Methods for quantum-classical hybrid optimization

Abstract

Disclosed are methods for quantum-classical hybrid optimization. In some embodiments, the method comprises providing a set of second variables that parameterize a Hamiltonian operator, and first variables that parameterize a quantum circuit executable on the quantum computing device; (a) executing, on a quantum computing device, the quantum circuit, to generate outcomes from read outs of the executions; (b) processing, by a classical computer, the outcomes by computing an expectation value of the Hamiltonian operator to obtain a cost function, and, using the cost function, computing implicit derivative information of the first variables with respect to the set of second variables; and (c) updating, by the classical computer, the first variables by using the implicit derivate information and updated second variables, wherein the updated second variables are generated by applying an update rule, and iteratively repeating steps (a)-(c) until the second variables reach final values, thereby evolving the quantum circuit and the Hamiltonian operator.

Description

METHODS FOR QUANTUM-CLASSICAL HYBRID OPTIMIZATIONTechnical Field

[0001] The present disclosure generally relates to computer implemented optimization methods. In particular, the disclosure relates to optimization methods implemented on hybrid classical-quantum computational devices and systems.Background

[0002] The world is full of optimization problems with industrial applications. Some examples include the traveling salesman problem, traffic optimization, protein folding, portfolio optimization, and training machine-learning models. Many of these problems may be mapped into a mathematical cost function of n bits f: {0,1}” — > HR to be minimized. However, the minimization of a large variety of cost functions is extremely challenging for classical computers.

[0003] Of course, there are myriad classical methods attacking the above-described problem depending on the case. Since the solution disclosed herein is based on the utilization of quantum computational devices, a description of methods previously used in this domain is presented below.

[0004] Adiabatic quantum computing and quantum annealing aim to solve optimization problems by mapping them first to the problem of minimization of a quantum Hamiltonian operator H = H[ (T)] such that the ground state of the quantum register of n quantum entities, such as quantum bits, i.e. qubits, is easy to find for 2(0). Then one prepares the ground state of W [2(0)] and evolves 2 in time t from 2(0) to 2(T) adiabatically, i.e., such that the state of the quantum register |l / / (t)) stays very close to the ground state of the instantaneous Hamiltonian W[2(t)] for all t e [0, T], If this succeeds, one has solved the original problem and can find the bit string minimizing the cost function by measuring |¥,(T). The caveat in this problem is that the energy gap that defines how fast one is allowed to change 2 in time closes with the problem size increasing, leading either to detrimental errors in the final answer |V / (T)) or slowing down of the algorithm to unreasonably long computational time. Quantum annealing aims to bypass this issue by introducing dissipation into the systems with the hope of excited states to decay to the ground state, but the dissipation also hinders the quantum coherence and tends to limit the quantum advantage obtained from the method. Note that the evolution of 2(t) can be implemented either in actual continuous time evolution (analog way) or using a series of quantum operations, e.g. quantum circuits (digital way).

[0005] Known variational optimization methods, on the other hand attack, this optimization problem by constructing a quantum circuit with m real-valued parameters x that prepares the state of the system to |'Ar(x)>. Subsequently, the expectation values {' / '(x) \H\ ' / '(X)} and their gradients Vx{ ' / '(x) | H\ ' / '(X)} are computed and the parameters x are updated according to known methods of non-linear optimization such as the method of deepest decent. After the minimization converges to x’, the hope is that |¥ / (x’)) provides the optimal solution to the original problem. If indeed |¥ / (x’)) minimized the expectation value (¥ / (X)|H|¥ / (X)}, the original problem would be solved. However, the issue here is that when the problem size increases, the gradients get increasingly small, and consequently the computational time of the algorithm increases to an intolerable level. This vanishing of the gradients is referred to as the Barren plateau.

[0006] Therefore, there is a need for an optimization method which truly enables harnessing the computation capabilities of the current hardware, which typically is optimized by combining a classical computer with a quantum computational device into a hybrid computational system, by taking into account the current limitations of quantum computational devices and efficiently dividing and determining computational tasks for the current quantum computational devices.Summary

[0007] One aspect of the disclosure relates to hybrid classical-quantum computational device implemented optimization methods according to the independent claim 1. Further embodiments and aspects of the disclosure are presented in the subsequent claims.

[0008] On objective of the disclosure is to develop a method for solving an optimization problem by optimally harnessing computational capabilities of quantum computational devices. One further objective is to enable, particularly on current and near-future quantum computers, the simultaneous optimization of both the problem itself and the parameters of the solution.

[0009] One feature of the disclosure is to map the optimization problem onto a quantum operator, develop the quantum operator to a continuum of problems starting from an easy problem and ending at the desired problem using a quantum computational device to calculate values and / or gradients of the expectation value of the current quantum operator, and use the quantum-computer-calculated results to steer the parameters of the solution such that it stays optimal when the problem is evolved into the desired, original optimization problem.

[0010] The advantages of the disclosure include optimized harnessing of computational capabilities of current and near-future hybrid classical-quantum computational devices tosolve practical optimization problems from various fields, such as traffic optimization, telecommunication and network optimization, and protein folding.

[0011] The disclosure may have further advantages that will become clear in the following detailed description.Description of the Drawings

[0012] Fig. 1 illustrates one example of a method according to certain embodiments of the disclosure.

[0013] Fig. 2 presents a flow chart of the method including optional steps of a number of embodiments of the method; and

[0014] Fig. 3 presents a graph comparing the performance of some embodiments of the disclosure.

[0015] Fig. 4 illustrates one example of a method according to certain embodiments of the disclosure.

[0016] Fig. 5 shows the evolution of the second variables y, the first variables, and the cost function of one example of a method according to certain embodiments of the disclosure.

[0017] Fig. 6 shows an architecture comprising apparatuses / devices according to various embodiments, including a classical computer and a quantum computing device configured to perform various disclosed methods.Detailed Description

[0018] The following description and drawings are illustrative and are not to be construed as unnecessarily limiting. The specific details are provided for a thorough understanding of the disclosure. However, in certain instances, well-known or conventional details are not described in order to avoid obscuring the description. In this specification, reference to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the disclosure. References to an embodiment can be, but are not necessarily, references to the same embodiment in the present disclosure.

[0019] The present disclosure relates to a method implemented on or by a hybrid classical-quantum computational device for solving an optimization problem by minimizing the expectation value of the energy of a Hamiltonian operator, onto which the optimization problem is mapped. In the present context, the terms “minimize” and “optimize” will be understood to refer to processes of reducing or improving as far as possible, which may not lead to the true minimum or optimum. In other words, minimization and optimization may beapproximate and not absolute. The same considerations apply to the terms “minimizing”, “minimized”, “optimizing”, “optimized” etc. Where these terms are used in the present description, they may therefore be used in a phrase such as “at least closely minimizing” or “at least closely optimizing”.

[0020] Many problems can be expressed this way. For example, finding the ground state of a molecule or a more complicated physical system is exactly a practical problem of this kind. Thus, a direct application of the optimization method would be designing e.g. new drugs and fertilizers which currently requires a lot of resources in a form of tedious synthesizing and testing of hundreds or thousands of candidate compounds. Being able to simulate the optimal structure and properties of complicated drug molecules would make their development substantially faster. Furthermore, many combinatorial problems (such as the max-cut problem) can be interpreted in a very natural way as being Hamiltonian energy minimization problems.

[0021] Moreover, as Hamiltonian minimization is an NP-hard problem, the vast majority of practical optimization or decision problems can be, in polynomial time, transformed into a Hamiltonian minimization problem whose solution yields a solution for the original problem. One very general source for technical applications of the disclosed methods is integer linear programming (ILP) problems, which can be mapped e.g. to a quadratic unconstrained binary optimization (QU BO) problem, which can subsequently be mapped onto a Hamiltonian operator. Integer linear programming is relevant, when optimizing resource allocation of resources that cannot be divided. Practical examples of ILP problems compriseroute planning: When allocating buses to serve a public transportation network, number of busses sent on a given route must be an integer; telecommunication and cellular network: Designing a network of lines to install with minimum costs so that a predefined set of communication requirements are met requires optimizing both the topology of the network along with setting the various line capacities which are constrained to be integer quantities in many cases. Furthermore, some possible additional restrictions can be modeled as linear inequalities with integer or binary variables; anddefense: Problems could be, for example, the optimal placement of fighter jets (or other, indivisible resource) in various airports (or other relevant locations).

[0022] Another example of solving a QUBO problem with the disclosed optimization method comprises turning a Markowitz model into a QUBO problem and then mapping this QUBO problem onto a Hamiltonian operator. Thus, minimizing the expectation value of theenergy of the Hamiltonian operator solves the original problem, which may be e.g. portfolio optimization problem expressed by the Markowitz model.

[0023] Fig. 1 illustrates a flow chart of a method 100 according to one embodiment of the disclosure. After starting 101, the method 100 comprises computing 110, by a quantum computational device, a set of optimal values of the set of first variables, by minimization of the expectation value of the Hamiltonian operator acting on a quantum state parameterized by the set of first variables. The Hamiltonian operator may be noted as H, set of first variables simply as x, the corresponding optimized set of values as xop, and the quantum state parametrized by the set of first values may be noted as | ' / '(x)). The expectation value of the Hamiltonian operator, (H), may be treated as a cost function of n bits f: {0,1 }n— > R to be minimized on a quantum register of n quantum entities.

[0024] The quantum state can be produced by a set of quantum operations, preferably quantum circuits, executed on an initial quantum state, which may be noted here as |0>, of the register of n quantum entities, preferably quantum bits, i.e. qubits. The set of quantum operations is parametrized by the set of first variables and may thus be noted as (x), creating the following relation: tX(x) 10> = |V / (x)>.

[0025] The quantum computational device may be a quantum computer, and, in general, the quantum computational device may perform different qubit operations, such as reading the state of a qubit, initializing the state of the qubit, and entangling the state of the qubit with the states of other qubits in the quantum computational device, etc. Existing implementation examples of such quantum computational devices include superconducting quantum computers, trapped ion quantum computers, quantum computers based on spins in semiconductors, quantum computers based on cavity quantum electrodynamics, optical photon quantum computers, quantum computers based on defect centers in diamond, etc.

[0026] The set of optimal values xop, and thus a set of first variables x in general, and the Hamiltonian operator are parametrized by a set of second variables, noted herein simply as y. The set of second variables y is defined such that the minimization of expectation values of the Hamiltonian operator, (¥ / (x)|H(yf)|¥ / (x)}, for a known set of final values of the set of the second variables, yf, at least closely solves the optimization problem, and for a known set of initial values of the set of the second variables, y, e ]RP, there exists a known or at least easily computable set of initial values of the set of first variables, x, e IRm. Above, p and m denote the dimensions of the parameters describing the parametrization of the Hamiltonian and of the quantum state, respectively. The set of initial values of the set of first variables at least closely minimizes the expectation values of the Hamiltonian operator, which is parametrized by a set of initial values of the set of the second variables:U&i) |0> = |^(%j)> = argmin|¥,w>{¥ / (%)|H(yi)|¥ / (%)).Such conditions can be easily met for any arbitrary cost function f, for example, by choosing H(yi) = ~and = I. Using the mathematical definitions hereinabove, the set of optimal values of the set of first variables may be defined as*oP(y) = argminx(¥ / (%)|H(y)|¥ / (%)}.

[0027] The Hamiltonian operator may refer e.g. to an explicit 2nx 2nHermitian matrix, that may be provided as a linear combination of Pauli strings, i.e. tensor products of one-qubit operators from the set {l, X, Y, Z}. The Hamiltonian operator may also be defined as a composition of a mixing Hamiltonian operator WM(y) and a problem Hamiltonian operator HP(y), i.e.#(y) = #M(y) + ^p(y).wherein the mixing Hamiltonian operator evolves to the desired problem Hamiltonian operator during the process of the optimization method, such that it fulfills the property / / (y,) = ^MOI)and f) = HP(yfIdeally, the ground state of / / (y,) is easy to prepare on the quantum computational device through a proper choice of quantum operations (7( ) and the set of initial values of the set of first variables, %j(y).

[0028] One practical way to prepare an initial state may be to choose the mixing Hamiltonian such that its ground state corresponds to a simple product state, which can be efficiently generated on a quantum computer. For example, if the mixing Hamiltonian is chosen as a sum of single-qubit Pauli- X operators, its ground state can be prepared by applying a Hadamard gate to each quantum entity, such as a qubit, initialized in the |0) state. More generally, the initial circuit parameters can be set such that the variational circuit U(xt) prepares this product state, for example by setting all rotation angles to in a layer of single-qubit Y -rotations. This approach ensures that the initial state is both theoretically justified and experimentally feasible, providing a reliable starting point for the subsequent optimization process (iteration).

[0029] As a next step, the method 100 comprises computing 120, by a classical computer, a set of gradient values of the computed set of optimal values of the set of the first variables, 7yxop(y), using data obtained from the quantum computational device.

[0030] The method 100 further comprises steering 130, by the classical computer, the set of the second variables y from the set of initial values yj to the set of final values yfusing the computed set of gradient values, Fyxop(y), while keeping the quantum state at least closely minimized for all sets of values of the realized sets of the second variables, i.e. keeping U(x) 10) = | ^(x)) at least close to |( / / [xop(y)]} for all values of realized y. One wayto do this is to just change the first set of variables in the direction of the gradient by some step size, as this keeps the quantum state at the ground state, at least approximately. However, in practice it might be advantageous to use more sophisticated gradient-based steering to achieve better results, as described hereinbelow.

[0031] The method steps described hereinabove may be iterated in the sense that the second variables are changed from the initial value to final value in several steps. At each step, the gradient values need to be computed in order to find the optimal values for the set of first variables for the next set of values of the set of second variables. Specifically, the optimization routine updates the set of first variables such that the cost function f remains sufficiently close to its global minimum as the set of second variables changes towards its final value.

[0032] In some embodiments, the path from y, to yfis chosen adaptively. For example, at each step, the method could choose the coordinate of y which leads to the smallest gradient Fy op(y),sothat the change to xop(y) is minimized at each step.

[0033] Furthermore, the method 100 comprises determining 140 a set of final, at least closely optimized values of the first variable, which at least closely minimizes the Hamiltonian operator, by reading out the final quantum state of the register of n quantum entities. In more mathematical terms, the method determines ^[xopCyf)]) which yields at least a close solution to the original problem of minimizing the cost function f: {0,1} — > R After determining a sufficiently optimized solution, the method finishes 199.

[0034] The different embodiments hereinbelow provide different optional details on how to efficiently compute the gradients and how to efficiently use them to sustain ^(x)) at |^[*op(yf)])-

[0035] Fig. 2 presents a flow chart introducing optional steps of a number of embodiments of the method.

[0036] In some embodiments, the optimization problem is formulated as a quadratic or higher-order, unconstrained or constrained, binary or integer, optimization problem, as described hereinbefore when providing applications for the disclosed method.

[0037] In case the Hamiltonian operator is not defined prior to starting the optimization algorithm, the method 100 may further comprise mapping 202, by the classical computer, the optimization problem onto the Hamiltonian operator to be minimized. The mapping is preferably carried out such that the said Hamiltonian operator is optimized to contain a small number of terms, interactions, or non-commuting operators, or a weighted combination of such. As a person skilled in the art appreciates, there are many methods of achieving such a mapping. Choosing a suitable mapping method is highly dependent on the original optimization problem. In principle, it is possible to map optimization problems onto aHamiltonian operator by pen and paper, but practical optimization problems are typically too complex to be mapped without a computer.

[0038] Furthermore, in case the quantum operations are not constructed prior to starting the optimization algorithm, the method 100 may further comprise constructing 203, by a classical computer, the set of quantum operations, preferably quantum circuits. Preferably, the set of quantum operations is optimized for the quantum computational device used to execute the set of quantum operations to yield high fidelity in the implementation of the set of quantum operations. Again, in principle, it is possible to construct quantum circuits by pen and paper, but practical optimization problems are typically too complex to be solved without using a computer for constructing the quantum operations.

[0039] In some embodiments, at least one of the quantum operations of the set of quantum operations, preferably quantum circuits, comprises at least one one-qubit quantum (logic) gate and / or at least one multi-qubit quantum (logic) gate. The one-qubit gates may comprise e.g. an identity gate I, one or more single-qubit Pauli gates (X, Y, Z) for single-qubit rotation, a NOT gate, a phase shift gate, and / or a Hadamard gate. The multi-qubit gates couple two or more qubits, and they may comprise e.g. a controlled NOT gate, one or more multi-qubit-controlled one-qubit rotation (Pauli) gates, a Toffoli gate, a parametrized, potentially multi-qubit-controlled SWAP gate, and / or a uniformly controlled unitary operation.

[0040] Furthermore, the at least one of the quantum operations of the set of quantum operations, preferably quantum circuits, may comprise alternating layers of at least one single-qubit gate layer and at least one multi-qubit gate layer. Here, the at least one singlequbit layer comprises at least one one-qubit rotation gate acting on all quantum entities, preferably quantum bits, of the register of n quantum entities with possibly different angles, and the at least one multi-qubit layer comprises at least one multi-qubit gate, such as listed hereinbefore, coupling at least two quantum entities, preferably quantum bits, of the register of n quantum entities. There may be, or may not be, one or more similar layers within the at least one quantum operation or within the set of quantum operations.

[0041] As an example, such a quantum operation may comprise d layers of one-qubit rotation gates acting on all qubits (with possibly different angles) in some configuration that may be different in different layers and h layers of multi-qubit gates in some configuration that may be different in different layers, wherein the one-qubit gate layers and the multiqubit gate layers alternate to make a stack of d+h layers. Here, the set of first variables would be all the one-qubit rotation angles, in total n*d angles. This is a general-purpose parametrized quantum circuit whose main advantage is that it can be made efficient oncurrent hardware, such as superconducting quantum computers. However, in practice problem-specific quantum operation parametrizations may yield more accurate results.

[0042] In some embodiments, at least a part of at least one quantum operation of the set of quantum operations corresponds to an exponential of a mixer Hamiltonian and / or a cost Hamiltonian, which are defined similarly as in quantum approximate optimization algorithm (QAOA).

[0043] In some embodiments the method 100 comprises constructing 205, by the classical computer, prior to computing the set of optimal values of the set of first variables %op(y), a set of quantum operators which partially or fully correspond to the Hamiltonian operator, wherein the set of quantum operators preferably yields a ground state or groundstate energy close to that of the Hamiltonian operator, and wherein the set of quantum operators is parametrized by a set of second variables y. The resulting set of quantum operators is used instead of the Hamiltonian operator in the subsequent method steps. The set of quantum operators may comprise e.g. a set of simple Hermitian operators, Pauli strings, matrix product operators, projection operators, operators obtained by similarity transformations of the mentioned operators, or sums thereof. For example, the Hamiltonian operator may be divided into manageable pieces by defining it with at least one simple Hermitian operator of which expectation values may be computed by the quantum computational device. One way to implement this is to express the Hamiltonian operator as a sum of Pauli strings, because the expectation value of a Pauli string can be computed with a quantum computer, essentially because any Pauli string can be implemented by a single-layer quantum circuit consisting entirely of one-qubit gates {X, Y, Z) discussed hereinbefore. In the subsequent method step 120, the computed expectation values of the simple Hermitian operators are used by a classical computer to compute the gradient values.

[0044] In some embodiments, the set of gradient values may be computed using a finite difference method, matrix product state and / or some other state approximating that of the quantum register, Fourier transformation, implicit differentiation, parameter shift rule, Heun’s method, path integrals, or a combination thereof. In any case, the Hamiltonian operator and the quantum state are preferably parameterized in such a way that it allows the usage of the chosen method to compute the gradients Fy op(y).

[0045] The method 100 may further comprise computing 211, by the quantum computational device, at least one set of second or higher order derivatives of the computed set of optimal values of the set of the first variables.

[0046] The first and the second order derivatives can be used to compute the gradient of the ground state Fy op(y) subsequently. In the optimization process, a collection ofparameter values y,..., ysinterpolating between ytand yfis chosen by some method, and the ground statexop(y / ) is computed at each parameter beginning from the known initial ground state, and computing the subsequent ones using the gradients ^,%op(y7). The point is that computing the expectation values of the Hamiltonian operator allows to evaluate an energy function E(y,x). To compute numerically the gradient (first or higher order), one could simply use the finite difference approximation of the derivative (again, first or higher order). The parameter shift rule is another way to numerically compute the gradients. It can only be done under some assumptions of the parametrization, but it has the advantage that it may be more numerically stable.

[0047] For certain families of quantum circuits (7(%), the derivatives of the cost function f (denoted f), with respect to the elements of % can be estimated with a quantum computer through the parameter shift rules. This allows one to use a quantum computer to formally obtain the implicit derivatives of the optimal circuit parameters %op(y) according to the expression for an implicit differentiation:dyxOp(0) = -[H(y,%)] \f'(y,x)\x=Xop(y).This expression in combination with the Hamiltonian operator composition, H(y,x) = HM(y,x) + HP(y,%), are the key elements of this embodiment of the method 100. Next, a particular optimization routine involving these elements is described.

[0048] A particular implementation described below connects the implicit derivatives dyxop(y) to an update rule for %op(y) through the Heun’s method, wherein %opis an approximately optimal set of first variables. The second parameter y is first linearly discretized into nstepsparts, i.e., ym= — — —, m = fo, 1,...,nsteps- 11. The predictor and ^steps- 1 1 Jcorrector components (i.e. terms) of the update rule of the Heun’s method at the m-th step respectively read■^-opjjjCym)—-^opjn ^Cym-l) T 6dyXOp(ym-l)>~ ~ Ay ~ >•^opm(ym)_-^opjjj ^Cym-i) i 2~ I. ^yxop(y„i-i) Ty-^op(ym)] with Ay = — - — and initial condition %op(0) = %op(0). The partial derivativesnsteps - 1 ”0 ”0dy%op(ym-i) and dyxop(ym) in the above update equations are estimated as in the implicit differentiation method, but now atx = ^0Pjn-1(ym-i) andx= *op (ym)> respectively. The update rule is repeated until the last step, yielding the first variables (e.g. the circuit angles)x = xop _1(1)-

[0049] The Hessian inversion in 3y%op(y) can be avoided by calculating the componentseTdyxop(y) = -uT(y, x)Vxf'(y, x) |,x~ ■*opwhere ejis a unit vector in the direction of the j-th component of x, and uj(y,x) is obtained by solving the linear system of equationsej= H(y,x)uj(y,x).

[0050] Here, the Hessian may refer to the matrix of second derivatives of the cost function with respect to the first variables, i.e., Hij=∂2f / ∂xi∂xj.

[0051] The inversion of the Hessian may refer to computing the matrix inverse of the Hessian, which may be used in implicit differentiation to solve for parameter updates. In practice, this may be performed numerically, and approximations such as diagonalization or regularization may be employed to ensure stability.

[0052] Since the Hessian is symmetric, the above linear system of equations can be solved approximately, e.g., through the minimal residual method. Different issues may affect the optimization routine in practice. First, the number of cost function evaluations in the full Hessian estimation scales quadratically with the number of circuit parameters M, whereas estimating gradients only scales linearly with M. Moreover, the Hessians may acquire large condition numbers through the optimization process, which may introduce significant numerical errors to x̃op(y).

[0053] These issues may be addressed for example by imposing the diagonal approximation to the Hessian and by adding a positive regularization term to its entries. Additionally, the number of quantum-computer calls can be further reduced if only subsets of the circuit parameters are updated at each step and by reusing previously estimated expectation values composing f and f. These actions ensure that the quantum optimization execution is feasible with the state-of-the-art quantum computers.

[0054] In some embodiments of the above implementation, the variables x̃op(ym) estimated through the update rule of the Heun’s method can be further optimized through the gradient descent method applied at the desired steps. Besides improving the quality of the results at a linear additional cost, this extends the algorithm flexibility. For example, implicit differentiation could be used as educated guesses for the initial variables (e.g. angles) of a gradient method.

[0055] The optimization routine described above can be used e.g. in quantum chemistry and industry applications, for example those that are in the scope of QU BO problems. Below, an example of how the optimization routine (referred to as ID) and its improved version (referred to as IDGD) may be used for solving the unweighted max cut problem of a graph G = (V, ε) is given. This is an example of QUBO problem mapped onto the problem Hamiltonian operatorHP(λ) = (ẐjẐk- 1).j,k e £

[0056] Fig. 3 shows the noiseless evolution of the cost function describing a complete unweighted graph with 12 nodes, where HP(λ) is given by the above equation, and the mixing Hamiltonian operator ĤM(θ) and variational circuits Û(φ) are chosen respectively as 12ĤM(λ) = -(1 - λ)∑Xj7=1andn 12exp(-iθjYj / 2)exp ( - 2~)’J=iwhere θ = (θ1,...,θ12), X̂j, Ŷj, and Ẑj are Pauli operators acting on qubit j. The results in Fig. 3 are compared to the ground-state energy of Ĥ(λ) obtained through numerical diagonalization. The approximation ratio of 0.937 obtained with ID increases to 0.986 with IDGD upon extra five gradient descent iterations at λ = λf= 1.

[0057] As a remark to the above example implementation of the discussed embodiment of the method 100, the success and feasibility of the embodiment depend on key factors, such as the number of discretization steps in A, the choices of mixing Hamiltonian ĤM(λ), and the family of circuits Û(θ). A deep understanding of the interplay among these features is necessary for maximizing the algorithm performance given the target optimization problems and quantum hardware constraints.

[0058] Furthermore, the method 100 may be used in conjunction with e.g. a variational quantum eigensolver (VQE) algorithm discussed hereinbefore, quantum adiabatic optimization algorithm discussed hereinbefore as well, short cuts to adiabaticity, counterdiabatic scheme, and / or a combination thereof in order to obtain more precise estimates of xop(yj). For example, the VQE algorithm may be employed to further optimize the set of optimal values of the set of first variables, wherein the initial set of values is defined from the set of the gradient values. Specifically, the gradient of the ground state ∇yxop(y) may be used to obtain a good starting ground for the value of xop(y / +1) from that of xop(y7), and this may then be optimized further by running VQE on xop(y7+1) until convergence.

[0059] In some embodiments, the method 100 may further comprise optimizing 221 the computed set of optimal values of the set of the first variables by employing at least one gradient and higher-order derivative-based optimization method. The at least one gradient and higher-order derivative-based optimization method may comprise at least one of thesteepest descent method, Newton’s method, stochastic gradient descent, coordinate descent, Frank—Wolfe algorithm, random coordinate descent, or nonlinear conjugate gradient method.

[0060] In addition, the method 100 may further comprise employing 204, 241, by the classical computer, a classical optimization algorithm to initially 204 or further 241 optimize the values of the set of first variables obtained by reading out the final quantum state of the register of n quantum entities. For example, if the final Hamiltonian operator encodes a QUBO problem, the quantum computational device may find a bit string that is close to the ground state. This string may then be used as a starting point of a classical optimization algorithm to find an even more accurate solution.

[0061] In some embodiments, the set of second variables corresponds to a loop in a parameter space. The benefit here is that employing a classical optimization method is employed for initially or further optimize the values of the set of first variables and / or the steepest decent method in addition to implicit differentiation method may enable getting close to the ground state by making a loop in the parameter space, since during the loop it is possible to perform a transition to the ground state thanks to the other algorithms from implicit differentiation.

[0062] Fig. 4 illustrates a flow chart of a computer-implemented method for quantum-classical hybrid optimization according to an embodiment 200 of a first aspect of the present disclosure. The method comprises between starting 201 and finishing 299 step 210 and iteratively executed steps 220, 230, and 240.

[0063] In step 210, a set of second variables that parameterize a Hamiltonian operator, and first variables that parameterize a quantum circuit, are provided preferably by a classical computer. The second variables may define, for example, interpolation parameters between a mixing Hamiltonian and a problem Hamiltonian forming the Hamiltonian operator, while the first variables may correspond to parameters such as rotation angles, phase shift angles, or unitary transformation parameters for quantum operations applied to quantum entities of a quantum computing device.

[0064] In step 220, the quantum circuit is executed on a quantum computing device to generate outcomes from read outs of the executions. The quantum circuit is parameterized by the first variables, and the execution may be repeated multiple times to obtain a statistical distribution of outcomes corresponding to measurement outcomes.

[0065] In the context of the disclosed method, the outcomes from read outs of the executions may be represented as bit strings (as may be for binary-encoded systems) or strings of digits (as may be for systems employing higher-dimensional encoding), depending on the application. Additionally or alternatively, each outcome corresponds to adistinguishable quantum state into which the quantum system collapses upon measurement. In other words, each outcome is associated with a quantum state of the system that is distinguishable within the chosen measurement basis.

[0066] In step 230, the outcomes obtained from the quantum computing device are processed by a classical computer. This processing comprises computing an expectation value of the Hamiltonian operator to obtain a cost function. Using the cost function, the classical computer further computes implicit derivative information of the first variables with respect to the set of second variables. The implicit derivative information may be determined, for example, by implicit differentiation techniques involving gradients, Hessians, or mixed partial derivatives of the cost function.

[0067] In step 240, the first variables are updated by the classical computer using the computed implicit derivative information obtained in step 230 and the updated second variables. The updated second variables are generated by applying an update rule, which may be predetermined or adaptive, and may involve interpolation, fixed increments, or selection based on a minimization criterion relating to the cost function or the first variables.

[0068] The method iteratively repeats steps 220 to 240 until the second variables reach final values, thereby evolving both the quantum circuit and the Hamiltonian operator. The method may be implemented by a hybrid system comprising a classical computer and a quantum computing device. The classical computer is configured to prepare the quantum circuit, transmit execution instructions to the quantum computing device, receive measurement results, and perform the processing and updating steps described above. The quantum computing device may be a gate-based quantum computer, optionally with pulse-level control.

[0069] A key technical effect of the disclosed quantum-classical hybrid optimization method is that, as the Hamiltonian operator is gradually evolved toward the target problem by updating the second variables, the quantum circuit parameters (first variables) are adaptively updated in tandem using implicit derivative information computed from quantum measurements. This coordinated update ensures that the quantum state remains close to the instantaneous minimum of the evolving Hamiltonian at each step, rather than drifting away from optimality. As a result, the method efficiently tracks and approximates the optimal solution throughout the evolution, avoiding the need for full re-optimization at each stage. This approach overcomes limitations of prior quantum and classical optimization techniques — such as slow convergence or loss of solution quality — by leveraging quantum-calculated gradients to maintain near-optimality, thereby enabling practical and scalable solutions to complex optimization problems that are otherwise intractable for classical algorithms.

[0070] In particular, the iterative process described herein includes performing steps 220-240 for the final values of the second variables. This ensures that the quantum circuit and the Hamiltonian operator are evolved and processed at the target setting, and that the solution obtained corresponds to the ground state or optimal configuration defined by the final values. By explicitly executing and processing the final iteration, the method guarantees that the optimization outcome is determined for the intended problem Hamiltonian.

[0071] In some embodiments, the method further comprises outputting an outcome selected from the outcomes obtained in the last iteration, or outputting the expectation value of the Hamiltonian operator obtained in the last iteration. The output may be used for further classical optimization or as a solution to the original optimization problem mapped to the Hamiltonian operator.

[0072] In some embodiments, in step 230, using the cost function comprises computing a derivative of the cost function with respect to the first variables, and using the derivative of the cost function with respect to the first variables to compute the implicit derivative information of the first variables with respect to the set of second variables. This can be implemented by applying classical or quantum-assisted differentiation techniques to the cost function, such as parameter-shift rules or finite differences, to obtain gradients with respect to the quantum circuit parameters. These gradients are then used to determine how changes in the second variables affect the optimal circuit parameters.

[0073] In some embodiments, the derivative of the cost function comprises one of gradients of the cost function with respect to the first variables or second-order derivative (Hessians) of the cost function with respect to the first variables. Utilizing higher-order derivative information allows for more sophisticated optimization strategies, such as Newton-type updates, which can accelerate convergence.

[0074] In some embodiments, in step 230, computing the implicit derivative information is computed by implicit differentiation and comprises the use of at least one of: a Hessian of the cost function with respect to the first variables; or a mixed partial derivative of the cost function with respect to the first variables and the updated second variables. This can be implemented by applying implicit differentiation formulas, where the Hessian and mixed partial derivatives are estimated using quantum measurements and classical computation, enabling the calculation of how the optimal circuit parameters (i.e. the first variables) change as the Hamiltonian evolves. Implicit differentiation provides a principled way to track the optimal parameters during Hamiltonian evolution, ensuring the quantum state remains near the minimum and reducing the need for repeated full optimization iterations.

[0075] In some embodiments, the Hamiltonian is decomposed into a set of quantum operators, such as Pauli strings, matrix product operators, projection operators, operators obtained by similarity transformations, or sums thereof. Decomposing the Hamiltonian into the set of quantum operators enables efficient quantum measurement and flexible adaptation to different hardware architectures, improving the practicality and scalability of the method.

[0076] In some embodiments, the update rule comprises interpolation between initial and final values of the second variables. This can be implemented by linearly or nonlinearly interpolating the second variables over a sequence of steps, gradually transforming the Hamiltonian from an initial to a final form. Interpolation-based update rules provide a controlled and smooth evolution of the Hamiltonian, which can help maintain adiabatic-like conditions and improve the reliability of finding high-quality solutions.

[0077] In some embodiments, the update rule is predetermined update rule, optionally wherein the predetermined rule comprises at least one of: a fixed increment of the second variables, a linear interpolation between initial and final values of the second variables, a discretized sequence of values for the second variables, a rule based on a predefined schedule, or a rule of a constant step. This can be realized by defining a fixed schedule for updating the second variables, such as incrementing by a constant value or following a precomputed sequence, which is applied at each iteration. Predetermined update rules simplify implementation and analysis, and can provide predictable convergence behavior, especially when the problem structure or hardware constraints favor fixed schedules.

[0078] In some embodiments, in step 230, applying the update rule comprises applying a step size parameter to the update of the second variables. This can be implemented by multiplying the computed update direction for the second variables by a tunable step size parameter, which may be fixed or adaptively chosen. Controlling the step size allows for balancing convergence speed and stability, enabling the method to adapt to different problem landscapes and hardware noise levels.

[0079] In some embodiments, in step 230, applying the update rule comprises incrementing the second variables by a value proportional to at least one of: a gradient or norm of the implicit derivative information. This can be realized by computing the gradient or norm of the implicit derivative information and using it to scale the update applied to the second variables. Gradient-based updates enable adaptive and data-driven evolution of the Hamiltonian parameters, which can accelerate convergence and improve solution quality by focusing on updates where they are most effective.

[0080] In some embodiments, the update rule is adaptive and comprises, in step 230, applying the update rule by selecting at least one of: a direction, a magnitude, or acoordinate, of the update for the second variables based on minimization criterion relating to a change in at least one of: the first variables or the cost function. This can be implemented by evaluating different possible update directions or magnitudes and selecting the one that advantageously reduces the cost function or optimizes the circuit parameters. Adaptive update rules allow the method to respond dynamically to the optimization landscape, improving hence efficiency of the optimization.

[0081] In some embodiments, the minimization criterion used for selecting the direction, magnitude, or coordinate of the update comprises at least one of: smallest magnitude, largest magnitude, a norm of the derivative information, a linear combination of the derivative information, a threshold-based criterion, a probabilistic or randomized selection, a rule-based selection, or an algorithmic heuristic based on the derivative information. This can be realized by implementing selection logic in the classical computer that evaluates these criteria at each iteration and chooses the update accordingly. Using flexible minimization criteria enables the method to tailor the optimization process to the specific problem structure or hardware constraints, potentially improving convergence and solution quality.

[0082] In some embodiments, the Hamiltonian operator is a composition, optionally a sum, of a mixing Hamiltonian and a problem Hamiltonian, and wherein the second variables interpolate between the mixing Hamiltonian operator and the problem Hamiltonian operator. This can be implemented by defining the Hamiltonian operator H(y) = WM(y) + HP(y), where the mixing Hamiltonian operator evolves into the problem Hamiltonian operator during the optimization process. Specifically, it may be so that Ĥ(yi) = ĤM(yi) for the initial set of second variables, and Ĥ(yf) = ĤP(yf) for the final set of second variables. This structure enables the method to start from an easily prepared ground state and gradually evolve toward the target problem, improving the likelihood of finding high-quality solutions.

[0083] In some embodiments, the interpolation by the second variables is such that, for the initial values of the second variables, the Hamiltonian operator corresponds to the mixing Hamiltonian operator, and for the final values of the second variables, the Hamiltonian operator corresponds to the problem Hamiltonian operator. This can be realized by setting the initial value of the interpolation parameter to zero (mixing Hamiltonian operator) and the final value to one (problem Hamiltonian operator). This ensures a well-defined and physically meaningful evolution path, facilitating initialization and final readout of the solution.

[0084] While the second variables often serve as interpolation parameters between a mixing and a problem Hamiltonian operator, more generally, they may represent any set ofparameters that define the evolution or structure of the Hamiltonian operator during optimization.

[0085] In some embodiments, the initial values of the second variables are set to zero, providing a clear and reproducible initialization for the method.

[0086] In some embodiments, the Hamiltonian operator is a composition, optionally a sum, of two or more Hamiltonian operators comprising at least a first Hamiltonian operator and a second Hamiltonian operator, and wherein the second variables control the interpolation between the two or more Hamiltonian operators.

[0087] When the Hamiltonian operator is a composition, optionally a sum, of two or more Hamiltonian operators comprising at least a first Hamiltonian operator and a second Hamiltonian operator, the Hamiltonian operator can be mathematically expressed as H(y) = ^jaj(y)Flj, where Hj are distinct Hamiltonian operators and a7(y)arefunctions of the second variables that determine the relative contribution of each component (operators) at a point during the interpolation. The second variables may be varied continuously or discretely, allowing smooth or in steps-wise transition between different Hamiltonian regimes. This flexible composition enables the quantum system to explore a broader solution space and adapt the optimization process to complex problem structures. In other words, such Hamiltonian operator provides an increased versatility in the optimization, allowing for tailored evolution paths that can improve convergence and solution quality for a wide range of problems.

[0088] In some embodiments, the interpolation by the second variables is such that, for the initial values of the second variables, the Hamiltonian operator is dominated by, or optionally corresponds to, the first Hamiltonian operator, and for the final values of the second variables, the Hamiltonian operator is dominated by, or optionally corresponds to, the second Hamiltonian operator.

[0089] This interpolation can be realized by defining the interpolation functions α1(y) and α2(y) such that |α1(yi)| ≫ |α2(yi)| and |α1(yf)| ≪ |α2(yf)|, and / or by setting α1(yi) = 1, α2(yf) = 0 and vice versa at the final values. This ensures that the quantum system starts at least approximately in the ground state of the first Hamiltonian operator and evolves towards the ground state of the second Hamiltonian operator as the second variables are varied and / or evolved. This allows that the system can be initialized in a state that is easy to prepare and then gradually transformed into a state that encodes the solution to the target optimization problem, thereby improving the reliability and efficiency of the overall optimization process.

[0090] In some embodiments, wherein the Hamiltonian operator comprises, during the interpolation, a catalyst Hamiltonian operator Hcatthat is distinct from the first and secondHamiltonian operators, the catalyst Hamiltonian operator being configured to perform at least one of the following: modify the energy states of the Hamiltonian operator, or facilitate transitions between quantum states during the evolution from the initial ground state to the final ground state of the Hamiltonian operator.

[0091] The catalyst Hamiltonian operator Hcatcan be included in the Hamiltonian operator such that H(y) =«j(y)Hj + acat(y)Hcat, where acat(y)Hcatis nonzero during at least a portion of the interpolation. The catalyst Hamiltonian may be chosen to not commute, or alternatively to commute, with the first or second Hamiltonian operators, or to introduce additional terms that alter the energy landscape, such as increasing the minimum energy gap or enabling transitions between quantum states that would otherwise be suppressed. Using such catalyst Hamiltonian operator provides an enhancement of the quantum evolution process: the presence of the catalyst Hamiltonian can accelerate the transition from the initial to the final ground state, reduce the likelihood of the system becoming trapped in local minima, and improve the overall efficiency and success probability of finding the optimal solution.

[0092] For the initial values of the second variables (yi) it is advantageous to set the coefficient αcat(yi) to negligible, such as zero, such that the catalyst Hamiltonian operator does not contribute to the Hamiltonian operator at the beginning of the interpolation. In this configuration, the system is governed solely advantageously mainly or alternatively fully by the first Hamiltonian operator, ensuring that the initial quantum state can be efficiently prepared without influence from the catalyst term. This approach facilitates initialization in a well-defined and easily accessible ground state.

[0093] Similarly, for the final values of the second variables (yf), it is advantageous to set the coefficient αcat(yf) to a negligible value, such as zero, so that the catalyst Hamiltonian operator does not contribute to the Hamiltonian operator at the end of the interpolation. In this configuration, the system is governed advantageously mainly, or alternatively fully, by the second Hamiltonian operator, ensuring that the final quantum state corresponds to the ground state of the target problem Hamiltonian, free from any residual influence of the catalyst term. This guarantees that the solution obtained at the end of the optimization process is determined solely by the intended problem Hamiltonian.

[0094] In some embodiments, at least one iteration of steps 220-240 comprises: in step 220, executing a plurality of quantum circuits, which are parameterized by different sets of first variables, to obtain a plurality of sets of outcomes from read outs of the executions of the different quantum circuits; and in step 230, processing, by the classical computer, the plurality of sets of outcomes by computing expectation values of the Hamiltonian operator corresponding to the different sets of first variables, selecting the set of first variablesassociated with the lowest cost function, and computing implicit derivative information of the selected set of first variables with respect to the set of second variables. This can be realized by running multiple quantum circuits in parallel or sequence, each with different parameter settings, and then selecting the best-performing set for further optimization. Evaluating multiple candidates in parallel increases the likelihood of escaping local minima and finding better solutions, enhancing the robustness of the optimization. The at least one iteration may be at least one of: the first iteration, one or more subsequent iterations, or both.

[0095] In some embodiments, the set of second variables is considered a bidimensional vector of Hamiltonian parameters y = (y1,y2)T∈ ℝ2, wherein the Hamiltonian operator can be defined asĤ(y) = (1 - y1)ĤM+ y2ĤP,wherein the method has the following update rule for:y(k+1)= [min(y1(k)+ β1(k)G1(k), 1)]min(y2« + ^2WG2w, l)]’where {βi(k)> 0} are step sizes and {Gi(k)} are the components of the bidimensional vector

[0096] The update rule described above is demonstrated here for solving the unweighted MaxCut problem on a complete graph with n = 6 nodes, each being represented by a different qubit. Below, we denote the Pauli operators for qubit j as Xj, Yj, and Zj. The corresponding problem can be mapped onto the problem Hamiltonian operatorĤP= (1 / 2) ∑(ẐjẐk- Î),j,k e £where £ denotes the set of edges of the graph and I is the identity operator. In this example we choose the mixing Hamiltonian operator asnĤM= -∑(X̂j+ Ẑj),7=1and the parametrized state |Ψ(x)⟩ = Û(x)|(0)⟩, wheren= ∏exp(-ixjYj / 2)J=i

[0097] Fig. 5 shows the evolution of the Hamiltonian parameters (the second variables) y, the optimal circuit angles %*(y) (the first variables), and the cost function f[x*(y),y]. In this setting, the ground state of ĤMis produced by the angles {xj* = π / 4}. With the update rule described above and fixed step sizes β1= β2= 0.05, the Hamiltonian operator convergesto ĤPin 10 iterations. For each iterated value of y, the corresponding Hamiltonian minimization problem was solved by simulated annealing in Mathematica. In this example, the state generated at the end of the protocol is a bitstring state |Ψ(x)⟩ minimizing the expectation value of ĤP. This is indicated in the center panel of Fig. 5 by the angle concentration at 0 and n starting at the 10-th iteration.

[0098] In certain scenarios, the exact preparation of the ground state of the initial Hamiltonian may not be feasible, for example, due to limitations in hardware, lack of analytical solutions, and / or insufficient prior knowledge of optimal circuit parameters. In such cases, executing a plurality of quantum circuits with different parameterizations allows for an empirical search over the parameter space, increasing the likelihood of identifying a circuit configuration that yields a low initial cost function value. This approach is particularly advantageous when the landscape of the cost function is complex and / or contains multiple local minima, as it enables the selection of a promising starting point for further optimization. In other words, by evaluating multiple candidates, the method provides robustness against suboptimal initialization and can be especially useful in practical implementations where exact ground state preparation is not possible. This strategy complements cases where the ground state is known, offering a flexible framework adaptable to both well-characterized and challenging optimization problems.

[0099] In some embodiments, the first variables comprise parameters that parameterize quantum operations applied to quantum entities of the quantum computing device, the parameters being selected from rotation angles, phase shift angles, and unitary transformation parameters. This can be realized by mapping the first variables to the control parameters of quantum gates in the circuit, such as rotation angles for single-qubit gates or parameters for multi-qubit unitaries.

[0100] In some embodiments, the derivative of the cost function with respect to the first variables is computed using a technique selected from the group consisting of: finite difference method, use of matrix product states, application of Fourier transformation, implicit differentiation, parameter shift rule, Heun’s method, path integrals, or a combination thereof.

[0101] In some embodiments, in step 240, updating the first variables comprises employing a variational optimizer to the first variables, wherein the initial values for the optimizer are defined from the derivative of the cost function with respect to the first variables, optionally wherein the variational optimizer is selected from the group consisting of a variational quantum eigensolver (VQE), or quantum approximate optimization algorithm (QAOA). This can be realized by initializing a classical or hybrid optimizer with the computedgradients and using it to iteratively refine the quantum circuit parameters. Employing advanced variational optimizers can accelerate convergence.

[0102] In some embodiments, the derivative of the cost function with respect to the first variables is further optimized by employing at least one of the following methods: steepest descent, Newton’s method, stochastic gradient descent, coordinate descent, Frank–Wolfe algorithm, random coordinate descent, or nonlinear conjugate gradient method. This can be implemented by selecting and applying one or more classical optimization algorithms to refine the quantum circuit parameters based on the computed derivatives. Using advanced classical optimization methods can improve convergence rates and / or solution quality, especially for complex or high-dimensional problems.

[0103] In some embodiments, computing the implicit derivative information comprises employing a diagonal approximation to the Hessian and / or adding regularization terms to Hessian entries. This can be realized by simplifying the Hessian matrix to its diagonal elements and / or stabilizing the inversion by adding regularization, reducing computational overhead and improving numerical stability. These approximations make the method more practical for large-scale problems and noisy hardware, while maintaining effective parameter updates.

[0104] In some embodiments, in at least one iteration, a subset of the first variables is updated instead of all the first variables. This can be implemented by selecting a subset of parameters to update at each step, for example using coordinate descent or block-wise updates. Partial updates reduce computational cost and can help avoid overfitting or instability, especially in high-dimensional parameter spaces, making the method more practical for large-scale problems.

[0105] The method for quantum-classical hybrid optimization may be a computer-implemented method for solving an optimization problem mapped to the Hamiltonian operator, executed on a hybrid system comprising a classical computer and a quantum computing device.

[0106] In some embodiments, the optimization problem is a combinatorial optimization problem, optionally selected from the group consisting of quadratic binary or integer, higher-order binary or integer, constrained binary or integer, or unconstrained binary or integer optimization problems.

[0107] In some embodiments, especially when the optimization problem is a combinatorial optimization problem, the method further comprises outputting an outcome, optionally as a bit string or a string of digits, selected from the outcomes obtained in the last iteration.

[0108] In some embodiments, especially when the method further comprises outputting an outcome, the outcome is selected based on statistical frequency, optionally such that theselected outcome has the highest statistical frequency among the outcomes obtained in the last iteration. This can be implemented, for example, by counting the occurrence of each outcome in the measurement results and selecting the most common one.

[0109] Alternatively, the optimization problem may be a continuous optimization problem, optionally selected from the group consisting of quadratic continuous, higher-order continuous, constrained continuous, or unconstrained continuous optimization problems. This can be realized by mapping the continuous problem to a parameterized Hamiltonian and applying the method to optimize over continuous variables. Supporting continuous optimization broadens the method’s applicability to scientific and engineering problems involving real-valued parameters.

[0110] In some embodiments, the method further comprises outputting the expectation value of the Hamiltonian operator obtained in the last iteration (i.e., with the final second variables, when the interpolation has reached the last item). This can be implemented by reporting the final computed expectation value as a measure of solution quality or as the solution itself for certain problems. Providing the expectation value enables quantitative assessment of the solution and can be used as input for further analysis or optimization.

[0111] In some embodiments, the quantum computing device is a gate-based quantum computing device, optionally having pulse level control. This can be realized by implementing the quantum circuit on a gate-based quantum processor, with optional access to pulse-level programming for fine-tuned control. Gate-based implementation ensures compatibility with current and emerging quantum hardware platforms, and pulse-level control allows for hardware-specific optimizations.

[0112] In some embodiments, the method further comprises performing a further optimization using the output outcome as a starting point, optionally for a classical optimization method.

[0113] Another aspect of the disclosure relates to a computer program product comprising instructions which, when executed by a hybrid classical-quantum computing device, cause the device to perform the steps of the method of the first aspect of the disclosure, i.e. the method described above. The computer program may be implemented in any suitable programming language and may be provided as source code, object code, or in an intermediate form such as bytecode. The instructions may be organized as one or more modules, functions, or classes configured to implement the method steps. For example, the computer program may comprise modules for constructing quantum circuits, transmitting execution instructions to a quantum computing device, receiving quantum circuit outcomes, and processing the outcomes as described in the first aspect.

[0114] The computer program may be distributed in various forms, including as a standalone application, a library, or a cloud-based service. In one embodiment, the program is provided as a downloadable package from a server or as part of a software development kit (SDK) for quantum computing. The program may also be embedded in a hybrid quantum-classical workflow platform, enabling integration with other computational tools. The computer program may be stored on a tangible medium or transmitted electronically via a network.

[0115] A hybrid classical-quantum computing device configured to perform the method according to the first aspect of the disclosure, i.e. the method described above.

[0116] In one embodiment, the hybrid classical-quantum computing device according to this disclosure comprises a classical computer 400 and a quantum computing device 300, as illustrated in Fig. 6. The quantum computing device 300 is arranged to execute quantum circuits and provide quantum circuit outcomes for subsequent processing. Within device 300, a quantum processor 302 is configured to perform quantum computations using quantum elements such as qubits or qudits. The quantum processor may be implemented using superconducting circuits, ion traps, photonic quantum devices, or neutral atom arrays, and is equipped with components for maintaining quantum elements in an environment suitable for quantum computation, such as cryogenic cooling systems. Quantum circuits, formed by sequences of quantum gates, operate on the quantum elements, with the gates including operations acting on single or multi-level quantum elements, based on a universal gate set, as well as state preparation and measurement operations.

[0117] Device 300 further comprises measurement means 304, which are configured to measure quantum states after execution of the quantum circuit and to produce quantum circuit outcomes, typically represented as bitstrings in the computational basis. Control means 306 are provided to receive instructions from the classical computer 400 and to apply quantum gates to the quantum elements according to those instructions. An internal quantum bus 312 provides a communication channel for transferring quantum information between the components of the quantum computing device.

[0118] The classical computer 400 is arranged to prepare quantum circuits, transmit execution instructions to the quantum computing device 300, and process quantum circuit outcomes received in return. Computer 400 comprises a processor 402, memory 404, and data storage 406, all operatively connected via a data bus 412. The processor is configured to execute instructions for implementing the steps of the method described above, while the memory and data storage are arranged to store quantum circuit definitions, configuration parameters, and computer program for performing the method. Input device 408 is providedfor receiving user commands, and output device 410 is arranged to display results and / or transmit processed data to other systems.

[0119] The two apparatuses 300 and 400 communicate via a secure interface 312, which may include network adapters, wired connectors, or wireless transceivers, and may use secure protocols such as TLS. This communication interface supports bidirectional exchange of data, including execution instructions transmitted from the classical computer 400 to the quantum computing device 400 and quantum circuit outcomes transmitted in the reverse direction. This arrangement enables integration into a hybrid quantum-classical workflow, where the classical computing apparatus coordinates quantum circuit execution and performs classical post-processing.

[0120] The system may be implemented as an integrated device or as a distributed solution, where the classical and quantum apparatuses are connected via a network or cloud platform. This architecture enables scalability and flexibility in deploying quantum-classical workflows and supports secure communication for data integrity. The system is configured to execute the hybrid workflow by coordinating quantum circuit execution and classical post-processing, and may include machine-readable media storing instructions for implementing the hybrid workflow.

[0121] As used herein, the terms “classical computer,” “classical computing apparatus,” “quantum computing device,” “quantum computing apparatus,” and “quantum computational device” etc. are intended to refer to their respective components as described above, and may be used interchangeably where appropriate. In particular, “quantum computational device” encompasses any apparatus capable of performing quantum computations, including quantum computers of various physical implementations. “Classical computer” or “classical computing apparatus” refers to any apparatus capable of performing classical computations and interfacing with a quantum computing device.

[0122] In one embodiment, there is provided a hybrid classical-quantum computing device for quantum-classical hybrid optimization, configured to perform the method as described in the first aspect above. The device comprises means for providing a set of second variables that parameterize a Hamiltonian operator, and first variables that parameterize a quantum circuit executable on a quantum computing device. The quantum circuit is constructed such that, when executed on the quantum computing device, it prepares a quantum state determined by the first variables and the current values of the second variables.

[0123] The device further comprises means for executing, on the quantum computing device, the quantum circuit to generate outcomes from readouts of the executions. Theoutcomes may be obtained by measuring the quantum state in a computational basis, resulting in a set of bitstrings or other outcome representations.

[0124] Additionally, the device comprises means for processing, by a classical computer, the outcomes by computing an expectation value of the Hamiltonian operator to obtain a cost function. Using the cost function, the classical computer is further configured to compute implicit derivative information of the first variables with respect to the set of second variables, for example by applying differentiation techniques such as parameter-shift rules, finite differences, or implicit differentiation.

[0125] The device also comprises means for updating, by the classical computer, the first variables by using the implicit derivative information and updated second variables, wherein the updated second variables are generated by applying an update rule. The update rule may be predetermined or adaptive, and may involve interpolation, fixed increments, or selection based on a minimization criterion relating to the cost function or the first variables.

[0126] The device is configured to iteratively repeat the steps of quantum circuit execution, outcome processing, and variable updating until the second variables reach final values, thereby evolving both the quantum circuit and the Hamiltonian operator.

[0127] The system may further comprise means for outputting a solution to the optimization problem, such as a bitstring selected from the outcomes obtained in the last iteration, or the expectation value of the Hamiltonian operator at the final iteration.

[0128] Still another aspect of the disclosure relates to a computer-implemented method, i.e., a classical computer-implemented method, for controlling quantum-classical hybrid optimization. The method comprises: providing a set of second variables that parameterize a Hamiltonian operator, and a quantum circuit comprising first variables that parameterize the circuit executable on the quantum computing device; (i) instructing a quantum computing device to execute the circuit; (ii) receiving outcomes from readouts of the executions of the circuit; (iii) processing the outcomes by computing an expectation value of the Hamiltonian operator to obtain a cost function, and, using the cost function, computing implicit derivative information of the first variables with respect to the set of second variables; and (iv) updating the first variables by using the implicit derivative information and updated second variables, wherein the updated second variables are generated by applying an update rule, iteratively repeating steps (i)–(iv) until the second variables reach final values, thereby controlling evolution of the quantum circuit and the Hamiltonian operator.

[0129] All embodiments described above in connection with the first aspect and other aspects of the disclosure, to the extent they relate to the classical computer, are equally applicable to this further aspect. Unless expressly excluded, features disclosed inconnection with one aspect may be combined with features of another aspect. For the sake of avoiding redundancy, those embodiments are not repeated here.

[0130] Still another aspect is provided. This aspect of the disclosure relates to a computer program comprising instructions which, when executed by a classical computer cause the system to perform the method described in connection with the previous aspect, i.e. the computer-implemented method. The computer program may be implemented in any suitable programming language, and may be provided as source code, object code, or in an intermediate representation such as bytecode. The instructions may be organized as one or more modules, functions, or classes configured to implement the steps of the method described above. For example, the computer program may include modules for constructing quantum circuits, transmitting execution instructions to a quantum computing device, receiving quantum circuit outcomes, and processing distributions of outcomes or updating the quantum circuit for the next iteration as described above.

[0131] Still another aspect is provided. This aspect aspect of the disclosure relates to a computer-readable medium storing a computer program comprising instructions which, when executed by a data processing system, cause the system to perform the method described in connection with the previous aspect, i.e. the computer program. The computer-readable medium may include any suitable storage device, such as a magnetic disk, an optical disk, a solid-state drive, or a semiconductor memory. Additionally or alternatively, the computer-readable medium may comprise a non-transitory medium such as flash memory or a read-only memory (ROM). The computer-readable medium may store the computer program in a format suitable for execution by a processor, such as executable binaries or interpreted scripts.

[0132] The embodiments described in the following detailed description are provided as non-limiting examples. A person skilled in the art will recognize that the underlying concepts of the disclosure can be implemented in ways other than those explicitly described. Unless expressly stated otherwise, features disclosed in connection with one embodiment may be combined with features of other embodiments, provided such combinations are technically compatible. References to certain embodiments in multiple instances do not imply that the reference is limited to a single embodiment or that the described characteristic is applicable only to that embodiment. Combinations may include features across different aspects of the disclosure.

Claims

1. Claims1. A method for quantum-classical hybrid optimization comprising:3.providing a set of second variables that parameterize a Hamiltonian operator, and first variables that parameterize a quantum circuit executable on the quantum computing device;4.(a) executing, on a quantum computing device, the quantum circuit, to generate outcomes from read outs of the executions;5.(b) processing, by a classical computer, the outcomes by computing an expectation value of the Hamiltonian operator to obtain a cost function, and, using the cost function, computing implicit derivative information of the first variables with respect to the set of second variables; and6.(c) updating, by the classical computer, the first variables by using the implicit derivate information and updated second variables, wherein the updated second variables are generated by applying an update rule,7.iteratively repeating steps (a)-(c) until the second variables reach final values, thereby evolving the quantum circuit and the Hamiltonian operator.

2. The method according to claim 1, wherein, in step (b), using the cost function comprises computing a derivative of the cost function with respect to the first variables, and using the derivative of the cost function with respect to the first variables to compute the implicit derivative information of the first variables with respect to the set of second variables.

3. The method according to claim 2, wherein the derivative of the cost function comprises one of gradients of the cost function with respect to the first variables or second-order derivative (Hessians) of the cost function with respect to the first variables.

4. The method according to any preceding claim, wherein, in step (c), computing the implicit derivative information is computed by implicit differentiation and comprises the use of at least one of: a Hessian of the cost function with respect to the first variables; or a mixed partial derivative of the cost function with respect to the first variables and the updated second variables.

5. The method according to any preceding claim, wherein the Hamiltonian is decomposed into a set of quantum operators, such as Pauli strings, matrix product operators, projection operators, operators obtained by similarity transformations, or sums thereof.

6. The method according to any preceding claim, wherein the update rule comprises interpolation between initial and final values of the second variables.

7. The method according to any preceding claim, wherein the update rule is predetermined update rule, optionally wherein the predetermined rule comprises at least one of: a fixed increment of the second variables, a linear interpolation between initial and final values of the second variables, a discretized sequence of values for the second variables, a rule based on a predefined schedule, or a rule of a constant step.

8. The method according to any preceding claim, wherein, in step (c), applying the update rule comprises applying a step size parameter to the update of the second variables.

9. The method according to any preceding claim, wherein, in step (c), applying the update rule comprises incrementing the second variables by a value proportional to at least one of: a gradient or norm of the implicit derivative information.

10. The method according to any preceding claim, wherein the update rule is adaptive and comprises, in step (c), applying the update rule by selecting at least one of: a direction, a magnitude, or a coordinate, of the update for the second variables based on minimization criterion relating to a change in at least one of: the first variables or the cost function.

11. The method according to claim 10, wherein the minimization criterion used for selecting the direction, magnitude, or coordinate of the update comprises at least one of: smallest magnitude, largest magnitude, a norm of the derivative information, a linear combination of the derivative information, a threshold-based criterion, a probabilistic or randomized selection, a rule-based selection, or an algorithmic heuristic based on the derivative information.

12. The method according to any preceding claim, wherein the Hamiltonian operator is a composition, optionally a sum, of a mixing Hamiltonian operator and a problem Hamiltonian operator, and wherein the second variables control the interpolation between the mixing Hamiltonian operator and the problem Hamiltonian operator.

13. The method according to claim 12, wherein the interpolation by the second variables is such that, for the initial values of the second variables, the Hamiltonian operator corresponds to the mixing Hamiltonian operator, and for the final values of the second variables, the Hamiltonian operator corresponds to the problem Hamiltonian operator.

14. The method according to claims 12 or 13, wherein for the initial values of the second variables are set to zero.

15. The method according to any claims 1 to 11, wherein the Hamiltonian operator is a composition, optionally a sum, of two or more Hamiltonian operators comprising at least a first Hamiltonian operator and a second Hamiltonian operator, and wherein the second variables control the interpolation between the two or more Hamiltonian operators.

16. The method according to claim 15, wherein the interpolation by the second variables is such that, for the initial values of the second variables, the Hamiltonian operator is dominated by, or optionally corresponds to, the first Hamiltonian operator, and for the final values of the second variables, the Hamiltonian operator is dominated by, or optionally corresponds to, the second Hamiltonian operator.

17. The method according to claims 15 or 16, wherein the Hamiltonian operator comprises, during the interpolation, a catalyst Hamiltonian operator that is distinct from the first and second Hamiltonian operators, the catalyst Hamiltonian operator being configured to perform at least one of the following: modify the energy states of the Hamiltonian operator, or facilitate transitions between quantum states during the evolution from the initial ground state to the final ground state of the Hamiltonian operator.

18. The method according to any preceding claim, wherein at least one iteration of steps (a)-(c) comprises:23.in step (a), executing a plurality of quantum circuits, which are parameterized by different sets of first variables, to obtain a plurality of sets of outcomes from read outs of the executions of the different quantum circuits; and24.in step (b), processing, by the classical computer, the plurality of sets of outcomes by computing expectation values of the Hamiltonian operator corresponding to the different sets of first variables, selecting the set of first variables associated with the lowest cost function, and computing implicit derivative information of the selected set of first variables with respect to the set of second variables.

19. The method according to any preceding claim, wherein the at least one iteration is at least one of: the first iteration, one or more subsequent iterations, or both.

20. The method according to any preceding claim, wherein the first variables comprise parameters that parameterize quantum operations applied to quantum entities of thequantum computing device, the parameters being selected from rotation angles, phase shift angles, and unitary transformation parameters.

21. The method according to any preceding claim, wherein the derivative of the cost function with respect to the first variables is computed using a technique selected from the group consisting of: finite difference method, use of matrix product states, application of Fourier transformation, implicit differentiation, parameter shift rule, Heun’s method, path integrals, or a combination thereof.

22. The method according to any preceding claim, wherein, in step (c), updating the first variables comprises employing a variational optimizer to the first variables, wherein the initial values for the optimizer are defined from the derivative of the cost function with respect to the first variables, optionally wherein the variational optimizer is selected from the group consisting of a variational quantum eigensolver (VQE), or quantum approximate optimization algorithm (QAOA).

23. The method according to any preceding claim, wherein the derivative of the cost function with respect to the first variables is further optimized by employing at least one of the following methods: steepest descent, Newton’s method, stochastic gradient descent, coordinate descent, Frank—Wolfe algorithm, random coordinate descent, or nonlinear conjugate gradient method.

24. The method according to any preceding claim, wherein computing the implicit derivative information comprises employing a diagonal approximation to the Hessian and / or adding regularization terms to Hessian entries.

25. The method according to any preceding claim, wherein, in at least one iteration, a subset of the first variables is updated instead of all the first variables.

26. The method according to any preceding claim, wherein the method for quantum-classical hybrid optimization is a computer-implemented method for solving an optimization problem mapped to the Hamiltonian operator, executed on a hybrid system comprising a classical computer and a quantum computing device.

27. The method according to claim 26, wherein the optimization problem is a combinatorial optimization problem, optionally selected from the group consisting of quadratic binary or integer, higher-order binary or integer, constrained binary or integer, or unconstrained binary or integer optimization problems.

28. The method according to any preceding claim, wherein the method further comprises outputting an outcome, optionally as a bit string or a string of digits, selected from the outcomes obtained in the last iteration.

29. The method according to claim 28, wherein the outcome is selected based on statistical frequency, optionally such that the selected outcome has the highest statistical frequency among the outcomes obtained in the last iteration.

30. The method according to claim 26, wherein the optimization problem is a continuous optimization problem, optionally selected from the group consisting of quadratic continuous, higher-order continuous, constrained continuous, or unconstrained continuous optimization problems.

31. The method according to claim 30, wherein the method further comprises outputting the expectation value of the Hamiltonian operator obtained in the last iteration.

32. The method according to any preceding claim, wherein the quantum computing device is a gate-based quantum computing device, optionally having pulse level control.

33. The method according to any claims 28 to 29, wherein the method further comprises performing a further optimization using the outcome as a starting point, optionally for a classical optimization method.

34. A computer program product comprising instructions which, when executed by a hybrid classical-quantum computing device, cause the device to perform the steps of the method according to any preceding claim.

35. A hybrid classical-quantum computing device configured to perform the method according to any of claims 1 to 33.

36. A computer-implemented method for controlling quantum-classical hybrid optimization comprising:42.providing a set of second variables that parameterize a Hamiltonian operator, and a quantum circuit comprising first variables that parameterize the circuit executable on the quantum computing device;43.(i) instructing a quantum computing device to execute the circuit;44.(ii) receiving outcomes from read outs of the executions of the circuit;45.(iii) processing the outcomes by computing an expectation value of the Hamiltonian operator to obtain a cost function, and, using the cost function, computing implicit derivative information of the first variables with respect to the set of second variables; and46.(iv) updating, the first variables by using the implicit derivate information and updated second variables, wherein the updated second variables are generated by applying an update rule,47.iteratively repeating steps (i)–(iv) until the second variables reach final values, thereby controlling evolution of the quantum circuit and the Hamiltonian operator.

37. A computer program comprising instructions which, when executed by a data processing system, such as a classical computer, cause the system to perform the method claims 36.

38. A computer-readable medium storing a computer program comprising instructions which, when executed by a data processing system, such as a classical computer, cause the system to perform the method according to claim 36.

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