Method and system for solving problems with multiple conflicting objectives

Abstract

To provide a computer-implemented method for obtaining a set of Pareto-optimal solutions to a given multi-objective optimization problem with several different objectives.SOLUTION: The method comprises generating a set of parameters on a classical computation component using a classical optimization algorithm. Based on the set of parameters, a quantum component generates a quantum state by executing a parameterized quantum circuit. A set of basis states is selected from the one quantum state. The classical computation component calculates an objective values set consisting of objective values for each solution in the set of classical solutions, and a new estimate for the Pareto set of solutions and the corresponding Pareto front by the classical optimizer is generated. The internal state of the classical optimization algorithm is updated based on the current set of solutions and the Pareto set and their corresponding sets of objective values.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to a computer-implemented method for obtaining a set of Pareto-optimal solutions for a given multi-objective optimization problem, and to a hybrid quantum-classical computer system implementing such a method. [Background technology]

[0002] Quantum optimization is a very active research area. The current generation of quantum computing hardware devices is still limited in size and susceptible to noise. Currently, the usefulness of so-called noisy intermedia scale quantum (NISQ) devices for applied problems is being tested and evaluated against a number of real-world applications.

[0003] Current state-of-the-art approaches to solving optimization problems using quantum computers are hybrid quantum-classical algorithms, as described, for example, in “Noisy intermediate-scale quantum (NISQ) algorithms, Rev. Mod. Phys. 94, 015004.” Examples are the variational quantum eigensolver (VQE) described in “The variational quantum eigensolver: a review of methods and best practices. Physics Reports, 986, 1-128”, the quantum approximate optimization algorithm (QAOA) described in “A Quantum Approximate Optimization Algorithm, arXiv:1411.4028 (2014)” or “From the quantum approximate optimization algorithm to a quantum alternating operator ansatz, Algorithms 12 (2019)”, as well as “Quantum machine learning beyond kernel methods”, arXiv preprint arXiv:2110.13162 (2021)”. All these methods consider problems where the goal is to find an optimal solution that minimizes a predefined cost function. Technically, they aim to solve single-objective optimization tasks. What is still missing are equivalent quantum methods that aim to solve multi-objective optimization problems.

[0004] There are actually some methods that use quantum hardware to consider multi-objective problems. In "Multiobjective variational quantum optimization for constrained problems: an application to Cash Management, arXiv:2302.04196, https: / / doi.org / 10.48550 / arXiv.2302.04196", a VQE quantum circuit and a combined classical NSGAII algorithm are applied to it. From the quantum state prepared by the quantum device, they classically calculate two objective functions: a cost function that, when minimized, achieves the actual goal, and the probability of finding a feasible solution at the final state. The goal is not to map the entire Pareto front, but to reach a single optimal solution point, where the probability of finding a feasible solution is very close to 1 and the cost function is minimized. In effect, the method is a method of constraint handling, using classical multi-objective optimization to efficiently explore a search space of feasible and infeasible solutions. The quantum aspects of quantum computing are not utilized to reach a solution to the multi-objective problem.

[0005] Even though addressing these problems is naturally becoming more realistic, one central feature of real-world problems remains largely unconsidered: Real-world problems often deal with multiple conflicting objectives, e.g., cost (the lowest possible), quality (the highest possible), user satisfaction (the highest possible), CO 2Footprint (the lower the better),...etc. These objectives are usually in a trade-off relationship, where a solution that has a very good value for one objective, e.g. high quality, is less favorable for another objective, i.e. also has a high cost. Thus, the goal of multi-objective optimization is to find a whole set of solutions (not just one) that gives a good approximation of the best feasible set of trade-off solutions, the so-called Pareto Front (PF). Each solution in the PF is equal or better in at least one objective than any other solution in the PF, while at the same time being worse in at least one other objective.

[0006] "Multi-Objective Routing Optimization for 6G Communication Networks Using a Quantum Approximate Optimization Algorithm" (Sensors 22(19), 7570 (2022) doi:10.3390 / s22197570) describes solving multi-objective problems by iteratively solving a large number of constrained single-objective problems. A quantum computer is used only to solve the single-objective problems. Each single-objective problem optimizes only one objective and imposes constraints on the other objectives. A quantum computer is not used to directly obtain a population of solutions.

[0007] There are several papers that use the so-called non-dominated quantum optimization (NDQO) and its variants such as multi-objective decomposition quantum optimization (MODQO) and non-dominated quantum iterative optimization (NDQIO), such as “Quantum-Assisted Routing Optimization for Self-Organizing Networks”, IEEE Access 2, 614-632 (2014) doi: 10.1109 / ACCESS.2014.2327596, “Non-Dominated Quantum Iterative Routing Optimization for Wireless Multihop Networks” (IEEE Access 3, 1704-1728 (2015) doi: 10.1109 / ACCESS.2015.2478793), or “Quantum-Assisted Joint Multi-Objective Routing and Load Balancing for Socially-Aware Networks” (EEE Access 4, 9993-10028 (2016) doi: 10.1109 / ACCESS.2016.2629671). All these methods aim to find a Pareto-optimal solution set for a multi-objective problem. They do so by utilizing quantum computing devices. They specifically design quantum circuits that use auxiliary qubits to evaluate the Pareto-dominance between individual solutions. A variant / extension of Grover search can then be used to search for a new suitable candidate solution, which can then be inserted into the current approximation of the Pareto front. These algorithms are specialized to solve one single-objective problem and evaluate the Pareto-dominance between a set of solutions using complex circuits.They cannot be directly applied to general multi-objective optimization problems and, in particular, cannot be run on NISQ devices, requiring fully fault-tolerant and error-corrected quantum hardware, which will not be available in the near to mid-term future. Summary of the Invention [Problem to be solved by the invention]

[0008] However, there remains a need to directly address the optimization of multiple competing objectives to enable efficient computation of multiple optimized solutions to the problem. Current technology does not allow multi-objective optimization problems to be processed natively on quantum hardware devices.

[0009] Note that in the following description, a "classical computing component" may be a classical computer, and a "quantum computing component" or "quantum component" may be a general-purpose or special-purpose quantum computing device. [Means for solving the problem]

[0010] The present invention addresses the above mentioned problems using a quantum-specific multi-objective quantum optimization technique according to the methods and computer systems defined in the independent claims on current and future quantum hardware devices to find solutions to practical real-world problems, in particular known from the domain of energy management.

[0011] The dependent claims define advantageous aspects and features.

[0012] The present invention involves utilizing quantum superpositions, which are intrinsic to quantum states, as a resource for encoding the complete population of solutions of a multi-objective optimization algorithm. Quantum circuits are designed to increase the diversity of the population and improve all objectives such that the objectives give a good approximation of the Pareto front.

[0013] The method and apparatus provide for, for a given multi-objective optimization problem with M distinct objectives:

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[0018] At each iteration t, the following steps are performed: a. The classical computation component uses a classical optimization algorithm to parameter γ t and sending it to the quantum component; b. In the quantum computing component, the parameter γ t By performing PQC using the set of

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[0033] After completing the iterations, the final result is a good set of Pareto optimal solutions.

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[0036] The present invention uses quantum superposition to encode the entire population of solutions into one quantum state. The quantum computing components are designed such that the layout of the quantum circuits collectively achieves a multi-objective evolution of Pareto optimal solutions with true quantum encoding.

[0037] According to a preferred embodiment, the quantum component comprises N L A unitary quantum operation U(γ) consisting of layers is

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[0044] According to another preferred embodiment, the quantum component has N L A unitary quantum operation U(γ) consisting of layers is

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[0050] The coefficients in these sums

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[0054] Each cost Hamiltonian H α is a function of one classical objective function C in the sense that the measurement basis state |x〉 is a simultaneous eigenstate of all the cost Hamiltonians and the eigenvalues ​​of the problem Hamiltonian are given by the values ​​of the corresponding classical objective function of the corresponding classical solution x. α , i.e. H α |x〉=C α (x)|x〉 and the mixed unitary

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[0058] According to another preferred embodiment, the quantum component has N L A unitary quantum operation U(γ) consisting of layers is

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[0072] More preferably, the mixing operator for each layer is given by a sum over all x angular momentum operators for each qudit variable, and the coefficients of each individual angular momentum operator are given by the freely selectable parameters

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[0075] As an alternative embodiment, the mixing operator for each layer is given by a sum over all generalized Pauli X-operators for each qudit variable, as described for example in Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, "Quudits and High-Dimensional Quantum Computing, Frontiers in Physics 8 (2020)," with one coefficient for each physical qudit, which is given by the freely selectable parameter

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[0079] In a preferred embodiment, the quantum state |ψ γ 〉, i.e., the probability of the Born rule being realized by repeatedly measuring the quantum state, P(x,γ)=|〈x|ψ γ 〉| 2 Sample x from each measured state |x m 〉, the corresponding classical solution x m A set of selected states

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[0082] The method and apparatus may further be applied to solving a multi-purpose bi-directional electric vehicle charging problem, where the goal is to define a discrete integer variable, , that encodes a full charge and service schedule for a fleet of electric vehicles.

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[0086] The method and apparatus according to one of the above-described configurations may be implemented by using an integer variable x={ω n,s This may be applied to obtain the efficient frontier of possible portfolio configurations, as formulated by the Markowitz model of modern portfolio theory, by finding the configuration of the portfolio over time, specified by {\displaystyle \mathbb {P\left(p\left(p\right)}\right)}, which is spanned by the portfolio configuration that has the optimal trade-off between maximizing return and minimizing risk, thus presenting a two-objective optimization problem.

[0087] The invention will now be further described with reference to the accompanying drawings. [Brief description of the drawings]

[0088] [Figure 1] FIG. 1 is a schematic diagram of a hybrid quantum-classical computer system illustrating method steps according to a preferred implementation of the present invention. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0089] The proposed computer-implemented method and hybrid quantum-classical computer system aim to solve combinatorial integer multi-objective optimization problems where there are two or more objectives to be minimized. Typically, these objectives cannot be simultaneously minimized by several optimal solutions, since they are in trade-off relationships with each other. Thus, the goal of a multi-objective optimization method is to find a whole set of distinct solutions, called a Pareto set, where each solution realizes one optimal trade-off between the objectives. Thus, the set of solutions must map all possible trade-offs between the objectives, called the Pareto front.

[0090] The proposed method and apparatus of the present invention allows such problems to be natively handled on current NISQ quantum hardware. A crucial aspect of the present invention is provided by representing the entire population of candidate solutions to an optimization problem as one quantum state and extracting the entire set of Pareto optimal solutions from this one quantum state. In accordance with the present invention, the proposed computer-implemented method and quantum circuit of the hybrid quantum-classical computer system increase the diversity of the population by utilizing all cost function Hamiltonians corresponding to all classical objectives.

[0091] According to this description, the term "classical" always refers to known calculations that were performed in the past using only digital computer systems.

[0092] The proposed computer-implemented method and hybrid computer system comprises a classical computing component and a quantum computing component, both configured to send and receive information to and from each other so as to achieve a shared computation to solve a multi-objective integer optimization problem. 1 , a 2 ,...,a N ) (where a i∈[0,...,L-1] is a finite integer) is represented by the degrees of freedom of the qubits (for L=2) or qudits (for L>2) of the quantum hardware device, i.e. |x〉=|α 1 ,α 2 ,...,α N 〉 It is.

[0093] Each of these states is a basis state of the measurement basis and satisfies the orthonormality condition:

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[0096] This encoding means that all possible solutions of the original multi-objective integer optimization problem are represented by the measurement basis states of the quantum computation component. One advantage of using a quantum computer to tackle this problem is that a general quantum state is naturally a superposition of all basis states, i.e., all superpositions of possible solutions of the problem.

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[0099] The present invention utilizes parameterized quantum circuits in quantum computing components, which are implemented with a predetermined fixed initial state |ψ 0 Such application of the unitary gate operation yields the final quantum state |ψ〉=U(γ)|ψ 0 〉 Leads to...

[0100] According to the present invention, this state is S Encode the entire population of candidate solutions;

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[0106] State selection mechanisms, such as full-state tomography, classical shadowing or measurement-based sampling, allow the entire set of solutions to be extracted from the quantum state.

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[0108] Suitable techniques are described in "Efficient quantum state tomography" (Nat Commun 1, 149 (2010) doi 10.1038 / ncomms114), "Predicting many properties of a quantum system from very few measurements" (Nat. Phys. 16, 1050-1057 (2020) doi 10.1038 / s41567-020-0932-7), or "Quantum Computation and Quantum Information" (Cambridge: Cambridge University Press. (2010) doi 10.1017 / CBO9780511976667). The unitary of the present invention is specifically designed to increase the contributions to the superposition from different states, which gives an optimal trade-off solution. That is, the population of candidate solutions represented by a quantum state is a Pareto set of solutions.

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[0111] In the proposed invention, the set of solutions is sent from the quantum computing component to the classical computing component, where for each solution a vector of objective values ​​is calculated, which is the Pareto front

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[0115] The final result after running the proposed method using the hybrid quantum-classical computing system of the present invention is a Pareto set of solutions.

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[0118] In the following, the invention is described in two different implementation examples that are typical for multi-objective problems:

[0119] The operation of a fleet of electric vehicles (EVs) constitutes a typical optimization problem with several challenges. A central aspect of this operation is to serve customers in such a way that customer demand can be met for all vehicles, while at the same time reducing the electricity costs for charging all EVs as much as possible.

[0120] In one formulation of this problem, the goal is to T N over time frames EV The goal of our study is to find an optimal charging schedule for a fleet of electric vehicles (EVs). The search variable x defines the charging schedule for the entire fleet of EVs. The schedule is EV N T It consists of discrete integer variables,

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[0126] The first objective (OBJ1) represents the total cost of electricity over the entire charging schedule. PV(s) represents the excess energy produced by a photovoltaic (PV) system in time slot s that is available for charging an EV, and the function P(s) gives the price for buying or selling energy from or to the grid in time slot s. P max represents the maximum unit price of electricity. If the total electricity demand is greater than the PV surplus, energy has to be bought from the grid, whereas if it is less than the surplus, the sign of the term in brackets is reversed and energy is sold to the grid, resulting in a negative cost, i.e., revenue.

[0127] The second objective (OBJ2) is to calculate the required energy E tot,nThis represents a measure of fairness between different EVs / customers by minimizing the difference between the charging energy provided to each EVn and its required value E tot,n , which strongly favors equal variances.

[0128] The third objective (OBJ3) seeks to use only surplus PV energy to charge EVs in each time slot s, so that this objective function seeks to minimize the difference between the surplus PV energy used in each time slot and the current total charging energy. Minimizing this objective is a means to make the service operation independent of the power grid.

[0129] The fourth objective (OBJ4) aims at an equal distribution of charging power over time frames, and in particular tries to reduce the peaks of charging power in all time frames. The rationale behind this is twofold: (i) from the company's point of view, the peak power over the billing period has a direct impact on the electricity price charged, and therefore strong demand peaks need to be avoided; (ii) strong fluctuations in electricity demand generally put a lot of pressure on the energy grid provider, which needs to balance it. Both aspects are met by reducing the overall variation, i.e. the distribution, of the total energy demand over time, as formulated in this objective.

[0130] The problem characterized by the objectives (OBJ1)-(OBJ4) in the example described above defines a combinatorial integer multi-objective optimization problem with M=4 objectives, all of which need to be minimized. However, the invention is not limited to four objectives, and fewer or more objectives may define a multi-objective optimization problem. These objectives are usually in trade-off relationships with each other, and therefore cannot be simultaneously minimized by several optimal solutions. Thus, the goal of the multi-objective optimization technique according to the invention is to provide a computationally efficient means for finding an entire set of different solutions, called a Pareto set, where each solution achieves one optimal trade-off between the objectives. Thus, the set of solutions must map all possible trade-offs between the objectives, called the Pareto front.

[0131] For the EV charging problem, the discrete search variables are the variables

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[0133] For the EV charging problem, the encoding into a quantum formulation is as follows: each qudit in the quantum computation component runs for one time window s ∈ [1,...,N T ], n∈[1,...,N EV ] one charging power level l n,s [0,...,L-1]. The quantum state of the measurement basis is

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[0137] The classical M objective functions C defined above 1 (x)~C 4 (x) is represented by M cost Hamiltonians whose functional form is given by quadratic polynomials in the search variables. This is equivalent to the classical objective function C 1 (x)~C 4 (x) and a search variable l that specifies the charge level of EVn in time window s. n,s the corresponding z angular momentum operator L of the qudit that encodes the corresponding variables z;n,s By replacing

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[0139] The measurement basis state is the z angular momentum operator L z;n,s and the eigenvalues ​​are the charge levels, i.e.,

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[0141] This gives us an explicit form for the corresponding cost Hamiltonian representing the objective function:

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[0143] The measurement basis states are also simultaneous eigenstates of these cost Hamiltonians (the latter being L z;n,s (since they are polynomials in ), the eigenvalues ​​are given by the values ​​of the objective function of the corresponding configuration.

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[0145] A quantum computation component that runs a parameterized quantum circuit (PQC) is a

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[0152] Mixed unitaries are N P N EV N T Freely selectable parameters

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[0156] N in each layer P Each of the phase terms is determined by one freely selectable parameter

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[0159] Coefficient a k,a ≧0 is the reference vector

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[0165] In a simple but representative case, a quantum circuit has only one layer, N L =1, in this case the quantum circuit prepares the following state:

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[0168] The claimed method executed on the claimed hybrid quantum-classical computer system includes the following steps: A classical optimization algorithm COA, such as stochastic gradient descent (SGD), is launched on the classical computation component. The search parameters of this classical optimizer are the 10 parameters of PQC, namely:

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[0170] Additionally, an archive A may be initiated in the classical computation component, where the search parameters, candidate solutions, and corresponding values ​​of the objective function are stored. Furthermore, the quantum device is configured such that the measurement basis represents the classical solutions as shown in (ENC), and the quantum circuit generates a parameterized quantum state (PQCEX) using the corresponding gate operations (MIXEX), (PQCEX), and (HAM1)-(HAM4). Then, an iteration counter is initialized to t=0, and the following steps are iterated: i) based on its internal state, a classical optimizer algorithm generates a set of search parameter values;

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[0172] ii) The quantum device derives the parameter set from the classical computation component.

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[0175] The quantum device then executes the quantum circuit. The resulting quantum state |Ψ(γ t )〉 is N S Encode the entire population of candidate solutions as a superposition,

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[0182] One way to practically extract these solutions from a quantum state may be by repeated sampling. In that case, a quantum state is prepared and sampled with the same parameter γ t are measured multiple times (i.e., N measure >N S ), where each measurement yields one ground state, i.e. one solution.

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[0184] Another method is to perform a full-state tomography protocol or classical shadow tomography (see, for example, “Efficient quantum state tomography”, Nat Commun 1, 149 (2010), doi 10.1038 / ncomms1147, or “Predicting many properties of a quantum system from very few measurements”, Nat. Phys. 16, 1050-1057 (2020). https: / / doi.org / 10.1038 / s41567-020-0932-7) and explicitly select the ground state with the largest probability.

[0185] In either case, the procedure is as follows: using the present invention, a quantum computing component computes a single quantum state |Ψ(γ t )〉 to the entire population of solutions

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[0187] iii) The classical computation component receives the population of solutions from the quantum computation component.

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[0192] From this set of non-dominated solutions, the hypervolume index

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[0195] Fitness score f t is then fed back into the classical optimization algorithm, COA←f(γ t ) The optimizer updates its internal state accordingly.

[0196] In addition, the search parameter γ t , the corresponding fitness value f(γ t ), the current estimate of the Pareto front

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[0200] iv) A convergence criterion is then checked in the classical computation component, for example whether the maximum allowed number of cost function evaluations has been reached or whether the fitness value f(γ t ) was not sufficient compared to a given threshold. a. If the convergence criterion is met, the optimization is stopped and as a final result, the final Pareto front and the corresponding Pareto set are calculated by selecting all non-dominated solutions from all approximations of the Pareto front for each iteration.

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[0202] The final result is a Pareto set of solutions.

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[0205] The proposed method and hybrid computer system utilizing PQC is effective in generating a good approximation of the Pareto front since all possible solutions typically exist in a superposition of the initial states of the quantum device. The phase operator of PQC assigns a different phase to each individual basis state based on the value of the reference vector used in the phase operator of the equation (PQCEX). In that way, the classical optimizer can tune the parameters of PQC to give the greatest diversity and include most non-dominated solutions in the quantum state, thereby maximizing the hypervolume index.

[0206] The set of Pareto-optimal solutions serves as a basis for realizing efficient control of EV charging and scheduling services. A solution selection routine may be implemented in which the best-fitting solution is automatically determined from the entire set of Pareto-optimal solutions. A natural solution selection routine utilizes current-art knee point detection methods, for example, as described in “Performance Analysis on Knee Point Selection Methods for Multi-Objective Sparse Optimization Problems” (IEEE Congress on Evolutionary Computation, 1-8 (2018) doi 10.1109 / CEC.2018.8477915), or “A Multi-objective Evolutionary Algorithm for “Finding Knee Regions Using Two Localized Dominance Relationships” (IEEE Transactions on Evolutionary Computation 25(1), 145-158 (2021) doi 10.1109 / TEVC.2020.3008877). The knee point naturally provides the best trade-off solution that is preferred.

[0207] The second application example concerns Markowitz portfolio optimization, whose goal is to manage a portfolio of financial assets over time so as to maximize expected returns and at the same time minimize risk (see, for example, “Computational aspects of alternative portfolio selection models in the presence of discrete asset choice constraints”, Quantitative finance 1(5), 489-501 (2001), “Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer”, IEEE Journal of Selected Topics in Signal Processing 10(6), 1053-1060, (2016) doi 10.1109 / JSTSP.2016.2574703, or “Dynamic portfolio optimization with real datasets using quantum processors and quantum-inspired tensor networks”, Phys. Rev. Research 4, 013006 (2022) for recent introductions). Since there is usually a trade-off between these two objectives, it is desirable to map the efficient frontier, i.e., the Pareto front of the best possible trade-off solutions.

[0208] Technically, the problem is solved by solving a set of problems over a fixed time window, s (s=1,...,N T ) in assets n (n=1,...,N Asset ) ratio ω n,sThe rebalancing, or changing, of the portfolio composition is typically done over a fixed, predetermined number of times, resulting in such discretized time steps. In addition, funds typically trade assets in discrete chunks, resulting in the ratio ω n,s is expressed in integer variables, i.e., ω n,s ∈[0,...,L-1].

[0209] The two objectives of this problem are

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[0211] Thus, the first term gives the cumulative total expected return for the entire portfolio over the time horizon considered.

[0212] In the second objective, Σ nm,s Let denote the covariance of assets n and m at time s, for the purposes of this discussion, represent the covariance of expected portfolio returns summed over all times. This provides an indication of risk because (a) a large covariance suggests that the true returns may differ significantly from the expected returns, whereas a small covariance suggests that the true returns are more similar to the expected returns, and (b) a large covariance between two assets suggests correlation between the assets, which in turn makes the portfolio less robust and therefore more risky.

[0213] The portfolio optimization problem for this two objectives is given by the proportion ω n,s The integer variables that describe the measurement basis states of the quantum device are

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[0217] This results in a cost Hamiltonian of

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[0219] The measurement basis states are also joint eigenstates of these cost Hamiltonians, with eigenvalues ​​given by the values ​​of the objective functions of the corresponding configurations.

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[0221] In this example, the quantum computing component that runs the parameterized quantum circuit PQC uses two layers to prepare the state: |ψ〉=U(γ 2 )U(γ 1 )|ψ 0 (PQC) Here, the unitary operation U is a function of a set of parameters γ = {γ 1 ,γ 2 The initial state of the circuit is given by an equal superposition of all possible states,

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[0227] Mixed unitaries are freely selectable parameters

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[0231] Each of the two phase terms in each layer is determined by one freely selectable parameter

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[0233] Putting it all together, the quantum circuit prepares the following state:

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[0236] The proposed method and apparatus performs the following steps:

[0237] First, a classical optimization algorithm COA such as Stochastic Gradient Descent (SGD) is started in the classical computation component. The search parameters of this classical optimizer are the eight parameters of PQC, i.e., D=8.

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[0239] Second, the quantum computation component is configured such that the measurement basis represents the classical solution as shown in (ENC), and the quantum computation component generates a parameterized quantum state (PQCEX) using (HAM1) and (HAM2) along with the corresponding gate operations (MIXEX) and (PQCEX). Then, an iteration counter is initialized to t=0, and the following steps are iterated: i) based on its internal state, a classical optimizer algorithm generates a set of search parameter values;

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[0241] ii) The quantum computation component is a parameter set

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[0244] The quantum device then executes the quantum circuit. The resulting quantum state |Ψ(γ t )〉 is N S The entire population of candidate solutions is encoded as a superposition.

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[0246] For this state, a full-state tomography (see, for example, “Efficient quantum state tomography”, Nat Commun 1, 149 (2010) doi 10.1038 / ncomms1147) is performed on the quantum device, thereby obtaining a list of all weights contributing to that state, T={|ξ t (x c )| 2 c is 1,...d H}, and select the solution with the largest amplitude from T to N at iteration t. S A set of solutions

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[0249] iii) The classical computation component receives the population of solutions from the quantum computation component.

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[0252] From this set of objective vectors, a set of non-dominated solutions is

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[0258] The fitness value is then fed back into a classical optimization algorithm, COA←f(γt ) The optimizer updates its internal state accordingly.

[0259] In addition, the search parameter γ t , the corresponding fitness value f(γ t ), the current estimate of the Pareto front

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[0263] iv) A convergence criterion is then checked in the classical computing device. This can be, for example, whether the maximum allowed number of cost function evaluations has been reached or whether the fitness value f over the last few iterations has been reached. t It is possible that the improvement in the a. If the convergence criterion is met, the optimization is stopped and as a final result, we obtain the final Pareto front and the corresponding Pareto set by selecting all non-dominated solutions from all approximations of the Pareto front at each iteration.

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[0265] The final result is a Pareto set of solutions.

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[0268] The proposed method utilizing PQC and a hybrid quantum-classical computer system is effective in generating a good approximation of the Pareto front since all possible solutions typically exist in a superposition of the initial states of the quantum device. The phase operator of PQC assigns a different phase to each individual basis state based on the value of the scalarization function used in the phase operator in equation (PQCEX). That way, the classical optimizer can tune the parameters of PQC to give the greatest diversity and include the majority of non-dominated solutions in the quantum state, thereby maximizing the hypervolume index.

Claims

1. A set of Pareto-optimal solutions for a given multi-objective optimization problem with M distinct objectives [0010] , wherein x represents a classical solution to the multi-objective optimization problem, the method comprising performing an iterative procedure, each iteration t having a. The classical computation component uses a classical optimization algorithm to find the parameter γ ｔ and sending it to the quantum component; b. In the quantum component, the initial quantum state |ψ ０ 〉, the parameter γ ｔ Quantum state [0025] generating said quantum state [0030] N Ｓ representing a set of distinct solutions; c. The one quantum state [0045] From N Ｓ ≧N ＰＦ A set of ground states [0050] and sending information about which ground state was selected to the classical computation component; d. computing, with the classical computation component, a set of objective values ​​comprising objective values ​​for each solution of a classical solution set; e) taking the set of solutions and the corresponding set of objectives with a classical optimizer to generate a new estimate of a Pareto set of solutions and a corresponding Pareto front; f. updating an internal state of the classical optimization algorithm based on a current set of solutions and the Pareto set and their corresponding sets of objective values; g. checking a convergence criterion, and if said convergence criterion is not met, repeating the process from step a onwards until said convergence criterion is met; A computer-implemented method in which the results after the most recent iteration is completed are taken as a set of Pareto-optimal solutions, for which a corresponding set of objective value vectors is an approximation of the true Pareto front.

2. 2. The method of claim 1, wherein in addition to updating the internal state of the classical optimization algorithm, an archive is updated based on the current solution set and the Pareto set and their corresponding sets of objective values, and the step of generating new estimates further incorporates values ​​from a previous iteration obtained from an archive.

3. The parameterized quantum circuit PQC has N quantum circuits each having a different set of selectable parameters. Ｌ A unitary quantum operation U(γ) consisting of layers is [006] Implement it as Each layer l has a separately selectable parameter set γ ｌ = {γ Ｘ；ｌ , γ Ｐ；ｌ }, and a topological operator, each of which has a unitary of [0070] and Topological unitary of each layer [0080] is a freely selectable parameter [0090] , M, and the cost Hamiltonian H α Each cost Hamiltonian H α is a function of one classical objective function C in the sense that the measurement basis state |x〉 is a simultaneous eigenstate of all the problem Hamiltonians, and the eigenvalues ​​of the problem Hamiltonian are given by the values ​​of the corresponding classical objective function of the corresponding classical solution x. α represents Mixed unitary for each layer [0010] is the freely selectable parameter γ Ｘ；ｌ，α , M parameterized mixing operators X α (γ Ｘ；ｌ，α 3. The method of claim 1, wherein the cost Hamiltonian is generated by:

4. The parameterized quantum circuit PQC has N quantum circuits each having a different set of selectable parameters. Ｌ A unitary quantum operation U(γ) consisting of layers is ##EQU00011## Implement it as Each layer l has a separately selectable parameter set γ ｌ ∈{γ Ｘ；ｌ , γ Ｐ；ｌ }, and a plurality of mixing operators and topological operators, ##EQU00012## and Topological unitary of each layer ##EQU00013## is a freely selectable parameter ##EQU14## and the cost Hamiltonian H α A specific sum over ##EQU00015## Each cost Hamiltonian H α is a function of one classical objective function C in the sense that the measurement basis state |x〉 is a simultaneous eigenstate of all the cost Hamiltonians and the eigenvalues ​​of the problem Hamiltonian are given by the values ​​of the corresponding classical objective function of the corresponding classical solution x. α represents Mixed unitary for each layer ##EQU00016## is the freely selectable parameter γ Ｘ；ｌ，ｋ , k=1, . . . , N Ｐ M parameterized mixing operators X ｋ (γ Ｘ；ｌ，ｋ 3. The method of claim 1, wherein the cost Hamiltonian is generated by:

5. The coefficients in these sums ##EQU00017## is the reference vector [0018] and the reference vector represents, for each k, [0019] and α ｋ，ｎ ≥ 0, and for each k, define a particular direction in the space of objective values, thereby reflecting one user preference. First, define the problem: [0020] 5. The method of claim 4, wherein the step is determined as part of

6. The parameterized quantum circuit PQC has N quantum circuits each having a different set of selectable parameters. Ｌ A unitary quantum operation U(γ) consisting of layers is ##EQU00021## Implement it as Each layer l has a separately selectable parameter set γ ｌ ∈{γ Ｘ；ｌ , γ Ｖ；ｌ }, and a plurality of mixing operators and topological operators, [0022] and Topological unitary of each layer [0023] is a freely selectable parameter ##EQU00024## and generated by an operator V that estimates a hypervolume in an object space spanned by the population of states when encoded by a quantum superposition of states; function [0025] is the set of solutions [0026] The objective vector of [0027] Given the hypervolume spanned by the function [0028] has a function value between zero and one and allows downscaling of the hypervolume of a particular solution set if some of the contributing states have individual amplitudes smaller than a threshold or the heterogeneity of the amplitude distribution exceeds a threshold; Each cost Hamiltonian H α is a function of one classical objective function C in the sense that the measurement basis state |x〉 is a simultaneous eigenstate of all the cost Hamiltonians and the eigenvalues ​​of the problem Hamiltonian are given by the values ​​of the corresponding classical objective function of the corresponding classical solution x. α represents Mixed unitary for each layer [0029] is at least one freely selectable parameter γ Ｘ；ｌ and a parameterized mixing operator X ｌ (γ Ｘ；ｌ 3. The method of claim 1, wherein the cost Hamiltonian is generated by:

7. The choice of the basis state determines the quantum state [0030] By repeatedly measuring the state |x ｍ 〉, the corresponding classical solution x ｍ A set of selected states [0031] To, N Ｓ Measuring different states of [0032] Until Ｓ 3. The method according to claim 1, wherein different states are added.

8. The choice of the basis state determines the quantum state [Equation 33] , where the set of weights for each basis state in the full-state tomography is [0034] is determined, and N Ｓ The basis states corresponding to the subsets of the largest weights are selected, and these are called the set of selected states [Equation 35] 3. The method according to claim 1 or 2, characterized in that:

9. The mixing operator of each layer is given by a sum over all x angular momentum operators or generalized Pauli X operators of each physical qudit, and the coefficient of each individual angular momentum operator of each layer is given by one freely selectable parameter [0036] 4. The method of claim 3, wherein k / α enumerates the mixing operators, l represents the sheave, and i enumerates all physics qudits.

10. the mixing operator of each layer is given by a sum over all x angular momentum operators or generalized Pauli X operators of each physical qudit, and there is only one global coefficient for the complete sum over all individual angular momentum or generalized Pauli X operators applied to each physical qudit of each layer, the global coefficient being determined by the freely selectable parameters [Equation 37] 4. The method of claim 3, wherein k / α lists the blending operators and l represents a layer.

11. Reference vector for the topological Hamiltonian [Equation 38] Coefficient α ｋ，α is determined, and said coefficient α is adjusted so as to ensure that a different set of reference vectors is used at each iteration, thereby allowing a speedup of the optimization process and a diversification of the Pareto solutions obtained. ｋ，α 5. The method of claim 4, characterized in that at each iteration step t, is adapted according to state-of-the-art procedures for reference vector adaptation.

12. The problem is a multi-objective bidirectional electric vehicle charging problem, where the goal is to find a set of Pareto-optimal configurations of discrete integer variables that encode a full-charging and service schedule for a fleet of electric vehicles, and the Pareto front [0039] 3. The method according to claim 1 or 2, characterized in that in order to achieve an approximation of , a number of objectives are taken into account, being at least two of the following: cost of electricity to meet the total charging demand, customer satisfaction, maximizing the use of renewable energy, minimizing electricity leaving the power grid, and minimizing peak power consumption.

13. 3. The method of claim 1 or 2, characterized in that it is applied to obtain an efficient frontier of possible portfolio configurations, as formulated by the Markowitz model of modern portfolio theory, by finding the configuration of a portfolio over time, specified by an integer variable specifying the amount of investment in asset n at time s, where the efficient frontier is spanned by portfolio configurations that have an optimal trade-off between maximizing return and minimizing risk, thus presenting a bi-objective optimization problem.

14. A hybrid quantum-classical computer system, A quantum component configured to realize a parameterized quantum circuit PQC by implementing a unitary quantum operation U(γ), the unitary quantum operation U(γ) being parameterized by a freely selectable parameter γ, the unitary quantum operation U(γ) being parameterized by a predefined initial state |ψ of N qubits or qudits. ０ By applying unitary operators to |ψ γ 〉, i.e., |ψ γ 〉=U(γ)|ψ ０ 〉, which can be decomposed into a measurement basis for quantum devices. [0040] , d Ｈ is the dimension of the Hilbert space, and each measurement basis state |x Ｃ 〉 is one classical solution x Ｃ a quantum component encoding A classical computing component provides a new parameter value γ to the quantum computing component and executes the PQC and determines the state |ψ γ and a classical computation component configured to execute a classical optimization algorithm by initiating the preparation of a set of classical solutions from the quantum module and then receiving a set of classical solutions as a result and computing all cost functions for each solution; A hybrid quantum-classical computer system, in which both components are used together to solve classical multi-objective optimization problems by iteration as defined in the method of claim 1 or 2.

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