Hybrid quantum computing architecture for solving linear systems of binary relations

Abstract

The present disclosure relates to a hybrid quantum computing architecture for solving systems of linear binary relations. A method for driving a quantum computing network to find a solution to a computational problem comprising a system of linear binary relations is provided, the method comprising: initializing a computational qubit, applying a set of quantum gates to the computational qubit and measuring a result state, and determining a solution to the system of linear binary relations associated with a variational parameter based on a plurality of candidate solutions for individual relations of the system of linear binary relations encoded in the result state, wherein a state of the register qubit is associated with a selected one of the binary relations and a state of the ancilla qubit is associated with a candidate solution for the selected one of the binary relations.

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 of a system of linear binary relations. 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 basis states, hereinafter referred to as |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” state or the “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 individual qubit states (so-called single-qubit gates) as well as on multiple qubits (so-called multi-qubit gates), e.g. entangling the states of two or more qubits and any combination thereof. For example, a single-qubit gate can induce a rotation of the spin state of an electron with a selectable value (e.g. p / 2). Multi-qubit gates can coherently act on two or more qubits, such as 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 probabilities of each possible outcome of the computation.

[0005] To compute solutions of problems that are considered difficult to handle on 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 solutions with relatively few computation steps.

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

[0007] Therefore, within the technical limitations of low numbers of qubits and short sequences of consecutive computation operations, in order to perform effective computations, it is often necessary to exploit quantum mechanical properties cleverly.

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

[0009] Tan et al.: “Qubit-efficient encoding schemes for binary optimisation problems,” arXiv preprint quant-ph / 2007.01774 teaches encoding schemes for solving problems of the Quadratic Unconstrained Binary Optimization (QUBO) type using a variational quantum algorithm, in which classical binary variables of the optimization problem are compressed into computational basis states of a register of qubits, i.e., states spanned by tensor products of the register states of the qubits. Solutions to the QUBO problem can be explored by measuring the conditional probability of measuring a certain state of an ancilla qubit and one of the computational basis states. The solutions are optimized using a classical optimizer, and the best performance is obtained via the COBYLA algorithm.

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

[0011] However, known systems and methods are limited by the hardware architecture and are mainly applied to toy problems that can find a solution on a classical computer in a shorter or comparable time frame. For example, the quantum algorithms implemented by Google AI Quantum with collaborators are limited to problems on graphs of 23 vertices (or less) that match the hardware geometry. The algorithms proposed by Tan et al. can exponentially cut down the number of qubits needed to solve a QUBO problem, so that problems with exponentially growing number of vertices can be solved. However, the scaling of the algorithms cannot reliably find the optimal solution because of its tendency to get stuck in local minima, which can be due to the use of an inadequate classical optimization algorithm. Furthermore, these algorithms are particularly suitable for mathematical problems of the QUBO class. At the same time, efficient solutions to sets of linear relations in discrete variables can make improvements in several fields such as network security.

[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 sets of linear binary relations in polynomial time and possibly finding solutions to problems that are currently difficult to handle on classical computers.

[0013] This object is solved by the method of driving a quantum computing network and the 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 for finding a solution to a computational problem using a hybrid quantum computing system, the computational problem comprising a set of linear binary relations. The system comprises a quantum computing network, the quantum computing network comprising a set of quantum gates. The set of quantum gates comprises a plurality of variational quantum gates having a variable action on a plurality of computational qubits of the quantum computing network. The variable action forms a set of variational parameters . The method comprises initializing a plurality of computational qubits, including a plurality of ancilla qubits and a plurality of register qubits, applying the set of quantum gates to the computational qubits and measuring a result state. The method further comprises determining a solution to the set of linear binary relations associated with the variational parameters based on a plurality of candidate solutions for the individual relations of the set of linear binary relations encoded in the result state. In the result state, the state of the register qubits is associated with a selected one of the binary relations and the state of the ancilla qubits is associated with a candidate solution for the selected one of the binary relations. The method also comprises iteratively improving the solution to the set of linear binary relations by a series of steps. The iterative method steps comprise determining the set of quantum gates with respect to the variational parameters a plurality of partial derivatives of the quantum gate set, and determining a gradient of a cost function of the set of linear bivariate relations based on the plurality of partial derivatives of the quantum gate set. The cost function associates a cost with a candidate solution of the set of linear bivariate relations encoded in a result state of the computational qubits. The cost includes a single relation penalty associated with a mismatch between the two sides of each of the relations after repeating measurements of the computational qubits and an inconsistency penalty associated with a conflict of values of the same variable in different linear bivariate relations. The iterative method step further includes updating the variational parameters based on the gradient

[0015] A method of driving a quantum computing network can determine a solution of a set of linear bivariate relations with a heuristic method similar to neural network operations. Variational parameters An initial (random) guess can be encoded and an evaluation of the quantum computing network with the variational parameters can be measured to determine a corresponding solution, where a set of register qubits can act as a pointer to a certain relation in the set of linear bivariate relations and where a set of ancilla qubits can act as a pointer to one of a set of predefined solutions of the respective relation. Thus, the problem can be encoded with a small number of qubits, which can scale only logarithmically with the number of relations. The result can be measured several times to determine a solution associated with the variational parameters .

[0016] Based on the solution, a cost function associated with the set of linear bivariate relations can be evaluated classically to attribute a cost to the solution, or in other words, to compute a measure of how good the solution is. The cost function can attribute a relation penalty to each measurement result state of the computational qubits of the quantum computing network and can merge a plurality of result states with the set of linear bivariate relations by an inconsistency penalty to exploit entanglement of the computational qubits. The cost function can be minimized by iteratively improving the variational parameters by a measurement gradient of the cost function.

[0017] 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.

[0018] In particular, the quantum computing network is optimized by determining partial derivatives with respect to the variational parameters using the quantum computing network, i.e. based on a direct measurement of a result state of the quantum computing network. Then, a gradient of the cost function can be determined by a classical computation based on the measured partial derivatives and the cost function, which is referred to as a measurement gradient of the cost function in the following.

[0019] The inventors found that optimizing the solution according to the gradient of the cost function can effectively find an efficient solution of a linear binary relation group in the above described architecture. In other words, based on the described architecture / encoding, after iteratively improving an initial guess according to the gradient of the cost function, the inventors found that the result into which the quantum computing network converges describes an efficient solution of a linear binary relation group. Surprisingly, the inventors found that based on the measured gradient update of the cost function It is also valid in the case where the solution (and thus the problem) is compressed to the computational basis states of the qubits, allowing for an exponential reduction of the number of computational qubits of the quantum computing network.

[0020] Therefore, a quantum architecture is designed that remedies the shortcomings of the prior art, such that a relatively low number of qubits can be used to perform computations while still efficiently finding solutions of complex problems in polynomial time. The described method can thus define a hybrid computing architecture, where function evaluations as well as partial derivative estimates can be computed on real quantum hardware implementing the quantum computing network, while the relationship between the measurement values and the problem can be determined on classical hardware.

[0021] 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 (e.g. organized into layers) sequentially and / or in parallel acting on the 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 the computational qubits, and the network can arise due to multi-qubit gates entangling multiple qubits.

[0022] For evaluating 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, a superposition state of each computational qubit is prepared, e.g. via applying a Hadamard gate.

[0023] The set of quantum gates can then act on the computational qubits to link the computational qubits in the quantum computing network, where the action of the (variational) quantum computing network is determined by the variational parameters Parameterization. The set of quantum gates can comprise a plurality of quantum gate layers, which can each comprise a cumulative action of a plurality of coherent operations on the state of the computational qubits, and can be applied to the computational qubits sequentially, e.g. one layer after the other. The cumulative action of the coherent operations in a layer should typically act on all computational qubits participating in the computation, or in other words, one quantum gate layer should directly influence the state of all computational qubits. 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 act directly on the state of all computational qubits. At the same time, the layers can be restricted in time or structure, 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 computational qubits used in the computation; and comprises at least one variational quantum gate, the number of which preferably substantially corresponds to the number of computational qubits or a multiple thereof. The skilled person will understand 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 computational qubits in the layer. Subsequently applying a plurality of quantum gate layers to the computational qubits can form a quantum computing network.

[0024] After the quantum gate layers have acted on the computational qubits, the computational qubits can be measured to obtain a characteristic result of the quantum computing network with respect to the initial state.

[0025] 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 of the computational qubits, such that one of the computational basis states is measured as a result. The computational basis states can be the orthogonal basis states of the Hilbert space spanned by the tensor product of the basis states of each computational qubit in the measurement basis.

[0026] Repeating the measurement can find the (conditional) probability of each result of the computation of the variational parameters Parameterization. The set of quantum gates can comprise a plurality of quantum gate layers, which can each comprise a cumulative action of a plurality of coherent operations on the state of the computational qubits, and can be applied to the computational qubits sequentially, e.g. one layer after the other. The cumulative action of the coherent operations in a layer should typically act on all computational qubits participating in the computation, or in other words, one quantum gate layer should directly influence the state of all computational qubits. 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 act directly on the state of all computational qubits. At the same time, the layers can be restricted in time or structure, 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 computational qubits used in the computation; and comprises at least one variational quantum gate, the number of which preferably substantially corresponds to the number of computational qubits or a multiple thereof. The skilled person will understand 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 computational qubits in the layer. Subsequently applying a plurality of quantum gate layers to the computational qubits can form a quantum computing network. q times to approximate the result state.

[0027] Since the number of evaluations can grow exponentially with the number of qubits, it can be advantageous to encode the problem with a minimum number of qubits.

[0028] ​In a preferred embodiment, the computational qubits comprise log N register qubits and m ancilla qubits, wherein the set is a set of N linear binary relations, wherein each of the N linear binary relations has at most m different variables.

[0029] In other words, the problem can be encoded by encoding pointers to the N relations into states of N rq = log(N) register qubits, such that for a sparse matrix having at most m different variables in each relation, the number of evaluations can scale only polynomially with the number of relations. The solution to this relation can then be encoded into the ancilla qubits.

[0030] Preferably, the relations and the solutions are encoded into computational basis states of the computational qubits, e.g. into computational basis states of the register qubits and the ancilla qubits, respectively. The computational basis states of the register qubits and the ancilla qubits can then point to the corresponding relations and the m variables in this relation, respectively, to encode the problem of finding a solution to a set of (sparse) linear binary relations.

[0031] For example, if the two register qubits each have basis states |0> and |1> in a measurement basis, the computational basis states can be |00>, |01>, |10> and |11>, and each computational basis state can be associated with one of the relations (e.g. the first, second, third and fourth relation in the set of binary relations). In other words, the N relations can be represented by N r q = log(N) register qubits.

[0032] The solution to a relation having at most m binary variables can be represented by a computational basis state of m ancilla qubits. For example, if the relations each have two variables, the computational basis state |00> can point to a solution where both variables are zero, and so on, such that a predefined correspondence between measurement states and a set of associated solutions to the m variables of this equation can be constructed.

[0033] The skilled person will appreciate that the m variables in the N relations can be different. For example, in the above example, the first relation can depend on variables x1, x2, the second relation can depend on variables x2, x3, and so on, and depending on the state of the register qubits, the state of the ancilla qubits can describe a solution to different variables. Thus, a solution to a sparse set of four relations having up to five variables (where each relation has at most two variables) can be encoded into a quantum computing network having only four computational qubits.

[0034] In a preferred embodiment, the computational fundamental state of the auxiliary qubit is associated with all possible candidate solutions of m variables in a corresponding linear binary relation, which is associated with the state of the register qubit.

[0035] Technicians will understand that the methods used to solve linear binary relation sets can also solve linear discrete relation sets, because N bit A discrete number of bits can be represented by the values ​​of N binary values, for example, using N bit The computational fundamental state of an auxiliary qubit.

[0036] In a preferred embodiment, the method further includes determining the maximum number m of variables in each of the N linear binary relations.

[0037] The maximum number of variables, m, can be classically determined so that a given set of linear binary relations can be encoded into a minimum number of computational qubits.

[0038] Then, the “solution” (i.e., the quantum computing network’s guess of the solution, which is encoded into the measurement result state of the computed qubit) can be weighted according to the cost function that can be used to determine the feedback to the quantum computing network.

[0039] In a preferred embodiment, the single relation penalty is a continuous and non-negative function of the mismatch between the two sides of each relation in the relation, wherein the single relation penalty is in particular an even-degree polynomial of the mismatch, and preferably based on the square of the mismatch.

[0040] For example, in a situation with N var Variable x k The form is N rel A system of linear equations (where a) k，i It is related to variable x k The coefficients of the multiplication and b i When the cost of a single relation for each equation is constant, the cost of that relation can be equal to the square of the mismatch between the two sides of the equation. Proportional. In a system of linear inequalities (e.g., showing...), In the case of (form), the cost of a single relation for each equation may be related to the expression. Proportional, where f(y) has the following properties: if y < 0, then it is zero; and if y > 0, then it is equal to y, or vice versa. For example, when the inequality is... In other words, in the case of an inequality in a set of linear binary relations, the mismatch of the inequality can be defined by a function f(y) that takes into account the respective type of inequality, e.g., the inequality is in the form of y < 0, y > 0, y < 0, or y > 0. In some examples, depending on whether the inequality is in the form of y < 0 or y > 0, or in the form of y < 0 or y > 0, a constant can be added to the mismatch, enabling to penalize the case of y = 0 in the first form. Thus, the candidate solutions for the linear relations can be weighted according to a quadratic cost function that has the advantageous properties of being continuous, differentiable, and non-negative, facilitating finding the solution using a quantum computing network. For example, the individual relation cost can be or

[0041] In a preferred embodiment, the cost comprises a computational cost expression C

[0042] C = C rel + λC inc

[0043] where C rel is the individual relation cost, where C inc is the inconsistency penalty, and where λ is a parameter chosen in a way that, for a given set of linear binary relations, the term λC inc is on average larger than C rel .

[0044] The inventors found that the inconsistency penalty should be scaled to be larger than the sum of the individual relation costs of a random (bad) solution to prevent the quantum computing network from getting stuck in a local minimum with conflicting relations. In other words, the inconsistency penalty can enforce that the values of the variables in the solution of all relations are consistent, and therefore the parameter λ should be scaled to prevent the solutions of the same variable in different relations from conflicting.

[0045] The parameter λ can depend on the size of the set of binary linear relations, the number of variables in each relation, and / or the number of shared variables between relations, and be chosen based thereon. Preferably, the quantum computing network is simulated / evaluated for a plurality of different random variational parameters to determine the relative magnitudes of C inc and C rel , and λ is chosen based on the costs associated with the results such that λC inc is on average larger than C rel .

[0046] ​In some embodiments, the parameter λ is determined empirically, such as by setting λ to an initial constant value (e.g. 1) and increasing λ stepwise when the solution includes conflicting solutions for individual relations, or decreasing λ gradually when the solution does not converge towards the correct solution, until a consistent solution for the relations is found.

[0047] In preferred embodiments, the inconsistency penalty is based on a probability of measuring different outcome states for the same variable when determining the state of the same variable based on the state of the computational qubits in different states of the register qubits.

[0048] For example, the inconsistency penalty C inc may be based on the following expression

[0049]

[0050] where I j is a subset of linear binary relations involving the same variable x j , and f(i, k) is a predetermined mapping between the state of the ancillary qubits and the relevant solution for the same variable x j in relation i encoded in the state of the register qubits, for a number N var of variables, the relevant solution for the same variable x j is associated with index k in the relation i, and P(x j = 1 / 0) is the probability of measuring the outcome state of the kth ancillary qubit to correspond to the value 1 / 0 in the relation i.

[0051] The above expression can be compatible with optimization strategies according to the measured gradient of the cost function, for example, the gradient of the inconsistency penalty C inc may be determined based on the partial derivatives of the quantum computing network. Furthermore, for binary probabilities (i.e. the probability of obtaining 1 or 0 is close to 1 / 0 solution), the inconsistency penalty can be minimal, and thus convergence towards binary solutions can be improved.

[0052] However, the skilled person will recognize that in practice other forms of C inc such as a cost proportional to can be chosen that also provide a non-negative and continuous cost function to penalize conflicting solutions for the same variable in different relations (e.g. based on conditional probabilities of obtaining different outcomes).

[0053] In preferred embodiments, the set of relations is a set of N linear binary equations, each having at most m different variables, wherein the individual relation cost C rel is based on, inter alia, the following expression:

[0054]

[0055] wherein, is the probability of measuring the state of the ancilla qubit to be S given that the state of the register qubit is i, where the state i is associated with a particular linear equation i, where a i，f(i，k) is a coefficient multiplying the kth variable in the linear equation i, where the candidate solution s k ∈ {0, 1} is associated with the state S, and wherein b i is a constant value of the linear equation i.

[0056] For example, for a linear equation with two variables, a solution of the linear equation can be encoded into 2 2 computational basis states |00>, |01>, |10>, and |11> of the ancilla qubit. Continuing the example, the state S can be expressed in formula as i.e., S ∈ {0, 1, 2, 3}, and can thus act as a pointer from the four computational basis states to the solution of the relation, where the kth variable has the state s k .

[0057] Then, the mismatch of the solution on both sides of the equation determined from the state of the ancilla qubit can be squared and multiplied with the probability of measuring the state S to determine the single relation cost C rel .

[0058] In the case of a linear binary inequality, the single relation cost C rel can be based on a similar expression

[0059]

[0060] where f(y) has the property that it is zero if the solution satisfies the inequality and equal to y if the solution does not satisfy the inequality.

[0061] The skilled person will appreciate that a set of linear binary relations can include both equations and inequalities, and the respective associated cost functions can be applied accordingly.

[0062] The skilled person will appreciate that a computational problem can include additional constraints that can be encoded as additional costs.

[0063] For example, for the knapsack problem, the problem can include constraints such as maximizing a linear function of the variables x k , where c k are coefficients, and N var is the number of variables in the set of linear binary relations or a subset thereof. In the method, the cost The additional constraint is encoded, and the cost can be added to the aforementioned cost C to incorporate the constraint into the feedback, where f(y) can be an odd function, such as y = x. Therefore, this method can also be used to find solutions to computational knapsack problems.

[0064] The cost function should typically be expressible as the probability of measuring a specific result for a computational qubit, thus enabling the calculation of the cost function based on the variational parameters of the quantum computing network. The gradient of the cost function is determined by the partial derivatives.

[0065] Quantum computing networks with regard to variational parameters The partial derivatives can be determined based on a direct measurement of the results of applying a quantum computing network to compute qubits, or in other words, the partial derivatives can be determined using a quantum computing network.

[0066] In a preferred embodiment, the quantum computing network is used to determine the set of quantum gates with respect to the variational parameters. The partial derivatives include:

[0067] Application of variational parameters with shift The set of quantum gates, the variational parameters of the shift The variational parameter includes the parameter shifted by a certain amount. A subset of, and variational parameters for determining the shift. The associated result state is used to evaluate the variational parameters. Partial derivatives of subsets of; and / or

[0068] Based on the state of the derived auxiliary qubit, the set of quantum gates and the additional quantum gate A are conditionally applied. k The additional quantum gate satisfies the equation Where K is a positive real value, and the resulting state is determined to evaluate the set of quantum gates with respect to the variational parameters. variable action θ j The partial derivatives of .

[0069] When a variational gate has two eigenvalues, it can be based on the shifted variational parameters. The result state is based on To evaluate the results f of applying a quantum computing network to compute the initial state of a qubit with respect to a variational parameter θ. j The partial derivatives of the variable quantum gate, where the shift quantity Where r is the eigenvalue of the variational quantum gate. Variational parameters A subset of can be a single variational parameter, i.e., the partial derivative can be determined with respect to each variational gate, or it can be multiple variational parameters. Thus, the partial derivatives of the quantum computing network can be directly computed by evaluating the results of the same quantum computing network architecture that was used to determine the solution, such that the architecture of the quantum computing network can be simplified.

[0070] In a preferred embodiment, the method comprises applying the quantum gate layer twice in sequence for each variational gate with shifted variational parameters comprising a subset of the variational parameters shifted by a symmetric shift amount for each variational gate to evaluate the partial derivative with respect to each variable action of the variational parameters before updating the variational parameters .

[0071] In a preferred embodiment, the two eigenvalues of the variational quantum gate are ±1 / 2 and the shift amount is ±π / 2.

[0072] For single qubit gates with eigenvalues ±1 / 2, such as one qubit rotation generators in 1 / 2{σ x , σ y , σ z}, the shift amount 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 case the variational gate is a single qubit rotation, the partial derivatives can be determined with higher accuracy than in case of a variational multi-qubit gate.

[0073] However, in general, the variational gate does not need to have only two eigenvalues. Rather, the variational gate can also be characterized by more than two eigenvalues. The partial derivatives of the quantum computing network can then still be obtained by evaluating the quantum computing network via adding an ancilla qubit and performing an adjusted quantum computation that is characterized by additional quantum gates A k acting on the computational qubits conditionally 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 gates A k can act on the qubits depending on the state of the ancilla qubit. The result of the quantum computation and the state of the ancilla qubit can then be measured to obtain the expectation values E0and E1for the state of the ancilla qubit being |0> and |1>, respectively, with probabilities p0and p1for each of the additional quantum gates A k The partial derivatives can then be determined from .

[0074] The efficiency of the gradient descent can depend on the accuracy of the gradient. While it is in principle possible to evaluate the quantum computing network only with respect to the variational parameters​ the partial derivative of a part of the variational gate, but thus for each step of optimizing the variational parameters the partial derivative of the cost function with respect to each variational parameter It can be advantageous to evaluate the partial derivative of the cost function with respect to each variational parameter of the quantum computing network separately.

[0075] For example, for each “gradient descent” step towards the optimal solution, the quantum computing network can be evaluated twice, i.e. twice per variational parameter T, and evaluated times to estimate the result of each evaluation in the computational basis of N q quantum bits.

[0076] In a preferred embodiment, the variational parameters

[0077] the variational parameters 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 will be referred to as “adaptive moment-based update function” hereinafter.

[0078] Since the adaptive moment-based update function depends on the moving average of the gradient of the cost function and the (element-wise) square of the moving average of the gradient of the cost function, the update of the variational parameters can be smoothed by the first and second moment of the gradient, enabling a descent towards the optimal solution. The inventors found that the update according to the adaptive moment-based update function outperforms methods based on (non-gradient) classical optimizers, such as the method used by Tan et al.

[0079] Preferably, the update function is substantially proportional to the moving average of the gradient of the cost function and substantially inversely proportional to the square root of the moving average of the squared gradient of the cost function, and the moving average of the gradient of the cost function and the moving average of the squared gradient of the cost function are most preferably exponentially decaying moving averages.

[0080] In a preferred embodiment, the update function at iteration step t is mathematically equivalent to:

[0081]

[0082] where m t is proportional to the moving average of the gradient of the cost function and v twhere is a small number relative to the expected magnitude of the update, and a is a learning rate hyperparameter, proportional to the moving average over the squared gradients of the cost function.

[0083] For example, e can be 10 -8 and a can be 0.01, so that e is 10 6 times smaller than the expected magnitude of the update.

[0084] Preferably, and where b1 and b2 are real values between 0 and 1, is the result state of the variational parameters based on the shift is the gradient determined at iteration step t, 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 m0 and v0 are zero.

[0085] The quotients 1 - b1 t and 1 - b2 t can be understood as bias correction terms, for correcting the initialization bias of the initial values of m t and v t which are initialized to zero (i.e., at t = 0), so that m t and v t can be respectively an exponentially decaying moving average of the gradient / gradient square of the cost function, with a decay rate given by b1 and b2. For example, b1 and b2 can be chosen as 0.9 and 0.999, respectively.

[0086] The inventors found that the update function based on the adaptive momentum can significantly improve the performance of quantum computing networks compared to other gradient descent algorithms. It is believed that the update function advantageously acts on the gradient component by using an exponentially moving average m t of the gradient to overcome the noise in the quantum system, while it advantageously acts on the learning rate component by dividing the learning rate a by an 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.

[0087] In a preferred embodiment, the set of quantum gates comprises a plurality of layers of quantum gates applied in succession, wherein the plurality of layers comprises the same arrangement of quantum gates in each layer.

[0088] 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 comprise in particular a plurality of multi-qubit quantum gates acting on all the computational qubits together.

[0089] The layers can comprise the same or different types of quantum gates and can be applied sequentially to the computational qubits. For example, each layer can feature the same quantum gate architecture, while different elements of the variational parameters can be applied to the variational gates of the layers. 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 .

[0090] In a preferred embodiment, each quantum gate layer comprises a set of variational quantum gates acting on each computational qubit, wherein the set of variational quantum gates is in particular a set of variational single-qubit gates.

[0091] By applying a variational quantum gate to each computational qubit in each layer, the number of layers used for convergence 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.

[0092] In a preferred embodiment, the number of variational quantum gates in each layer is substantially equal to the number of computational qubits.

[0093] The inventors found that by limiting the number and / or type of variational gates, the complexity of the cost function landscape can be avoided and can be constrained for convergence towards an optimized solution. In some embodiments, an advantageous trade-off can be found by providing a set of variational quantum gates acting on each computational qubit in each quantum gate layer, while the number of variational quantum gates in each layer is substantially equal to the number of computational qubits.

[0094] According to a second aspect, the invention relates to a hybrid quantum computing system for finding a solution of a system of linear binary relations. The system comprises a quantum computing network comprising a plurality of computational qubits, the plurality of computational qubits comprising a plurality of register qubits and ancilla qubits, and a set of quantum gates selectively acting on the computational qubits, the set of quantum gates comprising a plurality of multi-gates acting on the plurality of computational qubits, wherein the quantum gates comprise a plurality of variational quantum gates having a respective variable action on the computational qubits, wherein the variable action forms a set of variational parameters . The system further comprises a control system configured to initialize the computational qubits, and to apply the set of quantum gates to the computational qubits with the set of variational parameters computational qubits to determine a result state. The control system is further configured to determine, using the quantum computing network, partial derivatives of the set of quantum gates with respect to at least one variational parameter in the set of linear binary relations, and determine, based on the partial derivatives, a gradient of a cost function that associates a cost with a solution encoded in the result state. The states of the register qubits are associated with a selected one of the binary relations, and the states of the ancilla qubits are associated with candidate solutions of the selected one of the binary relations. The cost comprises a single relation penalty associated with a mismatch between the two sides of each of the relations after repeatedly measuring the computational qubits and an inconsistency penalty associated with a conflict of values of the same variable in different linear binary relations. The system is then configured to update the variational parameters based on the gradient

[0095] According to a third aspect, the application relates to a computer program or computer program product 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 the first and / or second aspect and / or to implement the system according to the third aspect, e.g. by sending control instructions to a control system of a quantum computer and / or a dedicated quantum computing network and / or by processing measurement results received from a quantum computer and / or a dedicated quantum computing network. BRIEF DESCRIPTION OF DRAWINGS

[0096] The features and numerous advantages of the method and hybrid quantum computing system according to the application will be best understood from the following detailed description of preferred embodiments, with reference to the drawings, in which:

[0097] Figure 1 schematically illustrates an example of a hybrid quantum computing system;

[0098] Figure 2 illustrates an example of a quantum computing network 20 with a plurality of quantum gates;

[0099] Figure 3 illustrates an example of a flowchart of a method of obtaining a solution of a set of linear binary relations;

[0100] Figure 4 illustrates an example of a flowchart of a method for iteratively improving a solution of a computational problem using a quantum computing network;

[0101] Figure 5 illustrates a flowchart of an iterative method for improving a solution of a computational problem using a quantum computing network;

[0102] Figures 6A to 6Ca simulation performance plot of a hybrid quantum computing architecture for solving a system of linear binary equations; and

[0103] Figure 7A 、 Figure 7B Fig. 6 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. DETAILED DESCRIPTION

[0104] 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 comprises a qubit register 12 comprising 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 basis state of the hybrid quantum computing system 10. The result can be received by a control system 18.

[0105] The control system 18 can be configured to repeatedly perform a sequence of computations. The sequence of computations can comprise initializing the qubits in the qubit register 12 before each computation, e.g. to the ground state of each qubit, e.g. to form an initial state |00...0> of the qubits. The control system 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 control system 18 can produce a superposition of all qubits, e.g. by applying a Hadamard gate to each qubit, and subsequently can apply the plurality of quantum gates 14 including variational quantum gates with variable actions. 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 control system 18 can classically compute the “energy” / “cost” of the solution using a cost function of the problem to be solved.

[0106] The control system 18 can then repeat the sequence of computations with adjusted variable actions based on the result in order to iteratively improve the solution of the system of linear binary equations associated with the measurement result. In particular, the control system 18 can repeat the sequence of computations by adjusting the operational parameters of the variational quantum gates in order to determine the gradient of the plurality of quantum gates 14 from the measured result, and can update the variable actions based on the estimated gradient in order to iteratively adjust the quantum computing network towards an improved solution.

[0107] The control system can comprise a single control unit, or can comprise multiple control units that can be functionally connected. The control unit can comprise a microcontroller, an ASIC, a PLA (CPLA), a FPGA, a quantum processing unit, or other control device (including a control device that operates based on software, hardware, firmware, or a combination thereof). The control device can comprise an integrated memory, or communicate with an external memory, or both, and can further comprise an interface for connecting to sensors, devices, apparatuses, integrated logic circuits, other control systems, etc., which can be configured to receive or transmit signals such as electrical signals, optical signals, wireless signals, acoustic signals, etc.

[0108] The multiple quantum gates 14 can be arranged in layers of similar or identical structure, and the control system 18 can subsequently apply layers of quantum gates 14 with their respective variational parameters. Preferably, each layer comprises multiple or all of the multi-qubit gates for entangling the states of the qubits, and the variational qubit gates that affect the states of all qubits.

[0109] Figure 2 An example of a quantum computing network 20 with multiple quantum gates 14 is illustrated. The quantum computing network 20 comprises a register of qubits 12 comprising multiple 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.

[0110] The qubits can initially be initialized to their ground state, e.g. |0>. The multiple quantum gates 14 can comprise multiple Hadamard (H) gates 22 that act 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 multiple (L) layers 24a, 24b of quantum gates 14 of identical structure, where the layers represent multiple quantum operations on the qubits in the register of qubits 12 that are subsequently applied.

[0111] Each layer 24a, 24b comprises multiple 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 across all layers 24a, 24b form a set of variational parameters of the quantum computing network 20. Figure 2The quantum computing network 20 described in the paper, with L layers 24a, 24b and N qubits, is characterized by using L*N variable actions (angles) as variational parameters. Each layer 24a, 24b includes multiple two-qubit gates that act on all adjacent qubit pairs in the qubit register 12, and the variable action can drive the variable single-qubit rotation of each qubit in each layer 24a, 24b.

[0112] Technicians will understand, Figure 2 The arrangement of the gates in the diagram is for illustrative purposes only, and the suitable geometry of the quantum computing network 20 may differ from the depicted representation. For example, in Figure 2 In this embodiment, CNOT gates acting on adjacent qubit pairs act on the qubits sequentially (in time). However, in a preferred embodiment, multiple multi-qubit gates can be applied to the qubits in parallel, for example, in two consecutive applications of two-qubit gates acting on odd / even pairs of adjacent qubits in parallel.

[0113] After applying quantum gate layers 14, 24a and 24b, to the qubit, the state of the qubit can be measured using measurement sensor 16. Measurement sensor 16 can be multiple single-qubit state detectors used to measure the state of each qubit after evolution according to multiple quantum gates 14. Repeated measurements allow the probability of each measurement result to be determined, and the results can be used to find solutions to a set of linear binary relations.

[0114] Figure 3 The diagram illustrates an example flowchart of a method for obtaining solutions to a set of linear binary relations. The method includes initializing the qubits in the qubit register 12 (S10), and applying a set of quantum gates 14 to the computation qubits and measuring the resulting state (S12). The method further includes determining variational parameters based on multiple candidate solutions to the various relations of the set of linear binary relations encoded in the resulting state. The solution to the associated linear binary relation set (S14).

[0115] Variational parameters It can be like Figure 2 The single-qubit rotation R illustrated in the figure y (θ i The variational angle θ i Or variable quantum gates The corresponding variational parameter θ iOther variational parameters. The resulting state can be derived from the state of each qubit in the measurement basis and can be associated with a solution to the set of linear binary relations. Specifically, the measurement state of the register qubits can be associated with a selected one of the binary relations and the measurement state of the ancilla qubits can be associated with a candidate solution to the selected one of the binary relations.

[0116] For example, for two register qubits with basis 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 one of the four relations. Thus, when the qubit register includes two register qubits, the method can be adapted to find a solution to a set of linear binary relations with 4 relations, and generally for a number N of register qubits to a problem with 2Nrelations. N

[0117] Continuing the example, for two binary variables in each relation, the computational basis states of the two ancilla qubits can represent the values of the two variables in the relation described by the register qubits as a tuple of results, e.g., the computational basis states can point to one of the solutions {(0, 0), (0, 1), (1, 0), (1, 1)}. Thus, when each relation has at most m binary variables, the solution to the 2mrelations can be encoded into m ancilla qubits. N

[0118] The relationship between the encoded variables and the solution can be obtained by measuring the conditional probabilities of measuring a certain register state of the register qubits and the state of at least one ancilla qubit. For example, for m ancilla qubits and N relations, the final result state can be given by:

[0119]

[0120] where the first state in the sum is the state of the ancilla qubits s0-s1and the state |i> corresponds to a computational basis state of the register qubits. For a relation i associated with a state i of the register qubits, the sampling of this state can give the component of the classical solution associated with a state S of the ancilla qubits as a probability

[0121] For example, the ancilla qubits can each describe the value of one of the variables, e.g., the state s k of the kth ancilla qubit can describe the value of the kth variable. The probability that the kth variable in the relation i has a value of 1 can then be obtained according to:

[0122] ​​

[0123] where the set of states S k is the state s k of the k-th ancilla qubit corresponding to the property of "1", and where f(i,k) is a mapping to the k-th variable x j of the relation i.

[0124] Thus, a measurement of the states of the qubits in the quantum bit register 12 after applying the layers 24a, 24b of quantum gates 14 can be used to obtain a (random initial) solution of the system of linear binary relations.

[0125] The quantum computing network 20 (i.e. the variational parameters parameterizing the action of the quantum computing network 20 on the quantum bits) can then be optimized in a feedback loop with the goal of minimizing an "energy" / "cost" associated with the solution.

[0126] Figure 4 An example of a flowchart of a method for iteratively improving a solution of a computational problem using a quantum computing network 20 is illustrated. The method comprises determining a plurality of partial derivatives of a set of quantum gates 14 with respect to variational parameters using the quantum computing network 20 (SI 6), determining a gradient of a cost function of the system of linear binary relations based on the plurality of partial derivatives of the set of quantum gates 14 (SI 8), and updating the variational parameters

[0127] To determine the partial derivatives of the layers 24a, 24b of quantum gates 14 with respect to the variational parameters the method can comprise applying the layers 24a, 24b of quantum gates 14 with shifted variational parameters The shifted variational parameters may comprise a subset of the variational parameters shifted by a certain shift amount. In particular, for a variational gate with two eigenvalues ±1 / 2 (e.g. a single qubit rotation according to a Pauli generator matrix, as in the example of Figure 2 and a variable action 0 j the shift amount can be π / 2. The partial derivative of the result f with respect to the variable action 0 j according to the evolution of the quantum bit register 12 in the quantum computing network 20 can then be determined according to:

[0128]

[0129] Thus, if the cost function of the system of linear binary relations is a differentiable function of the result state, the variational parameters The optimization can be performed towards solutions with lower cost based on the gradient of the cost function determined from the partial derivatives of the quantum gate 14 layers 24a, 24b.

[0130] For the purpose of illustration, consider the case of a system of N linear equations with N different variables in total, in the form: var

[0131]

[0132] where a i，j is the coefficient multiplying the jth variable in linear equation i, and where b i is the constant value of linear equation i. Assume that the coefficient matrix a i，j is sparse, such that each equation has at most m non-zero coefficients. The system of linear equations can thus be rewritten as:

[0133]

[0134] Using this representation, a single relational cost C rel associated with all N equations can be obtained based on the following expression:

[0135]

[0136] where, is the probability of measuring the ancilla qubit in state S given that the state of the register qubits is i, where state i is associated with a particular linear equation i, where the candidate solution s k ∈ {0,1} is associated with state S (e.g. the value associated with the state of the kth ancilla qubit), and where b i is the constant value of linear equation i.

[0137] However, the resulting single relational cost C rel only optimizes the equations individually, without taking into account the shared variables between the relations, which can be considered as a result of encoding the result into a relatively small number of qubits. To link the cost function to the relevant system of equations, the inventors introduce a second cost term, an inconsistency penalty C inc to increase the cost of result states with conflicting values for the same variable. For example, the inconsistency penalty C inc can be based on the following expression

[0138]

[0139] where I j is the set of indices of the same variable x j ​A subset of linear binary relations, f(i,k) is the state S of the auxiliary qubit and the same variable x in the relation i encoded in the state of the register qubit. j A predetermined mapping between associated solutions, for a number of N var The variable, the same variable x j The associated solution is associated with index k in relation i, and P(xj = 1 / 0) is the probability that the result state of measuring the k-th auxiliary qubit corresponds to the value 1 / 0 in relation i.

[0140] Since P(x) can be calculated according to equation (2) above, j =1 / 0) represents the probability of measuring a specific state of the computational qubit, therefore the inconsistency penalty C inc It can be used to determine the gradient, which is determined based on the partial derivatives of the resulting state to improve the solution of a set of linear binary relations.

[0141] Then, the cost function used to determine the quality of the solution can be based on the following expression.

[0142] C = C rel +λC inc (8)

[0143] Wherein, λ is a parameter chosen in the following manner: for a given set of linear binary relations, for random variational parameters λC inc On average, the items are greater than C. rel .

[0144] According to the following formula, based on the quantum computing network 20 with respect to the variational parameter θ q The partial derivatives are used to obtain the gradient of the cost function:

[0145]

[0146] Among them, according to equation (2), It can be determined according to the following formula:

[0147]

[0148] And according to equation (3), Based on the following formula, variational parameters with a symmetrical shift of π / 2 can be used. Result status The results will determine,

[0149]

[0150] In other words, the gradient of the cost function can be based on the variational parameters of the quantum computing network 20 for the shift. determined from measurements of the cost function, wherein the partial derivatives of the cost function are based on variational parameters for the shift The probabilities of the computational basis states of the qubit register 12 are measured.

[0151] The variational parameters can then be updated with the gradient towards the optimized solution To improve convergence based on the measured gradient of the cost function determined in this way, the variational parameters can be updated according to an adaptive moment-based update function

[0152] The adaptive moment-based update function can update the variational parameters based on an update function of the moving average over the measured gradient of the cost function and an update function of the moving average over the squared measured gradient of the cost function In particular, the variational parameters can be updated according to the following functions

[0153]

[0154] where m t is proportional to the moving average over the gradient of the cost function and v t is proportional to the moving average over the squared gradient of the cost function, a is a learning rate hyperparameter (e.g. 0.01) and e is a small number (e.g. 10 -8 ) relative to the expected magnitude of the update.

[0155] The moving averages can be exponentially decaying and can be determined according to and iteratively, wherein is the result state of the variational parameters at iteration step t, is the gradient determined at iteration step t, 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.

[0156] The skilled person will appreciate that the quotients 1 - β1 t and 1 - β2 t can be understood as bias correction terms to correct for an initialization bias of the initial values of m t and v t which are initialized to zero (i.e. at t = 0) such that m t and v tcan be exponentially decaying moving averages of the gradient / gradient squared 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.

[0157] The inventors found that an update function based on an adaptive norm can significantly improve the network performance and was found to outperform both a simple moving average of the gradient and an adaptive learning rate algorithm, respectively.

[0158] In some embodiments, determining an updated set of variational parameters can incorporate stochastic elements, for example by randomly selecting a subset of variational parameters at each iteration to determine the partial derivatives and estimate the gradient based on the randomly selected subset of variational parameters in a manner similar to stochastic gradient descent. Thus, the time for updating the variational parameters can be reduced compared to determining the partial derivatives for all variational parameters .

[0159] In some embodiments, an additional penalty can be added to the cost function for cost function regularization, or the magnitude of the change of the variational parameters can be bounded to incorporate additional constraints into the solution or to reduce the complexity of finding an optimized solution.

[0160] 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 qubits in the qubit register 12 applied to the quantum computing network 20 according to the cost function are characterized by a lower (minimal) cost. Thus, the quantum computing network 20 can be used to find a solution of a system of linear binary relations.

[0161] Figure 5 A flowchart of an iterative method for improving the solution of a computational problem using a quantum computing network 20 similar to the quantum computing network 20 illustrated in Figure 2 can be applied according to Figure 3 and Figure 4 .

[0162] Initially, the quantum computing network 20 can be evaluated with random variational parameters in an initial quantum computing network evaluation 26, which can be run on a quantum computer. The resulting results can then be analyzed by a cost function evaluation module 28, which can be run on a classical computer to determine the cost function with respect to the (initial random) variational parameters Associated cost. The cost can be passed to the convergence / threshold evaluation module 30, which can run on the same classical computer, comparing the cost to the target cost C target and checking if the cost / solution converges based on previous iterations. If the cost is below a threshold or has converged, the method can output the most likely solution for the variational parameters and the classical solution corresponding to the result of the evaluation of the quantum computing network 20. Preferably, the convergence / threshold evaluation module 30 checks if all variables of the most likely solution have the same value and if the solution satisfies all relations.

[0163] If the set has not yet reached the convergence criteria (e.g., when the set of binary linear relations has not been solved or when the cost has not converged or is above a threshold), a hybrid gradient evaluation 32 can be performed. The hybrid gradient evaluation 32 includes shifting the variational parameters and evaluating 34 the quantum computing network 20, which can run on the quantum computer, where the variable actions θ j are individually shifted by the symmetric shift amount and the result is passed to the 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 cost function to the update module 38 to update the variational parameters

[0164] 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 the quantum computer, which determines the result of the quantum computing network 20 for the updated variational parameters (quantum mechanically). The result 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 result converges or is below a predefined threshold.

[0165] Figures 6A to 6C Three plots illustrating the simulated performance of the hybrid quantum computing architecture described in connection with binary Figures 3 to 5 equations. The plots plot the normalized cost (solid line with open circles, left axis) of the solution of a set of eight linear binary equations (where each equation includes three variables (as indicated in the title)) as a function of the number of iterations of the hybrid quantum computing architecture described in connection with binary Figure 4 and Figure 5 described in connection with binary Figure 2The graph shows the number of iterations of the iterative method of the analog quantum computing network 20 as a function of the number of iterations. The graph further plots the ratio of the solved equations (solid line without open circles, right axis "solved ratio"), where the thin lines represent different linear equation systems (characterized by different coefficients) and the thick line represents the average solved ratio, both as a function of the iterations.

[0166] Figures 6A to 6C The number of shared variables differs among the 8 equations, i.e., in Figure 6A no variables are shared between the equations (the equations are independent), while for Figure 6C two variables are shared, so that the linear equation system can be written as follows:

[0167]

[0168] The system is solved in an analog quantum computing network 20 with six qubits (i.e., three register qubits and three ancilla qubits) and 20 layers of quantum gates 14. The analog quantum computing network 20 monotonically decreases towards the minimum, as can be seen from the decay of the cost and the increase of the solved equation ratio.

[0169] The inventors observed in their simulations that the solved equation ratio can further increase with the number of layers of quantum gates 14. The quantum computing network (QCN) gradient descent was performed using the "Cirq" simulator on a consumer-grade processing unit and is therefore limited in the number of simulated qubits / layers. However, since quantum hardware will be able to determine the gradient significantly faster, the inventors expect that by using a real quantum computer, the above quantum architecture will be able to solve problems comprising large linear binary relation systems (e.g., 10 5 relations or more) faster than classical devices.

[0170] The skilled person will appreciate that the quantum computing network 20 has been described with variational single-qubit gates having two eigenvalues 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, in principle other types of variational quantum gates can be used.

[0171] In a more general case, the partial derivatives can still be determined based on the evaluation of the quantum computing network 20 by conditionally applying the variational quantum gates and the additional quantum gates A k based on the state of the ancilla qubits.

[0172] Figure 7AThe illustration shows another example of a flowchart of a method for iteratively improving solutions to a quantum computing network 20 using a universal variable quantum gate. The method includes applying an additional quantum gate A with conditional application of states based on a second auxiliary qubit. k Quantum gates 14 layers 24a and 24b are defined, and the resulting states are determined to evaluate quantum gates 14 layers 24a and 24b with respect to variational parameters. variable action θ j The method further includes determining the gradient of the cost function based on the partial derivatives of quantum gate layers 24a and 24b (S24), and updating the variational parameters based on the gradient (S26).

[0173] Additional quantum gate A k The equation should be satisfied Where K is a natural number (e.g., 2). An additional quantum gate A can be applied conditionally after the corresponding variational gate, conditioned on the state of the auxiliary qubit. k Furthermore, the resulting state can be determined to evaluate the 14-layer quantum gate with respect to variational parameters. variable action θ j The partial derivatives of .

[0174] For example, such as Figure 7B As illustrated, the auxiliary qubit 42 in state |0> can be prepared in a superposition state by a first Adama gate 22a. The state of qubit 44 in qubit register 12 can then be conditionally subjected to a variable quantum gate 46 during the evaluation of the quantum computing network 20, for example, conditioned on the state of the auxiliary qubit 42 being |0>. Subsequently, the resulting state of qubit 44 in qubit register 12 can be conditionally subjected to an additional quantum gate 48, for example, conditioned on the state of the auxiliary qubit 42 being |1>. A second Adama gate 22b can then be applied to the state of the auxiliary qubit 42, and the state of the auxiliary qubit 42 can be measured by a measurement sensor 50.

[0175] The quantum computation result and the state of auxiliary qubit 42 can be measured to obtain the expected values ​​E0 and E1 of the auxiliary qubit states |0> and |1>, respectively, where the additional quantum gate A k Each of them has probabilities p0 and p1. Then, according to... Determine the partial derivatives.

[0176] Therefore, in some embodiments, the method may include a variable quantum gate 46 having more than two eigenvalues, and can be used with... Figure 7A , Figure 7B The iterative method illustrated in the diagram is a similar iterative method for optimizing the solution.

[0177] The preferred embodiments and the accompanying drawings are only intended to illustrate the present application and its associated benefits, and should not be understood as implying any limitation. The scope of the present application will only be determined by the appended claims.

[0178] List of reference signs

[0179] 10 system

[0180] 12 qubit register

[0181] 14 plurality of quantum gates

[0182] 16 measurement sensor

[0183] 18 control system

[0184] 20 quantum computing network

[0185] 22, 22a, 22b Hadamard gate

[0186] 24a, 24b layer

[0187] 26 initial quantum computing network evaluation

[0188] 28 cost function evaluation module

[0189] 30 convergence / threshold evaluation module

[0190] 32 hybrid gradient evaluation

[0191] 34 quantum computing network evaluation for shifted variational parameters

[0192] 36 partial derivative evaluation module

[0193] 38 update module

[0194] 40 quantum computing network evaluation for updated variational parameters

[0195] 42 ancilla qubits

[0196] 44 qubit register state

[0197] 46 variational quantum gate

[0198] 48 additional quantum gate

[0199] 50 ancilla measurement device

Claims

1. A method for finding a solution to a computational problem using a hybrid quantum computing system, the computational problem comprising a set of linear binary relations, the hybrid quantum computing system comprising a quantum computing network comprising a set of quantum gates implemented on quantum hardware, the set of quantum gates comprising a plurality of variational quantum gates having variable actions operating on a plurality of computational quantum bits of the quantum computing network, the variable actions forming a set of variational parameters wherein the quantum hardware comprises a quantum register and one or more measurement sensors, and wherein, The method comprises: initializing the plurality of computational qubits, the plurality of computational qubits comprising a plurality of ancilla qubits and a plurality of register qubits; applying the set of quantum gates to the computational qubits and measuring a result state; determining a solution of the associated set of linear binary relations based on the plurality of candidate solutions of individual relations of the set of linear binary relations encoded in the result state associated with the selected one of the binary relations and a state of the ancilla qubit associated with a candidate solution of the selected one of the binary relations. iteratively refining a solution of the set of linear binary relations by: - determining, using the quantum computing network implemented on the quantum hardware, a plurality of partial derivatives of the quantum gate set with respect to the variational parameters , determining, on classical hardware, a gradient of a cost function of the set of linear binary relations based on a plurality of partial derivatives of the set of quantum gates, wherein the cost function relates a cost to a candidate solution of the set of linear binary relations encoded in a result state of the computational qubits, wherein the cost comprises a single relation penalty associated with a mismatch between two sides of each of the relations after repeating the measurement of the computational qubits and an inconsistency penalty associated with a conflict of values of a same variable in different linear binary relations; and - updating the variational parameters based on the gradient as feedback to the quantum computing network implemented on quantum hardware.

2. The method of claim 1, wherein, the single relation penalty is a continuous and non-negative function of a mismatch between two sides of each of the relations, wherein the single relation penalty is an even polynomial of the mismatch and is based on a square of the mismatch.

3. The method of claim 1 or 2, wherein, The cost includes a cost expression C of the calculation wherein, is a single relationship cost, wherein, is an inconsistency penalty, and wherein, is a parameter chosen in such a way that, for a given set of linear binary relationships, the random variate , is greater than on average.

4. The method of claim 1 or 2, wherein, the inconsistency penalty is based on a probability of measuring different result states of the same variable when determining a state of the same variable based on states of the computational qubits in different states of the register qubits, wherein the inconsistency penalty is based on an expression wherein is a subset of linear binary relations of the same variable x j , f(i, k) is a state of the ancilla qubit and a relation encoded in a state of the register qubit i the same variable x j , a predetermined mapping between solutions of the relation N var of the same variable x j is associated with an index i in the relation k , and P (x j = 1 / 0) is a probability that the result state of the mth ancilla qubit corresponds to the value 1 / 0 in the relation k i .​ 5. The method of claim 1 or 2, wherein, The computed qubits include log N register qubits and m ancilla qubits, wherein the set is N a set of linear binary relations, wherein each of the N linear binary relations has at most m different variables.

6. The method of claim 5, wherein, the computational basis states of the ancilla qubits are associated with all possible candidate solutions of a respective linear binary relation of m variables associated with the states of the register qubits.

7. The method of claim 5, further comprising determining a maximum number of variables in each of the N linear binary relationships m .

8. The method of claim 5, wherein, The set of relationships is N a set of linear binary equations, each linear binary equation having at most m different variables, wherein the single relationship cost is based on the following expression wherein is the probability of measuring the ancilla qubit in state i given that the state of the register qubit is S wherein the state i is associated with a particular linear equation i wherein is a coefficient multiplying the i th variable in the linear equation k wherein the candidate solution S associated with the state and wherein b i is a constant value of the linear equation i .

9. The method of claim 1 or 2, wherein, determining, using the quantum computing network, a partial derivative of the set of quantum gates with respect to the variational parameters includes: - Apply variational parameters with shifts The set of quantum gates, the variational parameters of the shift The variational parameter includes the shifted value by a certain amount. A subset of, and variational parameters for determining the shift. The associated result state is used to evaluate the variational parameters. Partial derivatives of subsets of; and / or - conditionally applying the set of quantum gates and an additional quantum gate A based on a state of a derivative ancilla qubit k , the additional quantum gate satisfying the equation where K is a positive real value, and determining the resulting state to evaluate the partial derivative of the variable action of the set of quantum gates with respect to the variational parameters .​ 10. The method of claim 9, wherein, The method includes sequentially applying shifted variational parameters to each variational gate. The quantum gate layer is doubled, and the variational parameter of the shift is... The variational parameters include those shifted by a symmetrical shift for each variational gate. A subset of the subset is used to evaluate the variational parameters. The partial derivative of each variable action, thus in updating the variational parameters. The gradient was previously determined, wherein the two eigenvalues ​​of the variational quantum gate are And the shift amount is .

11. The method of claim 1 or 2, wherein, updating the variational parameters based on an update function of a moving average of a gradient of the cost and an update function of a moving average of a squared gradient of the cost wherein the update function is proportional to a moving average of a gradient of the cost function and inversely proportional to a square root of a moving average of a squared gradient of the cost function, wherein the moving average of the gradient of the cost function and the moving average of the squared gradient of the cost function are exponentially decaying moving averages.

12. The method of claim 11, 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, wherein and wherein and are real values between 0 and 1, is a shift-based variational parameter the result state at iteration step t is determined by the gradient, 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.

13. The method of claim 1 or 2, wherein, the set of quantum gates comprises a plurality of layers of quantum gates applied in succession, wherein the plurality of layers of quantum gates comprises a same arrangement of quantum gates in each layer, and wherein the quantum gates in each layer comprise a plurality of multi-qubit quantum gates that act together on all qubits in a register of qubits, wherein each layer of quantum gates comprises a set of variational single-qubit gates that act on each qubit in the register of qubits, wherein the set of variational single-qubit gates is a set of variational single-qubit gates, and / or wherein a number of variational single-qubit gates in each layer is equal to a number of qubits in the register of qubits.

14. A hybrid quantum computing system for finding a solution of a set of linear binary relations, the hybrid quantum computing system comprising: a quantum computing network implemented on quantum hardware, the quantum computing network comprising: a register of qubits comprising a plurality of qubits, the plurality of qubits comprising a plurality of register qubits and ancilla qubits, - a set of quantum gates selectively acting on the qubits of the qubit register, the set 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 control system configured to: initialize qubits in the register of qubits; - applying the set of quantum gates to qubits in the register of qubits with variational parameters to determine a resultant state; - The set of quantum gates is determined with respect to the variational parameters using the quantum computing network implemented on quantum hardware. The partial derivative of at least one variational parameter in; determine a gradient of a cost function based on the partial derivatives, wherein the cost function relates a cost to a solution encoded in the result state on classical hardware, wherein a state of the register qubits is associated with a selected one of the binary relations and a state of the ancilla qubits is associated with a candidate solution of the selected one of the binary relations, and wherein the cost includes a single relationship penalty associated with a mismatch between the two sides of each of the relationships after repeating measurements on qubits and an inconsistency penalty associated with a conflict of values of the same variable in different linear binary relationships; and - updating the variational parameters based on the gradient as feedback to the quantum computing network implemented on quantum hardware.

15. A computer program product 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 13 and / or implement the system according to claim 14.

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