Quantum differential encoder

Abstract

A method is described for solving a system of (non-)linear differential equations DE using a hybrid data processing system including a classical computer system and a quantum computer system, the method comprising receiving or determining, by the classical computer system, a quantum circuit formulation representing the system of DE, the quantum circuit being parameterized by variables x of the DE and including one or more points x j One or more test function values ​​f(x j ) and one or more function circuits and one or more points x j and one or more derivative circuits for determining one or more trial derivative values ​​about a point x in a variable space x of the DE, by a quantum computer system. j , executing a quantum circuit on the set of , receiving, by a classical computer system, hardware measurement data in response to the execution of the quantum circuit quantum, and determining, by the classical computer system, whether the quantum hardware measurement data forms a solution to the simultaneous (non-)linear DE based on the quantum hardware measurement data and the loss function.

Description

[Technical Field]

[0001] This disclosure relates to quantum differential encoders, and more particularly, to methods and systems for solving (non)linear sets of differential equations using a quantum computer, as well as computer program products that enable a data processing system including a quantum computer to solve (non)linear sets of differential equations. [Background technology]

[0002] The differential and integral calculus of infinitesimal changes forms a paradigm for modeling phenomena in various areas of science. Written as systems of differential equations (DEs) for one or more variables, the solutions typically allow us to describe the behavior of a function in space and time, given specified initial and boundary conditions. The applications of differential equations span all disciplines. In physics, they encompass mechanics, electrodynamics, fluid dynamics, thermodynamics, and quantum theory. In chemistry, DEs describe chemical reactions and molecular dynamics. In biology, they describe ecological environments and competition within ecosystems. DEs are also a useful tool in epidemiology. In economics and finance, they describe strategies for determining optimal pricing, hedging, and investment strategies.

[0003] For cases that cannot be solved analytically, advanced numerical methods are required to find solutions to differential equations. These can be classified into local and global methods. Local classical methods rely on discretization of the variable space and often require a fine grid to accurately represent the solution. Furthermore, since the derivative is approximated using numerical differentiation methods (finite difference method, Runge-Kutta method), small grid steps are required to qualitatively reproduce the result. This increases the computational cost as the number of points M (degrees of freedom) increases. Moreover, in grid-based calculations when the dimension of the problem (equal to the number of variables v) increases, the number of points required is approximately M. v This can lead to an "explosion," a phenomenon known as the "curse of dimensionality." Global (spectral) numerical methods for solving differential equations depend on expressing the solution with respect to a suitable basis set. This rewrites the problem as finding the optimal coefficients for a polynomial (e.g., Fourier or Chebyshev) approximation of the function to be solved. In some cases, spectral methods can find the optimal solution more efficiently, but to find the general function, the size of the basis set required for complex solutions can increase rapidly, and differentiation still requires numerical approximation, thus the same dimensional problem arises. Finally, unlike systems of linear algebraic equations, nonlinear differential equations can be rigid equations whose solutions become unstable in some ways. Such systems of equations are difficult to solve because the solution changes significantly over narrow intervals of parameters. Furthermore, difficult problems correspond to systems of equations with solutions that have high oscillations and discontinuities. Once again, this requires considering finer grids and larger basis sets, in conjunction with applying fine numerical differentiation techniques.

[0004] A prime example of a field where complex differential equations are widely used is fluid dynamics. Fluid dynamics studies the flow of Newtonian fluids, and based on the laws of conservation of energy and momentum, the corresponding Navier-Stokes differential equations are derived. These are highly nonlinear and rarely solvable by analytical methods, and are therefore dealt with in the field of computational fluid dynamics. Solving these equations is essential for many industrial applications, including, but not limited to, propulsion and aircraft manufacturing (aerospace industry), engine design (automotive industry), submarine design (naval industry), oil fields (drilling industry), blood flow in capillaries (health industry), weather forecasting, and climate modeling. On large classical computers, computation time is shared continuously, so efficiently solving systems of differential equations remains a particularly unsolved problem.

[0005] Quantum computers will change the way we process information. For some problems, they will bring about drastic increases in computational speed, ranging from a quadratic acceleration in searching unstructured data to exponential improvements in factoring large numbers used in cryptographic applications. Quantum machines, using coherent superpositions of qubits and binary strings, leverage quantum interference effects to amplify correct solutions, achieving this in fewer steps than possible on classical computers. Quantum computers are well-suited to chemical applications because they are naturally suited to simulating certain quantum processes. At the same time, quantum computers are not directly suited to all computational problems and can be considered specialized machines (similar to GPUs) that need to be reworked to suit the specific problem at hand. Designing these machines and their operational schedules is crucial for solving problems on quantum computers faster than any classical method available. This is equally true for work and applications in differential and integral calculus.

[0006] To date, there has been little research on whether quantum processors can solve systems of differential equations. This is mainly based on known quantum algorithms for solving systems of linear equations, such as the HHL algorithm [HHL 2009] and its improvements. More precisely, a quantum version of this problem is solved, where the variables are quantum states.

[0007]

number

[0008] As a matrix

[0009]

number

[0010] The constant term vector must be loaded as the quantum state |b>. This is called amplitude encoding, and any function is the quantum state for N qubits.

[0011]

number

[0012] It can be expressed as follows: Here, |k> represents a discretized point in space / time, and the amplitude u k The normalization of the quantum wave function gives the value of the function at these points (Σ k |u k | 2 Encodes =1). Amplitude encoding is also called "analog encoding" and is used in reference [Mitarai 2019], for example in the HHL protocol [HHL 2009], and in many cases can be considered conceptually the parent of various linear ODE solvers.

[0013] Some studies employ a similar approach by reducing DEs to algebraic equations using finite difference schemes (Euler's method). These require linearization, increasing the overhead in terms of the number of equations. Direct processing of nonlinear differential equations using quantum algorithms remains an unresolved challenge.

[0014] Generally, 2 N Compressing dimensional data into an N-qubit register is beneficial due to exponential memory savings. However, as highlighted in the review reference [Biamonte 2018], several problems arise. First, forming an exponentially compressed state from a constant vector is a serious problem, requiring advanced techniques such as quantum random access memory (QRAM), and potentially requiring exponentially increasing gate operations to prepare the general state, resulting in exponential scaling of algorithm execution time. In the case of ODE, this appears at the level of preparing arbitrary initial conditions. Second, reading the results is complex because sampling from the quantum state requires exponentially more samples of the number of qubits relative to a given numerical precision. Finally, even with fault-tolerant quantum devices, the required depth of protocols like HHL for industry-relevant problems is very large. Taking electromagnetic scattering simulation as an example, and considering the implementation cost of oracles, the circuit simulating an industry-relevant problem would require 10 8 10 qubits and 10 29 It was estimated that this would require a certain number of gates. Reference [Scherer 2017]. Even with extremely fast gates, this execution time is on a cosmic scale.

[0015] Current quantum devices are prone to generating noise and are not suitable for deep quantum circuits. However, since their Hilbert spaces increase exponentially with the number of qubits, they offer advantages over classical methods for some problems. A quantum processor using about 100 qubits can provide computational power beyond the reach of classical computers. This corresponds to noisy intermediate-scale quantum (NISQ) processors, which are dedicated devices that require co-designed algorithms tailored to the problems at hand. A particularly effective approach in this context is the variational quantum algorithm (VQA). Initially proposed for chemistry under the name variational quantum eigenvalue solver (VQE), this approach asks the quantum computer to prepare a low-energy state on the quantum device, but uses a classical computer to guide the optimization loop. This strategy enables quantum computation to be performed on relatively noisy devices and has enabled numerous advances that are not suitable for current deep protocols. This has led to the attention of general-purpose VQA, which has found applications in many fields including data science and quantum simulation.

[0016] One variational approach aimed at solving differential equations on a quantum computer is described in reference [Lubasch 2020]. The proposed approach allows for implementing non-linear terms using a "quantum non-linear processing unit" in exchange for adding complex auxiliary-based operations that increase the system size by a factor of K (where K is the degree of non-linearity of the differential equation). Importantly, this approach is based on the amplitude encoding of the function

[0017]

Number

[0018] and is based on the amplitude encoding of the function, with 2 NN bit strings (for N qubits) label the lattice points for the variable, and in addition, this method uses approximate numerical differentiation techniques. Furthermore, since the goal of the algorithm is to prepare the state |u(x)> as a function of the discretized variable x, the unsolved problem corresponds to reading out exponentially large amplitudes when complete functional behavior is required.

[0019] Another study relevant to the current context is referenced in [Gaitan 2020], which considers a numerical approach to fluid dynamics problems. The proposed approach relies on a finite difference method with lattice-based computation. The algorithm's workflow represents a classical CFD algorithm, with a single step delegated to quantum hardware. The latter corresponds to function averaging, as performed in Kacewicz's quantum ODE algorithm, with a core subroutine being the quantum amplitude estimation algorithm (QAEA). QAEA is a Grover search algorithm, which provides a quadratic quantum velocity after the desired function is prepared as a quantum state and two quantum Fourier transforms (direct and inverse) are performed. Estimating the complexity of the algorithm is difficult because function averaging is based on amplitude-encoded states and assumes that a black-box oracle for this operation is available. Also, unfortunately, the input problem (QRAM requirements) arises here.

[0020] Therefore, from the above, it is inevitable that improved methods and systems for solving different types of (non-)linear differential equations using quantum computers are needed in the art. In particular, there is a need in the art to improve the approximation of the trial function derivative, to efficiently encode solutions in quantum representations that should be easily loaded, processed, and retrieved, and to provide a framework that is compatible with short-term quantum hardware with limited circuit depth and is also scalable to fault-tolerant hardware. [Prior art documents] [Non-patent literature]

[0021] [Non-Patent Document 1] [HHL 2009] https: / / doi.org / 10.1103 / PhysRevLett.103.150502 [Non-Patent Document 2] [Gaitan 2020] https: / / doi.org / 10.1038 / s41534-020-00291-0 [Non-Patent Document 3] [Mitarai 2018] https: / / doi.org / 10.1103 / PhysRevA.98.032309 [Non-Patent Document 4] [Mitarai 2019] https: / / doi.org / 10.1103 / PhysRevA.99.012301 [Non-Patent Document 5] [Biamonte 2018] https: / / doi.org / 10.1038 / nature23474 [Non-Patent Document 6] [Lubasch 2020] https: / / doi.org / 10.1103 / PhysRevA.101.010301 [Non-Patent Document 7] [Scherer 2017] https: / / doi.org / 10.1007 / s11128-016-1495-5 [Overview of the project] [Problems that the invention aims to solve]

[0022] As those skilled in the art will understand, aspects of the present invention can be embodied as systems, methods, or computer program products. Accordingly, aspects of the present invention can take the form of embodiments that are entirely hardware, embodiments that are entirely software (including firmware, resident software, microcode, etc.), or embodiments that combine software and hardware aspects, all of which can be commonly referred to herein as “circuits,” “modules,” or “systems.” Functions described herein can be implemented as algorithms executed by a computer microprocessor. Furthermore, aspects of the present invention may take the form of computer program products embodied in one or more computer-readable media, in which computer-readable program code is embodied, for example, stored.

[0023] Any combination of one or more computer-readable media may be used. The computer-readable media may be a computer-readable signal medium or a computer-readable storage medium. The computer-readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination thereof. More specific examples (non-exclusive list) of computer-readable storage media would include electrical connections having one or more wires, portable computer diskettes, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fibers, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination of the foregoing. In the context of this specification, a computer-readable storage medium may be any tangible medium that can contain or store programs used by, or in combination with, instruction execution systems, apparatus, or devices.

[0024] A computer-readable signaling medium may contain data signals propagated, for example, in the baseband or as part of a carrier wave, using computer-readable program code embodied therein. Such propagated signals may take various forms, but are not limited to electromagnetic, optical, or any preferred combination thereof. A computer-readable signaling medium may be any computer-readable medium that is not a computer-readable storage medium and that can transmit, propagate, or carry programs for use in or in connection with an instruction execution system, apparatus, or device.

[0025] Program code embodied on a computer-readable medium may be transmitted using any suitable medium, including, but not limited to, wireless, wired, fiber optic cable, RF, or any suitable combination thereof. Computer program code for performing calculations for aspects of the present invention may be written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Java®, Smalltalk, C++, or similar, and traditional procedural programming languages ​​such as the C programming language or similar. The program code may run entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer via any type of network, including a local area network (LAN) or wide area network (WAN), or the connection may be to an external computer (for example, via the Internet using an Internet service provider).

[0026] Aspects of the present invention will be described below with reference to flowcharts and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present invention. It will be understood that each block in a flowchart and / or block diagram, as well as combinations of blocks in a flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general-purpose computer, a dedicated computer, or other programmable data processing device, in particular a microprocessor or a central processing unit (CPU), to generate a machine, which is executed via the computer's processor, other programmable data processing device, or other device, to generate means for implementing functions / operations specified in one or more blocks of the flowchart and / or block diagram.

[0027] These computer program instructions may also be stored in computer-readable media that can enable a computer, other programmable data processing devices, or other devices to function in a particular manner, and these instructions stored in computer-readable media produce products that include instructions that implement functions / operations specified in one or more blocks of a flowchart and / or block diagram.

[0028] Computer program instructions may also be loaded onto a computer, other programmable data processing device, or other device, so that the instructions executed on the computer or other programmable device may generate a computer implementation process by executing a series of operational steps on the computer, other programmable device, or other device, providing a process for implementing the functions / operations specified in one or more blocks of a flowchart and / or block diagram. In addition, instructions may be executed by any type of processor, including, but not limited to, one or more digital signal processors (DSPs), general-purpose microprocessors, application-specific integrated circuits (ASICs), field-programmable logic arrays (FPGAs), or other equivalent integrated circuits or discrete logic circuits.

[0029] The flowcharts and block diagrams in the figures illustrate the architecture, functions, and operations of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in the flowchart or block diagram may represent a module, segment, or part of code, which contains one or more executable instructions for implementing a specified logical function. It should also be noted that in some alternative implementations, the functions described in a block may be executed in a different order than that shown. For example, two blocks shown consecutively may actually be executed substantially simultaneously, or they may sometimes be executed in reverse order depending on the functions they relate to. It should also be noted that each block in the block diagram and / or flowchart, as well as combinations of blocks in the block diagram and / or flowchart, may be implemented by a dedicated hardware-based system that performs a specified function or operation, or a combination of dedicated hardware and computer instructions.

[0030] An objective of the embodiments in this disclosure is to reduce or eliminate at least some of the drawbacks known in the prior art. In particular, an objective of the embodiments in this application is to efficiently encode trial solutions of differential equations in quantum representations that are easily loaded, processed, and retrieved. A further objective of the embodiments is to compute the analytical derivative of a trial solution instead of numerical approximation. A further objective of the embodiments is to provide a framework compatible with the limited coherence time and available quantum circuit gate depth of short-term quantum hardware. An objective of the embodiments is also to take advantage of the increasing fault tolerance of anticipated future quantum hardware. An objective of the embodiments is also to enable obtaining a general solution to a general system of differential equations that includes nonlinear components and components representing complex oscillatory motion and / or nontrivial behavior. An objective of the embodiments is also to enable solving parametric differential equations where the parameters or terms of the equations are unknown with high precision, while making additional data points (measurements) available to perform partial regression and mitigate such uncertainties. The objective of the embodiment is also to enable the solution of parametric differential equations where it is unknown whether all terms of the equation have finite coefficients and accurately describe the data, and where trade-off relationships are required to ensure interpretability. [Means for solving the problem]

[0031] In one embodiment, the present invention may relate to a method for solving one or more differential equations DE, for example, a system of (non)linear differential equations, using a data processing system including a classical computer system and a quantum computer system. The method involves the classical computer system receiving or determining a formulation of a quantum circuit representing one or more DEs, where the quantum circuit is parameterized by a variable x of one or more DEs, and one or more points x j One or more trial function values ​​f(x) around x j One or more function circuits and one or more points x to determine ) jThe quantum computer system includes one or more derivative circuits for determining one or more trial derivative values ​​around a point x in the variable space X of one or more DEs. j This may include executing a quantum circuit on a set of values, receiving hardware measurement data by a classical computer system in response to the execution of the quantum circuit, and determining by the classical computer system, based on the quantum hardware measurement data and loss function, whether the quantum hardware measurement data forms a solution to one or more DEs.

[0032] In a further embodiment, the present invention may relate to a method for solving one or more differential equations DE using a data processing system including a classical computer system and a quantum computer system, the method wherein the classical computer system receives or determines a formulation of a quantum circuit representing an trial function for one or more DEs, the trial function being associated with one or more variables and a variable space, the quantum circuit including one or more function circuits for determining one or more values ​​of the trial function around one or more points in the variable space, one or more derivative circuits for determining one or more values ​​of the derivative of the trial function around one or more points, and one or more quantum variational circuits associated with one or more optimization parameters, and the classical computer system determines a set of points in the variable space of the trial function The execution of a quantum circuit includes executing the quantum circuit, which includes converting the quantum circuit into control signals for controlling the quantum elements of a quantum computer system and reading out the quantum elements to acquire hardware measurement data, and controlling the quantum computer system based on the control signals; the classical computer system receiving hardware measurement data in response to the execution of the quantum circuit, processing the hardware measurement data and feeding it into one or more trial functions and one or more derivatives of one or more trial functions; and the classical computer system determining a score indicating how well one or more measured trial functions satisfy the conditions of one or more DEs, based on one or more trial functions, one or more derivatives of one or more trial functions, and a loss function.

[0033] In one embodiment, this method may involve optimizing a loss function, wherein the optimization includes adjusting one or more optimization parameters and repeating the execution of a quantum circuit, processing of hardware measurement data, and determination of a score until the score satisfies predetermined optimization conditions.

[0034] In one embodiment, one or more derivative circuits can be obtained by the analytical differentiation of one or more function circuits with respect to one or more variables.

[0035] Accordingly, embodiments of this application enable the solution of differential equations, including nonlinear differential equations of general form, using a quantum computer in a substantially different manner from schemes known in the prior art. For a given system of nonlinear differential equations, a quantum circuit is constructed based on so-called differentiable quantum circuits (DQCs). These quantum circuits may be executed on a quantum computer, and cost functions, such as Hermitian operators like the Hamiltonian, may be used to measure observables that form an approximation of the solution to the system of nonlinear differential equations, which are used in classical optimization algorithms. Loss functions may be used to determine whether the approximation of the solution is sufficiently close to the solution to the system of nonlinear differential equations.

[0036] In one embodiment, determining whether quantum hardware measurement data form a solution to one or more DEs may be based further on one or more boundary conditions associated with one or more DEs.

[0037] In one embodiment, if one or more DEs represent one or more parameterized DEs, determining whether quantum hardware measurement data form a solution for one or more DEs may further depend on one or more boundary conditions and one or more data points associated with one or more DEs, such as measured data points.

[0038] In a further embodiment, determining whether quantum hardware measurement data form a solution for one or more DEs may be based on regularized data representing initial estimates for parameters used in one or more parameterized DEs.

[0039] In one embodiment, one or more DEs may include one or more parameterized DEs, and determining whether quantum hardware measurement data form a solution for one or more DEs is based on one or more boundary conditions and one or more data points associated with one or more parameterized DEs, e.g., measured data points, and optionally, regularized data representing initial estimates for one or more parameters used in one or more parameterized DEs.

[0040] In one embodiment, one or more DEs may include one or more parameterized DEs, the right-hand side RHS term of one or more parameterized DEs defines a parameterized linear combination of functionals, and determining whether quantum hardware measurement data form a solution for one or more DEs is based on one or more boundary conditions and one or more data points associated with one or more parameterized DEs, e.g., measured data points, and optionally, regularized data representing initial estimates for one or more parameters used in one or more parameterized DEs.

[0041] In one embodiment, a parameterized linear combination of functionals is a vector inner product.

[0042]

number

[0043] It can be defined as follows, and here,

[0044]

number

[0045] The parameters are defined in the form of a vector of coefficients, and F is a vector of functionals on f and x.

[0046]

number

[0047] The loss function optionally includes a regularization loss term based on a coefficient, preferably the norm of the coefficient.

[0048] In other words, the right-hand side RHS of this equation can be any linear combination of a set of library terms that are expected to be relevant to solving the differential equation. Then, during loss function optimization, the quantum circuit parameterization θ, and furthermore, the function coefficients...

[0049]

number

[0050] Both can be optimized.

[0051] In one embodiment, executing a quantum circuit may include converting each part of the quantum circuit into a series of pulses and applying the series of pulses to the qubits of a quantum computer.

[0052] In one embodiment, receiving hardware measurement data may include applying a readout pulse signal to a qubit in a quantum computer system and measuring quantum hardware measurement data.

[0053] In one embodiment, the quantum circuit may be parameterized based on at least one tunable optimization parameter, and if it is determined that the quantum hardware measurement data does not form a solution, the at least one tunable optimization parameter is adjusted to form a tuned quantum circuit, and further quantum hardware measurement data is determined based on the tuned quantum circuit. Furthermore, if one or more DEs include one or more parameterized DEs, at least one of one or more parameters of each of the one or more parameterized DEs can be adjusted.

[0054] In one embodiment, the function circuit may include a quantum feature mapping circuit for encoding the functional, preferably nonlinear, dependence of DE on one or more variables x into the quantum wave amplitudes of the qubits of the quantum computer system.

[0055] In one embodiment, the quantum feature map circuit may be differentiable to the sum of modified quantum circuits.

[0056] In one embodiment, each of one or more derivative circuits includes at least one modified quantum circuit.

[0057] In one embodiment, one or more function circuits and / or derivative circuits may further include quantum variational Ansatz circuits, which can be parameterized by tunable optimization parameters.

[0058] In one embodiment, one or more derivative circuits may represent the analytical derivative of one or more function circuits with respect to the variable x.

[0059] In one embodiment, the quantum circuit is a functional F[{d n f / dx n} n ,{f m (x) m The functional F may represent one or more solutions to a system of equations DE according to ]=0, and the functional F is determined by the system described by the (non)linear DE.

[0060] Accordingly, embodiments of this application utilize quantum feature-map encoding to overcome the complexity of amplitude encoding used in the prior art to prepare solutions at boundaries, enable the representation of analytical function derivatives without the inaccuracy characteristic of numerical differentiation (finite difference) using automatic differentiation of quantum feature-map circuits, and search for suitable solutions in the exponential space of the fitting polynomial, which is thus analogous to spectral and finite element methods with exponentially improved scaling, and avoid the data readout problem by encoding the solution into observable operators, so that the expectation can be calculated periodically.

[0061] In one embodiment, the quantum hardware measurement data may include measured trial function values ​​and measured trial derivative values, and the loss function may be used to determine whether the measured trial function values ​​and measured derivative values ​​form a solution to a system of differential equations.

[0062] In one embodiment, receiving hardware measurement data may include applying a readout pulse signal to a qubit in a quantum computer system and measuring quantum hardware measurement data.

[0063] In one embodiment, quantum hardware measurement data may be measured as the expectation value of a Hermitian cost operator, preferably a Hamiltonian operator.

[0064] In one embodiment, the quantum circuit may include a quantum feature map circuit that encodes the (non)linear function dependence of a differential equation variable x.

[0065] In one embodiment, the quantum feature map circuit may include a product-type feature map, a Chebyshev feature map, a Chebyshev sparse feature map, a Chebyshev tower feature map, a Fourier-type feature map, an evolution-enhanced feature map, and an amplitude encoding feature map, or any other feature map.

[0066] In one embodiment, the quantum circuit may include a variational quantum circuit that can be used to variationally optimize the quantum amplitude so that the cost function is minimized.

[0067] In one embodiment, the variational quantum circuit may include a hardware-efficient Ansatz or an alternating block Ansatz.

[0068] In one embodiment, the cost function is: point x j This can represent the function value of the differential equation in [location].

[0069] In one embodiment, the cost function may consist of chemistry-like general strings of single-qubit operators, general Ising Hamiltonian operators, or Pauli-string operators, which may or may not have variationally optimizable coefficients.

[0070] In one embodiment, the loss function may be based on the mean squared error, mean absolute error, Kullback-Leibler (KL) divergence, Jensen-Shannon divergence, or other error or divergence metrics.

[0071] In one embodiment, the boundary may be handled by a boundary pinning strategy, a floating boundary strategy, or an optimized boundary treatment.

[0072] In one embodiment, regularization prior information may be applied to aid convergence and reduce the risk of getting stuck at a local minimum.

[0073] In one embodiment, when the parameters of the equations are not precisely known or are known with low confidence, unregularized (real) information, data points, physical measurements, or other arbitrary priori known data may be used to solve a system of parametric differential equations. These data points may be considered in the total loss function to perform partial regression on the data while fitting the (system of) differential equations directly.

[0074] The present invention relates to quantum-classical hybrid systems for solving one or more differential equations, which include ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), higher-order DEs, higher-order DEs, single-dimensional or multi-dimensional DEs, one or more systems of DEs, elliptic / hyperbolic / parabolic PDEs, initial and / or boundary value problems, including Dirichlet and / or von Neumann boundary conditions, Cauchy types, and even Livin and mixed boundary types, and controlled problems, which are not limited to Hamilton-Jacobi-Bellman-Eikonaar boundary value problems, delay differential equations, differential-algebraic equations (DAEs), real and complex-valued functions, and nonlinear stochastic differential equations, including solutions to those of the Itoh and Stratonovich forms, for example.

[0075] The present invention may also cover one or more of the following embodiments.

[0076] In one embodiment, the variational quantum circuit may include a sequence of quantum gates that can be used to variationally optimize the quantum amplitude such that a loss function L is minimized and one or more exact solutions to one or more differential equations can be variationally obtained.

[0077] In one embodiment, the variational quantum circuit includes hardware-efficient Ansatz, with alternating layers of controlled-NOT operations following layers of parameterized quantum gate rotations, and the logic circuit requirements are natively mapped to quantum hardware information carrier connectivity.

[0078] In one embodiment, the variational quantum circuit includes an alternating block ansatz, preferably so that the circuit consists of gate blocks with a width M / 2 where M is greater than 1 and less than the number of quantum information carriers 1. The blocks are arranged in a checkerboard pattern and repeated several times.

[0079] In one embodiment, the loss function may define instructions to a classical computer regarding how to collect quantum measurement data and convert it into a single quantity ("loss") that quantifies the quality of the solution to a system of (non)linear differential equations. Preferably, the loss function includes a differential loss term and boundary terms.

[0080] In one embodiment, the distance definition of the loss is given by (ab) a single-valued parameter a and b. 2 It is quantified by the mean squared error (MSE) defined by [the relevant formula / method].

[0081] In one embodiment, the distance definition of the loss is quantified by the mean absolute error (MAE) defined by |ab| for single-valued parameters a and b.

[0082] In one embodiment, the distance definition of the loss is quantified using the Kullback-Leibler divergence, which is defined by the expected value of the logarithmic difference between the quantities under consideration.

[0083] In one embodiment, the distance definition of loss is quantified by the Jensen-Shannon information.

[0084] In one embodiment, boundary conditions for a system of one or more differential equations may be included as an additional consideration in the method. Preferably, these boundary conditions determine a specific solution to the differential equations.

[0085] In one embodiment, boundary conditions may be directly included in the loss function as additional terms.

[0086] In one embodiment, the boundary conditions may be included in the cost operator C along with fixed additional terms, presenting equivalent treatment of the boundary and derivative terms, both of which are encoded in the eigenspectrum of the cost operator.

[0087] In one embodiment, boundary conditions are included by repeatedly shifting the estimated trial solution based on boundary points or initial points in each iteration of the variational optimization sequence of a classical computer.

[0088] In one embodiment, boundary conditions may be included via a classical shift of the solution defined by a gradient descent method equivalent to the variational optimization. Preferably, this constitutes another variational parameter passed to the classical optimizer.

[0089] In one embodiment, adding a regularization procedure prevents the variational optimizer from getting trapped at a local minimum. Preferably, this includes one or more strategies of inputting prior information about a latent solution, a strategy of biasing the solution into a particular shape, and a strategy of searching for a solution within a region close to the known boundary values ​​of the differential equation.

[0090] In one embodiment, the regularization term decreases linearly in its weight contribution to the loss function.

[0091] In one embodiment, the regularization term may be smoothed out during a predefined training phase, which in one embodiment may correspond to an inverse sigmoid optimization schedule.

[0092] In one embodiment, the differential equation may be an ordinary differential equation (ODE).

[0093] In one embodiment, the differential equation may be a partial differential equation (PDE), and preferably, several quantum feature mapping circuits are linked together or arranged alternately.

[0094] In one embodiment, the differential equation may be a parameterized differential equation, where one or more elements in the equation are (hyper)parameters rather than functions (f(x), g(x, y)) or dependent / dimensional variables (x, y, z, t). To solve such an equation for a particular instance, additional data may be needed and used as boundary conditions for the parameters, and regularization may be given in the form of initial proposed values ​​for the parameters.

[0095] In one embodiment, the differential equation may be a higher-order DE in which one or more functions are differentiated multiple times.

[0096] In one embodiment, the differential equation may be of a higher order or nonlinear nature, and may be a power of one or more functions other than 0 or 1, or an argument to one or more nonlinear functions themselves.

[0097] In one embodiment, the differential equation may depend on a number of specific variables x, y, z, ..., also called dimensions.

[0098] In one embodiment, the differential equations considered may include a system of simultaneous differential equations.

[0099] In one embodiment, the differential equation may be elliptic, hyperbolic, or parabolic PDE.

[0100] In one embodiment, the differential equations considered, though not limited to them, describe initial and / or boundary value problems, including Dirichlet-type equations.

[0101] In an embodiment, the differential equation may have one or more boundary conditions of one of the following types: {Dirichlet, Neumann, Cauchy, Libin, mixed}.

[0102] In one embodiment, the differential equation may describe a controlled problem, including, but is not limited to, the Hamilton-Jacobi-Bellman equation and the Eikonal boundary value problem.

[0103] In one embodiment, the differential equation can form a system of (alternating) delay differential equations.

[0104] In one embodiment, the differential equation can describe a differential-algebraic equation (DAE).

[0105] In one embodiment, the differential equations include one or more (non)linear stochastic differential equations, including, but not limited to, those of the Itoh and Stratonovich forms.

[0106] Embodiments in this application can be implemented based on noisy quantum hardware having finite logic gate errors and finite coherence time.

[0107] In one embodiment, the method may be based on noisy quantum hardware in which subroutines of the algorithm are executed by multiple quantum devices operating in parallel and / or series, sending measurement data to a single classical computer that calculates a loss function value at each iteration.

[0108] In one embodiment, instead of measuring the cost function for each part of the loss function, we may rely on estimating the overlap between the left and right sides of a differential equation in a functional form, and the quantum information overlap of quantum hardware can be considered as a functional overlap.

[0109] In one embodiment, the quantum computer system may be implemented as a hardware quantum computer.

[0110] In one embodiment, a quantum computer system may include qubit-based quantum hardware in which quantum information carriers are embodied by qubits.

[0111] In one embodiment, the quantum hardware may include a continuous variable system, and the information carriers are defined by continuous quantum variables.

[0112] In one embodiment, the quantum computer system may be implemented as a software program for simulating a quantum computer system that includes quantum processing elements, such as qubits. In this embodiment, the software program may be a classical software program that runs a classical computer so that the quantum algorithm associated with the embodiments of this application can be developed, executed, and tested on a classical computer without requiring access to a hardware implementation of the quantum processor system.

[0113] In one embodiment, a quantum circuit may include a plurality of different quantum subcircuits. In one embodiment, one or more quantum subcircuits may include one or more quantum feature circuits, each of which is configured to map the variables of a differential equation to a Hilbert space associated with the qubits of a quantum computer.

[0114] In one embodiment, one or more quantum subcircuits may include one or more variational quantum circuits associated with one or more variational parameters for training the quantum circuits to approximate the solution of a differential equation.

[0115] In one embodiment, one or more quantum subcircuits may include one or more initializing quantum circuits configured to initialize at least a portion of the variational parameters of a quantum circuit based on one or more initialization parameters.

[0116] In one embodiment, at least some of the variational parameters of one or more variational quantum circuits may be initialized based on initialization values ​​classically calculated based on a classically simulable version of the quantum circuit that can be simulated on a classical computer.

[0117] In one embodiment, the classical simulation involves calculating the expectation value of the output of a classically simulable quantum circuit, where the expectation value is a function

[0118]

number

[0119] To define, to calculate,

[0120]

number

[0121] The dependency of the dependent variable is the initialization parameter

[0122]

number

[0123] It is determined to be a function of and the initialization parameters

[0124]

number

[0125] Fitting the optimal value for to the desired solution (or its estimate).

[0126]

number

[0127] Based on this, the quantum circuit is determined while maintaining other variational parameters within the classically simulable quantum circuit, and the initialization parameters

[0128]

number

[0129] This may include initializing based on the optimal value for [the function].

[0130] Subsequently, the quantum circuit can be variationally optimized based on variational parameters until a predetermined convergence of the quantum circuit's output with the actual solution is determined.

[0131] In one embodiment, one or more quantum feature maps, one or more variational quantum circuits, and a cost function may be constructed such that the circuit operators exhibit certain symmetries, such as real-value-conserving features. In this case, circuit differentiation can be performed with N+1 evaluations instead of 2N parameter-shift evaluations, (asymptotically) doubling the computation of function differentiation.

[0132] In one embodiment, the present invention may relate to a process for initializing a quantum circuit representing a function or equation. The quantum circuit comprises one or more quantum subcircuits, the one or more subcircuits may comprise one or more quantum feature maps, one or more variational quantum circuits, and one or more initializing unitary quantum circuits. In one embodiment, a classically similarable quantum circuit associated with the quantum circuit can be determined. To this end, in one embodiment, the relationships between at least some of the variational parameters of one or more variational quantum circuits may be defined such that the quantum circuit defines a classically similarable quantum circuit.

[0133] The expectation value at the output of a classically simulable quantum circuit can be determined by simulating the classically simulable quantum circuit using a classical computer, and the trial function can be calculated based on the expectation value.

[0134] Based on the trial function, values ​​for the variational parameters of one or more initialization quantum circuits can be determined. These values ​​for the variational parameters can be optimized by fitting the trial function to a desired function. The optimized initialization values ​​can then be used to initialize one or more variational quantum circuits of the quantum circuit.

[0135] Subsequently, the variational parameters of the quantum circuit can be variationally optimized until the output of the quantum computer converges toward the solution of the problem.

[0136] The initialization methods referenced above may be used in quantum learning methods performed on hybrid computer systems, including classical and quantum computers, as described with reference to Figure 1A of this disclosure, for example.

[0137] Accordingly, in a further embodiment, the present invention may relate to a quantum learning method based on a quantum circuit, the quantum circuit may include a plurality of different quantum subcircuits, each comprising one or more quantum feature circuits, each configured to map variables of a differential equation to a Hilbert space associated with qubits of a quantum computer, and one or more variational quantum circuits associated with one or more variational parameters to train the quantum circuit to approximate a solution. The method may include initializing at least a portion of the variational parameters of one or more variational quantum circuits based on initialization values ​​classically calculated based on a classically simulable version of the quantum circuit simulated on a classical computer, and variationally optimizing the quantum circuit based on the variational parameters until a predetermined convergence of the output of the quantum circuit with actual solutions is determined. The present invention may also relate to a system configured to perform such a quantum learning method.

[0138] In a further embodiment, the present invention may relate to a system for solving a system of (non)linear differential equations DE using a hybrid data processing system including a classical computer system and a quantum computer system, the system comprising a memory device containing computer executable instructions and a processor connected to the memory device, the processor being configured to perform an executable operation, the executable operation being to instruct the classical computer system to receive or determine a formulation of a quantum circuit representing the system of DE, the quantum circuit being parameterized by a variable x of DE, and one or more points x j One or more trial function values ​​f(x) around j One or more function circuits and one or more points x to determine ) j The instruction includes one or more derivative circuits for determining one or more trial derivative values ​​around a point x in the variable space x of the quantum computer system. j The processor includes instructing a set of quantum circuits to be executed, instructing a classical computer system to determine hardware measurement data in response to the execution of the quantum circuit quantum, and instructing the classical computer system to determine, based on the quantum hardware measurement data and loss function, whether the quantum hardware measurement data forms a solution to a system of (non)linear DEs.

[0139] The present invention may also relate to a computer program or group of computer programs comprising at least one software code portion, or a computer program product that stores at least one software code portion, wherein the software code portion is configured to perform any of the method steps described above when executed on a computer system.

[0140] The present invention may further relate to a non-temporary computer-readable storage medium that stores at least one software code portion, which is configured to perform any of the method steps described above when executed or processed by a computer.

[0141] The present invention is further illustrated with reference to the accompanying drawings, which outline embodiments according to the present invention. It will be understood that the present invention is not limited in any sense to these specific embodiments. [Brief explanation of the drawing]

[0142] [Figure 1A] This figure shows the network and optimization scheme of a data processing system for solving nonlinear DE according to one embodiment of the present invention. [Figure 1B] This figure shows the network and optimization scheme of a data processing system for solving nonlinear DE according to one embodiment of the present invention. [Figure 2A] This figure shows a differentiable quantum circuit (DQC) according to various embodiments of the present invention. [Figure 2B] This figure shows a differentiable quantum circuit (DQC) according to various embodiments of the present invention. [Figure 3A] This figure shows a quantum circuit according to various embodiments of the present invention. [Figure 3B] This figure shows a quantum circuit according to various embodiments of the present invention. [Figure 3C] This figure shows a quantum circuit according to various embodiments of the present invention. [Figure 4A] This figure illustrates exemplary variational quantum circuits according to various embodiments of the present invention. [Figure 4B] This figure illustrates exemplary variational quantum circuits according to various embodiments of the present invention. [Figure 5] This figure shows a schematic flowchart of a method for solving (non-)linear DEs using quantum computing according to one embodiment of the present invention. [Figure 6]This figure shows a method for solving a generalized (non)linear DE using quantum computing according to one embodiment of the present invention. [Figure 7] This is a hardware-level schematic diagram illustrating the application of logical operations to qubits using quantum circuits. [Figure 8A] This figure shows the differential quantum circuit computational solution obtained using three different feature maps. [Figure 8B] This figure shows the differential quantum circuit computational solution obtained using three different feature maps. [Figure 9] This figure shows the results for simultaneous strongly coupled equations. [Figure 10] This figure shows a schematic of the flow through convergence / divergence nozzles and numerical calculation results obtained by a standard classical solver. [Figure 11] This figure shows the DQC solution to the Navier-Stokes differential equation problem. [Figure 12] This figure shows a quantum circuit for a post-NISQ DQC-based scheme according to one embodiment of the present invention. [Figure 13A] This figure illustrates different types of differential equations that can be solved using the embodiments of this application. [Figure 13B] This figure illustrates different types of differential equations that can be solved using the embodiments of this application. [Figure 13C] This figure illustrates different types of differential equations that can be solved using the embodiments of this application. [Figure 14] This figure shows a generalized parameterized quantum circuit architecture with a generalized feature map and variational circuit layout, combined with an initialization subcircuit. [Figure 15] This diagram shows a workflow / strategy for initializing the variational parameters of a circuit based on classical preprocessing / fitting steps. [Figure 16]This figure shows an example of classical initialization illustrating a specific case of the workflow from Figure 15. [Figure 17] This figure illustrates an exemplary equation-solving method in which equations are parameterized using a functional library, illustrating the workflow and presenting concrete results that explain the underlying steps in the case of Figure 13C. [Modes for carrying out the invention]

[0143] Figures 1A and 1B illustrate a data processing system and optimization scheme for solving nonlinear differential equations according to one embodiment of the present invention. In particular, Figure 1A shows a system 102 comprising a first data processing system connected to a second data processing system 106, the first data processor system which may be implemented as a quantum computer system 104 including a quantum processor system 108 comprising quantum processing elements, e.g., gate-based qubits, and a controller system 110, e.g., a classical computer 106 comprising one or more classical processors, comprising input / output (I / O) devices forming an interface between the quantum processor and the second data processor. The controller system may include a system for generating control signals for controlling the quantum processing elements. The control signals may include, for example, a sequence of pulses, e.g., microwave pulses, voltage pulses, and / or optical pulses, used to manipulate the qubits. Furthermore, the controller may include output devices for reading out qubits, e.g., readout circuits, and control signals for reading out quantum processing elements, e.g., readout pulses for reading out qubits. In some embodiments, at least some of such readout circuits may be located with or integrated on a chip containing qubits.

[0144] The system may further comprise input 112 (of purely classical information) and output 114 (of purely classical information). The data processor system may be configured to solve nonlinear differential equations using a quantum computer. The input data may include information relating to the differential problem to be solved. This information may include the differential equation, boundary conditions, information for constructing a quantum circuit that can be run on the quantum computer, and information about the optimization process that needs to be performed to compute the solution to the nonlinear differential equation. The input data may be used by the system to classically compute values, such as a sequence of pulses, which can be used to construct a quantum circuit, in particular a differentiable quantum circuit, and to initialize and control qubit operations according to the quantum circuit. For this purpose, the classical computer may include a differentiable quantum circuit (DQC) generator 107. Similarly, the output data may include the ground state and / or excited state energies of the quantum system, the correlator operator expectation value, the optimization convergence result, the optimized quantum circuit parameters and hyperparameters, as well as other classical data.

[0145] Each of one or more quantum processors may include a set of controllable quantum processing elements, for example, a set of controllable two-level systems called qubits. The two levels are |0> and |1>, and the wave function of an N-qubit quantum processor is the two ground states of these qubits. N It can be considered as a superposition of qubits. However, the embodiments in this application are not limited to qubits and may include any multilevel quantum processing element suitable for performing quantum computation, such as a quantum trit. Examples of such quantum processors include noisy medium-scale quantum (NISQ) computing devices and fault-tolerant quantum computing (FTQC) devices.

[0146] A quantum processor can be configured to execute quantum algorithms according to gate operations of quantum circuits. A quantum processor is often implemented as a gate-based qubit quantum device, which allows for the initialization of qubits to their initial state, the interaction between qubits by sequentially applying quantum gates between different qubits, and the subsequent measurement of the qubit states. To this end, an input device may be configured to set the quantum processor to its initial state and control the gates that bring about the interaction between qubits. Similarly, an output device may comprise a readout circuit for reading out qubits, which can be used to determine a measure of energy associated with the expectation value of the Hamiltonian of the system taken across the prepared states.

[0147] In some embodiments, the first data processor system may be implemented as a software program for simulating a quantum computer system 104, which includes a quantum processor system 108. In that case, the software program may be a classical software program that runs a classical computer 106 so that quantum algorithms can be developed, executed, and tested on the classical computer without requiring access to the hardware implementation of the quantum processor system.

[0148] Embodiments of this application aim to solve general-form nonlinear differential equations using a quantum computer in a substantially different manner from schemes known in the prior art. For a given system of nonlinear differential equations, a quantum circuit is constructed based on so-called differentiable quantum circuits (DQCs). These quantum circuits may be executed on a quantum computer, and cost functions, such as Hermitian operators like the Hamiltonian, may be used to measure observables that form an approximation of the solution to the system of nonlinear differential equations, which are used in classical optimization algorithms. Loss functions may be used to determine whether the approximation of the solution is sufficiently close to the solution to the system of nonlinear differential equations.

[0149] An exemplary workflow of the optimization loop is shown in Figure 1B, illustrating an input step 120 for receiving information about a differential problem, such as a system of differential equations and boundary conditions. The optimization scheme may then be initialized 122, and the quantum circuit may be constructed based on a differentiable quantum circuit, which may represent a function and a derivative circuit. The construction of the quantum circuit may include a quantum feature map circuit, a variational Ansatz circuit that can be tuned by the optimization parameter θ, and a selection of cost and loss functions that may be used to exit the optimization loop.

[0150] Subsequently, the optimization scheme is, point x j This can be started by determining an approximate solution to a system of differential equations in the set of x. j Regarding this, a quantum circuit is executed on a quantum computer. As shown in the figure, the quantum circuit may include a function circuit 126 and a derivative circuit 128. The function circuit is at point x j Used to evaluate the function f around x, the derivative circuit is at point x j It can be used to evaluate the derivative df / dx of a function around x. The result of each quantum circuit execution is a measurement that is an observable of the quantum state of the quantum computer, which is at a particular point x j This represents an approximation of the function value or derivative value at . The loss function and measured values ​​can then be used to determine whether the measured values ​​form a sufficiently accurate approximation of the solution to the system of differential equations. If not, in a further step 130, a classical optimization scheme can be used to update the optimization parameter θ of the variational Ansatz circuit. The approximate solution is then obtained by running the quantum circuit based on the updated optimization parameter θ for each x j This will be decided.

[0151] It is pointed out that the differential equation described with reference to the scheme in Figure 1B is merely an example of several different types of differential equations that can be solved based on this scheme. Examples of such differential equations can be found throughout this application, and these may include parameterized differential equations. For example, in one embodiment, a parameterized differential equation may include one or more elements in the equation that are (hyper)parameters (alpha / beta) rather than functions (f(x), g(x,y)) or dependent / dimensional variables (x,y,z,t). In another embodiment, a parameterized differential equation may be based on a parameterized linear combination of functions. Examples of such differential equations are described with reference to Figures 13A–13C and Figure 17.

[0152] Figures 2A and 2B illustrate differentiable quantum circuits according to various embodiments of the present invention. In particular, the circuit in Figure 2A includes a feature map circuit 202, a variational quantum circuit 204, and a subsequent readout section 208. The quantum feature map is configured to encode data into a quantum register, for example, a classical variable x is transformed into a set of phases of rotations. The quantum feature map circuit undergoes unitary evolution on the qubits of a quantum computer.

[0153]

number

[0154] By activating this, the unitary evolution is a function of the differential equation parameter x, and furthermore, variational Ansatz

[0155]

number

[0156] 204, and operators

[0157]

number

[0158] Hamiltonian cost function for a set

[0159]

number

[0160] This is an observable-based readout 208, which includes the measured values ​​206. By combining these, the trial function f(x j ) can be calculated as a latent solution to the differential equation.

[0161] Therefore, the specific value of the variable x = x j The quantum circuit used to encode the function values ​​in the quantum computer is a feature map that encodes the x-dependency within the quantum computer.

[0162]

number

[0163] , and the subsequent variation Ansatz

[0164]

number

[0165] , and a set of operators

[0166]

number

[0167] This includes observable-based readouts for the measurement results, which are classically post-processed to provide a quantum function representation f(x) as a sum of expected values, and coefficients.

[0168]

number

[0169] This can be optimized using a quantum-classical hybrid loop, as explained with reference to Figure 6, for example. Measurements of different points on the optimization grid {X} are needed to construct the loss function circuit.

[0170] Figure 2B shows a structure similar to that illustrated in Figure 2A, but instead uses a derivative quantum circuit 210 to calculate the derivative of the function f with respect to x214, where x = x j It is evaluated as follows. The main difference from the circuit shown in Figure 2A is the parametric shift of the variables in the unitary 212. At a specific point x=x j The derivative of the obtained function f(x), evaluated by , can be estimated as the sum of the expectation values ​​for the derivative quantum circuit. The complete structure is derived from the differentiation of a feature map circuit, as described with reference to Figure 3, for example. The measured results of the function and derivative are classically post-processed, and from x to x j This can be combined for all j. Furthermore, the variational coefficient θ and cost structure coefficient α can be optimized in a quantum-classical hybrid loop to reduce the loss function values ​​for θ and α settings, as explained with reference to Figure 6.

[0171] Therefore, as can be seen from Figure 2, the trial function is the variable of the differential equation shown in Figure 2.

[0172]

number

[0173] It can be prepared as a quantum circuit parametricized by (or a set of variables). This description is for v variables

[0174]

number

[0175] Even in this case, it can be easily generalized. For the sake of simplicity, a simplified single-variable notation x is used. Quantum feature map encoding

[0176]

Number

[0177] By using

[0178]

Number

[0179] the pre-defined non-linear functions of variable

[0180]

Number

[0181] are cast into the amplitudes of the quantum state

[0182]

Number

[0183] By using

[0184]

Number

[0185] the pre-defined non-linear functions of variable

[0186]

Number

[0187] are cast into the amplitudes of the quantum state

[0188] Unlike amplitude encoding, quantum feature maps represent a latent space encoding that does not require access to each amplitude and is controlled by classical gate parameters. Quantum features map the real parameter x to the corresponding variable value. Next, a variational quantum circuit parameterized by a vector θ that can be adjusted in a quantum-classical optimization loop

[0189]

Number

[0190] is used. The resulting state for the optimal angle

[0191]

Number

[0192] contains x-dependent amplitudes sculpted to represent the required function. Finally, the real-valued function is read out as the expected value of a pre-defined Hermitian cost operator

[0193]

Number

[0194] and the function is

[0195]

Number

[0196] described as.

[0197] An optimization process based on variational circuit parameters θ and a loss function can be considered a quantum machine learning process, and a quantum circuit including a variational quantum circuit can define a plurality of interacting qubits that can be variationally tuned by (at least over) the variational circuit parameters θ. In this way, a plurality of interacting qubits can define a parameterized quantum circuit that can be trained based on variational circuit parameters θ, training data, and a loss function to approximate a function f(x) for some value of the variable x.

[0198] The derivative of a quantum feature map circuit is given by:

[0199]

number

[0200] It is often defined as a circuit with a modified action derivative.

[0201]

number

[0202] This makes it possible to express it as the sum of the product of the derivatives. In this way, the function derivative can be expressed using the product rule for differentiation. Thus, in the case of quantum feature maps generated by a string of Pauli matrices or arbitrary syntactic matrices, the parameter shift rule is used and the function derivative is the sum of the expected values.

[0203]

number

[0204] It is often expressed as such. however,

[0205]

number

[0206] It is defined through a parameter shift operation, where the index j is taken across the individual quantum operations used within the feature map encoding. By applying the parameter shift law again, the second derivative d 2 u(x) / dx 2 However, each generator can be obtained with four shifted terms.

[0207] Importantly, automatic differentiation (AD) techniques can be used to perform quantum circuit differentiation. AD, in contrast to numerical differentiation, allows for the expression of an exact analytical formula for a function derivative using a simple set of calculation rules. Since automatic differentiation yields the analytical derivative of the circuit at any point in the variable x, this scheme is independent of the cumulative errors arising from approximating the derivative. It is noteworthy that all known prior art schemes for quantum ODE solvers involve numerical differentiation using Euler's method and finite difference schemes, which have shortcomings in approximation errors, and often require fine discretized lattices. Embodiments of the present invention mitigate this problem.

[0208] One of the objectives of the present invention is that quantum circuits are generally F[{d n f / dx n} n ,{f m (x) m The task is to define the conditions for the solution of the differential equation written as ]=0, where the functional F[·] is given by the problem. This requires that the derivative and nonlinear functions must give a net zero contribution. Thus, solving the differential equation implies the loss function L θ [d x This can be written as an optimization problem using the set of points {x}. i In}

[0209]

number

[0210] It minimizes the constraint and also ensures that the boundary conditions are met. Optimal angle

[0211]

number

[0212] After finding the solution, a solution can be generated from equation (1) as a function. Therefore, an embodiment of this application is as follows: 1. Using quantum feature mapping encoding, we overcome the complexity of amplitude encoding used in conventional techniques to prepare solutions at the boundary. 2. Using automatic differentiation of quantum feature mapping circuits, it becomes possible to represent analytical function derivatives without the inaccurate errors characteristic of numerical differentiation (finite difference), 3. Using variational quantum circuits, we search for suitable solutions in the exponential space of the fitting polynomial, which is thus analogous to spectral and finite element methods with exponentially improved scaling. 4. The solution is encoded into an observable operator, and therefore the expectation can be calculated periodically, thus avoiding the data retrieval problem.

[0213] Regarding the latter, unlike amplitude encoding in HHL and related methods, obtaining a complete solution from the amplitude incurs an exponential cost and requires tomographic measurements.

[0214] One of the objectives of this invention is to construct circuits that can operate on computationally limited quantum processors, meaning that the gate depth (the number of operations performed in series) is limited to a specific, restricted amount. The gate depth largely defines the training procedure used in classical optimization loops. To mitigate this depth reduction problem, it is also possible to approach the ideal quantum computing regime by utilizing parallel training strategies for quantum circuits and quantum state encodings.

[0215] The following describes quantum circuits that can be used to construct differentiable circuits as solutions to differential equations. These quantum circuits include quantum feature maps and their derivatives, variational quantum circuits (Ansatz), cost functions defining trial functions, and loss functions used in optimization loops. In addition, boundary processing techniques, regularization schemes, and fully optimized schedules are described.

[0216] Quantum feature maps are used in unitary circuits.

[0217]

number

[0218] This is a typical nonlinear function with respect to the variable x.

[0219]

number

[0220] It is parameterized by acting on the state, which maps the x-dependency so that it translates to the quantum state amplitude.

[0221]

number

[0222] This is achieved through latent space mapping, also known as latent space mapping. Different methods of feature map encoding exist. Below, various examples are described, including Chebyshev quantum feature maps, which enable approximation of highly nonlinear functions. The procedure for feature map differentiation is also described as an important step in constructing quantum circuits that find solutions to differential equations.

[0223] In the first embodiment, a product feature map using qubit rotation may be used.

[0224] Figures 3A to 3C show quantum circuits according to various embodiments of the present invention. In particular, Figure 3A shows the basic form of a quantum feature map, which is illustrated here as an example of a "product" type feature map, and a single qubit rotation (here

[0225]

number

[0226] The selected function acts on each qubit individually and is parameterized by a function of the variable x. Such an operation can be referred to as a layer of rotational operations. Reading from left to right in chronological order, the rest of the circuit is summarized outlinely, including the application of variational Ansatz 304 and cost function measurement 306, as described in more detail with reference to Figure 6. For nonlinear feature mapping encoding, a nonlinear function

[0227]

number

[0228] This can be used as the angle of rotation. The product feature map consists of multiple stacks and different functions.

[0229]

number

[0230] This can be further generalized. For example, several feature maps can be concatenated to represent a multivariable function.

[0231] Figure 3B illustrates an example of a derivative quantum circuit for the product feature map in Figure 3A. The derivative with respect to the variable x is the shifted phase.

[0232]

number

[0233] The differential chain rule is followed, including qubit operations 312 by . Here, the expectation value of the derivative is given by each x-dependent curl 310 1-4 It is written as the sum of separate expectation values ​​with shifted phases, repeated over and over.

[0234] Figure 3C shows an example of a generalized product feature map, where the layer of rotation is Hamiltonian

[0235]

number

[0236] It follows a unitary evolution generated by 314. For complex multi-qubit Hamiltonians, the encoded states may contain exponentially many unique x-dependent amplitudes. The time interval τ can be set variationally or annealed to values ​​from zero to a finite value in an optimization procedure.

[0237] Preferably, the product feature map has a nonlinear dependency on the encoded variable x. In its simplest form, this may correspond to a single layer of rotations. Such a product feature map is expressed by the equation

[0238]

number

[0239] Often described by N'≦N is the number of qubits used by the quantum computer for encoding. Furthermore,

[0240]

number

[0241] The Pauli procession

[0242]

number

[0243] This is the Pauli rotation operator for (α = x, y, z respectively), and this is the phase

[0244]

number

[0245] This acts on the qubit j. Here, if we consider rotations with respect to different j, the symbol

[0246]

number

[0247] This represents the tensor product. This type of feature map circuit is also used in quantum circuit learning. The next step is to assign a nonlinear function to the rotation. In one embodiment, the nonlinear function is such that only real amplitudes are generated.

[0248]

number

[0249] and α = y can be selected. The unitary operator in equation (5) is,

[0250]

number

[0251] This is rewritten, and the amplitude depends on the variables encoded as cos[(arcsin x) / 2] and sin[(arcsin x) / 2]. When applied to the initial state |φ>, this feature map makes the variables

[0252]

number

[0253] And it can be encoded as an Nth-degree polynomial formed by the product [QCL]. The redundancy from many qubits thus forms a basis for function fitting [SS]. The basis can be continuously improved by adding more rotations, and it should be noted that this basis selection is suitable for linear and quadratic functions, but often lacks expressive power.

[0254] The product feature map is several layers of rotation.

[0255]

number

[0256] , various nonlinear functions

[0257]

number

[0258] , and generalized to a specific subset of qubits N,

[0259]

number

[0260] It could be written as follows.

[0261] The following examples show how quantum feature maps can be differentiated, for example, in the example of equation (4), α=y rotation and all layers are considered. The derivative for a unitary operator generated by an arbitrary syntactic matrix (in this case, a Paulist ring of length 1) is:

[0262]

number

[0263] It can be written like this, Here, Euler's formula is used, and the derivative can be rewritten in the form of a unitary sum, and the x-dependent rotation is shifted by 1. Next, this formula follows the steps of the standard parameter shift law for any observable for the encoded state.

[0264]

number

[0265] This can be generalized to the expected value of .

[0266]

number

[0267] It is written as follows: Here

[0268]

number

[0269] and

[0270]

number

[0271] is the sum of the shifted unitaries

[0272]

number

[0273] That is the case.

[0274] The corresponding derivative quantum circuit (DQC) is shown in Figure 3B, where the differentiation of the cost function with respect to the feature map is performed using the chain rule (highlighted rotation). Similar strategies are used for generalized multilayer feature maps and nonlinear maps.

[0275]

number

[0276] This can be applied to different selections. Finally, the feature map generator (encoding Hamiltonian

[0277]

number

[0278] If the matrix is ​​not an articulated matrix, it can be rewritten as a sum of unitary operators, and its derivative can be measured as a sum of overlaps using a SWAP test.

[0279] In another embodiment, a nonlinear quantum feature map, which may be called a Chebyshev feature map, may be used. This feature map belongs to the product feature map family and drastically changes the basis for the functional representation. As a component, single-qubit rotation

[0280]

number

[0281] It can be used, but the nonlinearity

[0282]

number

[0283] It is introduced with n=0,1,2,... and the encoding circuit is

[0284]

number

[0285] It is described as follows.

[0286] Here, the coefficient n[j] is generally thought to depend on the qubit position. A seemingly small change, multiplying by the coefficient 2, takes a surprisingly long route. That is, when we expand the rotation using Euler's formula,

[0287]

number

[0288] To obtain.

[0289] The resulting decomposition of equation (11) is T n (x) and U n This corresponds to a unitary operation with matrix elements defined through Chebyshev polynomials of degree n of the first and second kinds, denoted as (x). Formally, Chebyshev polynomials are Chebyshev differential equations.

[0290]

number

[0291] This represents the solution to, Here y(x)=Acos(n arccos(x))+Bsin(n arccos(x)) (13) ≡AT n (x)+BU n (x), |x|<1, Here, A and B are constants. For low-degree Chebyshev polynomials of the first kind, T0(x)=1, T1(x)=x, T2(x)=2x 2 -1, T3(x)=4x 3 It is explicitly written as -3x, and for higher degrees, the recursive relation is used T n+1 (x) = 2xT n (x)-T n-1 (x) (14) can be estimated to be (x) (14).

[0292] Similarly, for Chebyshev polynomials of the second kind, U0(x)=1, U1(x)=2x, U n+1(x) = 2x and U n (x)-U n-1 It can be written as (x). Important properties of Chebyshev polynomials are chaining, nesting, and simple differential laws. The chaining properties for polynomials of the first and second kinds are 2T, respectively. m (x)T n (x=T m+n (x)+T |m-n| (x) and

[0293]

number

[0294] It is written as follows. The derivative is dT n (x) / dx=nU n-1 It can be obtained with (x). Nesting is in relation to T n (T m (x))≡T nm This corresponds to (x). Finally, different types of polynomials are when j is even.

[0295]

number

[0296] When j is odd

[0297]

number

[0298] It can be transformed as follows. Finally, note that the Chebyshev polynomial, when defined in the x=(-1,1) region, often represents an oscillatory function, and its derivative diverges at the boundary of this interval.

[0299] The power of the described representation can be inferred from approximation theory. This means that any smooth function is

[0300]

number

[0301] It is stated that it can be approximated as follows. The Chebyshev polynomial is uniform L ∞ They form an optimal set of basis functions in the sense of a norm. This is why Chebyshev polynomials form the basis of spectral algorithms for solving ODEs and why they enhance quantum simulations.

[0302] The following examples consider two types of Chebyshev quantum feature maps. The first version is:

[0303]

number

[0304] This corresponds to a sparse Chebyshev feature map defined as such, where the encoded order is homogeneous and equal to 1. Here, the chaining T n (x) and U n (x) should be remembered, and note that after a state with a Chebyshev polynomial as a prefactor is created, the basis system will become even larger by concatenating elements. Below, the distinction of sparseness is dropped, and equation (15) is simply called the Chebyshev feature map. The second version corresponds to the Chebyshev tower feature map, which is

[0305]

number

[0306] It is often defined as such, where the encoded order increases with the number of qubits, forming a structure resembling a tower of polynomials as n=j increases. Here again, the basis set becomes greatly richer as the polynomials chain together and change form between two types and their degrees. This is a choice used when great representability is needed without increasing the system size and the number of rotations. Equation (16) enables the representation of general functions and can be further improved by using a sheath of rotations as in equation (6).

[0307] Product feature maps can induce a nonlinear mapping between a variable x and quantum states described by the tensor product of separate single-qubit wave functions. These states are confined to a subspace of product states. To utilize the forces of the entire Hilbert space of the system, we approach the case of amplitude encoding, which requires independently different amplitudes to be input, including a subspace of entangled states. To further enrich the described feature maps, it is proposed to enhance the product feature maps (and especially layered Chebyshev maps) with additional entanglement layers represented by Hamiltonian evolutions. That is, after one set of single-qubit rotations, another unitary

[0308]

number

[0309] This can be considered, and this acts on time τ, and the Hamiltonian

[0310]

number

[0311] It is generated by . The circuit sketch is shown in Figure 3C. As a complex many-body Hamiltonian

[0312]

number

[0313] By selecting this option, it is ensured that exponentially large amplitudes are generated. Quantum simulations of the dynamics are known to result in increases such as the volume of entanglement.

[0314] One of the important choices is,

[0315]

number

[0316] The question is when to address the hard problems from the NP-hard complexity class, as has been proposed. Then, by using two layers of rotation plus evolution, the embedding becomes difficult to simulate classically but can be implemented as unitary evolution on a quantum computer. Evolution-enhancing feature maps are also applied to qubits in J, evolution

[0317]

number

[0318] It can also be viewed through the prism of a recently proposed Fourier feature map, which is a class of quantum feature maps based on . A Fourier map is a function that

[0319]

number

[0320] This allows it to be encoded as a Fourier series defined by the difference of its eigenvalues. The evolved reinforced feature map then combines the Chebyshev basis and the Fourier basis, and complex

[0321]

number

[0322] It is encoded within a complete Hilbert space.

[0323] In another embodiment, the variable may be encoded in the data using a feature map by converting it to a canonical amplitude encoding form. This relates x, written in binary form, to the computational base state of its binary representation. The corresponding feature map for encoding the binary variable x.

[0324]

number

[0325] teeth,

[0326]

number

[0327] It is written as, however {x j} is the binary value for parameter x at the j-th digit. The derivative of the amplitude encoding feature map then depends on the product rule for N rotations, and also includes the binary derivative of the variable from the product rule.

[0328] In a further embodiment, the variable is x int =Σ j x j ·2 j It can be converted to decimal representation as follows: In the reverse procedure, each binary number x j This is the remainder x of repeated division. j =mod(x int ,2 j ) can be identified in this way.

[0329]

number

[0330] is, x intIt can be rewritten as a function of x = x int This shows how to differentiate a circuit that has this feature map.

[0331] Amplitude encoding feature maps offer a powerful technique when dealing with functions of discrete variables and functions encoded as quantum wave functions (rather than expectation values). These can be useful means of compressing data into quantum registers.

[0332] To construct the solution to a differential equation as a quantum circuit, we need to manipulate the latent space basis functions to obtain the desired form for both the derivative and the function. This is a variational circuit.

[0333]

number

[0334] This is achieved through a process typically referred to as quantum Ansatz. Various architectures are described in detail below.

[0335] In one embodiment, a variational circuit of one or more layers of parameterized rotation

[0336]

number

[0337] These layers may be selected. These layers may be followed by a layer of CNOT operations. This is known as Hardware Efficient Ansatz (HEA), proposed for variational quantum encoder VQE schemes for chemical applications. The structure of the HEA quantum circuit corresponds to a connected layer of single-qubit rotation and global entanglement layers for all N qubits or at least most of them.

[0338] Figures 4A and 4B illustrate exemplary variational quantum circuits according to various embodiments of the present invention. In particular, Figure 4A shows a so-called hardware-efficient form of variational Ansatz 402, which allows any single-qubit rotation to be implemented.

[0339]

number

[0340] It includes parameterized rotation layers that form a pattern. The variational angle θ is set individually for each rotation, 406, 408. The rotation layers are applied to the initial qubit states |ψ(x)>404 coming from the preceding quantum circuit, the feature map circuit. Following the variational rotations, one or more entanglement layers may follow, which may include controlled NOT (CNOT) operations between nearest qubits. Block 710 of "rotation-plus-entangler" is repeated d times to form a complete variational circuit 412.

[0341]

number

[0342] Forms 403.

[0343] Figure 4B shows a more general form of the alternating block Ansatz 416, incorporating, for example, the hardware-efficient form circuit from Figure 4A. The variational circuit can consist of blocks with a width of M qubits (M / 2 for boundary qubits). The blocks can be selected in a hardware-efficient form, as illustrated at depth b. The blocks are arranged in a checkerboard pattern, and n b This can be repeated a certain number of times. The goal of this alternating blocking strategy is to locally entangle the qubits while avoiding global entanglement operations that typically result in vanishing gradients during the optimization of θ.

[0344] In further embodiments, alternating block ansatz (ABA) may be used, where separate subblocks interleaved in a checkerboard configuration, as shown in Figure 4B, are used instead of a global entanglement layer. Each subblock has a hardware-efficient configuration, as shown in Figure 4A, for a given depth b. For the first layer, the width of the subblock (number of active qubits) may be equal to M so that [N / M] blocks are used (or less than M if N / M is not an integer). The next layer may contain (or consist of) the same subblocks but be shifted by [M / 2] so that the edge subblocks span all qubits. The checkerboard configuration described is d layers This can be repeated. The motivation behind ABA is to gradually form correlated states by first locally entangle the qubits and then interleaving the subblocks. Furthermore, during the optimization of θ, global entanglement operations, which usually result in vanishing gradients, are avoided. This helps to improve the trainability of the circuit while maintaining high expressiveness.

[0345] It should be noted that the choice of Ansatz can be sensitive to the choice of cost function operator, as determined by symmetry. That is, as a result of the cost function choice, noncommutative generators for variational Ansatz may be affected.

[0346]

number

[0347] teeth

[0348]

number

[0349] The generator needs to be selected in such a way that the result is

[0350]

number

[0351] This is a rewriting form of a general unitary. Here, a generator refers to a quantum operator that acts in a controlled manner on one or more qubits. This is a completely generic concept and basically applies to any circuit, gate, etc. This ensures that it covers the entire range of the solution space. Also, symmetry is considered and the Hilbert space for the search can be reduced. In many cases, the generator can be selected so that only real amplitudes are produced. Adaptive strategies or genetic searches can also be used.

[0352] Furthermore, to search for optimal circuit parameters, stochastic gradient descent schemes, particularly their adaptive versions represented by Adam, can be used. For this purpose, variational circuits

[0353]

number

[0354] One or more gradients can be measured using an automatic differentiation approach. By selecting an Ansatz parameterized by a single-qubit rotation, it becomes possible to apply a parameter shift law, opening up the option of overlap measurements as a more general strategy.

[0355] To read out information, a Hermitian operator is selected, thereby allowing the measurement of observability. Generally, different selections are available. In one embodiment, the Hermitian operator is the magnetization of a single qubit j,

[0356]

number

[0357] This may include a function with a range of [-1,1]. In another embodiment, the total magnetization of the system

[0358]

number

[0359] Hermitian operators including (those with equal weights) may be selected.

[0360] In a further embodiment, the cost may be selected as a quantum Hamiltonian having a provable complex spectrum and belonging, for example, to the ergodic phase. Such a Hermitian operator is an Ising Hamiltonian with additional transverse and longitudinal magnetic fields.

[0361]

number

[0362] Often includes, Ising bond J j,j+1 and

[0363]

number

[0364] It may be assumed to be non-uniform. For cost functions having several non-commutative groups of observables, a Hamiltonian averaging procedure may be used, and term-by-term measurements may be performed. Instead of the nearest Hamiltonian, in one embodiment a spin-glass type cost function may be used, which is formal

[0365]

number

[0366] It may have.

[0367] These are known to include instances of NP-hard problems, enabling high expressibility of circuits describing DE solutions. Finally, general cost functions can include a large set of Paulistrings, similar to those in quantum chemistry, for example.

[0368] Along with measuring individual cost functions, a function expressed as a classical weighted sum of observables may also be considered. Such a cost function is formal

[0369]

number

[0370] Often, it is considered to have the following characteristics: α l ∈R is a weighting coefficient,

[0371]

number

[0372] is a cost function that can be selected from the pool of operators described above. The coefficients can be tunable, and the cost can be adjusted so that gradient descent (represented by Adam in our case) has an optimal form. This procedure further improves the strength of the hybrid quantum classical workflow.

[0373] To solve a system of differential equations, a means is needed to measure (for example, with respect to distance) how well a differentiable quantum circuit fits the conditions for solving the problem under consideration. A classical optimizer then updates its parameters to reduce this distance. This distance corresponds to the difference between the differential equation and zero, evaluated on a set of points, as well as the coincident initial and boundary conditions. This can be reformulated as an optimization problem for derivatives and loss functions of a function, evaluated on a grid of points.

[0374] The following example format

[0375]

number

[0376] A loss function parameterized by the variation angle θ may be used. Loss contribution due to coincidence with differentiation

[0377]

number

[0378] Loss contribution due to satisfying boundary conditions

[0379]

number

[0380] It can be divided. The differential loss is

[0381]

number

[0382] Defined as, L(a,b) is a function that describes how the distance between two arguments a and b is measured. The loss can be estimated on a grid of M points and normalized by the grid size. The functional F is of the form F[d x This corresponds to the differential equation written as u,u,x]=0. This is the f and d at the training grid points. x It can be evaluated by combining the values ​​of f. This functional contains information about all differential equations when dealing with the system, and explains the contributions from all equations. The boundary loss contribution is

[0383]

number

[0384] It was written as follows: This includes the distance between the function value at boundary x0 and a given boundary value u0. Note that x0 can be an initial point or a set of boundary points. The boundary pinning coefficient η can be used to control the weight of the boundary term in the optimization procedure. In particular, a larger η > 1 can be used to ensure that the boundary is preferred and represented with higher precision.

[0385] A different selection of losses defined by three distance definitions L may be used. In one embodiment, the loss type corresponding to the mean squared error (MSE) L(a,b)=(ab) 2 (twenty three) It may be used.

[0386] MSE is simple but performs well enough in numerical simulations. In a further embodiment, the mean absolute error (MAE) may also be used as a loss defined by the distance L(a,b)=|ab|.

[0387] In further embodiments, several more complex metrics may be used, including variations of the Kullback-Leibler (KL) divergence and the Jensen-Shannon divergence. These are well-known loss functions that are routinely used in statistical modeling.

[0388] The choice of loss function determines how the optimizer perceives distances between vectors and therefore affects convergence. MSE (Most Significantly Evaluated Loss) places greater emphasis on large distances and less emphasis on small distances, strongly suppressing terms with large L values. MAE (Most Significantly Evaluated Loss) and KL (Most Significantly Evaluated Loss) do not place such emphasis and may have slower convergence. However, as we approach the optimal solution, they can achieve higher accuracy than MSE. KL has an additional incentive to keep the magnitude of the first argument low, which works well for differential loss terms when we want to match the differential equation to zero.

[0389] When constructing a quantum circuit that satisfies a differential equation, it is necessary to ensure that, together with the matching derivative, an initial value problem or a boundary value problem is solved. Generally, this corresponds to setting the function values to the set of required initial or boundary points and thus resembles the quantum circuit learning task [QCL]. At the same time, there are several ways in which a DQC-based function f θ (x) can be constructed, and various performances and specific advantages / disadvantages arise when solving a particular problem.

[0390] Information about the boundary can be included as part of the loss function defined by Equation (20). For the MSE loss function type, Equation (22) for the boundary part is in the form

[0391]

Number

[0392] and can be written as where x0 represents the set of boundary points (or initial points), u0 is the vector of boundary values, and η is the pinning coefficient as described above.

[0393] In one embodiment, information about the boundary can be included in the expected value of the cost function. This can be referred to as pinning boundary processing. This simply involves selecting the cost operator

[0394]

Number

[0395] and representing the solution in the form

[0396]

Number

[0397] which corresponds to.

[0398] Next, the initial value u0 is matched through the boundary term of the loss function. The strength of pinning boundary processing lies in the equivalent processing of the boundary term and the derivative term, both of which

[0399]

number

[0400] It is encoded with its intrinsic spectrum. At the same time, it is typically generated randomly, with an initial value θ init The boundary values ​​starting from the value represented by the cost operator need to be adjusted.

[0401]

number

[0402] It can be assumed that this is adjusted by shifting f(x) with a constant multiple identity term that is added, where α0 is θ init ~A function for random [0, 2π]

[0403]

number

[0404] It is typically set so that it is close to the value of u0 when evaluated at x=x0.

[0405] In a further embodiment, the boundary handler may accommodate iteratively shifting the estimated solution based on boundary points or initial points. In this method, boundary information does not require a separate boundary loss term and is not encoded in the expected value of the cost function. Instead, it is iteratively set within the parameterization range of the function. In this method, additional terms in the function and derivatives are computed, and the boundary information is

[0406]

number

[0407] It is contained within itself. The function is

[0408]

number

[0409] It is often parameterized as follows: f b ∈R after each iteration step

[0410]

number

[0411] These are parameters that are adjusted as follows.

[0412] This effectively solves a differential equation where the solver is shifted to an arbitrary position and then to the desired initial conditions as shown in equation (26).

[0413]

number

[0414] This makes it possible to find the initial value θ. This method of boundary treatment guarantees an exact match to the given initial value and does not require a separate boundary term in the loss function, so the derivative loss term does not have to compete with the boundary loss. Furthermore, it can be made possible to match the cost function to a solution shifted by any amount, thus simplifying the selection of the optimal angle and initial value θ init This eliminates the dependency on [the specified value]. However, this method requires having precise initial values, which can be problematic in certain situations.

[0415] In a further embodiment, a boundary handling technique defined by a gradient descent procedure, which depends on the classical shift of the solution but is equivalent to variational angle optimization, may be used. This removes the need to include boundary information in the cost expectation value, but the information still needs to be included in the loss function, whether through boundary loss terms or regularization. Thus, the solution of the form

[0416]

Number

[0417] is sought, where f c ∈R is the variational parameter associated with the quantum ansatz angles and is updated appropriately via a classical optimizer. Thus, the gradient with respect to f c must also be calculated in addition when using this boundary handler. The strength of this method is that, due to the classical shift, even if the random initial angles start far from

[0418]

Number

[0419] the initial value u0, the optimizer can quickly and easily update f c to correct this. However, in some cases, the boundary terms and differential terms in the loss may compete with each other.

[0420] If the goal is to find a variational spectral representation of the solution of a differential equation using a large basis system, the optimization procedure benefits from having a good initial estimate or a "pre-trained" DQC. This can be achieved by introducing a regularization procedure and also helps to avoid the optimizer being trapped in a local minimum.

[0421] Modifications to the regularization procedure include: 1) inputting prior information about potential solutions; 2) biasing DQC-based solutions into a specific form; and 3) exploring solutions in the region near boundary values ​​and inputting points from the initial training into the next session. The inputs for steps 1) and 2) are variables

[0422]

number

[0423] The regularization points for are the corresponding function values ​​for R points.

[0424]

number

[0425] It is included together with the derivative value. Similarly, regularization based on the derivative value can be considered. The additional contribution to the loss function is the regularization point.

[0426]

number

[0427] A simple strategy derived from this can be employed. This loss is defined such that the DQC-based function matches the regularization value at the corresponding grid point. This has a form similar to the boundary loss contribution. Using the MSE loss as an example, the regularization contribution is

[0428]

number

[0429] It was written as follows: Here, ζ(n j) is introduced as an iteration-step-dependent regularization weight, and thus indicates the optimization schedule. Generally, more emphasis should be placed on the regularization-based training in the initial stages, and the regularization should decrease to zero as the number of iterations increases. This allows us to first utilize prior information to set a rough solution or preferred function behavior, and then perform precise derivative loss optimization in the later training stages.

[0430] One possible choice for the optimization schedule is the regularization weight ζ(n j ) = 1 - n j / n iter This corresponds to linearly decreasing n j is the current iteration count, n iter is the maximum number of iterations. This strategy works for small learning rates and large iterations, and such optimizers have enough "time" to adapt to the constantly changing loss landscape. Another option is an inverse sigmoid optimization schedule, where smooth reduction of regularization weights is performed in a predefined training phase. This schedule is

[0431]

number

[0432] It is often parameterized as follows: n drop δ represents the number of iteration steps at which the regularization weight decreases. j This assigns a transition rate. This allows DQC to initially focus almost entirely on regularization optimization, and then later switch to focusing on gradient optimization.

[0433] Finally, a workflow for constructing a differential equation solver based on derivative quantum circuits using the elements and strategies described above is outlined. Figure 5 shows a schematic flowchart of a method for solving a general (non)linear DE using quantum computing according to one embodiment of the present invention. The process can be started by specifying inputs to the solver 502, which include the given problem specified as a system of various types of (non)linear differential equations, along with their respective boundary conditions. In addition, a set of regularization points may be added to ensure that the optimized solution is selected in the desired qualitative form. Next, a schedule for derivative quantum circuit optimization may be set up, including the selection of the quantum circuit configuration. For this purpose, a quantum feature map 504 may be defined. Furthermore, an Ansatz for the variational quantum circuit may be defined, including its depth 506. In addition, a cost function type may be selected, as well as whether variational weights are considered. Then, a loss function 510 may be selected, which may include a scheme for matching boundary terms 512 and derivatives. Furthermore, a classical optimizer (essentially an optimizer) for the variational angle and weights (with associated hyperparameters) needs to be defined, including the number of iterations and termination conditions. The optimizer refers to an algorithm configured to optimize the cost function or loss function as a function of the variational parameters. The quantum circuit is then variationally optimized in a quantum-classical hybrid loop, after which the solution can be sampled from the optimized quantum states 514.

[0434] Figure 6 shows a method for solving a general (non)linear DE using a quantum computer according to one embodiment of the present invention. In particular, after determining the quantum circuit and optimization schedule, several initialization steps need to be performed.604 First, for each equation variable x, a set of points {X} (a regular or randomly drawn grid) can be specified.606 The variational parameter θ is set to an initial value (for example, as a random angle). Next, the variational quantum state for the cost function is defined.

[0435]

number

[0436] Expected value over time<C(x,θ)> However, the selected point x j Using quantum hardware, 610 can be estimated. Then, considering the boundary conditions, a potential solution at this point can be constructed.

[0437] The derivatives of the quantum circuits can be determined, and their expected values ​​d<C(x,θ)> / dx represents point x j And the estimated cost function is 610 for all x in {X}. j It is often assumed that the function values ​​and derivative values ​​are collected by repeating step 606, and the loss function is constructed for the entire grid and the system of equations (forming the necessary polynomials and intersection terms by classical post-processing), as shown in 612. Regularization points are also added, forcing the solution to take specific values ​​at these points. The goal of the loss function is to assign a "score" to how well the potential solution (parameterized by the angle of variation θ) satisfies the differential equations, coincident derivative terms and function polynomials to minimize the loss.

[0438] With the aim of increasing the score (and decreasing the loss function), we also calculate the gradient of the loss function 612 with respect to the variational parameter θ. By using a gradient descent procedure (or in principle any other classical optimization procedure 614), in step 616 the variation angle is calculated for each iteration n j = 1 to the next iteration n j +1

[0439]

number

[0440] It can be updated as follows (where α is the "learning" speed). The steps described above can be repeated until the termination condition is reached. The termination condition can be selected as follows: 1) Maximum number of iterations niter It reaches. 2) The loss function value is smaller than the pre-specified value. 3) The loss gradient is less than a certain value.

[0441] After exiting the classical loop, the solution is the angle θ that minimizes the loss. opt It is selected as a circuit having the following characteristics. Finally, the complete solution is the optimal angle

[0442]

number

[0443] This is extracted by sampling the cost function. In particular, this can be done for any point x, since DQC constructs a solution that is valid beyond (and between) the point where the loss is originally evaluated.

[0444] Figure 7 is a hardware-level schematic illustrating the application of logic operations to qubits using a quantum circuit according to an embodiment of this application, for example, a quantum circuit as shown in Figure 6. The Ansatz and variational unitary, 702 and 704, can be decomposed into a sequence of logic gate operations. These logic gate operations are transformations in quantum Hilbert space on the qubits. To transform the internal states of these qubits, a classical control stack, i.e., a quantum computer controller, is used to send pulse information affecting one or more qubits to a pulse controller. The controller may send such sequences of pulses independently to each qubit in time. For example, an initialization pulse may be used to initialize a qubit to the |0> state 702. Then, for example, a series of 1-qubit pulses may be sent to the qubit array 704, which may represent the application of a single-layer feature map. Then, a 2-qubit pulse sequence may be used to effectively entangle multiple qubits 706. The duration, type, intensity, and shape of these pulses determine the quantum logic operation achieved. The way in which the qubits should interact is defined by the quantum feature mapping circuit and the quantum variational circuit. Reference number 708 indicates a “gap” in the depicted time axis, meaning that the sequence of gates can be repeated in a similar manner in the direction of the time axis 712. At the end of the pulse sequence, the qubits are measured 710.

[0445] The following explains how the algorithm is actually executed. For this purpose, a differential equation with a known analytical solution is selected and can be compared to one obtained by the derivative quantum circuit. We select a single ODE for the initial value problem, which is

[0446]

number

[0447] It was written as follows: Here, λ and κ, as well as u0, set the value of the function u at x=0. Equation (31) is the damped oscillation function. u(x)=exp(-κλx)cos(λx)+const (32) It has a solution of the form of .

[0448] Here, the constant const is determined by the initial conditions. This problem is a single ODE and is fairly simple, but in order to reproduce the damped oscillation solution, a reachable basis is required, which must contain both the oscillation function and the increasing / decreasing function. As λ and κ become large, the function begins to oscillate and dampen rapidly, and the solution becomes difficult to represent.

[0449] To demonstrate how the proposed method works, a derivative quantum circuit is used to solve equation 31 using the optimization of a differentiable quantum feature map. In particular, two cases are selected with parameters λ=8 and λ=20, and κ=0.1 and u0=1. Note that the parameters are chosen to make the construction of the DQC difficult, and λ=20 is a complex case as the resulting solution is highly nonlinear and oscillatory. It is often considered that an optimization grid of 20 points is considered, starting from x=0, with a maximum time of 0.9 (dimensionless units are used). To find the solution, it is often considered that a quantum register with N=6 qubits is used, and the cost function is the total magnetization in the Z direction.

[0450]

number

[0451] It is selected as follows. For the variational circuit, a standard hardware-efficient Ansatz described in this application may be selected, and the depth may be set to d=5. Optimal angle θ optTo explore this, adaptive stochastic gradient descent is performed using Adam with automatic differentiation made available by the analytical derivative. In particular, the workflow is coded using well-known packages that enable fast and efficient implementation. A full quantum state simulator is used in a noise-free environment. In this example, we use the handling of floating boundaries.

[0452] Circuit-based solutions were explored and their performance compared using three different feature maps described in this application. These correspond to the product feature map, a sparse version of the Chebyshev feature map, and the tower Chebyshev feature map. The results for these feature maps are labeled Prod, Cheb, and ChebT, respectively. Several metrics may be used to evaluate performance. The first metric is the perfect loss (L F This is expressed as ( ). This refers to the loss calculated from the differential equation and any boundary or regularization term, and this is

[0453]

number

[0454] The second metric is the differential loss L. D This corresponds to a portion of the perfect loss excluding the regularization contribution. Finally, the third metric is the quality of the solution (L Q The quality of the solution is the distance of the current DQC-based solution from the known true solution. This is calculated by evaluating the DQC-based solution and the true solution in a set of points and using the MSE loss type, which is

[0455]

number

[0456] This is equivalent to the quality of the solution, which is a useful way to compare how two different training setups perform, especially when training to solve the same differential equation.

[0457] Figures 8A and 8B show the DCQ computational solutions obtained by comparing them with the true solutions (solid lines) for λ=8 (802) and λ=20 (804), respectively, using three different feature maps (dashed lines). Graphs 808 and 810 show the optimization results for graphs 802 and 804 with n iterations, respectively. j The perfect loss L is shown as a function of L. F (Solid curve) and quality L of the solution Q (Dashed curve) is shown. Graph 806 shows DQC-based solutions to the damped oscillator problem for four different Ansatz depths d, with the true analytical solution plotted by the solid curve. Graph 812 shows the perfect loss L as a function of the number of iterations for the solutions presented in Graph 806. F (Solid curve) and quality L of the solution Q (Dashed curve) is shown.

[0458] For λ=8, it is observed that both Chebyshev feature maps converge and approach the true solution. The more expressive Chebyshev tower feature map takes longer to converge but reaches a solution closer to the true solution. The less expressive product feature map fails to converge, and the loss quickly reaches a horizontal state.

[0459] For λ=20, the true solution is more oscillatory, has stronger damping, and is difficult to represent. The product feature map still fails to converge, but now the simpler Chebyshev feature map also fails to converge. The perfect loss for both cases quickly reaches a horizontal state. The highly expressive ChebT feature map continues to work well. This confirms that it is important to choose a feature map that is well expressible for the problem, and that more simulations with more qubits provide a means to drastically increase power, but more expressive feature maps can result in longer training times.

[0460] Benchmarking of evolution-reinforced feature maps has shown that adding the evolution operator τ helps represent functions with large oscillations by forming a large pool of fitting functions, as implemented in quantum circuit learning. At the same time, DQC in this case requires a finer training grid to track the gradient of the oscillation function and has slower convergence, making training more difficult. Firstly, τ[n j ] is the number of iterations n j It is intended that τ can be set as a fluid parameter that increases from zero as it grows (similar to the annealing procedure in adiabatic quantum computing). This ensures easy trainability at the beginning, and then allows for tuning θ for a larger set of fitting functions, resulting in additional precision. Another approach is to consider τ as a variational parameter, starting at some small value, and perform gradient descent using numerical differentiation or by measuring the overlap of wave functions in a SWAP test.

[0461] Finally, variational circuits

[0462]

number

[0463] The effects of Ansatz depth are compared. Here, we have d=3, 6, 12, 24, λ=20, and a Chebyshev Tower feature map, but the rest of the training setup remains the same as previously considered. For lower depths, it has been observed that the solver converges slowly and does not reach the same high accuracy as at higher depths. As the depth increases, more layers of parameterized gates are included in the variational Ansatz, and therefore the number of variational angular parameters increases. This increases the number of gate operations required in each iteration and the number of parameters that the classical optimizer needs to update, resulting in longer time per iteration. If the Ansatz depth continues to increase, the problem of a barren plateau may eventually arise. Subsequently, when the gradient vanishes, the solver struggles to improve the parameters, although these problems were not observed with more than 400 variational parameters at d=24.

[0464] Based on a single ODE example, a system of differential equations can be considered by taking two strongly coupled differential equations as an example. These equations may describe the evolution of competing modes u1(x) and u2(x) as functions of the variable x, in this case corresponding to time. The associated rate equations are

[0465]

number

[0466] It was written as follows: Here, .'' 1,2 is a coupling parameter, and u 1,0 u 2,0 These are the initial conditions. The larger |λ1| is compared to |λ2|, the more strongly the two equations are coupled. This is intuitively clear when we consider |λ1|≧|λ2|, and therefore u2 contributes more to the equation for the derivative of du1 / dx than u1, and vice versa.

[0467] To move from considering a single differential equation to considering a system of equations, it is necessary to encode multiple functions using a quantum circuit, as described with reference to embodiments of this application. In this particular example, a simple parallel encoding may be chosen, where separate cost functions and Ansatz are considered. In this case, each function has a separate set of parameters to be optimized. The loss is modified accordingly to include information about separate contributions from two coupled differential equations that are optimized simultaneously. We encode each function using a differentiable feature map combined with a separate variational Ansatz parameterized by a set of angles θ1 and θ2, before determining the type of boundary evaluation.

[0468]

number

[0469] Obtained, Here

[0470]

number

[0471] These are fundamentally different cost functions for each equation. Regarding the loss function, we consider the sum of the MSE losses for the first and second differential equations. This loss is,

[0472]

number

[0473] It is written that F1 and F2 are as written in equations (33) and (34), and L depends on the choice of loss as detailed in the Methods section. Depending on the chosen boundary evaluation method, there may be additional boundary loss terms. If present, these contribute to the loss function as the sum of boundary terms for u1 and u2. Note that we are considering the loss as the sum of separate contributions that are joined together, both of which we are trying to minimize simultaneously with the same weights. The joining of the two equations can lead to a race between the two loss terms (a parameter update that would lead to a loss reduction for one DE may lead to an increase in the loss for the other). This consequently increases the probability of converging to a local minimum rather than a global minimum, which can be mitigated if the loss contributions are weighted in any way.

[0474] We set the problem setting parameters λ1=5, λ2=3, and the initial condition u 1,0 =0.5, ~u 2,0 We define = 0. We set up a training scheme using a 6-qubit register, cost selection of total magnetization in the Z direction for both u1 and u2, hardware efficient variational Ansatz with depth d=5, ADAM optimizer with a learning rate of 0.02, and feature map selection of Chebyshev Tower feature maps. We test the performance for three boundary evaluation types: pinning boundary, floating boundary, and optimized boundary.

[0475] Figure 9 shows the results for a system of strongly coupled equations. The first graph 902 shows the resulting estimated functions for a system of strongly coupled equations with λ1=3 and λ2=5 for pinning boundaries (Pin), floating boundaries (Float), and optimized boundaries (Optim). The second graph 904 shows the loss function value as a function of the number of iterations for training, as shown in graph 902. In graph 904, the solid line represents the perfect loss L F The dashed line indicates the quality of the solution L. Q This indicates.

[0476] The pinning boundary handler and the optimization boundary handler are executed similarly, and the analytical solution u is obtained within 250 iterations. 1,2 It slowly converges to (x). These two approaches have similar convergence with respect to perfect loss, but differ in terms of the quality of the solution. When using floating boundary types, a function close to the true solution is obtained (L Q The value is approximately 10 -4 This difference in convergence speed is a result of boundary information that affects the loss, requiring matching for the pinning boundary handler and the optimization boundary handler, whereas floating boundaries automatically match the initial conditions and no loss boundary terms are needed. The result of the competition of terms in the loss function can be seen in the initial oscillations of the perfect loss, as shown in the figure.

[0477] One field where solvers for complex differential equations are quite necessary is fluid dynamics. In this case, several prominent models are difficult to tackle due to their nonlinearity and the possible discontinuous solutions in the form of shock waves. Examples include the Burgers equation and the Navier-Stokes equation. We will focus on the latter and show how it can be approached with a DQC solver.

[0478] The Navier-Stokes equations describe the flow of incompressible fluids. This highly nonlinear system of partial differential equations is used to model fluids, magnetic plasmas, turbulence, and more. It is widely used in the aerospace industry and weather forecasting. It can be derived from general principles; that is, we consider fluid motion following Newton's laws and simply track the fluid mass passing through an (infinitesimal) volume. The general form of the Navier-Stokes equations is formal

[0479]

number

[0480] It can be presented as follows: Here, (ν x ν y ν z) is the instantaneous velocity in the (x,y,z) direction, τ is the stress tensor, f is the volume force per unit mass acting on the fluid element, ρ is the density, and p is the pressure. Here, V is the velocity field.

[0481] The components of the Navier-Stokes equations described above are continuity equations for density ρ that obey the laws of conservation of energy and conservation of momentum. Finally, when we use the conservation of energy, we can rewrite the equations in terms of one of the thermodynamic state functions, which may be temperature, pressure, or enthalpy.

[0482] The Navier-Stokes equations, while general in themselves, are usually solved in specific, limited cases. Specifically, they can address spatial contraction (2D, quasi-1D, 1D), isotropic / anisotropic media properties, and fluid properties (viscous or inviscous flow). Figure 10 shows a schematic of the flow through a convergence-divergence nozzle and numerical calculation results obtained by a standard classical solver. The physical layout of the system under consideration is shown in 1002. An input subsonic flow is considered in 1004, which is compressed at the center of the nozzle and exhibits supersonic flow at the output under specific boundary conditions in 1006. The equations can be derived for inviscous fluids under a quasi-1D approximation. These

[0483]

number

[0484] It was written as follows: Here, equation (40) corresponds to the continuity equation, equation (41) represents the conservation of energy, and equation (42) derives from the conservation of momentum. A(x) corresponds to the spatial shape of the nozzle and is essentially the potential as a function of the transverse coordinate x. γ describes the ratio of specific heat capacities and is equal to 1.4 in the case involving airflow. Dimensionless variables were used here.

[0485] The nozzle shape issue is, A(x) = 1 + 4.95(2x - 1) 2, 0≦x≦3 (43) It is set up as follows: Boundary conditions ρ(x=0,t)=1, T(x=0,t)=1, V(x=0)=0.1 (44) Specify as, This is for the specificity of solving the initial value problem. Furthermore, initial conditions must be specified when solving a mechanical problem. These can be selected as follows: ρ(x,t=0)=1-0.944x (45) T(x,t=0)=1,-0.694x (46) V(x,t=0)=(0.1+3.27x)T(x,t=0) 1 / 2 (47)

[0486] First, the steady-state problem expressed by equations (40)~(42) under the above conditions is considered, and the time derivative is set to zero. Interestingly, when attempting to solve the system for a steady-state solution using various classical methods implemented in Mathematica's NDSolve or Julia's software package, the calculations do not converge. This demonstrates that it is difficult because the system is rigid. The solution can become unstable depending on the initial values, especially the input velocity. To understand this problem, the system of steady-state Navier-Stokes equations is formally expressed

[0487]

number

[0488] It would be beneficial to rewrite it as follows.

[0489] Immediately, each function of RHS is given by T(x)=V(x) at point x. 2It was observed that the solution diverged to the following state. This resulted in singular behavior, which breaks classical solvers, including those with rigid handling. At the same time, a complete dynamic solution and its extrapolation as t→∞ are possible. Graphs 1008 and 1010 in Figure 10 show the numerical results achieved by a standard classical solver here for comparison and benchmarking with the performance of the DQC solver. System variables (density, temperature, and velocity) are shown as a function of time at the center of the nozzle in 1008. Furthermore, the steady-state solution 1010 is plotted as a function of spatial dimension x.

[0490] A two-stage optimization approach may be employed to construct the solution, where the solution is first obtained for a small x and then generalized to the nozzle end. For the circuit, a 6-qubit quantum register with Chebyshev quantum feature maps can be considered, keeping in mind x ∈ [0,1]. The cost function is the total magnetization

[0491]

number

[0492] Three functions {ρ(x), T(x), V(x)} are considered, so the three equations contribute combinatorially to the loss function. Floating boundary treatment may be used, and each curve is selected as ρ(0)=1, T(0)=1, V(0)=0.1 and adjusted according to the starting values. The variational Ansatz is assumed to be the standard hardware efficient form with depth d=6. Adam is used as the classical optimizer, and the learning rate is set to 0.01.

[0493] Figure 11 shows the DQC solution to the Navier-Stokes differential equation problem. In the first stage, we define the region (x min ,x maxWe train DQC at x=(0,0.4), and the circuit represents the region close to the initial point x=0. As expected, in the subsonic region, the flow velocity increases towards the center of the nozzle, and the temperature and density decrease slowly. Training is set up for 20 equally distributed points of x, with no prior regularization and an adjustable boundary. We use Adam with a learning rate α=0.01 for n iter We optimize DQC for 200 iterations. We find high-quality solutions based on gradient information in the imposed solution region x < 0.4, as shown in Graph 1102 in Figure 11, where the gray area represents the generalized solution for points without pre-training. Note that the loss function for the strongly coupled Navier-Stokes equations has a complex landscape. Different terms in the loss can compete with each other, so regularization techniques can be used to select the overall shape of the solution.

[0494] In the second training phase, we proceed to search for a complete solution including the divergent nozzle portion for x > 0.5. In this session, we select a 40-point grid, and the regions (0,0.4) and (0.6,0.9), each with 20 points, are used. As previously described, a key problem with convergent-divergent nozzles in the case of subsonic-supersonic transitions is the divergence near the middle of the nozzle. This poses a significant problem for classical solvers, as it prevents the finding of a stationary solution. The divergent contribution from this region also affects the loss function, complicating training. However, this problem is mitigated by excluding the region (0.4,0.6) near the nozzle throat from training. Continuing to use the same Ansatz and boundary handling, and supplying the variational angles from Stage 1 as initial parameters, a sub-best optimal solution is obtained. We employ weak regularization to ensure that the required subsonic-supersonic solution is formed. In other words, we use 20 points in the (0,0.4) region, benefiting from a previously found solution (in general, we can access as many points as needed), and then add 5 more points in the x∈(0.6,0.9) region that represent a weak bias toward the desired solution. Training is performed n iterThe simulation was performed with a learning rate of =600 and α=0.005, and the regularization switch-off function ζ(j) was set to smoothly remove the regularization weights around j=150. The complete solution is shown in graph 1104 in Figure 11. This converges to the expected long-term behavior for the system, and the function derivative by DQC agrees well with the nonlinear contribution. The increase in airflow velocity, as well as the decrease in temperature and density, in the convergence region is quantitatively reproduced. Details of the optimization run can be inferred from the loss plotted in graph 1106 in Figure 11. This is regularization (n j This shows different regions in the presence of <150) and the quality of the solution L Q The value remains low, but the perfect loss L F L represents the regularization contribution and the weakly improved derivative contribution. D Thanks to this, it improves. In the region where regularization was switched, we observed a steady improvement in all metrics, which indicates that DQC is being efficiently trained to approach the true solution, and this also L Q This can also be seen from the decrease in x. In particular, compared to many methods that depend on the sparse discretization of x, we found a solution along the entire length of the nozzle.

[0495] The embodiments described in this application define a general framework for solving general (system of) differential equations using differentiable quantum circuits on gate-based quantum hardware. This method uses quantum feature mapping circuits to encode function values ​​into latent space, allowing for consideration of spectral decomposition of trial solutions to differential equations. Differential quantum devices can accurately represent nonlinear solutions using fewer qubits by utilizing a large number of spectral basis functions that are exponential with respect to the number of qubits.

[0496] Analytical circuit differentiation can be used to represent function derivatives appearing in differential equations. These embodiments also demonstrate how analytical circuit differentiation can be used to represent function derivatives appearing in differential equations, and also show constructed loss functions aimed at improving prepared trial solutions. These embodiments reveal a novel method for solving general complex (non)linear systems, and as an example, a solution to the Navier-Stokes equations is presented, but the same method can be applied to all fields in which differential equations appear. Furthermore, it is well known that the analytical gradient used here is less susceptible to noise than the numerical gradient evaluated on variational quantum circuits. Noisy black-box optimization algorithms for NISQ applications would perform well with this method.

[0497] Figure 12 shows a quantum circuit for a post-NISQ DQC-based scheme according to one embodiment of the present invention. One is quantum feature mapping encoding.

[0498]

number

[0499] The process may begin by applying this to the starting state and showing this state as the wave function representation of the solution. This corresponds to the higher-dimensional latent space representation of the obtained function. The differential state may be obtained from the feature map derivative, for example, as shown in equation (7).

[0500]

number

[0501] And so, Unitary operator

[0502]

number

[0503] This corresponds to the shifted feature map circuit. Importantly, equation (A1) represents a unitary linear combination (LCU) acting on the initial state. Although the LCU is itself a non-unitary operation, its operation can be efficiently simulated using controlled operations by using auxiliary registers.

[0504] First, a general implementation of LCU calculation is shown. The unitary summation action on the initial state is

[0505]

number

[0506] As, the actual coefficient

[0507]

number

[0508] To implement this, we N A =[log2(L)]Requires an additional register for the qubit. Also, the operator

[0509]

number

[0510] ga Ansira

[0511]

number

[0512] It acts as,

[0513]

number

[0514] This is necessary to form a superposition state and bit string

[0515]

number

[0516] This brings about a unitary sum state preparation with the flags set. Then, each bit string is N A Controlled operations acting on ancilla qubits and N-qubit system registers

[0517]

number

[0518] The unitary operator in the sum is flagged. The effective function of LCU is the operator

[0519]

number

[0520] This corresponds to the combined application of these. When acting on the initial state, this

[0521]

number

[0522] Bringing about, Here, ⊥> is the orthogonal state to which the system is projected after the LCU implementation fails. A After performing the measurement of the ancilla qubit, the system returns to the normalized LCU state after 0 bits have been read out on the auxiliary register.

[0523]

number

[0524] It collapses. Calculation

[0525]

number

[0526] The probability of implementing this is p = 1 / c 2 It depends on the sum of the expansion coefficients.

[0527] The general LCU framework is used, which is known to result in exponential acceleration of quantum simulations and matrix inversions. Figure 13 illustrates how this technique can be applied to the DQC method. Training of the latent space solution state |f(x)> is performed on a set of points.

[0528]

number

[0529] Superimposing functions in

[0530]

number

[0531] It can be carried out by considering it as such. The state of auxiliary register 1204 flags the application of a feature map at a specific point. Similarly,

[0532]

number

[0533] The derivative evaluated on a grid of points is,

[0534]

number

[0535] It was written as follows: The register is slightly enlarged to accommodate the sum of the unitaries representing the circuit derivatives. This procedure can be generalized to higher-order derivatives. Importantly, the number of points increases exponentially, and the number of states within {|i>}

[0536]

number

[0537] The ability to encode as it increases is possible. After the differentiated state 1230 on register 1202 is prepared, the same procedure can be applied to the RHS of the differential equation F[f,x] on register 1228. In general, this consists of functions and variables of different orders.

[0538]

number

[0539] It is often defined as follows: l is a set

[0540]

number

[0541] This is a superindex for pairs of degrees over . The next goal is to write information about the structure of F[·] as a quantum register. For linear functions, a simple unitary feature map is applied. For functions with higher powers, we use the QNPU, which needs to be used to multiply the states in a unitary manner. Finally, monomials of the variables act as weights. Using the same procedure as before,

[0542]

number

[0543] Obtained, This is implemented using auxiliary register 1204 (which may differ from the derivative part). Finally, the bipartite cost function L = L diff +L bound Training needs to be performed using . First, the function of the latent space is their overlap L diff,θ =(1-| <dψ v,θ |F v,θ >| 2 ) (A7) and

[0544]

number

[0545] Matching can be achieved by maximizing as follows: However, note the θ parameterization. Maximizing this overlap can be achieved by using the SWAP test protocol outlined in 1220, 1222, 1224, 1225, and 1226. By matching the two parts simultaneously at all points, the solution to the system of differential equations can be found.

[0546] Figures 13A–13C illustrate different types of differential equations that can be solved using the embodiments of this patent application. In each of these examples, the objective is to “solve” (generalize) the model so that the value of function 1312 can be determined at any point in the region x 1316. In Figure 13A, no input data is provided except for the precise knowledge of the differential equation 1306 and the boundary conditions 1311. In other words, there is no assumed uncertainty within the equation parameter 1308. The trial solution 1310 is trained to fit the differential equation and boundary conditions by including them as terms in the loss function, according to the scheme described with reference to Figures 1–7 above, until a solution 1314 is found.

[0547] Figure 13B illustrates an example of a parameterized differential equation, where one or more elements in the equation are (hyper)parameters (α / β) rather than functions (f(x), g(x,y)) or dependent / dimensional variables (x,y,z,t). Such parameters may also be called coefficients. To solve such an equation for a particular instance, additional data is needed and used as boundary conditions for the parameters, and regularization may be given in the form of initial proposed values ​​for the parameters.

[0548] The suggested parameter values ​​in the terms of Equation 1320 may be provided as approximations (rather than exact values). The initially proposed parameter values ​​may be provided as regularized data for use in the training process. An example of a loss contribution that makes this effectively usable is:

[0549]

number

[0550] And, Here, zeta can be some function similar to the epoch-dependent equation 30. The parameter 1320 is α i Initialized to (0), any deviation is first punished by this loss term in order to push the solver first to find a solution close to the original estimated DE form. In the last epoch of optimization, this loss contribution can be completely turned off so that the DE hyperparameter α is fully variationally optimized to fit the observations.

[0551] Subsequently, the goal of solving the differential equation under such conditions is achieved by relying on already known experimental data / measurement points 1322. These may be only a few points, but should be known with some non-negligible certainty. In one embodiment, these data points can be added as representing parameter boundary conditions in a regression strategy where differential equation matching, boundary condition matching, and prior knowledge-data matching are guaranteed. In this way, a generalized solution to the equation can be found, while at the same time, the values ​​of the equation parameters can be fine-tuned with greater precision given the data. Thus, these embodiments enable solving parameterized differential equations using tunable variational equation parameters under conditions equivalent to circuit parameters.

[0552] Figure 13C illustrates another example of a parameterized differential equation. In particular, this figure illustrates a parameterized differential equation similar to Figure 13B, extended to include terms that may have coefficients equal to zero, but whose non-zero or zero nature is not known beforehand. Such a parameterized differential equation is, therefore, in general form

[0553]

number

[0554] It is often written as, Here, the dots represent the inner product of vectors, and the parameters are the vectors of coefficients.

[0555]

number

[0556] This can be expressed as follows, where F is a vector of functionals in terms of f and x. In other words, the right-hand side RHS of this equation may be any linear combination of a set of library terms that are expected to be relevant to solving the differential equation. In optimizing the loss function, then the function parameterization θ and further the function coefficients are determined.

[0557]

number

[0558] Both are optimized in the same way as in Figure 13B. The difference here is that we need to know from the information of the data loss term that all terms in a given library are "correct," rather than having a priori certainty. The left-hand side LHS can be one or more known terms with f or x. At least one term must be known, otherwise the system is undecided even with data.

[0559] In Figure 13C, the data points and the trial function for y may be the same as in Figure 13B, but the differential equation being modeled is different. Nevertheless, it is assumed that one of the terms 1326 exists, and it may also include hyperparameters 1328 and 1336 similar to those described with reference to Figure 15B, as well as other terms 1330, 1332, and 1334 whose coefficients and / or presence / absence are desirable to learn, although it is not a priori known whether they should be included. The goal is to find coefficients such that the equation fits the data.

[0560]

number

[0561] The goal is to learn and solve differential equations in the process. In this case of DQC, the loss function is given by equation (21).

[0562]

number

[0563] It looks similar to that.

[0564] Additionally, there is an additional loss term aimed at weighted regularization.

[0565]

number

[0566] This could be introduced.

[0567] This loss term is,

[0568]

number

[0569] In the optimization over a certain period, unimportant or unlikely terms are suppressed, and in a sense

[0570]

number

[0571] Ensuring increased sparsity in the structure is crucial. This is useful because, naturally, most processes can be described with relatively few terms; the real issue is often rather which of those terms appear and to what extent.

[0572] Figure 17 illustrates an example of the process of solving a parameterized differential equation using DQC according to one embodiment of the present invention. The figure illustrates a workflow including the DQC step. This example concerns a differential equation that models a damped harmonic oscillator 1702. The function u represents the position of the oscillator, and t is the time variable. Using the example, we assume that not only are the coefficients unknown, but we do not know exactly which terms will appear. Instead, a real-world experiment on a damped oscillator is performed, and the observed data shows how the function u behaves as a function of time. These measured data are circular data points in plot 1704. These data are then fitted with a parameterizable function u, which can be represented in DQC. Functional block 1706 includes quantum circuits such as quantum feature maps and variational quantum circuits that represent a function and its derivative for arbitrary variables, as described with reference to, for example, Figures 2 and 3.

[0573] Trial model 1708 may be used to narrow down the generality, and the trial model has at least one term, in this example the double derivative with respect to time ∂u tt It can be shown that it definitely exists within the model. Furthermore, the function

[0574]

number

[0575] The library is often used, and the purpose of this scheme is to learn its coefficients, where a coefficient of 0 means that no term exists. The loss function is θ1710 and so that both the function shape and model fit can be optimized.

[0576]

number

[0577] Both of 1711 are differentiable. Graph 1712 illustrates how, within approximately 150 iterations of the algorithm, DQC was able to somehow find the correct parameters for the example of a damped harmonic oscillator. The final result is the coefficients of the trial model 1708 shown in 1714. We can see that only the linear term and its derivative with respect to time are non-zero coefficients, and that these coefficients also accurately represent the values ​​used to generate the data in this case. This demonstrates that DQC can be used to effectively solve differential equations based on partial knowledge of the underlying model, combined with data / observations of the physical or observed process.

[0578] In one embodiment, when the parameters of the equations are not precisely known or are known with low confidence, unregularized (real) information, data points, physical measurements, or other arbitrary priori known data may be used to solve a system of parametric differential equations. These data points may be considered in the total loss function to perform partial regression on the data while fitting them directly to the (system of) differential equations. Such loss term contributions may be formally expressed as an example.

[0579]

number

[0580] It may have.

[0581] Here, u i This is the data point x i These are known function values ​​given in . The importance of these data can be adjusted by η, but they are epoch-independent and therefore exhibit behavior similar to boundary loss terms.

[0582] Figure 14 is a schematic diagram of a quantum circuit according to one embodiment of the present invention. In particular, this figure illustrates a parameterized quantum circuit similar to the example described with reference to Figures 2A and 2B, but in this embodiment, the quantum circuit may include a plurality of different quantum subcircuits, each associated with a different functionality. For example, the subcircuit may be one or more quantum feature maps

[0583]

number

[0584] ,

[0585]

number

[0586] It may include 1408 and 1412, each mapping the variables of the differential equation to the Hilbert space associated with the qubit. Furthermore, the subcircuit is variational unitary

[0587]

number

[0588] ,

[0589]

number

[0590] This may also include one or more quantum subcircuits configured as 1406, 1410, which are one or more variational parameters

[0591]

number

[0592] ,

[0593]

number

[0594] These variational parameters are associated with (and can be represented as vectors). These variational parameters may be used to train a quantum circuit that is parameterized to approximate a solution to a differential equation, i.e., this is the trial function.

[0595]

number

[0596] And,

[0597]

number

[0598] and

[0599]

number

[0600] This applies to different values ​​of .

[0601] The calculation of the trial function f1426 is performed using a predefined Hermitian cost operator, as the function is described with reference to equation (1).

[0602]

number

[0603] This can be summarized by equation 1424, which shows that it is the expected value of . The function is one or more vectors that define the variables of the differential equation.

[0604]

number

[0605] , vectors that define classical optimization parameters

[0606]

number

[0607] , one or more variational parameters

[0608]

number

[0609] ,

[0610]

number

[0611] It may depend on the variational parameter. In some embodiments, the quantum feature map is a variational parameter.

[0612]

number

[0613] It may also depend on one or more initialization units.

[0614]

number

[0615] ,

[0616]

number

[0617] 1404 and 1416 can be used to initialize the variational parameters of a quantum circuit.

[0618] The newline 1414 means that any of the components shown may or may not be repeated, or similar components may be introduced as needed for the application. The qubit may be initialized in state 1402, and after applying the qubit operations represented by the quantum circuits 1404-1416, the output may be measured in X, Y, or Z basis 1418 to calculate the expectation of operator 1422, which sums to a function representation 1420.

[0619] As explained with reference to Figure 6, the variational parameters need to be set to random initial values ​​such as angles. However, starting with a completely random circuit may result in trial functions that deviate significantly from the correct solution and do not converge to a good approximation, or do not converge to a good approximation quickly enough.

[0620] Therefore, in one embodiment, instead of random values, a predetermined set of angles may be used to initialize the variational parameters. In particular, in one embodiment, one or more initialization quantum circuits, i.e., initialization unitaries, may be computed using a classical computer. In particular, variational parameters

[0621]

number

[0622] At least some of these can be determined based on one or more initialization quantum circuits that are based on classically simulable quantum circuits, i.e., quantum circuits that can be classically simulated on a conventional computer. A classically simulable quantum circuit is defined as a quantum circuit that can be classically simulated on a conventional computer, where a Clifford gate is an element of the Clifford group produced by three gates: Hadamard, CNOT, and S gate. Clifford gate quantum circuits, i.e., quantum circuits consisting only of Clifford gates, can be efficiently simulated on a classical computer. Other examples of classically simulable quantum circuits are quantum circuits that can be described in terms of single-qubit operations, such as single-qubit rotations or match gate circuits.

[0623] If, for some setting of all variational parameters, the entire circuit can be described by a classically simulable quantum circuit, such as a Clifford gate quantum circuit, then the entire quantum circuit can be efficiently evaluated on a classical computer. Therefore, in one embodiment, a feature-mapped quantum circuit

[0624]

number

[0625] Variational quantum circuits

[0626]

number

[0627] , and initialization quantum circuits

[0628]

number

[0629] These may be designed in a parameterized manner so that they can be deterministically set up as classically simulable quantum circuits. In that case, the variational angles can be classically calculated in an efficient manner on a classical computer. For example, the quantum circuit can be formal

[0630]

number

[0631] Select such that it has a relationship between the variational parameters, for example,

[0632]

number

[0633] By setting this, such a variational quantum circuit represents the Clifford operation, and therefore the identity operation, which is classically simulable.

[0634] Therefore, by efficiently computing the non-trivial x dependence of the output f=f(x) on a particular variational parameter setting of a quantum circuit, f can be fitted to a dataset in an efficient manner, provided that some (free) variational parameters do not cause the quantum circuit to move outside the space of classically simulable quantum circuits, such as Clifford gate space. In particular, if the number of basis functions and tunable parameters of the classically simulable quantum circuit is only polynomial, the dataset can be fitted with sufficient precision in at most polynomial time. After computing the optimal variational parameter values ​​using a classical computer, the circuit can be initialized based on those values. Then, all variational parameters can be further optimized by causing the quantum circuit to move outside Clifford gate space.

[0635] Figure 15 shows a workflow for initializing the variational parameters of a variational quantum circuit based on a classical preprocessing or fitting step according to one embodiment of the present invention. In the first step, a representation of the quantum circuit may be determined or provided, the quantum circuit being one or more variational unitary quantum circuits associated with one or more quantum feature maps and one or more variational parameters θ, and an initialization parameter θ init It may include one or more initialized unitary quantum circuits associated with (step 1502). Then, the relationships between at least some of the variational parameters may be set such that the entire quantum circuit is a classically simulable quantum circuit. The classically simulable quantum circuit can be simulated using a classical computer and the expected value of the output can be calculated (step 1504). Furthermore, the initialization parameter θ init as a function of

[0636]

number

[0637] The dependency of the dependent variable can be determined (step 1508). Then, the initialization parameter θ init An optimal value for can be determined. For example, a fitting algorithm such as regression can be used to best fit f to the desired solution (estimate of θ) (step 1510). The quantum circuit is then optimized using the classically calculated θ. init The values ​​can be initialized (step 1512), but the other angles are maintained in a state that allows the quantum circuit to be a classically simulable quantum circuit. Finally, all parameters are variationally optimized in a variational optimization manner until the output converges toward the actual solution (step 1514).

[0638] Figure 16 shows an example of an initialization strategy as described with reference to Figures 14 and 15. This figure includes a quantum circuit that describes the execution of qubit operations in time. In this example, the quantum circuit describes that the qubits can be initialized in their ground state 1602. Then, a single R y The rotation may be applied to each bit indexed n, and each rotation has its own rotation angle

[0639]

number

[0640] It is associated with the unitary circuit.

[0641]

number

[0642] 1606 can be applied to all qubits. Furthermore, feature-map quantum circuit 1608

[0643]

number

[0644] This can be applied to each qubit, and the prefactor n before the arccos function is determined by the qubit index. Then another unitary circuit

[0645]

number

[0646] 1610 is applied to each qubit, and finally, different parameters

[0647]

number

[0648] Each R has y A layer of rotation can be applied. Then, a cost function, for example, a Z operator 1618 corresponding to the total magnetization cost function, can be determined by independently measuring the output 1614 of all the qubits.

[0649] Next, at least some of the variational parameters can be set such that the entire quantum circuit is a classically simulable quantum circuit. For example, some of the variational parameters are

[0650]

number

[0651] and

[0652]

number

[0653] When initialized in this way, these circuits reduce to the identity operator I. This makes quantum circuits classically simulable because there is virtually no entanglement between qubits. In this way, each expectation value can be calculated separately. For n qubits, the expectation value f n (x) = <C_n> can be expressed as 1620, which is simply a cos(n arccos(x)) term with a constant and some coefficient (as a function of x), all before the Z operator α n This is the result of multiplying by the coefficients. This is all f n When these are added together, the entire expression has a constant β0 and coefficient β n is parameter

[0654]

number

[0655] and α nThis means that it becomes f(x) 1622, which is some trivial function of that, and defines the sum of all Chebyshev polynomials up to degree n.

[0656] The Chebyshev polynomial function 1622 of degree n can then be fitted to a particular approximation of the function or data of interest, e.g., the function f(x), with minimal classical overhead (providing a scaling of O(n)). Variational parameters

[0657]

number

[0658] and α n After finding the optimal settings for , the fully quantum circuit can be initialized on an actual quantum computer based on those values. Then, the hybrid quantum-classical variational feedback loop in Figure 6A is initialized for angle

[0659]

number

[0660] ,

[0661]

number

[0662] ,

[0663]

number

[0664] ,

[0665]

number

[0666] Often used to manipulate vectors independently, thereby unitarily

[0667]

number

[0668] and

[0669]

number

[0670] These are no longer identity operators. These operations are for all Chebyshev terms T n The mixing begins, and thereby n T started (by chaining and nesting) n Other degrees also appear along with the terms. Although the embodiment in Figure 16 is intended to illustrate an initialization scheme in a specific setting, it should be noted that the initialization schemes described with reference to Figures 14-16 are valid for any type of feature map, variational ansatz, and cost function design, as long as the entire cost function can be computed classically efficiently.

[0671] As already explained with reference to Figures 2 and 6, in order to perform quantum machine learning such as DQC and solve related problems, the data (quantum or classical) must first be encoded in a quantum register. For classical data, the quantum feature map, where the variable x is embedded through rotations or the (nonlinearly transformed) phase of Hamiltonian evolution, is represented by unitary operations. The data can then be manipulated by a trainable parameterized circuit according to a variational quantum algorithm workflow, such as the workflow shown in Figure 6. The variational parameters, often the phases of the unitary operators forming the quantum circuit, are variationally tuned based on the training objective defined by the loss function. As with classical machine learning, this is a non-convex optimization problem and requires a stable optimization schedule. In the field of classical deep learning, this has been achieved by utilizing stochastic gradient descent and automatic differentiation (AD). Given the complexity of quantum optimization and the problem of noisy operations (both physical and statistical) on data embedded in Hilbert space, AD is essential for finding the optimal parameters for the quantum circuit. Automatic differentiation is also necessary for feature map differentiation and solving differential equations.

[0672] First, the function is a circuit measurement for some operator C.

[0673]

number

[0674] It can be designed as follows (Equation B1).

[0675] Part of the circuit is,

[0676]

number

[0677] It can be assumed that it is created using a feature map.

[0678]

number

[0679] This is the so-called generator of features. The formal derivation of this equation is the functional derivative of the generator and the (effective) cost function with respect to the commutator.

[0680]

number

[0681] It forms (Equation B2).

[0682] generator

[0683]

number

[0684] It is diagonalized, and so this is diagonalized, and the eigenvalue λ j The sum of projection matrices P having

[0685]

number

[0686] This is the result (Equation B3).

[0687] Combining equations B2 and B3, we can obtain an equation for the derivative without loss of generality.

[0688]

number

[0689] This is obtained (Equation B4).

[0690] A more compact expression of this equation

[0691]

number

[0692] It is often provided as (Formula B5), however,

[0693]

number

[0694] And, However, this assumes that all terms are real (since the function value is real, the derivative must also be real). For s unique gaps in the generator spectrum, there are s terms. For each gap s, there are two terms: a real term and an imaginary term. The relationship between the shift of the function evaluation itself and all of its higher-order derivatives is given by equation

[0695]

number

[0696] This is shown by (Equation B6).

[0697] This is the basic parameter shift law (Equation B7) when two shifts are equal but opposite in sign.

[0698]

number

[0699] It can be easily demonstrated that it can be recovered.

[0700] In a typical shift, the same symmetry disappears, requiring two more measurements. Below, for several feature maps and circuits used in DQC, it is shown that symmetry can be used to substantially reduce the number of circuit evaluations. Typically, a feature map represents a tensor product of rotations (parallel sequences) parameterized by the same variable x. An example is:

[0701]

number

[0702] This corresponds to the product feature map defined as (Equation B8), N is the number of qubits used for encoding.

[0703]

number

[0704] The Pauli procession

[0705]

number

[0706] or

[0707]

number

[0708] It is a Pauli rotation operator for the phase

[0709]

number

[0710] This acts on qubit j. For simplicity, we assume α=z, and other cases can be similarly considered by placing additional Hadamard and phase gates before and after the feature map.

[0711] Equation B8 is,

[0712]

number

[0713] It is often written in the form of a generator, and unitary operators are generators

[0714]

number

[0715] It is generated by (Equation B9), which corresponds to the sum of the Pauli operators. Since it is the total effective magnetization, the generator in Equation B9

[0716]

number

[0717] It has a spectrum with N+1 eigenvalues ​​Λ = {-N, -N+2, ..., 0, 2, ...}. Some eigenvalues ​​are multiple degenerate, and the number of unique positive gaps is

[0718]

number

[0719] The feature map can also be differentiated by applying the parameter shift law individually to each rotation, so the derivative can be measured using at most 2N circuit evaluations.

[0720] First,

[0721]

number

[0722] Note that the eigenstates coincide with the computational ground state and have all real amplitudes. Next, consider an input state |φ> that has real amplitudes, and an Ansatz that preserves this structure.

[0723]

number

[0724] ,

[0725]

number

[0726] This is a good idea, |Ψ θ > remains a real number. Finally, a cost function with real-valued matrix elements can be considered, which can be formed by strings of Pauli X and Z operators, and even strings of an even number of Pauli Y operators. Given the real structure of the (variational) state and readout, the matrix elements in equation B7

[0727]

number

[0728] is any set of shifts

[0729]

number

[0730] For this, it is a real number, and the difference considered in equation B7 is,

[0731]

number

[0732] It is often expressed in this way. Imaginary part of matrix elements

[0733]

number

[0734] This remains zero. Since the number of unknown coefficients is reduced by symmetry, it is sufficient to perform the measurement with only N+1 shifts (which may include the original function evaluation), reducing the budget for analytical derivative measurements from the 2N evaluations previously required for a single-qubit parameter shift law. This halves the measurement time for the function derivative and speeds up the solution of systems of differential equations.

[0735] So far, certain choices have allowed us to reduce the number of measurements required. Similarly, we can imagine situations where only the imaginary part is non-zero, or where the matrix elements take complex values ​​but have a specific structure connecting the real and imaginary parts (like the Kramers-Kronig relation familiar to physicists).

[0736] The technology of this disclosure may be implemented in a variety of devices or apparatus, including wireless handsets, integrated circuits (ICs), or sets of ICs (e.g., chipsets). Various components, modules, or units are described in this disclosure to highlight functional aspects of devices configured to perform the disclosed technology, but implementation by different hardware units is not necessarily required. Rather, as described above, various units may be combined within a codec hardware unit, or provided by an interoperable collection of hardware units, including one or more processors as described above, in conjunction with suitable software and / or firmware.

[0737] The terms used herein are intended to describe only specific embodiments and are not intended to limit the scope of the invention. As used herein, “one (or sometimes omitted)” and “it (or sometimes omitted)” (the singular articles “a,” “an,” and “the” in the original English text) are intended to include the plural unless the context clearly indicates otherwise. Furthermore, the phrases “equip, include” and / or “equip, include,” where used herein, mean the presence of the described features, integers, processes, operations, elements, and / or components (or constituents) and do not exclude the presence or addition of one or more other features, areas, integers, processes, operations, elements, components (or constituents), and / or groups thereof.

[0738] All means or steps of the following claims, plus corresponding structures, materials, activities, and equivalents of functional elements, are intended to include structures, materials, or activities for performing a function in combination with other claim elements, such as those specifically designated in the claims. The description of the invention is presented for illustrative and explanatory purposes, but is not intended to be exhaustive or to limit oneself to the disclosed forms of the invention. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the invention. The embodiments have been selected and described to best illustrate the principles and practical applications of the invention, so that those not skilled in the art can understand the invention in terms of various embodiments, along with various modifications suitable for a particular intended use. [Explanation of symbols]

[0739] 102 System 104 Quantum Computer Systems 106 Second Data Processing System 107 Differentiable Quantum Circuit (DQC) Generator 108 Quantum Processor Systems 110 Controller System 112 (Input of purely classical information) 114 Output (of purely classical information) 126 Function Circuits 128 Derivative Circuits 202 Feature Map Circuit 204 Variational Quantum Circuits 208 Readout section 210 Derivative quantum circuit 212 Unitary 304 Variational Ansatz 3101-4 x-dependent rotation 312-qubit operation 402 Variational Ansatz 404 |ψ(x)> 416 Alternating block Ansatz 502 Solver 504 Quantum Feature Map 510 Loss Function 514 Quantum State 612 Loss Function 614 Classical Optimization Procedures 706 qubits 712 Timeline Graphs 802 and 804 806 Graph Graphs 808 and 810 812 Graph 902 Graph 1 904 Second graph Graphs 1008 and 1010 1102 Graph 1104 Graph 1106 Graph 1202 Register 1204 Auxiliary Register 1230 Differentiated state 1308 Equation Parameters 1310 Trial solution 1311 Boundary conditions 1312 Functions 1314 solution 1316 area x 1320 formula 1322 Already known experimental data / measurement points 1328, 1336 hyperparameters Sections 1330, 1332, 1334 1402 |φ>state 1404~1416 Quantum circuit 1418 X, Y, or Z basis 1420 Function Representation 1422 Operator 1424 formula 1426 Trial function f 1602 Ground state 1608 Feature Map Quantum Circuits 1614 output 1618 Z operator 1622 f(x) 1702 Attenuated Harmonic Oscillator 1704 Plot 1708 Trial Model 1712 Graph

Claims

1. A method for solving one or more differential equations DE using a data processing system including classical computer systems and quantum computer systems, A step of receiving or determining a formulation of a quantum circuit representing an experiment function for one or more DEs by the classical computer system, wherein the experiment function is associated with one or more variables and a variable space, and the quantum circuit includes one or more function circuits for determining one or more values ​​of the experiment function around one or more points in the variable space, one or more derivative circuits for determining one or more values ​​of the derivative of the experiment function around one or more points, and one or more variational quantum circuits parameterized by one or more optimization parameters, A step of executing the quantum circuit on a set of points in the variable space of the trial function using the classical computer system, wherein the execution of the quantum circuit includes the steps of converting the quantum circuit into control signals for controlling the quantum elements of the quantum computer system and reading out the quantum elements to obtain hardware measurement data, and controlling the quantum computer system based on the control signals. The classical computer system receives the hardware measurement data in response to the execution of the quantum circuit, processes the hardware measurement data and puts it into one or more trial functions and one or more derivatives of the one or more trial functions, The steps include: determining a score indicating how well the measured trial function satisfies the conditions of the one or more DEs, based on the one or more trial functions, the one or more derivatives of the one or more trial functions, and the loss function, using the classical computer system; A method comprising the steps of optimizing the loss function, wherein the optimization includes the steps of adjusting one or more optimization parameters, and repeating the execution of the quantum circuit, the processing of the hardware measurement data, and the determination of the score until the score satisfies the optimization conditions.

2. The method according to claim 1, wherein the loss function is further based on one or more boundary conditions associated with one or more DEs.

3. The method according to claim 1 or 2, wherein the one or more DEs include one or more parameterized DEs, and the loss function further comprises one or more boundary conditions and one or more data points associated with the one or more parameterized DEs.

4. The method according to claim 1 or 2, wherein the one or more DEs include one or more parameterized DEs, the right-hand side RHS term of the one or more parameterized DEs defines a parameterized linear combination of functionals, and the loss function further comprises one or more boundary conditions and one or more data points associated with the one or more parameterized DEs.

5. The parameterized linear combination of the functionals is expressed as a vector inner product. [Math 1] Define, [Math 2] The parameters are defined in the form of a vector of coefficients, and F is a vector of functionals on f and x. [Math 3] The method according to claim 4.

6. The method according to any one of claims 1 to 5, wherein the one or more DEs include one or more parameterized DEs, and one or more parameters in the one or more parameterized DEs are included as optimization parameters in the optimization of the loss function.

7. The method according to any one of claims 1 to 6, wherein the control signal for controlling the quantum elements of the quantum computer system comprises a series of pulses, and the control signal for reading out the quantum elements comprises the step of applying readout pulses to the quantum elements of the quantum computer system.

8. The method according to any one of claims 1 to 7, wherein the function circuit includes a quantum feature map circuit for encoding the function dependence of the trial function on one or more variables into the quantum wave function amplitude of the quantum element of the quantum computer system.

9. The aforementioned simultaneous DE is F[{d ｎ f / dx ｎ } ｎ , {f ｍ (x) ｍ The method according to any one of claims 1 to 8, which determines a functional F represented by ] = 0.

10. The method according to any one of claims 1 to 9, wherein the quantum hardware measurement data is measured as the expectation value of the Hermitian cost operator.

11. The method according to any one of claims 1 to 10, wherein the quantum circuit comprises a plurality of different quantum subcircuits, each of which includes one or more quantum feature map circuits, and each quantum feature map circuit is configured to map one or more variables of the DE to a Hilbert space associated with the quantum element of the quantum computer system.

12. The method according to any one of claims 1 to 11, wherein at least some of the one or more optimization parameters of the one or more variational quantum circuits are initialized based on initialization values ​​classically calculated based on a classically simulable version of the quantum circuit simulated on a classical computer using classical simulation.

13. The quantum circuit further includes one or more initialization quantum circuits configured to initialize at least a portion of the one or more optimization parameters of the quantum circuit based on one or more initialization parameters, and the classical simulation is, A step of calculating the expected value of the output of the classically simulable version of the quantum circuit, wherein the expected value is a function [Math 4] The steps to define, [Math 5] The dependency of the dependent variable is the initialization parameter 【Number 6】 The step of determining as a function of, The initialization parameter [Number 7] Fitting the optimal value for to the desired solution (or its estimate). [Number 8] The steps to be decided based on, While keeping other variational parameters fixed to define the classically simulable version of the quantum circuit, the quantum circuit is modified by the initialization parameter [Number 9] The method according to claim 12, further comprising the step of initializing based on the optimal value for the given value.

14. The method according to any one of claims 11 to 13, wherein the one or more quantum feature quantum circuits, the one or more variational quantum circuits, and the cost function are configured to exhibit symmetry.

15. A system for solving one or more differential equations DE using a hybrid data processing system including classical and quantum computer systems, wherein the system is The system comprises a memory device containing computer executable instructions and a processor connected to the memory device, the processor being A step of receiving or determining a formulation of a quantum circuit representing an trial function for one or more DEs by the classical computer system, wherein the trial function is associated with one or more variables and a variable space, and the quantum circuit includes one or more function circuits for determining one or more values ​​of the trial function around one or more points in the variable space, one or more derivative circuits for determining one or more values ​​of the derivative of the trial function around one or more points, and one or more variational quantum circuits associated with one or more optimization parameters, A step of executing the quantum circuit on a set of points in the variable space of the trial function using the classical computer system, wherein the execution of the quantum circuit includes the steps of converting the quantum circuit into control signals for controlling the quantum elements of the quantum computer system and reading out the quantum elements to obtain hardware measurement data, and controlling the quantum computer system based on the control signals. The classical computer system receives the hardware measurement data in response to the execution of the quantum circuit, processes the hardware measurement data and puts it into one or more trial functions and one or more derivatives of the one or more trial functions, The steps include: determining a score indicating how well the measured trial function satisfies the conditions of the one or more DEs, based on the one or more trial functions, the one or more derivatives of the one or more trial functions, and the loss function, using the classical computer system; A classical computer system configured to perform a feasible operation comprising the steps of optimizing the loss function, wherein the optimization includes adjusting one or more optimization parameters, and repeating the execution of the quantum circuit, the processing of the hardware measurement data, and the determination of the score until the score satisfies the optimization conditions.

16. The system according to claim 15, further comprising the steps of the method described in any one of claims 2 to 14.

17. A non-temporary computer-readable storage medium for storing a computer program comprising at least one software code portion, wherein the software code portion is configured to perform a step according to any one of claims 1 to 14 when executed on a classical computer system, the classical computer being part of a data processing system that includes the classical computer system connected to a quantum computer system.

18. A quantum learning method for approximating solutions to mathematical problems using data processing systems, including classical and quantum computer systems, The method includes a step of receiving or determining a quantum circuit by the classical computer system, wherein the quantum circuit comprises a plurality of different quantum subcircuits, each of which is one or more quantum feature map circuits, each quantum feature map circuit configured to map variables of the solution to the mathematical problem to a Hilbert space associated with a quantum element of the quantum computing system, and one or more variational quantum circuits associated with one or more variational parameters to train the quantum circuit to approximate the solution to the mathematical problem, the method being The steps include: initializing at least some of the variational parameters of one or more variational quantum circuits using the classical computer system based on initialization values ​​classically calculated based on a classically simulable version of the quantum circuit simulated on the classical computer; A quantum learning method comprising the steps of optimizing a quantum circuit using a variational method that includes changing the variational parameter, using the classical computer system and the quantum computer system, until the output of the quantum circuit converges to the solution of the mathematical problem.

19. The quantum learning method is a method for training the quantum circuit to approximate a solution to one or more differential equations DE, wherein the step of variationally optimizing the quantum circuit includes a step according to any one of claims 1 to 14, the quantum learning method according to claim 18.

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