WO2023105083A1

WO2023105083A1 - Method for solving an optimization problem

Info

Publication number: WO2023105083A1
Application number: PCT/EP2022/085375
Authority: WO
Inventors: Enrique Solano; Narendra HEGADE; Nancy Korina BARRAZA AGUILERA
Original assignee: Kipu Quantum Gmbh
Priority date: 2021-12-10
Filing date: 2022-12-12
Publication date: 2023-06-15

Abstract

The invention relates to a computer-implemented method, to a computer program and to a computer device as described herein. The computer-implemented method for solving an optimization problem comprises the steps of - providing a quantum annealer for solving the optimization function configured to produce a desired final Hamiltonian, wherein the quantum annealer comprises at least one schedule function, - performing a general parametrization of the schedule function for at least one tunable parameter, and - optimizing the parametrization.

Description

METHOD FOR SOLVING AN OPTIMIZATION PROBLEM

The invention relates to quantum computing. Specifically, the invention is defined by the appended independent claims. Embodiments of the invention are defined in the dependent claims.

The invention also relates to a method and/or to a computer-implemented method, in particular to an analog version of the quantum approximate optimization algorithm (QAOA) suitable for quantum annealers.

In the method/algorithm, the schedule function, which defines the adiabatic evolution, is optimized. This is achieved by choosing a suitable parametrization of the schedule function based on interpolation methods for a fixed time. The method/algorithm provides an approximate result of optimization problems coherently, avoiding negative issues affecting current quantum annealers.

Description

Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems.

The object of the present invention is to provide an improved means for solving an optimization problem that is performed by a computer. In particular, optimization problems may be fast and high-fidelity solutions of minimizing the ground state energy of chemical molecules.

This object is achieved by the invention. In certain embodiments, the tunable parameters which are suitable for current quantum annealers are optimized.

In one aspect, the invention refers to a method, in particular a computer-implemented method, for solving an optimization problem.

In one embodiment, the invention refers to a method, in particular a computer-implemented method, for solving an optimization problem, comprising the steps of

- providing a quantum annealer configured to produce a desired final Hamiltonian (also referred to as a problem Hamiltonian), wherein the quantum annealer comprises at least one schedule function, - performing a general parametrization of the schedule function for at least one tunable parameter, and

- optionally, optimizing the parametrization.

In certain embodiments, the quantum annealer comprises one schedule function or two different schedule functions that are different from each other.

Quantum annealing starts from a quantum-mechanical superposition of all possible states (candidate states) with equal weights. Then the system evolves following the time-dependent Schrodinger equation, a natural quantum-mechanical evolution of physical systems. The amplitudes of all candidate states keep changing, realizing a quantum parallelism, according to the time-dependent strength of the transverse field, which causes quantum tunneling between states. If the rate of change of the transverse field is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian. If the rate of change of the transverse field is accelerated, the system may leave the ground state temporarily but produce a higher likelihood of concluding in the ground state of the final problem Hamiltonian, i.e., diabatic quantum computation. The transverse field is finally switched off, and the system is expected to have reached the ground state of the classical Ising model that corresponds to the solution to the original optimization problem.

Quantum annealing is a computation method that may be used to find a low-energy state, typically preferably the ground state, of a system. Quantum annealing is used for problems where the search space is discrete (combinatorial optimization problems) with many local minima. Similar in concept to classical annealing, the method relies on the underlying principle that natural systems tend towards lower energy states because lower energy states are more stable. However, while classical annealing uses classical thermal fluctuations to guide a system to a low-energy state and ideally its global energy minimum, quantum annealing may use quantum effects, such as quantum tunneling, to reach a global energy minimum more accurately and/or more quickly than classical annealing. In quantum annealing thermal effects and other noise may be present to aid the annealing. However, the final low-energy state may not be the global energy minimum. Adiabatic quantum computation, therefore, may be considered a special case of quantum annealing for which the system, ideally, begins and remains in its ground state throughout an adiabatic evolution. Thus, those of skill in the art will appreciate that quantum annealing systems and methods may generally be implemented on an adiabatic quantum computer. Throughout this description, any reference to quantum annealing is related to adiabatic quantum computation unless the context requires otherwise. Quantum annealing uses quantum mechanics as a source of disorder during the annealing process. The optimization problem is encoded in a Hamiltonian HP, and the algorithm introduces quantum effects by adding a disordering Hamiltonian Ho that does not commute with Hp.

A quantum annealer is a programable quantum device that is capable of interpolating an initial Hamiltonian Ho at time t=0 with a final or problem Hamiltonian Hp at a time t=T.

An appropriate design of quantum annealer for solving the optimization function configured to produce a desired final Hamiltonian (problem Hamiltonian) comprises a tunable schedule function.

The quantum annealer may comprise at least one schedule function that is a time-dependent tunable parameter allowing for adiabatic parameters.

An optimization problem is a mathematical problem where the idea is to find the parameters that minimize or maximize a given multivariable function, normally called the cost function.

Optimization algorithms may include simulated annealing, parallel tempering, Markov Chain Monte Carlo techniques, branch and bound algorithms, and greedy algorithms, which may be performed by a classical computer. Optimization algorithms may also include algorithms performed by a quantum computer, such as quantum annealing, quantum approximate optimization algorithm (QAOA) or other noisy intermediate-scale quantum (NISQ) algorithms, quantum implemented fault-tolerant optimization methods, or other quantum optimization algorithms.

A general parametrization of the schedule function may be performed using several ways, including several time-dependent tunable parameters in the adiabatic quantum dynamics.

The final Hamiltonian (also referred to here as problem Hamiltonian) is the Hamiltonian that is addressing a time t=T in adiabatic quantum computing, quantum annealing, and by quantum annealers. The ground state of this Hamiltonian codifies the solution of an optimization problem.

A schedule function is a function that interpolates the initial and final Hamiltonian in adiabatic quantum computing and quantum annealing, and needs to be experimentally realizable by a quantum annealer. Parameters are “tunable” if they are experimentally manipulate during the course of the method.

Tunable coupling refers to the tunable parameter that is related to the physical interaction between two or more informational units (quantum bits).

An expectation value is the mean value obtained after an experimental measure of a physical quantity several times in a quantum experiment.

A quantum processor is a programable quantum devices composed of several informational units (qubits) that can be tuned in order to perform quantum algorithms.

In certain embodiments, the invention is used in chemistry to solve optimization problems. Such chemical optimization problems may be optimization problems for finding the ground state (the lowest energy state) of a chemical molecule.

In some embodiments of the invention, the method comprises the step of performing a general parametrization of the schedule function. The term “general parametrization” refers to accessible arbitrary time-dependent schedule functions. The method may also comprise the step of optimizing this parametrization. A way of optimizing the parameters of the schedule function is the use of a gradient-based optimizer. The optimization of the invention may be a combinatorial optimization and/or is performed using a classical gradient-based optimizer, optionally with a simulated annealing algorithm.

An arbitrary time-dependent schedule function is a general solution of choosing a schedule function, without the idea of optimization. The time-dependent schedule function can be (1) any arbitrary function which only needs to fulfill the initial (t=0) and final (t=T) boundary conditions; and (2) accessible to be a smooth time-dependent function (for example avoiding a sudden dump during the time evolution).

In some embodiments of the invention, the method not only contains one schedule function, but one, two or three different schedule functions (referred to as a first, a second, and a third schedule function). In analog QAOA one can define one or two functions, one of them is used to manipulate the initial Hamiltonian and the other to manipulate the final Hamiltonian. If only one function f is defined, then the second one is defined as 1-f. In another embodiment, the invention refers to a method, in particular a computer-implemented method, for solving an optimization problem, comprising the step of

- providing a quantum processor with tunable coupling for encoding an optimization problem.

T unable parameters of the schedule function may be the magnetic flux, the voltage, and/or the coupling strength. The tune of the schedule function can be experimentally reached by the manipulation of the bias voltage and magnetic flux through the difference to produce the schedule function defined by a given set of parameters.

The parametrization of at least one given schedule function into tunable parameters leads to the fixation of random initial values for each parameter, and the fixation of the running time (according to the coherence time) of the quantum processor that is used in the method.

In certain embodiments of the method, the quantum annealer is provided for running on a quantum computer (quantum processor) with tunable coupling.

Quantum computers (quantum processors) may include quantum annealing processors, digitized quantum processors, gate-based processors, or adiabatic quantum computation. On successive iterations the incremented optimization algorithm may provide samples, including quantum annealing, gate model based processors, etc.

In certain embodiments, the invention refers to a method for solving an optimization problem, comprising the step of:

- Using an arbitrary number of qubits in a quantum processor device with tunable coupling for adiabatically encoding an optimization problem, in particular wherein a time dependent Hamiltonian which at time zero is given by an uncoupled Hamiltonian is at a final time given by the final or problem Hamiltonian.

An arbitrary number of qubits refers to any integer value of qubits. The minimum number of qubits is 1. In certain embodiments, the number of qubits is between 1 and 100. In other embodiments, the number of qubits may be between 1 and 10.

In certain embodiments, the invention refers to a method for solving an optimization problem, comprising the step of: - Parametrizing the schedule function for the tunable parameters by determining initial parameters randomly, and the final running time according to the coherence time of the quantum processor.

This can be done by considering continuity in the schedule function and its first derivate, as well as monotonicity, to ensure that experimental restriction in the parameters also means a bound in the function.

A tunable experimental parameter may be magnetic field or voltage.

- Measuring the expectation value of the final Hamiltonian, and updating the initial parameters of the schedule function according to an optimization algorithm to minimize the expectation value of the final Hamiltonian, in particular, wherein the optimization algorithm is gradient-free or gradient-based plus simulated annealing algorithm.

The measurement of the expectation value can be performed using an standard methods that is available in a given hardware. Updating the initial parameters is done by the standard initialization methods of a given hardware.

In certain embodiments, the invention refers to a method for solving an optimization problem, wherein the generation of the schedule function comprises:

- Choosing an arbitrary number N of parameters, with N being a natural number excluding 0, in particular wherein N is in the range of 1 to 50, preferably N is in the range of 15 to 25, in particular wherein each of the N of parameters is initialized in a random number from 0 to 1 , and labelled as pj for the j-th parameter, in particular wherein each of the N of parameters defines a point in a two-dimensional space.

In certain embodiments of the method of the invention, the schedule function is given by the piecewise function that interpolate the points

J, in particular wherein he interpolation function comprises a monotonic cubic piecewise interpolation method or Hermite cubic polynomial interpolation.

In certain embodiments of the method of the invention, the optimization algorithm is a constrained algorithm configured to avoid that the updated parameters surpass the experimental capabilities of the quantum device. In that case, the optimization bound for each parameter ^, should be lj < p - < hj.

In certain embodiments, the method of the invention comprises measuring an expectation value of the final Hamiltonian and updating the schedule function parameters according to a gradient-free or gradient-based optimization algorithm to minimize the expectation value of the final Hamiltonian. This may be performed using the standard methods which are available in a given hardware.

- Measuring the expectation value of the final Hamiltonian, and updating the initial values for each of the parameters of the schedule function according to an optimization algorithm to minimize the expectation value of the final Hamiltonian. The optimization algorithm used may be gradient free or gradient based.

For the process described above, the quantum processor generates an evolution to arrive at the final Hamiltonian. The evolution generated by the quantum processor is not adiabatic in certain embodiments of the method of the invention.

In other embodiments, the invention refers to a method wherein the generation of the at least one schedule function comprises:

- Choosing a number N of parameters, with N being a natural number excluding 0, in particular wherein N is in the range of 1 to 20, preferably N is in the range of 8 to 12, in particular wherein each of the N of parameters is initialized in a random number from 0 to 1 , and labelled as pj for the j-th parameter, in particular wherein each of the N of parameters defines a point in a two-dimensional space (for example in the lambda - s plane, where lambda is the value of the schedule function and s the normalized time t/T.)

In other embodiments, the invention refers to a method wherein the schedule function is given by the piecewise function that interpolate the points

(1, 1)] in particular wherein he interpolation function comprises a monotonic cubic piecewise interpolation method.

In other embodiments, the invention refers to a method wherein the optimization algorithm is a constrained algorithm configured to avoid that the updated parameters surpass the capabilities of the quantum processor. The experimental realization of the at least one schedule function depends of the magnetic flux and bias voltages, as in a physical devices the flux and the voltage are limited by experimental features, thereby limiting also the schedule function (creating that a maximum and minimum possible value). This limitation is overcome by constraining the variational parameters as described herein.

In other embodiments, the invention refers to a method comprising measuring an expectation value of the final Hamiltonian and updating the schedule function parameters according to an optimization algorithm to minimize the expectation value of the final Hamiltonian. The optimization algorithm used to minimize the expectation value of the final Hamiltonian may be gradient-free or gradient-based.

In other embodiments, the method described herein, in particular a method of operation in a computational system, comprises: receiving an optimization problem; and for a number of iterations i to a number n; where n is a positive integer: causing a solver in the form of a quantum annealer that has a schedule function that is parameterized for at least one tunable parameter to be executed by at least one quantum processor to generate a plurality of samples as potential solutions to the optimization problem; causing, by at least one controller, a performing of at least one post-processing operation on the plurality of samples by at least one post-processing quantum processor-based device to generate a set of post-processing results; determining whether to modify the optimization problem based at least in part on the set of post-processing results; upon determining to modify the optimization problem based at least in part on the set of postprocessing results, the i^th iteration further comprising: causing the optimization problem to be modified; and initiating an (i+1)^th iteration.

A solver is a mathematical-based set of instructions executed via hardware circuitry that is designed to solve mathematical problems. Some solvers are general purpose solvers, designed to solve a wide type or class of problems. Other solvers are designed to solve specific types or classes of problems. A non-limiting exemplary set of types or classes of problems includes: linear and non-linear equations, systems of linear equations, non- linear systems, systems of polynomial equations, linear and non-linear optimization problems, systems of ordinary differential equations, satisfiability problems, logic problems, constraint satisfaction problems, shortest path or traveling salesperson problems, minimum spanning tree problems, and search problems.

There are numerous solvers available, most of which are designed to execute on classical computing hardware, that is computing hardware that employs digital processors and/or digital processor-readable nontransitory storage media (e.g., volatile memory, non-volatile memory, disk based media). More recently, solvers designed to execute on non-classical computing hardware are becoming available, for example solvers designed to execute on analog computers, for instance an analog computer including a quantum processor.

A method of operation in a computational system, may be summarized as including: receiving a problem and performing a number of iterations i to a number n, where n is a positive integer. Each iteration includes causing a solver to be executed by at least one processor to generate a plurality of samples as potential solutions to the problem; causing, by at least one controller, a performing of at least one post-processing operation on the plurality of samples by at least one post-processing non-quantum processor-based device to generate a set of postprocessing results; and determining whether to modify the problem based at least in part on the set of post-processing results. Upon determining to modify the problem based at least in part on the set of post-processing results, the ith iteration further includes causing the problem to be modified and initiating an (i+1)th iteration.

Causing the solver to be executed by at least one processor to generate a plurality of samples as potential solutions to the problem may include causing the problem to be optimized by at least one heuristic optimizer executed by at least one processor to generate a plurality of samples as potential solutions to the problem.

Determining whether to modify the problem based at least in part on the set of post-processing results may include comparing a result to a determined satisfaction condition and/or comparing the number of iterations performed to a determined limit.

In some implementations of the above-described method of operation in a computational system, the at least one processor is a quantum processor.

In another aspect, the invention refers to a system comprising

- a quantum processor with tunable coupling and free energies,

- a memory to save the results of the measurements of the final Hamiltonian expectation values, and

- a classical processor to perform the classical optimization. In another aspect, the invention refers to a use of the method described herein in quantum chemistry, quantum finance, or quantum machine learning.

Examples for an application of the invention in quantum chemistry is the search for the state of a molecule with the lowest energy (ground state). The method is general and can be applied to any molecular ground state calculations. However, the size of the molecule may be in the range of few tens of atoms to a few hundred atoms. Molecules can be, for example, organic or inorganic molecules. For example, the method can be applied to a hydrogen molecule (H2) or a more complex molecule, such as CH4, CO2, O3, peptides, etc. In certain embodiments, the molecule has at least two atoms. In some embodiments, the molecule has two to one hundrer atoms, in particular two to ten atoms. The molecule may have more than ten atoms, for example, 20, 30, 40, 50, 60, 70, 80, 90, 100 or more atoms.

Examples for an application of the invention in quantum finance is the prediction of a financial crash in the market.

Examples for an application of the invention in quantum machine learning is the training of restricted Boltzmann machines.

In another aspect, the invention refers to a computer program having a program code for performing the method of one of the beforementioned claims, when the computer program is executed on a computer, a processor, a quantum-processing unit and/or a programmable hardware component.

In another aspect, the invention refers to a computation device comprising: an interface for communicating with a quantum-processing unit; and one or more processors configured to perform the beforementioned method using the quantum-processing unit.

In another aspect, the invention refers to an apparatus comprising: a quantum device; and one or more computing devices communicatively coupled with the quantum device; the one or more computing devices being configured to at least cause the apparatus to: provide a quadratic unconstrained binary optimization problem defined by an equation with a cost function for optimization (for example of trading trajectories of an asset portfolio); and introduce a first set of data into the problem, the first set of data comprising historical financial data for a first period of time, the historical financial data at least comprising prices of considered assets; the quantum device being configured to at least cause the apparatus to solve the quadratic unconstrained binary optimization problem for the first period of time, thereby obtaining optimal trading trajectories for the first period of time; and the one or more computing devices being configured to at least further cause the apparatus to: provide a quantum or classical machine learning algorithm that provides a recommended composition of an asset portfolio based on a set of inputs; train the machine learning algorithm by both inputting the optimal trading trajectories obtained by the quantum device for the first period of time and minimizing a predetermined error function for each time unit of the first period of time for which there is historical financial data in the first set of data; introduce a second set of data into the machine learning algorithm, the second set of data comprising financial data for a second period of time that is posterior to the first period of time, the financial data at least comprising prices of the considered assets; and provide a recommended portfolio composition for the second period of time by running the trained machine learning algorithm with the second set of data introduced therein.

In another aspect, the invention relates to a data processing apparatus/device/system comprising means for carrying out the method of the invention as described herein.

In particular, the invention relates to a system for performing the method of the invention as described herein, comprising

- a quantum processor with tunable coupling,

- a memory to save the results of the measurements of expectation values of a final Hamiltonian, and

- a classical processor for performing a classical optimization

In another aspect, the invention relates to a computer program (product) comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method of the invention as described herein.

In another aspect, the invention relates to a computer-readable data carrier having stored thereon said computer program (product). Figures

Figure 1 Parametrized schedule function The parameters pj define the points (j/N, Pj) , which are interpolated using a piecewise monotonic cubic interpolation.

Figure 2: Performance of the AQAOA for the one qubit case given by Hamiltonian (14). The horizontal axis is the total time T considered for the algorithm in units of w¹. We calculate the relative error (a) and the fidelity (b), considering one (blue stars), two (orange stars), four (green stars), and eight (red stars) parameters for the algorithm. The black triangles show the performance using a linear schedule function. Figure (c) shows the schedule function for one (blue line), two (orange line), four (green line) and eight (red line) free parameters (dots) for T = 3.

Figure 3: Performance of the AQAOA for the hydrogen molecule case given by Hamiltonian (15). The horizontal axis is the total time T consider for the algorithm in units of w ¹. We calculate the relative error (a) and the fidelity (b), considering one (blue stars), two (orange stars), four (green stars), and eight (red stars) parameters for the algorithm. The black triangles show the performance using a linear schedule function. Figure (c) shows the schedule function for one (blue line), two (orange line), four (green line) and eight (red line) free parameters (dots) for T = 5.

Figure 4: Performance of the AQAOA for an Ising chain (blue) and homogeneous Heisenberg chain (orange) for a different number of sites, (a) Total running time T necessary for fidelity over 0.99 using the same number for parameters as sites in the chain, (b) The relative error ^e«for the same cases than in (a).

Figure 5: An embodiment of the invention. From left to right, the figure shows the following steps:

1. Initialization of a number of qubits in a given quantum hardware, 2. Optimization of a schedule function, in particular with a suitable CD term, 3. Dynamic evolution under the schedule function, 4. The measurement of the expectation value of the final Hamiltonian which provides the solution to the optimization problem. Examples

An analog version of the quantum approximate optimization algorithm suitable for current quantum annealers is described as an embodiment of the invention. The central feature of this algorithm is to optimize a schedule function, which defines the adiabatic evolution. This is achieved by choosing a suitable parametrization of the schedule function based on interpolation methods for a fixed time. This algorithm provides a result of optimization problems coherently, avoiding the issues affecting current quantum annealers.

I. Introduction

In nature, the dynamics of several relevant systems can be derived from the solution of an optimization problem. Consequently, the development of efficient optimization algorithms has been a central field for computer science, and naturally, this interest in optimization algorithms also arises in quantum computing. One of the most important approaches for solving optimization problems is by employing quantum annealers.

Accordingly, a quantum system is adiabatically driven from an initial Hamiltonian (at time t = 0), with a ground state that is easy to prepare, to a final Hamiltonian (at time t = 7), whose ground state codifies the solution of the optimization problem. If the evolution time (T) is sufficiently large, the adiabatic theorem ensures that after the evolution, the system will be in the ground state of the instantaneous Hamiltonian.

Nevertheless, the adiabatic evolution demands a large execution time. This time is beyond the coherence time for current quantum annealers, which turns the process incoherent; thus, the possible quantum advantage is unclear.

On the other hand, hybrid quantum-classical algorithms have received due to the possibility of being implemented in current noisy intermediate-scale quantum (NISQ) devices. These algorithms are focused on the minimization of a cost function. The cost function is codified in the expectation value of quantum observables, in general, the Hamiltonian, and it is computed by a quantum processor using parametrized quantum states prepared via parametrized quantum gates. Finally, the minimization is obtained by using a classical optimization algorithm over the parameters of the quantum gates. Some examples of these classes of algorithms are the variational quantum algorithms, the digital quantum approximate optimization algorithm (QAOA), the adaptive random quantum eigensolver, and the digitized counterdiabatic QAOA among others.

An analog version of QAOA by the suitable parametrization of a schedule function by parts followed by a classical optimization is disclosed herein in different embodiments. This algorithm is suitable for use in quantum annealers, finding optimal protocols for coherent evolution of the annealer quantum processor, exploiting the full potential of such devices.

II. Analog QAOA

An adiabatic algorithm is a time evolution given by the following time-dependent Hamiltonian

where Hi is the initial Hamiltonian which ground state is easy to prepare, Hf is the problem Hamiltonian which ground state codifies the solution of the optimization problem, Tis the total evolution time and A(x) is the schedule function with /\(0) = 0 and /\(1) = 1.

The time evolution is given by

and after digitizing the time, we obtain

where Af = T/N. Now, by the use of first order Trotter expansion,

In quantum annealers, Hi is called the mixer Hamiltonian and has the forrr/7, = and Hf is diagonal in the computational basis in general. On the other hand, the digital QAOA algorithm is given by the unitary evolution parametrized by 2/V real numbers

where

are the parameters to optimize with the cost function given by

where 1 ) is the ground state of Hi.

From equations (Eqs.) (4) and (5) one can see that the digital version of QAOA is basically an optimization of the schedule function A(x) in its digital form.

Now, for the analog QAOA (AQAOA), the time evolution is given by a time dependent Hamiltonian in Eq. (1), using a schedule function defined by parts (see Fig. 1).

and

It means that

is given by a cubic interpolation function of the points

_wjth This leads to the following

unitary evolution and the cost function

where s the ground state of the initial Hamiltonian /-/,.

Hi and Hf can be the Hamiltonians that quantum annealers can implement currently, where the only requirement is the manipulation of the schedule function in the form of a cubic interpolation function. The total evolution time T plays the role of the circuit depth that in this case is independent of the number of parameters to optimize, which is a fundamental difference to the digital QAOA. It is required that this evolution be coherent, such that T needs to be smaller than the coherence time of the quantum annealer.

In the next section, the AQAOA is tested numerically for a cubic interpolation for different problem Hamiltonians

is not considered. As in the digital version of QAOA, the parameters are completely free and only bound by the experimental setup limitations. Then, this class of interpolation allows that Q In certain embodiments, the

method an adiabatic path is not generally followed.

The AQAOA only focuses on minimizing the cost function and does not consider the distance to the adiabatic evolution path. Finally, it is requested that the interpolation be soft, which means continuity in the first derivative, and monotonic, which means that the second derivative will not change of sign among two contiguous points.

III. Numerical results

The performance of the algorithm of the invention can be calculated using the relative error of the energy obtained by the AQAOA, it means

where is the ground state of the final Hamiltonian Hr. Also, one can consider the fidelity

Nevertheless, the last could fail as a good performance measure for degenerate ground states because the algorithm/method of the invention focuses on minimizing the energy and does not consider a specific ground state.

In the examples provided here, the following initial Hamiltonian is used

where is the x-Pauli matrix of the jth quibit, w, is the initial frequency gap for all the qubits, and n is the total number of qubits.

A. One qubit case

For the first example, the following final Hamiltonian for one qubit is used

For simplicity, the same frequency for the initial and final Hamiltonian (w,= w/) is chosen in this example. The performance of the algorithm is collected in Figure 2 (a) for the relative error 06 and Figure 2 (b) for the fidelity. Each point (star) of the figures represents the performance of the algorithm for different total algorithmic time T and a different number of parameters in the minimization process, namely, one parameter (blue stars), two parameters (orange stars), four parameters (green stars) and eight parameters (red stars). Moreover, the performance of the algorithm is compared with the performance of adiabatic quantum computing using a common schedule function, i.e. , the linear schedule function A(t) = t/T (black triangles). However, other schedule functions may also be employed in other embodiments.

From these two figures, one can see that a time of T = 4 [w/¹] is enough for this embodiment of the algorithm to find the solution of the problem with only two parameters, while the linear function needs approximately double the time. On the other hand, the algorithm performance using one parameter for the optimization process follows the same performance as the linear schedule function. If one considers a superconducting flux qubit

~ 2 [GHz], then for a time T ~ 2 [ns], our algorithm produces the correct solution using two parameters for a single qubit Hamiltonian. Finally, Figure 2 (c) shows optimal schedule functions for one (blue), two (orange), four (green), and eight (red) free parameters. It is pointed out that the end points (0,0) and (1 ,1) are fixed. The oscillations in the schedule functions mean that an adiabatic evolution is not followed, which is also observed by the fact that we have zones where W > 1 < 0 |_t i_mp|i_es that the adiabatic theorem does not constrain the time for the final result obtained by the method/algorithm. B. Exemplary application: Hydrogen Molecule

The next example is a non-stoquastic Hamiltonian which describes a hydrogen molecule with a bond length of 0.2 [A],

with go =2.8489, g1 = 0.5678$, g2 =-1.4508, =0.6799, and g^ = gs = 0.0791 .

The performance of the algorithm is shown in Fig. 3 (a) for the relative error

and Fig. 3 (b) for the fidelity. In this case, the same number of parameters is considered as in the previous case, and again we compare the performance of the algorithm with the performance using the linear schedule function (black triangles in the figure). From these two figures, we can see that for a time of T=8 [cui^-1 ] the AQAOA algorithm can find the solution of the problem with only two parameters with a fidelity larger than 0.99. In this case, the performance of the algorithm using only one parameter for the optimization process improves the performance of the linear schedule function, obtaining fidelities over 0.95 for a time T= 10 [wr¹]. If we consider a quantum annealer based on flux qubits, the method/algorithm gets the correct solution using two parameters for T ~ 4 [ns]. Finally, Fig.3 (c) shows optimal schedule functions for a different number of parameters.

C. Ising and Heisenberg Hamiltonians

Finally, the scaling in the total algorithmic time T for two kinds of nearest-neighbor Hamiltonians is performed. First, we consider the stoquastic Hamiltonian of an Ising chain given by

and second the non-stoquastic Hamiltonian of a homogeneous Heisenberg chain which reads

For simplicity, in both cases co, = cor = 2 J is considered.

parameters are used or the optimization process for the chain of N sites, i.e. if we consider a chain of <=5 sites for the Ising or Heisenberg Hamiltonians, 5 parameters are used in the parametrization of the schedule function. Furthermore, we are interested in the time T needed to get a solution with fidelity over 0.99. These results are collected in Fig.4 (a), which shows T for an Ising chain (blue dots) and homogeneous Heisenberg chain (orange dots) as a function of the number of sites Moreover, Fig 4 (b) shows the corresponding relative error ft for the same cases than in Fig.4 (a). For the homogeneous Heisenberg chain, the time T decreases with the number of the qubits in the system, which means that with a time T =3 [co,^-1] the algorithm can obtain results with fidelities larger than 0.99 with M parameters for a chain of N sites. On the other hand, the time Tfor the Ising chain increases with the number of sites and reaches a maximum value in Ar = 8 sites, with a time T ~ 10 [co,^-1]. Finally, in a superconducting circuit platform, one of the most popular and advanced architectures for quantum computing, the frequency co, is in the order of a few [GHz], For most superconducting circuit setups co, > 1 [GHz] => w, < 1 [ns], and considering that typically the superconducting circuits devices have a coherent time in the scale of microseconds, i.e. larger than 10³ w/’¹.

Therefore, the disclosed AQAOA algorithm achieves fast and large-fidelity approximations for optimization problems suitable for quantum annealers in a coherent way.

Taken together, an analog version of the quantum approximate optimization algorithm suitable for quantum annealers is provided. The algorithm is based on a general parametrization of the schedule function, which produces any function if enough parameters are considered. By optimizing this parametrization in the same way as in the standard QAOA algorithm, fast and large-fidelity solutions are obtained. As examples, the algorithm is tested numerically for different cases, first, a single qubit Hamiltonian, second the ground state energy of the hydrogen molecule, and the Ising and homogeneous Heisenberg chain for a different number of sites, from 2 to 10, obtaining in all the studied cases high fidelities with a relatively low number of parameters, i.e, the same number of parameters than the qubits in the Hamiltonian. Furthermore, the algorithm experimental implementation depends in certain embodiments on two features: first, a quantum annealer capable of producing the desired final or problem Hamiltonian Hf, and second, a quantum annealer with a schedule function that can be manipulated. The experimental limitations on the final Hamiltonian will determine the classes of problems that can be solved, and depending on the experimental manipulability of the schedule function, the optimization process will require more or fewer constraints.

The method/algorithm can be implemented efficiently in analog devices such as coherent quantum annealers, thereby exploiting the quantum nature of the device.

Claims

1. Method, in particular a computer-implemented method, for solving an optimization problem, comprising the steps of

- providing a quantum annealer for solving the optimization function configured to produce a desired final Hamiltonian, wherein the quantum annealer comprises at least one schedule function,

- performing a general parametrization of the schedule function for at least one tunable parameter, and

- optimizing the parametrization.

2. Method of claim 1 , wherein the quantum annealer comprises one schedule function or two different schedule functions that are different from each other.

3. Method of claim 1 or 2, wherein the tunable parameter is chosen from the group consisting of: magnetic fluxes and voltages.

4. Method of any of claims 1 to 3, wherein the optimization is a combinatorial optimization and/or is performed using a classical gradient-based optimizer, optionally with a simulated annealing algorithm.

5. Method of any of claims 1 to 4, comprising the step of

- providing a quantum processor with tunable coupling, wherein the quantum annealer is provided for running on a quantum processor.

6. Method of claim 5, comprising the step of:

- using a number of qubits in the quantum processor with tunable coupling for adiabatically encoding the optimization problem, in particular wherein a time dependent Hamiltonian which at time zero is given by an uncoupled Hamiltonian is at a final time given by the final Hamiltonian.

7. Method of any of claims 1 to 6, comprising the step of:

- Parametrizing the schedule function for the tunable parameters by determining initial parameters randomly, and the final running time according to the coherence time of the quantum processor.

8. Method of any of claims 1 to 7, comprising the step of: - Measuring the expectation value of the final Hamiltonian, and updating the initial parameters of the schedule function according to an optimization algorithm to minimize the expectation value of the final Hamiltonian, in particular, wherein the optimization algorithm is gradient-free or gradient-based plus simulated annealing algorithm.

9. Method of any of claims 1 to 10, wherein the generation of the schedule function comprises:

- Choosing a number N of parameters, with N being a natural number excluding 0, in particular wherein N is in the range of 1 to 50, preferably N is in the range of 15 to 25, in particular wherein each of the N of parameters is initialized in a random number from 0 to 1 , and labelled as pj for the j-th parameter, in particular wherein each of the N of parameters defines a point in a two dimensional space.

10. Method of claims 1 to 9, comprising measuring an expectation value of the final Hamiltonian and updating the schedule function parameters according to a gradient-free or gradient-based optimization algorithm to minimize the expectation value of the final Hamiltonian.

11. Method of claims 1 to 13, in particular method of operation in a hybrid quantum-classical computational system, the method comprising: receiving an optimization problem; and for a number of iterations i to a number n; where n is a positive integer: causing a solver in the form of a quantum annealer that has a schedule function that is parameterized for at least one tunable parameter to be executed by at least one quantum processor to generate a plurality of samples as potential solutions to the optimization problem; causing, by at least one controller, a performing of at least one post-processing operation on the plurality of samples by at least one postprocessing quantum processor-based device to generate a set of postprocessing results; determining whether to modify the optimization problem based at least in part on the set of post-processing results; upon determining to modify the optimization problem based at least in part on the set of postprocessing results, the i^th iteration further comprising: causing the optimization problem to be modified; and initiating an (i+1)^th iteration.

12. Use of the method of claims 1 to 15 in quantum chemistry, quantum finance, or quantum machine learning.

13. A system for performing a method of any of claims 1 to 11 , comprising

- a quantum processor with tunable coupling,

- a classical processor for performing a classical optimization

14. A computer program having a program code for performing the method of any of claims 1 to 11 , when the computer program is executed on a computer, a processor, a quantum processor and/or a programmable hardware component.

15. A computation device comprising: an interface for communicating with a quantumprocessing unit; and one or more processors configured to perform the method of claims 1 to 11 using the quantum processor.