can coherently act on two or more qubits, such as performing a coherent CNOT operation on the states of two qubits. Multiple quantum gates can be applied in parallel or sequentially to the qubits of a quantum computer to perform a computation. Eventually, after applying a sequence of quantum gates, the states of the qubits can be repeatedly measured to determine the probability of each possible outcome of the computation.

[0005] To compute solutions to problems that are considered difficult to handle on classical computers, quantum computers can exploit special properties of quantum mechanical states, in particular superposition and entanglement of different quantum states, to find solutions in relatively few computation steps.

[0006] However, superposition / entangled states of quantum mechanical systems are inherently unstable (e.g. subject to decoherence) and control and measurement of these systems are subject to fidelity margins, such that state-of-the-art quantum computers are currently limited in the number of controllable quantum mechanical systems (qubits) and the number of control actions (quantum gates) that can be performed consecutively.

[0007] Therefore, to perform efficient computations, it is generally necessary to exploit quantum mechanical properties ingeniously within the technical limits of low numbers of qubits and short sequences of consecutive computation operations.

[0008] Google AI Quantum and Collaborators: “Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor”, preprint quant-ph / 2004.04197 on arxiv.org, shows an implementation of a quantum approximate optimization algorithm for discrete binary optimization problems like the MaxCut problem of a connected vertex graph. The quantum processor applies a sequence of quantum gate layers to a register of qubits and iteratively optimizes control parameters based on classical feedback from a quadratic fit of multiple evaluations of the computation.

[0009] Tan et al.: “Qubit-efficient encoding schemes for binary optimisation problems, preprint quant-ph / 2007.01774 on arxiv.org, teach encoding schemes for solving quadratic unconstrained binary optimization (QUBO) type problems using variational quantum algorithms, in which classical binary variables of the optimization problem are compressed into a computational ground state of a register of qubits, i.e. a state spanned by a tensor product of register states of qubits. Solutions of the QUBO problem can be explored by measuring the conditional probability of a certain state of an ancilla qubit and one of the computational ground states. The solutions are optimized using a classical optimizer, in which the COBYLA algorithm yields the best performance.

[0010] Schuld et al.: “Evaluating analytic gradients on quantum hardware”, Physical Review A, 99(3), teach evaluating gradients of a composite quantum gate by probing the results of adjusted sets of composite quantum gates to determine the partial derivatives of the composite quantum gate with respect to single-parameter gates in a series of unitary evolutions. SUMMARY

[0011] However, known systems and methods are limited by the hardware architecture and mainly apply to toy problems that can find a solution on a classical computer in a shorter or comparable time frame. For example, the quantum algorithm implemented by Google AI Quantum with collaborators is limited to problems on graphs of 23 vertices (or less) that match the hardware geometry. The algorithm proposed by Tan et al. can exponentially cut down the number of qubits needed to solve a QUBO problem, so that problems with exponentially higher number of vertices can be solved. However, the scaling of the algorithm cannot reliably find the optimal solution because it has a tendency to get stuck in local minima, which can be due to the use of an inadequate classical optimization algorithm.

[0012] In view of this state of the art, it is an object of the present invention to provide an efficient quantum algorithm for solving a quadratic unconstrained binary optimization problem in polynomial time, where it is possible to find a solution for problems that are hard to handle on a classical computer.

[0013] This object is solved by a method of driving a quantum computing network and a hybrid quantum computing system according to the independent claims. The dependent claims relate to preferred embodiments.

[0014] According to a first aspect, the invention relates to a method of driving a quantum computing network for determining an extremum of a cost function of a solution of a quadratic unconstrained binary optimization (QUBO) problem. The quantum computing network comprises a plurality of qubits and further comprises a plurality of layers of quantum gates acting on the qubits, the plurality of qubits comprising a plurality of register qubits and at least one ancillary qubit in a qubit register. The method comprises initializing the qubits and sequentially applying the layers of quantum gates to the qubits, wherein each layer comprises a multi-qubit quantum gate acting on a plurality of qubits and a plurality of variational quantum gates having two eigenvalues and a variable action on the qubits, wherein the variable actions of the variational quantum gates of the layers of quantum gates form a set of variational parameters . The method further comprises determining an output state of the quantum computing network to obtain a solution associated with the set of variational parameters , wherein each binary variable of the quadratic unconstrained binary optimization problem is associated with a computational ground state of the register qubits and the solution is obtained by evaluating probabilities of measuring the computational ground states corresponding to the binary variables. The solution is iteratively improved by sequentially applying the layers of quantum gates with altered variational parameters , the altered variational parameters comprising a subset of the variational parameters altered by an alteration amount; determining an output state of the altered variational parameters to evaluate probabilities of measuring the computational ground states corresponding to the binary variables; and iteratively improving the solution by sequentially applying the layers of quantum gates with altered variational parameters partial derivatives of a subset of the variational parameters an output state of the quantum computing network determines a gradient of the cost function; and the variational parameters are updated based on an update function of a moving average over the gradient of the cost function and a moving average over the squared gradient of the cost function

[0015] A method of driving a quantum computing network can determine a solution of a QUBO problem with a heuristic approach similar to neural network operations. Variational parameters An initial (random) guess can be encoded and an evaluation of the quantum computing network with variational parameters can be measured to determine a corresponding solution. Based on the solution, a cost function of the QUBO problem can be classically evaluated to give a cost of the solution, or in other words, a measure of how good the solution is. By iteratively improving the solution by updating the variational parameters in a way that can be similar to gradient descent of neural networks, the quantum computing network gradually approaches an optimized solution.

[0016] In contrast to the state of the art, the quantum computing network is optimized by determining partial derivatives of the quantum computing network with respect to the variational parameters based on a direct measurement of the quantum computing network. Based on the partial derivatives of the quantum computing network, a gradient of the cost function is determined by classical computation, which is referred to as measured gradient of the quantum computing network. However, without some additional strategies, gradient descent in quantum networks cannot provide stable performance compared to gradient methods in classical deep (neural) networks.

[0017] According to the present invention, the variational parameters are updated based on an update function of a moving average over the gradient of the cost function and a moving average over the squared gradient of the cost function This update function is referred to as update function based on adaptive moments. The inventors found that the update function based on adaptive moments enables gradient descent based on the measured gradient of the quantum computing network and significantly improves the gradient descent towards an optimal solution.

[0018] In fact, the inventors found that the update according to the update function based on adaptive moments can compete with and even outperform the methods used by Tan et al. using (gradient-free) classical optimizers. Surprisingly, the update function based on the measured gradient of the quantum computing network is also valid in case the solution and thus the problem is equally compressed to the computational ground state of the qubits, allowing for an exponential reduction of the number of qubits.

[0019] In other words, by combining the following: encoding the solution as a computational ground state of qubits, evaluating measurement gradients of a quantum computing network, and an update function based on an adaptive matrix, a beneficial quantum architecture is designed that remedies the shortcomings of the prior art, such that computations can be performed using a relatively low number of qubits while still efficiently finding solutions to complex problems in polynomial time. The method can thus define a hybrid computing architecture, where function evaluations and gradient estimations can be computed on quantum hardware, while the relationship between measurements and the problem can be determined on classical hardware.

[0020] The skilled person will understand that the term “quantum computing network” should not be understood as being limited to a linked (physical) network, but can refer to a plurality of quantum gates (organized in layers) sequentially and / or in parallel acting on qubits to link the states of the qubits via multi-qubit operations. In other words, the network can be established by a cascade of quantum gates acting on a register of qubits, and links can occur in the network due to multi-qubit gates entangling multiple qubits.

[0021] To evaluate the quantum computing network, the qubits can be initialized to an initial state, such as the ground state of each qubit. In some embodiments, after initializing the qubits to their ground state, the superposition state of each qubit in the register of qubits is ready, e.g., via applying a Hadamard gate.

[0022] The quantum gate layers can then act on the qubits to link the qubits in the quantum computing network, where the action of the variational quantum computing network is governed by variational parameters Parametrization. A quantum gate layer can comprise a cumulative action of multiple coherent operations on the states of the qubits in the qubit register. The cumulative action of the coherent operations in a layer should typically act on all qubits in the qubit register that participate in the computation, or in other words, the quantum gate layer should directly affect the state of all qubits in the qubit register. Each layer should comprise at least one multi-qubit gate and at least one variational quantum gate (which can in principle be the same gate). Preferably, both the multi-qubit gate and the variational gate of a layer directly act on the state of all qubits in the qubit register. At the same time, the layer can be temporally or structurally limited, e.g. a layer in a sequence of coherent operations can be defined by the shortest sequence of quantum gates that satisfies the following criteria: acts on most or all qubits in the qubit register used in the computation; and comprises at least one variational quantum gate, which is preferably a plurality of variational quantum gates substantially corresponding to the number of qubits in the qubit register or a multiple thereof. The skilled person will appreciate that multiple quantum gates in a layer can be applied to the qubits in parallel to shorten the sequence of coherent operations on the state of the qubits in the layer. Subsequent application of multiple quantum gate layers to the qubits can then form a quantum computing network.

[0023] In a preferred embodiment, the quantum gate layers comprise the same quantum gate arrangement in each layer, and wherein the quantum gates in each layer in particular comprise a plurality of multi-qubit quantum gates that together act on all qubits in the qubit register.

[0024] The layers can contain quantum gates of the same or different types and can be applied to the qubit register sequentially. For example, each layer can feature the same quantum gate architecture, while the variational parameters of the variational gates of the layers can be applied differently. In other words, the layers can feature the same quantum gate architecture, but the action of the quantum gates on the qubits in each layer can differ due to the variational parameters of the variational gates.

[0025] In a preferred embodiment, each quantum gate layer comprises a set of variational quantum gates that act on each qubit in the qubit register, wherein the set of variational quantum gates in particular is a set of variational single-qubit gates.

[0026] By applying a variational quantum gate to each qubit of the qubit register in each layer, the number of layers used to converge towards a solution can be reduced, such that the quantum computing architecture can be executed with a shorter sequence of quantum gates and is less sensitive to noise.

[0027] In a preferred embodiment, the number of variational quantum gates in each layer is substantially equal to the number of qubits in the qubit register.

[0028] The inventors found that by limiting the number and / or type of variational gates, the complexity of the cost function landscape can be constrained to achieve convergence towards an optimized solution. In some embodiments, a favorable trade-off can be found by providing a set of variational quantum gates acting on each qubit of a register of qubits in each layer of quantum gates, while the number of variational quantum gates in each layer is substantially equal to the number of qubits in the register of qubits.

[0029] After the quantum gates have acted on the qubits, the qubits can be measured to obtain a characteristic result of the quantum computing network with respect to a known initial state. The result of the quantum mechanical computation can be linked to a classical solution of the problem via the computational basis states of the qubits. The computational basis states can be orthonormal basis states of a Hilbert space spanned by the tensor product of the basis states of each qubit.

[0030] In a preferred embodiment, the computational basis state of the register qubits is a tensor product of the computational basis of a plurality of the register qubits, in particular a tensor product of the computational basis of all register qubits.

[0031] For example, if two qubits each have basis states |0> and |1>, the computational basis states can be |00>, |01>, |10>, and |11>, and each of the computational basis states can be associated with one classical binary variable. Thus, N c classical variables can be represented by the computational basis state of a register of qubits having a number N q = log(N c ) of register qubits.

[0032] In a preferred embodiment, the qubits of the quantum computing network are arranged in log(N c ) register qubits and a number N a of ancilla qubits for solving a quadratic unconstrained binary optimization problem having N c classical binary variables.

[0033] For example, the classical variables can be encoded as N q = log(N c ) register qubits and one ancilla qubit for a minimal encoding sufficient to solve the MaxCut problem on a complete graph, for example after evaluation of a quantum computing network having variational parameters After evaluation of the quantum computing network having variational parameters

[0034]

[0035] where the computational basis states |i> of the register qubits are associated with a classical variable and the amplitudes b i (or a i ) encode the state of the classical variable. For example, by measuring each computational basis state the classical solution of the QUBO problem corresponding to the variational parameters can be determined.

[0036] In some embodiments, the qubits of the quantum computing network can feature at least two ancilla qubits to increase the order of correlation between the captured classical variables and thus extend the types of problems solvable by the quantum computing architecture to different types of QUBO problems. Tan et al. (“Qubit-efficient encoding schemes for binary optimisation problems”, preprint on arxiv.org) have presented in Chapter IV “TWO-BODY CORRELATIONS” a general encoding scheme for encoding QUBO type problems with more than one ancilla qubit and the corresponding teachings with respect to the encoding are incorporated herein by reference.

[0037] In preferred embodiments, the solution is obtained by evaluating the conditional probability of measuring an ancilla state of the at least one ancilla qubit and the conditional probability of measuring one of the computational basis states of the register qubits corresponding to the binary variable.

[0038] The measurement of the ancilla qubits and the register qubits can project the complex quantum mechanical state of the register qubits onto the computational basis such that one of the computational basis states is measured as a result. Repeated measurements can find the (conditional) probability of each result of the computation of the variational parameters For example, for each quantum computing network (parameterized by the variational parameters , the times in the register of qubits can be measured to approximate the result state.

[0039] The cost function can give the cost of the solution (measurement result) obtained by the quantum computing network according to the cost function of the QUBO problem.

[0040] For example, the cost function can be the MaxCut problem on an arbitrary graph, i.e. the problem of finding a cut that runs through a connected set of nodes such that the total weight of the edges between the two separated sets is as large as possible. This problem can be mapped to several equivalent problems, like the optimization of a chip layout in VLSI circuit design, and thus, finding the correct solution is of technical importance. The MaxCut problem can be parameterized as a QUBO problem that minimizes the cost function:

[0041]

[0042] where d ij is the weight of the edge between the i-th node and the j-th node in the graph. The solution is a binary string of the nodes that shows the corresponding relation to one of the two sets Further, the elements of the corresponding QUBO matrix can be Q ij = 2d ij (i > j) and Q ii = -∑ j d ij . Thus, after determining the probability of the result state of the quantum computing network, the cost can be calculated according to the respective cost function.

[0043] Generally, the method according to the first aspect with one ancilla qubit can be able to solve a QUBO problem with a cost function of

[0044]

[0045] where is a quadratic function with the property that if then where is the integer rounding. However, by increasing the number of ancilla qubits, the method can be applied to solve more general QUBO problems with increased correlation between the variables.

[0046] The cost function can then be minimized by iteratively improving the solution according to the measured gradient of the quantum computing network. Specifically, the quantum mechanical network can be repeatedly evaluated to determine the partial derivatives of the quantum gate layers with respect to the variational parameters, and the gradients can be classically calculated from the measured partial derivatives. The variational parameters can then be updated with an update function based on adaptive moments. Since the update function based on adaptive moments depends on the moving average over the gradient of the cost function and the (element-wise) square of the moving average over the gradient of the cost function, the update of the variational parameters can be smoothed by the first and second moments of the gradient, enabling a descent towards the solution of the optimization.

[0047] In preferred embodiments, the update function is substantially proportional to a moving average over the gradient of the cost function, and substantially inversely proportional to the square root of a moving average over the squared gradient of the cost function, where the moving average over the gradient of the cost function and the moving average over the squared gradient of the cost function are in particular exponentially decaying moving averages.

[0048] For example, the update function can be substantially proportional to a normalized moving average over the gradient of the cost function, normalized by the absolute value of the moving average over the gradient of the cost function.

[0049] In preferred embodiments, the update function at iteration step t is mathematically equivalent to:

[0050]

[0051] where m t is proportional to a moving average over the gradient of the cost function, and v t is proportional to a moving average over the squared gradient of the cost function, a is a learning rate hyperparameter, and e is a small number relative to the expected magnitude of the update.

[0052] For example, e can be 10 -8 and a can be 0.01, so that e is 10 6 percent of the expected magnitude of the update.

[0053] In preferred embodiments, and where b1 and b2 are real values between 0 and 1, is the gradient determined at iteration step t based on the output state of the altered variational parameters and is the element-wise square of the gradient determined at iteration step t, while m t-1 and v t-1 are the previous values determined at time step t-1 of m t and v t and m0 and v0 are zero.

[0054] The quotients 1 - b1 t and 1 - b2 t can be understood as bias correction terms, to correct for the initialization bias of the initial values of m t and v t that are initialized to zero (i.e. at t = 0), so that m t and v tThe exponentially decaying moving average of the gradient / gradient square of the cost function, respectively, where the decay rates are given by β1 and β2. For example, β1 and β2 can be chosen as 0.9 and 0.999, respectively.

[0055] The inventors found that the update function based on the adaptive metric significantly improves the performance of the quantum computing network compared to other gradient descent algorithms. It is believed that the update function advantageously acts on the gradient component by using the exponentially moving average m t of the gradient to overcome noise in the quantum system, while it advantageously acts on the learning rate component by dividing the learning rate a by the exponentially moving average v t of the squared gradient to optimize the update magnitude in terms of the landscape of the cost function imposed on the variational quantum computing network.

[0056] The efficiency of the gradient descent can also depend on the accuracy of the gradient. While it is in principle possible to evaluate only the partial derivatives of the quantum computing network with respect to the variational parameters / parts of the variational gates, it can be advantageous to evaluate the partial derivatives of the quantum computing network individually for each of the variational parameters / parts of the variational gates for each step of optimizing the variational parameters .

[0057] In a preferred embodiment, the method comprises applying the quantum gate layer with the altered variational parameters twice sequentially for each variational gate, the altered variational parameters comprising a subset of the variational parameters altered by a symmetric alteration amount for each variational gate, to evaluate the partial derivative with respect to each variable action of the variational parameters for determining the gradient before updating the variational parameters .

[0058] The subset of the variational parameters may be a single variational parameter (i.e. the partial derivative can be determined with respect to each variational gate) or can be a plurality of variational parameters The symmetric alteration amount should depend on the eigenvalues of the variational quantum gate(s). Typically, the partial derivative can be estimated by altering the gate by an alteration amount where r is the eigenvalue of the variational quantum gate. Then, the partial derivative of the result f of applying the quantum computing network to the initial state of the quantum bit register with respect to the variational quantum gate with the variational parameters θ may be evaluated according to j . Thus, the partial derivative of the quantum computing network can be directly computed by evaluating the result of the same quantum computing network architecture that is used to determine the solution, such that the architecture of the quantum computing network can be simplified.

[0059] In a preferred embodiment, the two eigenvalues of the variational quantum gate are ± 1 / 2 and the variational change is ±π / 2.

[0060] For single qubit gates with eigenvalues ± 1 / 2, such as 1 / 2 {σ x , σ y , σ z} one qubit rotation generators, the variational change should be ±π / 2. Single qubit rotations are inherent to most implementations of quantum computers, have two eigenvalues, and are typically characterized by a higher fidelity than multi-qubit gates. Thus, in the case where the variational gate is a single qubit rotation, the partial derivatives can be determined with higher accuracy than in the case of a variational multi-qubit gate.

[0061] For each “gradient descent” step towards the optimal solution, the quantum computing network can be evaluated 2*T*2 Nq times, i.e. twice for each variational parameter T, and evaluated 2 Nq times to estimate the result of each evaluation in the computational basis of N q quantum bits. Since N c classical variables can be compressed to a number N q = log(N c ) quantum bits, the number of evaluations can scale only polynomially with the number of classical variables. Thus, an efficient quantum computing architecture for solving QUBO problems can be provided.

[0062] According to a second aspect, the invention relates to a method of driving a quantum computing network for determining an extremum of a cost function of a solution of a quadratic unconstrained binary optimization problem. The quantum computing network comprises a plurality of quantum bits and further comprises a plurality of layers of quantum gates acting on the quantum bits. The method comprises initializing the quantum bits and applying quantum gate layers sequentially to the quantum bits, wherein each layer comprises a multi-qubit quantum gate entangling a plurality of quantum bits and a plurality of variational quantum gates G having a variable action on the quantum bits, wherein the variable action θ j of the variational quantum gates of the quantum gate layer forms a set of variational parameters The method further comprises determining an output state of the quantum computing network to obtain an estimate of the cost function for the set of variational parameters associated with the set of solutions of a quadratic unconstrained binary optimization problem, wherein each binary variable of the quadratic unconstrained binary optimization problem is associated with a computational ground state of the register qubits, and the solution is obtained by evaluating the conditional probability of measuring the computational ground state corresponding to a binary variable. The solution is iteratively improved by applying a quantum gate layer and an additional quantum gate A k that is conditionally applied based on a state of an ancilla qubit, which satisfies the equation where K is a positive real value; and determining an output state to evaluate a partial derivative of the variable action θ of the quantum gate layer with respect to a variational parameter j Based on the partial derivative of the quantum gate layer, a gradient of a cost function is determined; and the variational parameter is updated based on an update function of a moving average over the gradient of the cost function and a moving average over the squared gradient of the cost function

[0063] In general, the variational gate does not need to have only two eigenvalues. Instead, the variational gate can also be characterized by more than two eigenvalues. Then, the partial derivative of the quantum computing network can still be obtained by evaluating the quantum computing network via adding an ancilla qubit and performing an adjusted quantum computation that is characterized by an additional quantum gate A k that conditionally acts on the register of qubits depending on the state of the ancilla qubit. The Hadamard gate can put the ancilla qubit into a superposition state, and the variational quantum gate G or the additional quantum gate A k can act on the qubits depending on the state of the ancilla. Then, the outcome of the quantum computation and the state of the ancilla can be measured to obtain the expectation values E0and E1for the state of the ancilla being |0> and 1>, respectively, where the probabilities p0and p1for each of the additional quantum gates A k are determined according to The partial derivative.

[0064] According to a third aspect, the invention relates to a hybrid quantum computing system for determining an extremum of a cost function of a solution of a quadratic unconstrained binary optimization problem. The system comprises a quantum computing network comprising a register of qubits comprising a plurality of qubits, a plurality of quantum gates, and a controller. The plurality of quantum gates selectively act on the qubits in the register of qubits and comprises a plurality of quantum gates that act on a plurality of qubits in the register of qubits and a plurality of variational quantum gates having a respective variable action on a qubit in the register of qubits, wherein the variable actions form a set of variational parameters The controller is configured to initialize the qubits in the register of qubits and to apply the quantum gates with the variational parameters The layer sequence is applied to the qubit register, where each layer includes a variational quantum gate for determining the output state. The controller is further configured to apply the modified variational parameters. The sequence, the altered variational parameters Including variational parameters of a certain change. A subset of; and determining the modified variational parameters. The associated output state and the modified variational parameters are determined. The output state is used to evaluate the variational parameters. Partial derivatives of subsets; and / or conditional application of quantum gate layers and additional quantum gates A based on the states of auxiliary qubits. k The additional quantum gate satisfies the equation Where K is a real positive value; and the output state is determined to evaluate the quantum gate layer with respect to variational parameters. variable action θ j The controller is further configured to: determine the gradient of the cost function based on the partial derivatives, wherein the cost function associates the cost with a solution encoded in the output state, wherein each binary variable of the quadratic unconstrained binary optimization problem is associated with the computational ground state of the qubit, and the solution is obtained by evaluating the conditional probability of measuring the computational ground state corresponding to the binary variable; and update the variational parameters based on an update function of the moving average on the gradient of the cost function and the moving average on the squared gradient of the cost function.

[0065] According to a fourth aspect, the present invention relates to a computer program or computer program product comprising machine-readable instructions, wherein when the computer program is executed by a processing unit, the machine-readable instructions cause the processing unit to implement the method described in accordance with the first aspect and / or the second aspect and / or implement and / or control the system described in accordance with the third aspect. Detailed Implementation

[0066] The features and numerous advantages of the method and hybrid quantum computing system according to the invention will be best understood from the detailed description of the preferred embodiments with reference to the accompanying drawings, in which:

[0067] Figure 1 An example of a hybrid quantum computing system is illustrated schematically;

[0068] Figure 2 An example of a quantum computing network 20 with multiple quantum gates is illustrated;

[0069] Figure 3 This diagram illustrates an example of a flowchart for methods used to obtain solutions to the QUBO problem;

[0070] Figure 4 Figure illustrates an example of a flowchart of a method for iteratively improving a solution to a computational problem using a quantum computing network;

[0071] Figure 5 Figure illustrates a flowchart of an iterative method for improving a solution to a computational problem using a quantum computing network;

[0072] Figure 6A 、 Figure 6B Figure illustrates two graphs of simulated performance of a hybrid quantum computing architecture;

[0073] Figure 7 Figure illustrates a graph of simulated performance of a hybrid quantum computing architecture with a quantum computing network; and

[0074] Figure 8A 、 Figure 8B Figure illustrates another example of a flowchart of a method for iteratively improving a solution, corresponding parts of a quantum computing network, and a quantum computing architecture.

[0075] Figure 1 An example of a hybrid quantum computing system 10 for implementing and driving a quantum computing network is schematically illustrated. The system 10 includes a qubit register 12 containing a plurality of qubits. A plurality of quantum gates 14 can act on the qubits in the qubit register 12 to perform a computation. The result of the computation can be measured by a measurement sensor 16, which projects the state of the qubits onto a computational ground state of the hybrid quantum computing system 10. The result can be received by a controller 18.

[0076] The controller 18 can be configured to repeat the sequence of computations. The sequence of computations can include initializing the qubits in the qubit register 12 prior to each computation, such as to the ground state of each qubit, for example, to form an initial state of the qubits |00...0>. The controller 18 can then apply the plurality of quantum gates 14 to the qubits in the qubit register 12 to drive coherent evolution of the qubits. Initially, the controller can produce a superposition of all the qubits, such as by applying a Hadamard gate to each qubit, and subsequently can apply the plurality of quantum gates 14 including variational quantum gates with variable action. After the coherent evolution, the state of the qubits in the qubit register 12 can be measured with the sensor 16. Based on the measurement result, the controller 18 can classically compute the "energy" / "cost" of the solution using a cost function of the QUBO problem to be solved.

[0077] The controller 18 can then repeat the sequence of calculations with adjusted variable actions based on the results, such as to iteratively improve a solution to a QUBO-type problem associated with the measurement results. In particular, the controller 18 can repeat the sequence of calculations with adjusted operational parameters of the variational quantum gates in order to determine gradients of the plurality of quantum gates 14 from the measured results, and can update the variable actions based on the estimated gradients in order to iteratively adjust the quantum computing network towards an improved solution.

[0078] The plurality of quantum gates 14 can be arranged in layers of similar or identical structure, and the controller 18 can subsequently apply the layers of quantum gates with their respective variational parameters. Preferably, each layer comprises a plurality of multi-qubit gates for entangling the states of a plurality or all of the qubits, and a variational quantum gate affecting the states of all qubits.

[0079] Figure 2 An example of a quantum computing network 20 with a plurality of quantum gates 14 is illustrated. The quantum computing network 20 comprises a register of qubits 12 comprising a plurality of qubits with states Ψ1- Ψ N The evolution of each qubit state is illustrated as a horizontal line extending from the register of qubits 12 towards the measurement sensor 16.

[0080] The qubits can initially be initialized to their ground state, e.g. |0>. The plurality of quantum gates 14 can comprise a plurality of Hadamard (H) gates 22 acting on each qubit in the register of qubits 12 after the qubits have been initialized to put each qubit in a superposition state. The quantum computing network 20 can then comprise a plurality (L) of layers of quantum gates 24a, 24b of identical structure, wherein each layer represents a plurality of quantum operations on the qubits in the register of qubits 12 that are subsequently applied.

[0081] Each layer 24a, 24b comprises a plurality of multi-qubit gates, e.g. CNOT gates (depicted as vertical lines and hollow circles on the respective horizontal line of the “control qubit”). Further, in each layer 24a, 24b, a variational single-qubit gate drives each qubit to rotate R i (θ) around the y-axis with a variable angle θ y (θ) around the y-axis with a variable angle θ N The set of variational parameters of the quantum computing network 20. Figure 2 The quantum computing network 20 depicted in the middle with L layers 24a, 24b and N qubits has L*N variable actions (angles) as variational parameters To the features, wherein each layer 24a, 24b comprises a plurality of two-qubit gates acting on all pairs of adjacent qubits in the register 12 of qubits 12, and the variable action can drive a variable single-qubit rotation of each qubit in each layer 24a, 24b.

[0082] The skilled person will appreciate that Figure 2 The arrangement of gates in FIG. 4 is for illustrative purposes only, and suitable geometries of the quantum computing network 20 can differ from the depicted representation. For example, in FIG. 5, the CNOT gates acting on pairs of adjacent qubits act on the qubits in (temporal) sequence. However, in preferred embodiments, multiple multi-qubit gates can be applied to the qubits in parallel, for example in two successive applications of two-qubit gates acting in parallel on an odd / even number of pairs of adjacent qubits. Figure 2

[0083] After applying the layers of quantum gates 24a, 24b to the qubits, the state of the qubits can be measured by the measurement sensors 16. The measurement sensors 16 can be a plurality of single-qubit state detectors for measuring the state of each qubit after evolution according to the plurality of quantum gates 14. Repeated measurements can allow the probability of each measurement outcome to be determined, and the results can be used to find a solution to the QUBO problem.

[0084] Figure 3 An example of a flowchart illustrating a method for obtaining a solution to a QUBO problem is shown in FIG. 6. The method comprises initializing qubits in a register 12 of qubits (S10), and applying layers of quantum gates 24a, 24b to the qubits in sequence, wherein each layer 24a, 24b comprises a plurality of multi-qubit quantum gates acting on a plurality of qubits and a plurality of variational quantum gates having a variable action on a set of variational parameters forming a variational parameter for the qubits (S12). The method further comprises determining an output state of the quantum computing network 20 to obtain a solution associated with the set of variational parameters

[0085] The variational parameters may be variational angles θ Figure 2 of single-qubit rotations R y (θ i ) as illustrated in FIG. 4, or variational parameters θ i of variational quantum gates as illustrated in FIG. 5. i ​​other variational parameters. The output state can then be in the computational basis of each qubit. For example, for two register qubits with ground states |0> and |1>, the computational basis states of the register qubits can be |00>, |01>, |10>, and |11>, and each of the computational basis states can be associated with a classical binary variable. Thus, when the qubit register comprises two register qubits, the method can be adapted to find a solution of a (discrete) binary optimization problem with 4 variables, and generally for a number N q of qubits to a problem with 2 Nq variables.

[0086] The relationship between the encoded variables and the solution can be obtained by measuring the conditional probability of a certain register state of the register qubits and the state of the at least one ancilla qubit by measurement. For example, for one ancilla qubit, and the minimal encoding of classical variables to the quantum register, the final state can be given by the equation

[0087]

[0088] where the first state in the sum is the state of the ancilla qubit, and the states |i> correspond to the computational basis states of the register qubits. Sampling over such states gives the components of the classical solution Such probabilities can be related to the cost function of the QUBO problem via the QUBO matrix Q via evaluating the following equation on a classical computer

[0089]

[0090] Thus, the measurement of the states of the qubits in the qubit register 12 after applying the layers of quantum gates 24a, 24b can be used to obtain a (random initial) solution of the QUBO problem.

[0091] The quantum computing network 20, i.e. the variational parameters that parameterize the action of the quantum computing network 20 on the qubits can then be optimized in a feedback loop with the goal to minimize the cost of the solution.

[0092] Figure 4 An example of a flowchart of a method for iteratively improving a solution of a computational problem using the quantum computing network 20 is illustrated. The method comprises sequentially applying the layers of quantum gates 24a, 24b with altered variational parameters , the altered variational parameters comprising a subset of the variational parameters that are altered by an alteration amount (S16); and determining an output state of the altered variational parameters to evaluate on the variational parameters The partial derivatives of a subset of (S18). The method further includes: based on the modified variational parameters. The output state determines the gradient of the cost function (S20); and the variational parameters are updated using an update function based on the moving average of the gradient of the cost function and the moving average of the squared gradient of the cost function. (S22).

[0093] Specifically, for a given set of two eigenvalues ​​± 1 / 2 (Based on the single-qubit rotation of the Pauli generator matrix, such as...) Figure 2 (Example) and variable action θ j The variational gate can have a change of π / 2. Then, based on... The result of the evolution of the qubit register 12 in the quantum computing network 20 is determined with respect to the variable action θ. j The partial derivatives of .

[0094] In order to improve the combination Figure 3 The cost function described is a weighted solution based on observable probabilities. The gradient of the cost function can be evaluated using the following equation.

[0095]

[0096] Where the observable probability of the cost function The partial derivatives are given by the following equation:

[0097]

[0098] The above equation contains a variable action θ j The partial derivatives of the variational gate can be obtained from the variational parameters with altered parameters. The measured expected value is determined, and the determination is based on

[0099]

[0100] as well as

[0101]

[0102] Based on variational parameters that have been symmetrically altered by π / 2 Measure the state of a certain register The expected value of |i>.

[0103] In other words, it is possible to base the quantum computing network 20 on the modified variational parameters. measurements of the cost function, where the partial derivatives of the cost function are based on measurements of the probability of measuring the computational ground state of the register qubits measurements of the probability of measuring the computational ground state of the register qubits and measurements of the conditional probability of measuring the ancilla state (e.g., |1>) of the at least one ancilla qubit and the computational ground state of the register qubits measurements of the probability of measuring the computational ground state of the register qubits and measurements of the conditional probability of measuring the ancilla state (e.g., |1>) of the at least one ancilla qubit and the computational ground state of the register qubits

[0104] The variational parameters can then be updated using the gradient towards the optimized solution However, because the measured gradients of the quantum computing network 20 determined in this way do not provide stable performance, additional optimization of the gradients can generally be needed, such as an update function based on adaptive moments.

[0105] The update function based on adaptive moments can update the variational parameters based on an update function of a moving average over the measured gradients of the cost function and a moving average over the squared measured gradients of the cost function In particular, the variational parameters can be updated according to the function

[0106]

[0107] where m t is proportional to the moving average over the gradients of the cost function, and v t is proportional to the moving average over the squared gradients of the cost function, a is a learning rate hyperparameter (e.g., 0.01), and e is a small number relative to the expected magnitude of the update (e.g., 10 -8 ).

[0108] The moving averages can be exponentially decaying and can be according to and are determined iteratively, where is the gradient determined at iteration step t based on the output state of the modified variational parameters , and is the element-wise square of the gradient determined at iteration step t, while m t-1 and v t-1 are the previous values of m t and v t determined at time step t-1, and m0and v0are zero.

[0109] One skilled in the art will appreciate that the coefficients 1 - β1 t and 1 - β2 t can be understood as bias correction terms to correct for m t and v tinitialized to zero (i.e. at t = 0) initialization bias, such that m t and v t may be exponentially decaying moving averages of the gradient / gradient square of the cost function, respectively, where the decay rates are given by β1and β2. For example, β1and β2may be chosen as 0.9 and 0.999, respectively.

[0110] The inventors have found that the adaptive matrix-based update function significantly improves network performance and is individually superior to both the simple moving average of the gradient and the adaptive learning rate algorithm.

[0111] In some embodiments, determining an updated variational parameter based on the update function can incorporate random elements, such as by randomly selecting a subset of the variational parameters to determine the partial derivatives and estimate the gradient based on the randomly selected subset of the variational parameters in a manner similar to stochastic gradient descent. Thus, the time for updating the variational parameters may be reduced as compared to determining the partial derivatives for all of the variational parameters .

[0112] In some embodiments, an additional penalty can be added to the cost function for cost function regularization, or the magnitude of the change in the variational parameters may be bounded to incorporate additional constraints into the solution or to reduce the complexity of finding an optimized solution.

[0113] By iteratively repeating the method illustrated in Figure 4 , the variational parameters should be optimized towards the optimized variational parameters , where the relevant measurement results (solutions) of the quantum bits in the quantum bit register 12 applied to the quantum computing network 20 are characterized by a lower (minimal) cost according to the cost function. Thus, by relying on the measurement gradient of the quantum computing network 20, a solution to the QUBO problem can be found with the quantum computing network 20 without the constraints of a classical update function.

[0114] Figure 5 illustrates a flowchart of an iterative method for improving a solution to a computational problem using a quantum computing network 20 similar to the quantum computing network 20 illustrated in Figure 2 , which can apply the methods according to Figure 3 and Figure 4 .

[0115] Initially, a random variational parameter The quantum computing network 20, which can run on a quantum computer, is evaluated. The resulting outcome can then be analyzed by a cost function evaluation module 28, which can run on a classical computer, to determine the cost associated with the (initial random) variational parameters The cost can be passed to a convergence / threshold evaluation module 30, which can run on the same classical computer, to compare the cost to a target cost C target or check if the cost / solution has converged based on previous iterations. If the cost is below a threshold or has converged, the method can output the classical variables (x) corresponding to the result of the evaluation of the quantum computing network 20 for the variational parameters to the quantum computing network 20.

[0116] If the cost has not converged or is above a threshold, a hybrid gradient evaluation 32 can be performed. The hybrid gradient evaluation 32 includes an evaluation 34 of the quantum computing network 20, which can run on a quantum computer, with each variable action θ being individually varied by a symmetric variation amount, and the result is passed to a partial derivative evaluation module 36, which can run on a classical computer, and computes the partial derivative of the cost function with respect to the respective variable action θ j The hybrid gradient evaluation then outputs the measured gradient of the quantum computing network 20 to an update module 38 to update the variational parameters j based on the measured gradient of the quantum computing network 20.

[0117] The quantum computing network 20 with the updated variational parameters can then be evaluated in a quantum computing network evaluation 40, which can run on a quantum computer, that determines the outcome of the quantum computing network 20 for the updated variational parameters (quantum mechanically). The outcome of the quantum computing network evaluation 40 can be passed to the cost function evaluation module 28 and the convergence / threshold evaluation module 30 to iteratively repeat the optimization process until the cost of the outcome converges or is below a predefined threshold.

[0118] Figure 6A 、 Figure 6B Two graphs illustrating the simulated performance of the hybrid quantum computing architecture described in connection with Figures 3 to 5 These graphs plot the normalized cost of the solution as a function of the number L of layers 24a, 24b of the quantum computing network 20, which is trained on the MNIST dataset with 60,000 training images and 10,000 test images. Figure 2The same examples shown in Fig. 1 are used for the MaxCut problem on "sun-graphs" with different numbers of nodes / vertices (as can be seen in the legend).

[0119] A sun-graph is a simple toy graph, where the first node is connected to all other nodes, while the other nodes are not connected among each other. The sun-graph has the obvious solutions x = [1, 0,..., 0] and x = [0, 1,..., 1 ], because the maximum cut isolates the first node and thus cuts through all edges, but the algorithm does not know the solution and has to start from a random guess.

[0120] Figure 6A Gradient descent is implemented based on the measured gradients of the quantum computing network 20, where no further optimization is applied, i.e. the update function described above is not applied, but the variational parameters are updated according to Each point of the curve is the result of T = 300 steps of gradient descent with a learning rate of a = 0.01 averaged over 20 random sun-graphs. The accuracy of the solution increases with the depth of the circuit, i.e. with the number L of layers 24a, 24b of the quantum computing network 20, however for larger problems the gradients can be unstable even for simple toy graph problems.

[0121] Figure 6B Gradient descent is implemented based on the measured gradients of the quantum computing network 20, but where the variational parameters are updated according to the update function based on the adaptive metric described in conjunction with Figure 4 Again, each curve is obtained with T = 300 steps of gradient descent with a learning rate of a = 0.01 averaged over 20 random sun-graphs. It can be noted that starting from L = 4 layers 24a, 24b, the cost function of all numbers of classical variables (nodes) n c reaches 0, which indicates that the relatively shallow quantum computing network 20 found the exact solution. It should be noted that for the problem with 8192 nodes, the quantum computing network 20 only comprises 14 qubits and four layers 24a, 24b of quantum gates, while optimizing 56 parameters.

[0122] Figure 7 Fig. illustrates the simulated performance of a hybrid quantum computing architecture with a quantum computing network 20 similar to the one used in Figure 6B Fig. 1. However, in Figure 7 Fig. 1, the quantum computing network 20 is applied to solve the MaxCut problem on random complete graphs, which have 256 nodes and edge weights randomly chosen from a continuous range of [0.01, 1 ]. The number of steps of gradient descent (i.e. the learning rate a) is varied, while the number of layers 24a, 24b of the quantum computing network 20 is fixed to L = 4. Figure 4 ​The energies of the solutions were plotted for the number of iterations of the illustrated method, and compared to the performance of the IBM ILOG CPLEX optimization software package as a classical solver.

[0123] On six Intel i9 CPUs with 64 GB RAM, the CPLEX optimization software package can accurately solve MaxCut on a 32-node graph in a few minutes, however, finding an exact solution can become challenging when the size of the graph is > 64 nodes.

[0124] For each graph, the quantum computing network depth L was fixed (number of layers L is given in the legend), and the results obtained by running the CPLEX optimization software package for 1 hour and 5 hours on six Intel i9 CPUs were compared in order to find an approximate solution. The quantum computing network (QCN) gradient descent was performed using the “Cirq” simulator on the same processor(s). The simulation ran for about 30 minutes to simulate the actions of a quantum computing network 20 with 10 layers 24a, 24b, and for about 3 hours to simulate the actions of a quantum computing network 20 with 20 layers 24a, 24b for 1000 steps. From Figure 7 It can be clearly seen that in principle, deeper quantum circuits provide more accurate solutions with fewer iterations, which illustrates the general advantage of deeper quantum computing networks 20 for finding more accurate solutions.

[0125] The skilled person will appreciate that a 400-step gradient descent to search for the MaxCut of a 256-node graph in a 20-layer QCN can be performed on a simulated quantum processor in 20 minutes, while CPLEX provides significantly worse accuracy on the same timescale. Therefore, the hybrid quantum computing architecture described above is superior to a classical solver when simulating a quantum computing network 20 on the same classical hardware, even without access to quantum hardware. Therefore, the hybrid quantum computing architecture is expected to provide further advantages when run on quantum hardware.

[0126] The skilled person will also appreciate that the quantum computing network 20 has been described with variational single-qubit gates for illustrative purposes. While single-qubit gates are typically characterized by a higher fidelity than multi-qubit gates, and while it can also be advantageous to limit the number of variational quantum gates in each layer 24a, 24b to improve the convergence of the solution, other types of variational quantum gates can in principle be used.

[0127] In a more general case, the partial derivative can still be determined based on the evaluation of the quantum computing network 20, by conditionally applying the variational quantum gate and the additional quantum gate A k based on the state of the ancillary qubit.

[0128] Figure 8A Another example of a flowchart of a method for iteratively refining a solution of a general variational quantum gate, the quantum computing network 20 is illustrated. The method comprises applying an additional quantum gate A k with a conditional application based on a state of a second ancilla qubit to the quantum gate layer 24a, 24b; and determining a partial derivative of an output state to evaluate a variable action θ of the quantum gate layer 24a, 24b with respect to a variational parameter j (S24). The method further comprises determining a gradient of a cost function based on the partial derivative of the quantum gate layer 24a, 24b (S26); and updating the variational parameter (S28) based on an update function of a moving average over the gradient of the cost function and a moving average over a squared gradient of the cost function.

[0129] The additional quantum gate A should satisfy the equation k where K is a natural number (e.g. 2). The additional quantum gate A may be applied after the respective variational gate has been conditionally applied with a state of an ancilla qubit, and a partial derivative of an output state to evaluate a variable action θ j of the quantum gate layer with respect to a variational parameter can be determined.

[0130] For example, as illustrated in Figure 8B , an ancilla qubit 42 in a state |0> can be prepared in a superposition state by a first Hadamard gate 22a. The state of a qubit 44 in the qubit register 12 can then be conditionally subjected to the action of a variational quantum gate 46 during evaluation of the quantum computing network 20, e.g. conditioned on the state of the ancilla qubit 42 being |0>. Subsequently, the resulting state of the qubit 44 in the qubit register 12 can be conditionally subjected to the action of an additional quantum gate 48, e.g. conditioned on the state of the ancilla qubit 42 being |1>. A second Hadamard gate 22b can then act on the state of the ancilla qubit 42 and the state of the ancilla qubit 42 can be measured by a measurement sensor 50.

[0131] The quantum computing result and the state of the ancilla 42 can be measured to obtain expectation values E0and E1, respectively, for the state of the ancilla qubit being |0> and |1>, where the probabilities p0and p1of each of the additional quantum gates A k in the respective additional quantum gate A can be determined according to

[0132] Thus, in some embodiments, the method can comprise a variational quantum gate 46 with more than two eigenvalues, while the partial derivative can be determined with respect to Figure 8A , Figure 8BThe iterative method illustrated in the middle is similar to the iterative method to optimize the solution.

[0133] The preferred embodiments and the description of the drawings are only illustrative of the application and the associated advantages, and should not be understood as implying any limitation. The scope of the application will only be determined by the appended claims.

[0134] List of reference signs

[0135] 10 system

[0136] 12 qubit register

[0137] 14 plurality of quantum gates

[0138] 16 measurement sensor

[0139] 18 controller

[0140] 20 quantum computing network

[0141] 22, 22a, 22b Hadamard gate

[0142] 24a, 24b layer

[0143] 26 initial quantum computing network evaluation

[0144] 28 cost function evaluation module

[0145] 30 convergence / threshold evaluation module

[0146] 32 hybrid gradient evaluation

[0147] 34 quantum computing network evaluation for altered variational parameters

[0148] 36 partial derivative evaluation module

[0149] 38 update module

[0150] 40 quantum computing network evaluation for updated variational parameters

[0151] 42 ancilla qubits

[0152] 44 qubit register state

[0153] 46 variational quantum gate

[0154] 48 additional quantum gate

[0155] 50 ancilla measurement device

Claims

1. A method of driving a quantum computing network for determining an extremum of a cost function of a solution of a quadratic unconstrained binary optimization problem, the quantum computing network comprising a plurality of qubits and further comprising a plurality of layers of quantum gates acting on the qubits, the plurality of qubits comprising a plurality of register qubits and at least one ancilla qubit in a qubit register, the method comprising: - initializing the qubits; - applying the quantum gate layers sequentially to the quantum bits, wherein each layer comprises a multi-qubit quantum gate acting on a plurality of quantum bits and a plurality of variational quantum gates having two eigenvalues and a variable action on the quantum bits, wherein the variable actions of the variational quantum gates of the quantum gate layers form a set of variational parameters ; and - determining an output state of the quantum computing network to obtain a solution associated with a set of the variational parameters where each binary variable of the quadratic unconstrained binary optimization problem is associated with a computational ground state of the register qubit and the solution is obtained by evaluating a probability of measuring a computational ground state corresponding to the binary variable. and wherein the solution is iteratively improved by: - applying for each variational gate sequentially a quantum gate with altered variational parameters of the quantum gate layer twice, the altered variational parameters comprising a subset of the variational parameters that are altered symmetrically altered amount for each variational gate ; - determining the changed variational parameter of the output state to evaluate the partial derivative with respect to each variable action of the variational parameter ​ - determining a gradient of the cost function based on the output state of the altered variational parameters ; and - updating the variational parameters based on an update function of a moving average over the gradient of the cost function and a moving average over the squared gradient of the cost function .

2. The method of claim 1, wherein, the quantum gate layers comprising in each layer the same arrangement of quantum gates, and wherein the quantum gates in each layer in particular comprise a plurality of multi-qubit quantum gates acting together on all qubits in the qubit register.

3. The method of claim 1, wherein, each layer of quantum gates comprises a set of variational qubit gates acting on each qubit in the qubit register, wherein the set of variational qubit gates in particular is a set of variational single-qubit gates.

4. The method of claim 1, wherein, the number of variational qubit gates in each layer is substantially equal to the number of qubits in the qubit register.

5. The method of claim 1, wherein, the computational ground state of the register qubits is a tensor product of computational bases of the plurality of register qubits, in particular a tensor product of computational bases of all register qubits.

6. The method of claim 1, wherein, The qubits of the quantum computing network are arranged in N number of register qubits and a number N a N number of ancilla qubits for solving a quadratic unconstrained binary optimization problem with N c number of classical binary variables.

7. The method of claim 1, wherein, the solution is obtained by evaluating a conditional probability of measuring an ancilla state of the at least one ancilla qubit and measuring one of the computational ground states of the register qubits corresponding to the binary variables.

8. The method of claim 1, wherein, the update function is substantially proportional to a moving average over the gradient of the cost function and substantially inversely proportional to a square root of a moving average over the squared gradient of the cost function, wherein the moving average over the gradient of the cost function and the moving average over the squared gradient of the cost function in particular are exponentially decaying moving averages.

9. The method of claim 1, wherein, the update function at iteration step t is mathematically equivalent to: wherein is proportional to a moving average over the gradient of the cost function, and is proportional to a moving average over the squared gradient of the cost function, is a learning rate hyperparameter, and is a small number relative to the expected magnitude of the update.

10. The method of claim 9, wherein, And wherein, and are real values between 0 and 1, is the gradient determined at iteration step t of the output state based on the altered variational parameters and is the element square of the gradient determined at iteration step t, while and are respectively and the previously determined values at time step t-1 of and are zero.

11. The method of claim 1, wherein, The two eigenvalues of the variational quantum gate are ± and the change is ± .

12. A hybrid quantum computing system for determining an extremum of a cost function of a solution of a quadratic unconstrained binary optimization problem, the system comprising: a quantum computing network, the quantum computing network comprising - a qubit register comprising a plurality of register qubits, - a plurality of quantum gates selectively acting on the qubits of the qubit register, the plurality of quantum gates comprising a plurality of multi-qubit gates acting on a plurality of qubits of the qubit register, wherein the quantum gates comprise a plurality of variational quantum gates having a respective variable action on a qubit of the qubit register, wherein the variable actions form a set of variational parameters ; and a controller configured to - initialize qubits in the qubit register; applying the quantum gate to the register of qubits in a sequence of layers having the variational parameters wherein each layer comprises a variational quantum gate for determining an output state; - Each variational gate application has modified variational parameters. The sequence is repeated twice, the modified variational parameters. The variational parameters, including the amount of symmetric change for each variational gate. A subset of; and determining the modified variational parameters. The associated output state, and determine the modified variational parameters. The output state is used to evaluate the variational parameters. Each variable action The partial derivatives; - determine a gradient of a cost function based on the partial derivatives, wherein the cost function associates a cost with a solution encoded in the output state, wherein each binary variable of the quadratic unconstrained binary optimization problem is associated with a computational ground state of the register qubits and the solution is obtained by evaluating a conditional probability of measuring a computational ground state corresponding to the binary variable; and - updating the variational parameters based on an update function of a moving average over the gradient of the cost function and a moving average over the squared gradient of the cost function .

13. A computer program comprising machine readable instructions which, when the computer program is executed by a processing unit, cause the processing unit to implement the method according to any one of claims 1 to 11 and / or to implement and / or control the system according to claim 12.

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