Shot-Efficient Quantum Solver for Differential Equations

Abstract

Systems and methods for solving differential problems using a hybrid computer system including a classical computer system and a quantum processor are disclosed. The method includes receiving or determining one or more variational quantum circuits that define a parameterized quantum model of an execution function, and an integral formula of a differential equation. The integral formula defines an equation having a differential order smaller than the differential order of the differential equation, and the integral formula includes at least some of the boundary conditions. The integral formula includes a path integral having a first integral limit corresponding to a point on the boundary. The method includes determining a solution to the differential problem based on a loss function associated with the parameterized quantum model and the integral formula of the differential problem. The evaluation of the loss function for a point under consideration includes calculating an integral based on one or more collocation points between the point on the boundary and the point under consideration. The step of determining the solution includes varying variational parameters to determine a set of optimal parameters that define the solution to the differential equation.

Description

Technology Field

[0001] The present disclosure relates to quantum encoders, and, but is not exclusive, to methods and systems for solving a set of (non)linear differential equations using a hybrid quantum computer in particular, and to a computer program product that enables a hybrid quantum computer to perform such methods. Background Technology

[0002] Quantum computing is an emerging field that offers the potential for implementing algorithms for specific problems involving speed enhancements known as the quantum advantage over classical computing. These speed enhancements stem from the inherent properties of quantum computers, such as superposition, entanglement, and exponentially expanding computational spaces, which fundamentally lead to different modes of computation. Fields where quantum advantages are expected to be found include those where quantum mechanical phenomena play a role, such as specific chemical applications and molecular simulations. Other examples are provided by fields with computationally large problems, such as generative modeling and numerous differential equations.

[0003] Differential equations form the basis for modeling numerous systems, such as electromagnetic waves, the evolution of heat distributions, population dynamics, the optimization of industrial processes, and the time-evolution of probability distributions. Consequently, there has been significant recent interest in skewing neural networks toward solving differential equations. A common means of representing differential equations within neural networks is to use Physics-Informed Neural Networks (PINNs), which utilize the (potentially higher-order) derivatives of the outputs with respect to the network's inputs to construct appropriate normalization terms toward the prescribed differential equation.

[0004] Quantum algorithms for solving differential equations are often based on so-called quantum neural networks or Quantum Physics-Informed Neural Networks (qPINNs), which are based on variational solvers. Variational solvers utilize circuits with adjustable parameters to represent an execution function. Subsequently, a loss function representing the problem is optimized to receive parameter values ​​for the execution function to approximate the solution of the differential equation. Algorithms within this category are often suitable for near-future implementations.

[0005] However, a limiting factor within such quantum algorithms is the so-called shot rate—that is, the rate at which measurements can be taken. For example, many algorithms provide the solution as an expectation value, which implies that a sufficient number of measurements must be taken to determine an expectation value with the desired accuracy. Nevertheless, most demonstrations of this technique assume an ideal simulated quantum circuit where each expectation value can be used directly. In actual quantum hardware, such ideal expectations are not accessible and must be obtained by averaging over many shots. This introduces additional computations, and the cost of this has not been considered in many of the published studies. The requirements for higher-order derivatives for physics-informed regularizers are particularly high in terms of circuit iterations (shots) compared to the lower-order derivatives required for supervised learning.

[0006] Therefore, in related technical fields, there is a need for algorithms to solve differential equations that reduce the number of quantum measurements ('shots') without a decrease in accuracy. means of solving the problem

[0007] The purpose of the embodiments of the present disclosure is to provide a system and method for solving differential equations that avoid or at least reduce the disadvantages of the prior art.

[0008] In a first aspect, embodiments of the present disclosure relate to a method for solving a differential problem using a hybrid computer system comprising a classical computer system and a quantum processor. The differential problem defines a differential equation and boundary conditions. The method includes receiving or determining information about one or more quantum circuits that define operations of a parameterized quantum model of an trial function parameterized by variational parameters, and an integral formulation of the differential problem, in particular, a differential equation. The integral formulation defines an equation having a differential order smaller for at least one independent variable than the differential order of the differential equation for at least one independent variable. The integral formulation may include at least some of the boundary conditions. The integral formulation may include an integral having a first integral limit corresponding to a point on the boundary and a second integral limit corresponding to a point within the domain, preferably a path integral. The method includes determining a solution to the differential problem based on a loss function based on the parameterized quantum model and the integral formulation of the differential problem. The evaluation of the loss function for a point within the domain (the point under consideration) may involve the calculation of an integral, the calculation of the integral being based on one or more collocation points between the point on the boundary and the point within the domain (the point under consideration) and / or one or more previously evaluated points within the domain. The step of determining the solution includes the step of varying variational parameters to determine the set of optimal parameters that define the solution to the differential equation.The step of changing variational parameters includes: converting operations of one or more quantum circuits into control signals for reading and controlling quantum elements (e.g., qubits) of a quantum processor; applying operations of one or more quantum circuits to quantum elements based on the control signals; determining classical measurement data by estimating one or more expected values ​​based on a plurality of measurements of the quantum elements; calculating a loss value based on a loss function and classical measurement data; and adjusting variational parameters based on the loss value.

[0009] A differential equation is associated with at least one independent variable. The domain defines the space associated with the independent variable in which the differential equation is defined; that is, the domain is the set of potential input values ​​for the differential equation.

[0010] In the integral formulation of a differentiation problem, the integral has a different second limit for each point where the loss function is evaluated, and the second limit corresponds to a point within the domain under consideration. Such an integral can be referred to as a Volterra integral.

[0011] Collocation points include a set of values ​​(predetermined or randomly selected) of independent variables used for optimization (typically within the domain or at least adjacent to the domain).

[0012] Current (variable) quantum algorithms are typically sets of collocation points parameterized trial function Evaluate one or more derivatives of. The function value (or its derivative) at a given point is typically expressed as an expected value, and its accuracy increases with the number of measurements.

[0013] The derivatives can be computed using either of two methods: an analytical method by differentiating a feature map (e.g., possibly using a generalized parameter shift rule), or a method using a finite-differencing-based method.

[0014] The calculation of analytic derivatives requires the determination of at least two expected values ​​per quantum element (e.g., qubits or qudits) (more for higher-order derivatives, and also more for non-idempotent feature maps). Furthermore, for a given desired accuracy, since the derivative is determined as the difference between two expected values, a large number of shots are required to determine the derivative compared to the determination of the base function. The methods according to the embodiments described herein require fewer (possibly none) determinations of derivatives compared to prior art methods, thereby improving the number of required shots to at least twice the number of quantum elements, and typically Improving the drainage system, here N represents the number of data encodings, and d represents the order of the derivative. Consequently, the methods according to the embodiments described in this application can substantially reduce the number of shots (e.g., by a factor of tens to hundreds for each 'removed' derivative). In particular, for certain classes of ordinary differential equations and partial differential equations where substantially all derivatives can be removed, this can lead to a corresponding computational speed-improvement of a factor of tens to hundreds and a corresponding reduction in resources and energy.

[0015] Compared to methods using analytic derivatives, finite-difference-based methods use fewer shots to determine local derivatives when assuming the same number of collocation points. However, depending on the function's behavior, significant errors can be introduced. These can lead to poor convergence or the need for a larger number of collocation points, which ultimately requires more shots. Additionally, inherent noise (and the resulting uncertainty within the function values) is multiplied by a small number (the magnitude of the finite difference) of the multiplicative inverse, which can lead to very large errors. Therefore, if the derivative is to be known with the required accuracy, the accuracy of the function for which the derivative is taken must be much higher, and generally, this requires a much higher shot count.

[0016] In the methods according to the embodiments, the calculation of at least one derivative is effectively replaced by the calculation of an integral extending from the boundary to the evaluated (collocation) point. The integral can be calculated as the (weighted) sum of the function values. As mentioned above, the need to calculate fewer derivatives already reduces the shot requirement. Furthermore, since the integral calculated at the point effectively combines the results of all previous points, the uncertainty caused by measurement noise is reduced accordingly. This can particularly reduce the shot requirement (of a given required accuracy) for function values ​​located at a certain distance from the initial value. This can be particularly advantageous if at least two boundary values ​​are known for the variable being integrated.

[0017] In particular, these latter advantages are not limited to quantum implementations but also apply to classical implementations. This can lead to faster convergence and / or more accurate solutions for classical neural networks, spiking neural networks, stochastic neural networks, etc.

[0018] Accordingly, the embodiments of the present disclosure demonstrate a method for constructing a global formulation of physics-informed losses that is particularly suitable for solving (ordinary) differential equations in near-future quantum computers in a shot-efficient manner. The resulting approach reduces the order of derivatives required to calculate the loss compared to classical physics-informed neural networks (PINNs) and quantum physics-informed neural networks (qPINNs). For initial value problems within ordinary differential equations (ODEs) and some partial differential equations (PDEs), current methods completely eliminate the need for (higher-order) automatic derivatives and thus (at least) in shot efficiency It provides improvements, here N n is the number of data-encodings of the quantum neural network. The formulation of the loss function naturally includes boundary conditions and physical-information losses within a single optimization term.

[0019] It should be noted that the relative performance of quantum neural network algorithms within the infinite-shot limit (as is frequently used in simulations) does not necessarily correspond to the relative performance within the finite-shot limit (i.e., in the case of actual hardware implementations).

[0020] In one embodiment, the step of changing variation parameters includes one or more of the steps of determining or estimating the gradient of loss values ​​for the variation parameters and adjusting the variation parameters based on the determined or estimated gradient, or determining a set of initial sets of variation parameters based on associated loss values ​​and determining an adjusted set of variation parameters based on the initial sets of variation parameters and associated loss values.

[0021] The step of estimating the gradient can be performed with very few estimates (e.g., one, two, three, or four). These methods are sometimes referred to as 'gradient-free' methods because they omit most of the computational load involved in actually determining the gradient. These methods require relatively few shots and are therefore particularly suitable for implementations of near-future quantum computers.

[0022] In one embodiment, the step of calculating the loss value includes the step of estimating the integral using at least one of a regular Monte Carlo-based method, a quasi-Monte Carlo-based method, a quasi-Monte Carlo-based method, a midpoint rule-based method, or a quadrature-based method. These are all methods suitable for calculating the integral described herein. However, other numerical integration methods may also be used. Quasi-Monte Carlo-based methods have been proven to be particularly efficient for numerically evaluating integrals.

[0023] In one embodiment, the parameterized quantum model is a quantum neural network, and the quantum neural network preferably includes a feature map and a variational circuit for encoding independent variables, and preferably the function value of the trial function is associated with an observable (e.g., expectation value) of one or more quantum circuits, or the function value is associated with a functional of an output state of one or more quantum circuits (e.g., superposition with a zero state, real component, or imaginary component).

[0024] In one embodiment, the parameterized quantum model is a quantum-kernel based model, and the evaluation of the preferred quantum kernel includes the step of determining the superposition between a first quantum state associated with a first input variable and a second quantum state associated with a second input variable, and preferably, the superposition is determined based on a Hadamard test or a SWAP test.

[0025] In one embodiment, the parameterized quantum model is an amplitude-encoding based model, and preferably, one or more quantum circuits define operations for encoding primitive elements of differential equations (e.g., terms such as f(x), f(x) / dx, g(x), etc.) within a plurality of quantum states of quantum elements of a quantum processor, and each quantum state of the plurality of quantum states corresponds to a set of coefficients associated with basis functions.

[0026] The feature map may be, for example, a Fourier feature map or a Chebyshev feature map. In some embodiments, the feature map may include one or more learnable parameters.

[0027] In one embodiment, the differential equation is an ordinary differential equation, a partial differential equation, a stochastic differential equation, or a delay differential equation, and / or the differential problem is an initial value problem.

[0028] In one embodiment, the differential problem defines an additional differential equation. In such an embodiment, the method may further include the step of solving the additional differential equation using an additional method for solving the differential problems. The additional method may be based on differential quantum circuits, based on quantum kernels, or based on amplitude-encoding of a set of coefficients associated with basis functions representing an trial function.

[0029] This combination with other methods may be particularly relevant to multi-dimensional partial differential equations (PDEs) or highly non-linear ordinary differential equations, where it is impossible to eliminate all derivatives in the domain by integrating them.

[0030] In one embodiment, the step of executing one or more quantum circuits by a quantum processor includes translating each of the one or more quantum circuits into a series of signals (e.g., pulses, typically electrical or electromagnetic pulses such as voltage pulses, light pulses, or microwave pulses) and using the series of signals to compute quantum elements within a quantum register of the quantum processor.

[0031] In one embodiment, the step of receiving hardware measurement data includes the step of applying a read-out signal to quantum elements of a quantum processor and the step of measuring quantum hardware measurement data in response to the read-out signal.

[0032] Generally, the embodiments described herein may be implemented as a series of operations that can be executed, for example, in a digital operation mode, an analog mode, or a digital-analog mode of a neutral atom quantum computer. In particular, the hybrid quantum computer described herein may include one or more of a gate-based qubit device, a digital / analog quantum device, a neutral atom-based quantum device, an optical qubit device, and / or a Gaussian boson sampling device.

[0033] In one aspect, the embodiments relate to a method for solving differential problems, and the differential problem defines differential equations and boundary conditions.

[0034] The method includes the step of receiving or determining information regarding a parameterized model of an trial function parameterized by variational parameters and an integral formula of a differential equation, wherein the integral formula defines an equation having a differential order smaller for at least one independent variable than the differential order of the differential equation for at least one independent variable, and the integral formula is based on boundary conditions.

[0035] The method includes a step of determining a solution to a differential problem based on a parameterized model and a loss function associated with the integral formulation of the differential problem, the step of determining a solution to the differential problem includes a step of training a neural network to determine a set of optimal neural network parameters, and the trained neural network represents a solution to the differential equation.

[0036] In one embodiment, the step of training a neural network includes the step of optimizing variational parameters variationally. The step of optimizing variational parameters variationally may include the step of evaluating a parameterized model of an execution function using first values ​​for the variational parameters, the step of calculating a loss value based on the evaluated parameterized model and a loss function, and the step of adjusting the variational parameters based on the loss value.

[0037] In one aspect, the present disclosure relates to a system for solving a differential problem, wherein the differential problem defines a differential equation in a domain and boundary conditions on the boundary of the domain, the system comprises a classical computer system and a special purpose processor, and the system is configured to perform the steps of receiving or determining information about a parameterized model of an execution function parameterized by variational parameters, and receiving or determining an integral formula of the differential problem, in particular a differential equation, wherein the integral formula defines an equation having a differential order smaller for at least one independent variable than the differential order of the differential equation for at least one independent variable, the integral formula includes at least some of the boundary conditions, and the integral formula includes a path integral having a first integral limit corresponding to a point on the boundary and a second integral limit corresponding to a point within the domain, and is configured to perform the step of determining a solution to the differential problem based on a loss function associated with the parameterized model and the integral formula of the differential problem, wherein the evaluation of the loss function for a point within the domain includes the calculation of an integral, the calculation of the integral is based on one or more collocation points between a point on the boundary and a point within the domain, and determining a solution The step includes training a neural network to determine the optimal set of variational parameters, and the trained neural network represents the solution to the differential equation.

[0038] In one embodiment, the system is a hybrid quantum computer system comprising a classical computer system and a quantum processor.

[0039] In this embodiment, the special purpose processor is a quantum processor, and the step of changing variational parameters includes: translating operations of one or more quantum circuits into control signals for controlling and reading quantum elements of the quantum processor; applying operations of one or more quantum circuits to quantum elements based on the control signals; determining classical measurement data by estimating expected values ​​based on a plurality of measurements of quantum elements; calculating loss values ​​based on classical measurement data and a loss function; and adjusting variational parameters based on loss values.

[0040] Generally, a computer system can be configured to execute any of the steps of the method described above.

[0041] One aspect of the present disclosure relates to a hybrid quantum computer system or a classical computer system comprising a computer-readable storage medium in which computer-readable program code is implemented therein and a processor coupled to the computer-readable storage medium, preferably a microprocessor, wherein, in response to a step of executing the computer-readable program code, the processor is configured to perform any of the methods described herein.

[0042] One aspect of the present disclosure relates to a computer program or a collection of computer programs comprising at least one software code portion, or a computer program product storing at least one software code portion, and is configured to execute any of the methods described herein when the software code portion is run on a classical computer system or a hybrid quantum computer system.

[0043] One aspect of the present disclosure relates to a non-transient computer-readable storage medium storing at least one portion of software code, and the portion of software code is configured such that, when executed or processed by a computer, the computer or hybrid computing system performs any of the methods described herein.

[0044] As understood by those skilled in the art, aspects of the present invention may be implemented as systems, methods, or computer program products. Accordingly, aspects of the embodiments may take the form of hardware embodiments in whole, software embodiments in whole (including firmware, resident software, microcode, etc.), or embodiments combining software and hardware aspects that may generally be referred to herein as "circuits," "modules," or "systems." The functions described in this disclosure may be implemented as algorithms executed by a microprocessor of a computer. Additionally, aspects of the embodiments may take the form of computer program products implemented within one or more computer-readable media(s) on which computer-readable program code is implemented (e.g., stored).

[0045] Any combination of one or more computer-readable media(s) may be utilized. A computer-readable media may be a computer-readable signal medium or a computer-readable storage medium. A computer-readable storage medium may be, for example, any of an electric, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or element, or a suitable combination thereof, but is not limited thereto. More specific examples (non-comprehensive list) of computer-readable storage media include: an electrical connection having one or more wires, a portable computer diskette, a hard disk, random access memory (RAM), read-only memory (ROM), replaceable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disc read-only memory (CD-ROM), optical storage device, magnetic storage device, or any suitable combination thereof. In the context of this document, a computer-readable storage medium may be any type of medium capable of containing or storing a program used by or connected to an instruction execution system, apparatus, or device.

[0046] A computer-readable signal medium may include a propagated data signal on which computer-readable program code is executed, and may include, for example, a portion of a baseband or carrier wave. Such propagated signal may take any of various forms, including but not limited to electromagnetic forms, optical forms, or any suitable combination thereof. A computer-readable signal medium may be any computer-readable medium other than a computer-readable storage medium capable of transmitting, propagating, or communicating a program used by or connected to a command execution system, apparatus, or device.

[0047] Program code executed on a computer-readable medium may be transmitted using any suitable medium including, but not limited to, wireless, wired, optical fiber, cable, RF, etc., or any suitable combination thereof. Computer program code for performing operations for aspects of the embodiments may be written in any combination of one or more programming languages, including functional or object-oriented programming languages ​​such as Java, Scala, C++, Python, etc., and conventional procedural programming languages ​​such as the "C" programming language or similar programming languages. The program code may be executed entirely on a user's computer, partially as a standalone software package on a user's computer, partially on a user's computer and partially on a remote computer, or entirely on a remote computer, server, or virtualized server. In the latter case, the remote computer may be connected to the user's computer through any network including a Local Area Network (LAN) or a Wide Area Network (WAN), or a connection to an external computer may be established (e.g., via the Internet using an Internet service provider).

[0048] Aspects of the embodiments are described below with reference to block diagrams and / or flowcharts of computer program products, apparatuses (systems), and methods according to the embodiments. It will be understood that each block of the flowcharts and / or block diagrams and combinations of blocks within the flowcharts and / or block diagrams may be implemented by computer program instructions.

[0049] These computer program instructions may be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing mechanism, in particular a microprocessor or a central processing unit (CPU) or a graphics processing unit (GPU), to create a machine, and the instructions executed through the processor of the computer, other programmable data processing mechanism, or other device accordingly create a means for implementing the functions / operations embodied in the blocks or blocks of a flowchart and / or block diagram.

[0050] These computer program instructions may also be stored in a computer-readable medium capable of instructing a computer, other programmable data processing mechanism, or other device to function in a specific manner, and accordingly, the instructions stored in the computer-readable medium produce an article of manufacture comprising instructions that implement a function / operation embodied in a block or block of a flowchart and / or block diagram.

[0051] Additionally, these computer program instructions may be loaded into a computer, another programmable data processing mechanism, or other device to enable a series of operation steps to be performed on a computer, another programmable mechanism, or other device to generate a computer implementation process, and accordingly, instructions executed on a computer, another programmable mechanism provide processes for implementing functions / operations embodied in blocks or within blocks of a flowchart and / or block diagram.

[0052] The flowcharts and block diagrams in the drawings illustrate the operations, functions, and architectures of possible embodiments of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in the flowcharts or block diagrams may represent a part, module, or segment of code containing one or more executable instructions for implementing a specific logical function(s). Additionally, it should be noted that in some alternative embodiments, the functions mentioned in the blocks may occur differently from the order mentioned in the drawings. For example, two blocks shown in succession may, in fact, be executed substantially simultaneously depending on the related functionality, or the blocks may occasionally be executed in reverse order. Each block in the block diagrams and / or flowcharts and combinations of blocks in the block diagrams and / or flowcharts may be implemented by special-purpose hardware-based systems or by a combination of special-purpose hardware and computer instructions that perform the specific functions or operations.

[0053] Further embodiments may be illustrated with reference to the attached drawings, which will schematically illustrate embodiments according to the invention. It will be understood that the invention is not limited in any way to these specific embodiments. Identical reference numerals refer to identical or at least similar elements. Brief explanation of the drawing

[0054] Figure 1a schematically illustrates a known method for solving differential equations, Figure 1b shows an example of the basic form of a quantum feature map, and Figure 1c shows an example of its derivative. FIG. 2 is a flowchart of a method according to an embodiment compared with a method according to the prior art. FIG. 3(a) illustrates a method for solving differential problems using a variational quantum circuit using a known method, and FIG. 3(b) and FIG. 3(c) illustrate a method for solving differential equations using a variational quantum circuit according to an embodiment. Figure 4 is a flowchart of a method according to an embodiment. Figure 5 is a quantum circuit diagram of the quantum circuit used in the shots experiments shown in Figures 6 and 7. FIGS. 6a to 6f are graphs showing experimental results comparing a method according to the prior art and a method according to an embodiment. FIGS. 7a to 7c are graphs showing experimental results according to a method and an embodiment according to the prior art. FIGS. 8a to 8c are graphs showing additional experimental results comparing a method according to the prior art and a method according to an embodiment. FIGS. 9a to 9c are graphs showing additional experimental results comparing a method according to the prior art and a method according to an embodiment. FIGS. 10a to 10c are graphs showing experimental results comparing a method according to the prior art and a method according to an embodiment. FIGS. 11a to 11c illustrate differentiated feature maps according to embodiments. FIGS. 12a through 12e show hardware-level schematic diagrams illustrating the application of logical operations to qubits using quantum circuits. FIGS. 13a and FIGS. 13b illustrate systems illustrating digital-analog implementation examples of a quantum algorithm and a quantum feature map, respectively. FIG. 13c illustrates pulses corresponding to a quantum feature map. FIGS. 14a and FIGS. 14b present quantum circuits based on digital-analog implementation examples of quantum circuits. FIGS. 15a and FIGS. 15b present quantum circuits based on digital-analog implementation examples of quantum circuits. FIG. 16 presents a workflow for generalized circuit differentiation to be used in the embodiment. FIGS. 17a and FIGS. 17b are block diagrams illustrating an exemplary classical data processing system and an exemplary hybrid data processing system that can be used to implement the software products and methods described in the present disclosure, respectively. Specific details for implementing the invention

[0055] In addition to applications for neural networks within conventional silicon-based hardware, physical-information loss functions represent promising means for solving differential equations using quantum neural networks (QNNs) in near-future quantum computers, particularly so-called noisy intermediate-scale quantum (NISQ) computers. However, theoretical studies on solving differential equations on quantum computers tend to assume that the expected value of a quantum circuit can be calculated directly, which is known as the infinite-shot limit. This is possible only in the simulation of quantum circuits. When using actual quantum computers, expected values ​​must be determined through repeated sampling (taking 'shots'). This cost is largely ignored in current research on the use of quantum neural networks for solving differential equations. As discussed in more detail below with reference to Figures 10a through 10c, the (relative) performance of an algorithm within the infinite-shot limit does not always represent the (relative) performance of an algorithm within a finite-shot implementation.

[0056] Physical-information normalization involves taking the derivatives of neural network outputs with respect to input features. In classical neural networks, the approach to higher-order derivatives can be optimized by mixing various modes of automatic differentiation (e.g., considering forward-mode rather than reverse-mode automatic differentiation, or through Taylor-mode automatic differentiation). Conversely, in the case of quantum neural networks, the derivatives with respect to circuit inputs are typically cost It entails, and here N is the number of data encodings into the quantum circuit, and d is the order of the derivatives. This scaling is similar to the repeated use of forward-mode automatic differentiation in classical neural networks, and is a strategy known as suboptimal.

[0057] To reduce the number of shots required for variational solutions of differential equations, the present disclosure presents formulations of physical-information loss functions suitable for driving quantum neural networks on actual (NISQ) quantum hardware. Line integrals of physical-information losses are constructed from the domain boundaries where the suitable loss functions are defined. This formulation eliminates the need for higher-order automatic derivatives when solving initial-value problems of ordinary differential equations (ODEs); therefore, this formulation is particularly interesting as a candidate for solving differential equations on near-future quantum hardware. In addition to reducing the order of automatic derivatives required for ordinary differential equations (ODEs), current methods also avoid the need to individually balance boundary contributions and interior loss contributions, as in general physical-information optimization. The methods also normalize the global characteristics of neural networks through line integrals, which introduces an alternative optimization scheme for the local properties of pointwise gradient optimization exhibited by conventional physical-information neural network (PINN) optimizations. This is discussed in more detail below with reference to Figures 3 (a) to (c).

[0058] Although the methods are illustrated using quantum neural networks, they are equally applicable to any neural network regardless of the underlying hardware. Examples are applications of propagated line integrals within classical and spiking neural networks.

[0059] The present disclosure demonstrates means for solving differential equations, particularly ordinary differential equations and partial differential equations of a specific class, using quantum neural networks in a manner suitable for near-future quantum hardware. The proposed methods perform well compared to benchmark solutions in terms of both shot requirements and root mean square error (RMSE). Experimentally, the relative performance of quantum neural networks at finite-shot and infinite-shot limits can be unpredictable, and as discussed with reference to FIGS. 9 and 10, this highlights the need to consider finite shots within the quantum neural network when designing quantum algorithms.

[0060] The performance of this method may depend to some extent on the selection of collocation points. Given that the propagated physical-information loss functions depend on numerical integration, it may be beneficial to draw inspiration from (quasi) Monte Carlo integration techniques, such as importance sampling, to improve the proposed techniques.

[0061] Figure 1a illustrates a known method for solving differential equations. The method shown is based on solving the problem variationally using differentiable quantum circuits (DQCs). As shown in the figure, the parameterized target function A quantum model for is represented by a quantum circuit, and the quantum circuit is an independent variable x A feature map for encoding into the Hilbert space of the quantum register (102). (104_1), and learnable parameters Variational Ansatz parameterized by Includes (106). Therefore, it can be written as follows:

[0062] [Mathematical Formula 1]

[0063]

[0064] The derivative of the target function is obtained by differentiating the feature map and using the same variational Ansatz.

[0065] [Mathematical Formula 2]

[0066]

[0067] Feature Map This is also referred to as a differentiated feature map.

[0068] Learnable parameters of the solution to the differential equation It is found by variationally optimizing. Quantum circuits may include a series of quantum gates (layers of) that can be executed in quantum registers. Here, the only dependency of the input variable (independent variable) is on the feature map, and the variational Anzatz is the input variable Independent of, conversely, feature maps are trainable parameters It is independent of.

[0069] The training of the quantum model can be based on an optimization scheme, and after the initialization of variational parameters, gate operations of the quantum circuit can be executed, and the state of the quantum register can be measured using a cost function. Based on the measured states (108_1, 108_2), a loss value can be calculated based on the loss function (block 110). The loss function is based on a differential equation. For example, if the differential equation is as follows:

[0070] [Mathematical Formula 3]

[0071]

[0072] ( Is x (in the case of a announced function), the loss function is Contribution based on measurements (108_1) representing and Includes a contribution based on measurements (108_2) representing. Scaling factors a and b and a known function It can be included by the (classical) optimizer without any quantum measurements. The feature map is a single one at a time x - Since only the values ​​are encoded, the loss function is multiple collocation points It includes the sum of. For the example differential equation above, the loss function is as follows:

[0073] [Mathematical Formula 4]

[0074]

[0075] refers to some cost operators used for measuring quantum states, and represents the appropriate norm. (In practice, the loss function may also include contributions for boundary conditions, but these have been omitted here for the sake of simplification.)

[0076] If the loss function does not satisfy one or more optimization conditions, the variational parameters can be updated by the optimizer (112). Then, the quantum circuit can be executed using the updated variational parameters, the state of the quantum register can be measured, and a new loss value can be calculated. These steps can be repeated until one or more optimization conditions are satisfied. The state of the optimized quantum circuit can represent an optimized approximation of the desired target function.

[0077] Therefore, the logic gates of the quantum circuits used within the quantum machine learning scheme are learnable parameters learned by the classical optimizer to minimize the loss function It is parameterized by a set of. In general, parameterized quantum circuits can be used as quantum machine learning models and can be applied to applications such as learning or generating unknown distributions from a set of input features or solving differential equations.

[0078] FIG. 1b shows an example of the basic form of a quantum feature map, illustrated here as an example of a 'product' type feature map. In this example, single-qubit rotations that operate individually on each qubit of the feature map (104_1) (here Implemented using (selected as), and qubit rotations are variables x The function of It is parameterized by. Throughout these applications, qubits are used as standard examples of quantum elements, but most arguments are equally applicable to systems having other quantum elements (e.g., qudits). Function is preferably a non-linear function. In the case of non-linear feature map encoding, a non-linear function θ can be used as the angle of rotation. Applying a single rotation operation to each qubit in the register (122) can be referred to as a single layer of rotation operations. A multiplication feature map consists of multiple multiplication layers and other functions This can be further generalized. For example, multiple feature maps can be concatenated to represent a multivariable function. Measurement of a quantum state requires one measurement per encoding qubit. Depending on the quantum hardware, qubits may be measured one by one or all at once within a single measurement.

[0079] FIG. 1c illustrates an example of a derivative quantum circuit for the multiplication feature map of FIG. 1b. Variable x The derivative with respect to follows the chain rule of differentiation, and the shifted phase (e.g., It includes qubit operations (132) having ). Here, the expected value of the derivative is written as the sum of individual expected values ​​having shifted phases, and each x - It is repeated for dependent rotations (130_1-n). (Rotations with shifted phases are shaded.) Therefore, the number of measurements to determine the derivative value is (at least) twice the number of encoding qubits compared to a single measurement for the basis function.

[0080] When data reuploading is used (as described in more detail below with reference to Figs. 14 and 15), the number of measurements increases further. The same applies to more complex feature maps that may be necessary to approximate the solution of the differential equation in sufficient detail. In general, the derivatives for circuit inputs are measurement costs It involves, N is the number of data encodings into the quantum circuit, and d is the degree of the derivatives.

[0081] The methods described herein enable reducing the order of derivatives by at least one order for at least one input parameter. For linear (or linearizable) ordinary differential equations, derivatives can be completely eliminated (except for derivatives on the boundaries of higher-order differential equations).

[0082] FIG. 2 is a flowchart of a method according to an embodiment compared with a method according to the prior art. Step (202) includes receiving or determining a differential problem. Generally, the differential problem includes a differential equation and boundary conditions (e.g., initial values). In principle, the differential equation may be any type of differential equation (e.g., ordinary differential equation (ODE), partial differential equation (PDE), stochastic differential equation (SDE), delayed differential equation (DDE), etc.). The differential equation may be linear or non-linear. The differential equation is in a domain It can be defined across, and boundary conditions boundary It can be defined on (in some parts).

[0083] Generally, differential equations can be written as follows.

[0084] [Mathematical Formula 5]

[0085]

[0086] The boundary conditions are as follows.

[0087] [Mathematical Formula 6]

[0088]

[0089] For the sake of simplification, the diagram is limited to first derivatives. A more general consideration is provided below.

[0090] Step (204) is a parameterized quantum model of the execution function It includes the step of receiving or determining information about one or more quantum circuits that define the operations of. Here, x represents the independent variable(s) of a differential equation. Quantum model variable parameters It can be parameterized by. For example, the quantum model may be a so-called quantum neural network (QNN) as described above with reference to Fig. 1a. In that example, the quantum model x - Dependency feature map and x and independent parameterized variational Anzatz It includes. The quantum model may also be a kernel model, an amplitude-encoding-based model, or any other suitable quantum model.

[0091] Step (206) includes the step of receiving or determining a loss function as described in the prior art. In these methods, the loss function typically has two terms, an interior loss term based on a differential equation (for the domain inside) , and boundary loss term based on boundary conditions is. The loss term based on the differential equation typically [is] the differential equation By writing in the form, and collocation points It is obtained by determining the distance from zero of the differential equation evaluated for the trial function. Usually, it is as follows.

[0092] [Mathematical Formula 7]

[0093]

[0094] In addition, a (similarly derived) loss term is calculated for the boundary conditions. An example is as follows.

[0095] [Mathematical Formula 8]

[0096]

[0097] Consequently, the evaluation of the loss function requires all derivatives of the trial function existing in the differential equation.

[0098] Conversely, step (208) includes the step of receiving or determining a loss function as described in the present disclosure. A differential equation Instead of writing it in the form, the highest-order derivative is separated, and the differential equation is It is written in the form. Then, both sides of the equation are integrated to derive the following, and

[0099] [Mathematical Formula 9]

[0100]

[0101] Here is a term that depends on boundary conditions. The loss function typically represents the distance between the left and right sides of [Equation 9]. For the trial function, e.g.:

[0102] [Mathematical Formula 10]

[0103]

[0104] Here, It refers to a suitable norm, and typically or Depending on the differential equation, the integration can be repeated to eliminate additional derivatives. Thus, the differential equation includes derivatives up to the nth order, and unlike the loss function of the prior art, which likewise includes quantum circuit derivatives up to the nth order, the current loss function includes only derivatives up to the n-1th order (at most for at least one independent variable). This reduces the number of measurements on the quantum circuit during each optimization epoch. More general considerations as well as more specific examples are given below.

[0105] Step (210) minimizes the loss function to obtain optimal parameters It includes a step of determining. This includes a step of changing variational parameters until a stop criterion is satisfied and a quantum model It requires a step of evaluating (and / or the derivatives of the quantum model required by the loss function). Typical examples of stopping conditions are as follows: reaching a predetermined number of steps, obtaining a loss value smaller than a predetermined threshold, and / or having a change in loss value smaller than a predetermined threshold (possibly on the average over several previous optimization steps). As discussed in more detail below, various optimization methods may be used. For example, gradient-based methods that calculate or estimate the gradient of the quantum model with respect to variational parameters may be used. Methods that use a very small number of measurements (e.g., one or two) to estimate the gradient may sometimes be referred to as gradient-free methods. As an additional example, group-based methods may be used. A group-based method may include the step of determining an initial ('parent') group of initial sets of variational parameters having associated loss values, and the step of determining an additional ('offspring') group of adjusted sets of variational parameters based on the initial sets of variational parameters and associated loss values.

[0106] The optional step (212) is one or more points Optimized parameters for quantum model having It includes a step of evaluating the solution using. These evaluation points do not need to coincide with collocation points.

[0107] FIG. 3(a) illustrates a method for solving differential problems using a variational quantum circuit (VQC) employing a known method. Differential problems are differential equations and boundary conditions Includes FIt should be noted that, as in the case of comprehensive partial differential equations, it can possibly be a very comprehensive function that includes derivatives with respect to other variables. Similarly, the function It itself may include derivatives with respect to t (effectively representing a higher-order differential equation) and / or derivatives with respect to other variables. For simplification, Dirichlet boundary conditions are used, but they work similarly for other types of boundary conditions.

[0108] A variational quantum circuit is a parameterized trial function It is encoded and is represented by a solid line (302) in the drawing. Given variation parameters For the set of, the execution function is each of the multiple collocation points It is evaluated in (304_0-6). In this example, the collocation points are evenly distributed, but in other embodiments, the selection of collocation points may have a random component. The evaluation of the trial function is a probabilistic process, and for each collocation point, multiple measurements (306_1-8) must be taken to obtain the expected value (308). Also, for each collocation point, the slope (310) must be calculated (approximately represented by tangent line segments (310_1-8)). As described above with reference to FIGS. 1a through 1c, this has a high computational cost for quantum circuits and requires a large number of shots. Again, the measurements to obtain the expected value (312) must be repeated countless times.

[0109] Therefore, the quantum processor has multiple collocation points Values ​​for each It calculates. These values ​​are subsequently used by classical computers to calculate loss values ​​based on the quantum processor's measurements and differential equations. Typically, the loss function is given as follows, and

[0110] [Mathematical Formula 11]

[0111]

[0112] w represents the weights for the relative contributions of the interior and boundary. Based on the loss function, the parameters until some stopping criterion is met This can be changed, and the entire process can be repeated.

[0113] Similar considerations can be applied to other types of variational models, including classical neural networks, spiking neural networks, and other physical- or biological-information models.

[0114] FIG. 3(b) illustrates a method for solving differential equations using a variational quantum circuit (VQC) according to an embodiment. Similar to known cases, the variational quantum circuit is a (possibly identical) parameterized trial function It is encoded and is represented by a solid line (322) in the drawing. Again, variation parameters For a given set of, multiple collocation points to obtain the expected value (308) (304_0-6) In each case, the trial function is evaluated iteratively. However, in this case, there is no need to calculate the gradient. Instead, the differential equation is integrated 'manually' to obtain the integral form of the differential equation. The integral form depends on both the differential equation and the boundary condition(s). This integral form includes a definite integral along the path from the boundary to the evaluated collocation point. This is also known as the Volterra integral. This integral can be calculated numerically and is generally in the form of a sum (possibly weighted) over all collocation points on the path between the boundary and the collocation point under consideration.

[0115] This is illustrated in Fig. 3(c). Function (given as part of a differentiation problem or derived therefrom) is indicated by the solid line (332). Boundary conditions (0 in this example) and collocation points under consideration (in this example, A definite integral having integration limits obtained from ), that is is boundary ( ) and collocation points ( between ) It is equal to the area under the curve (334) defined by. This integral can be approximated using function values ​​for collocation points between the boundary and the collocation point under consideration. For example,

[0116] [Mathematical Formula 12]

[0117]

[0118] Here, it is assumed that the collocation points are in a specific order, that is , These are weighting factors. For example, for equidistant collocation points, typically 0 <j<i Regarding, And, j =0 and j = i Regarding Other numerical integration methods may use differently defined sums (e.g., Monte Carlo-based methods, quasi-Monte Carlo-based methods, quantum Monte Carlo-based methods, midpoint rule-based methods, or quadrature-based methods, etc.).

[0119] As shown in Fig. 3(c), My inherent uncertainty is This leads to corresponding uncertainty, and consequently, the values ​​determined based on the measurements do not exactly match the theoretical curve. Since the integral is based on all collocation points between the boundary and the collocation point under consideration, this uncertainty can be averaged out. In other words, the value of the integral to calculate the area Since it depends on the average of the points to the left, the noise within each collocation point can be averaged out by the noise of other collocation points. This represents an additional advantage in terms of optimizing quantum neural networks with a finite number of shots.

[0120] FIGS. 4a and 4b illustrate a method according to an embodiment. FIG. 4a illustrates the steps of various methods, and FIG. 4b provides a practical example. A first step (402) includes receiving information related to a differential problem. The differential problem includes a differential equation defined over a domain and at least one boundary condition defined over at least part of the boundary of the domain. The differential equation may be an ordinary differential equation, a partial differential equation, or an equation of a mixed type. The differential equation may be a first-order equation or a higher-order equation. The differential equation may be a linear differential equation, a non-linear equation, or a combination (e.g., linear for a first variable and non-linear for a second variable).

[0121] For example, step (422) includes receiving a linear first-order ordinary differential equation.

[0122] [Mathematical Formula 13]

[0123]

[0124] In general, differential problems involve first-order ordinary differential equations or can be formulated as first-order ordinary differential equations.

[0125] [Mathematical Formula 14]

[0126]

[0127] Boundary conditions here defines the initial value.

[0128] Step (404) includes the step of receiving or determining an integral equation based on a differential equation and at least one boundary condition. An integral equation is an equation having a differential order smaller for at least one independent variable than the differential order of the differential equation for at least one independent variable. Thus, a first-order ordinary differential equation leads to an integral equation without derivatives. In general, linear (or linearizable) higher-order ordinary differential equations can lead to an integral equation without derivatives (possibly, excluding boundaries). Other differential equations can lead to an integral-differential equation containing at least one integral and one or more derivatives.

[0129] The integral formula is based on boundary conditions. For example, the integral equation may include boundary values ​​(initial values), and / or one or more integral limits may be based on one or more boundary conditions. Generally, the integral equation can be obtained by integrating the differential equation one or more times over one or more independent variables. Typically, the integral formula includes at least some of the boundary conditions, and the integral formula includes a path integral having a first integration limit corresponding to a point on the boundary and a second integration limit corresponding to a point within the domain.

[0130] For example, step (424) includes the step of determining an integral equation based on the differential equation received in step (422).

[0131] [Mathematical Formula 15]

[0132]

[0133] In general, both sides of the differential equation [Equation 14] can be integrated to derive the following:

[0134] [Mathematical Formula 16]

[0135]

[0136] (This is essentially the fundamental theorem of calculus.)

[0137] Step (406) includes receiving or determining information about one or more quantum circuits that define operations of a parameterized quantum model of an execution function parameterized by variational parameters. By optimizing the variational parameters, a solution to the differential problem is obtained.

[0138] For example (step (426)), a variational quantum circuit can be defined as follows.

[0139] [Mathematical Formula 17]

[0140]

[0141] Based on [Equation 16], the solution must follow the following equation.

[0142] [Mathematical Formula 18]

[0143]

[0144] Here silver represents the parameters for solving the differential equation.

[0145] For this purpose, step (408) includes receiving or determining a loss function associated with the integral formulation of a differential equation. The evaluation of the loss function includes the calculation of the integral.

[0146] For example, the loss function can be defined as follows.

[0147] [Mathematical Formula 19]

[0148]

[0149] Here, refers to a suitable norm (e.g., 2-norm). Typically, am.

[0150] In general, integrals are approximated by sums. In principle, any of the known suitable numerical methods may be used to evaluate integrals. For example, general Monte Carlo-based methods, quasi-Monte Carlo-based methods, quantum Monte Carlo-based methods, midpoint rule-based methods, or quadrature-based methods may be used. Which integral is suitable depends on the details of the problem, such as the dimensionality and smoothness of the integrand.

[0151] [Mathematical Formula 20]

[0152]

[0153] Conversely, this can be written in a recursive manner as follows, and

[0154] [Mathematical Formula 21]

[0155]

[0156] The evaluation of the function value is based on previous function values ​​along the path from the boundary to the point where the function was evaluated.

[0157] Combining [Equation 19] and [Equation 20] yields the following:

[0158] [Mathematical Formula 22]

[0159]

[0160] Therefore, the evaluation of the loss function for a point under consideration within the domain involves the calculation of an integral, and the calculation of the integral is based on one or more collocation points between the point on the boundary and the point under consideration.

[0161] Considering the variational quantum model and the loss function, variational parameters are optimized in steps (412) and (414). In principle, any suitable optimization method may be used. Roughly speaking, two classes of optimization methods are distinguished as gradient-based optimization methods (step (412)) and gradient-free optimization methods (step (414)), such as SPSA, genetic algorithms, evolutionary strategies, etc.

[0162] For example, step (432) shows the calculation of the analytic derivative of the variational quantum circuit with respect to the variational parameter(s), the step of determining the slope of the loss function with respect to the variational parameters, and the step of changing the variational parameters based on the determined slope of the loss function. As another example, step (434) shows the step of changing the variational parameters with random components to obtain new variational parameters, the step of comparing a current loss value based on the current variational parameters with a new loss value for the new variational parameters, and the step of updating the variational parameters if the new loss value is lower than the current loss value.

[0163] Optimization may be stopped when appropriate stopping criteria are met (e.g., when the loss value is less than a threshold and / or when the change in the (possibly averaged) loss value is less than a threshold). Corresponding optimized parameter values It can be used later to obtain a solution function and follows the following.

[0164] [Mathematical Formula 23]

[0165]

[0166] An optional step (416) includes a step of evaluating the solution, and a step of using a quantum model having optimized parameters for one or more selected input variables (typically within a domain). For example (step (436)), a function value this It can be evaluated as, represents the optimized variational parameter.

[0167] Some specific examples are provided below. As mentioned above, the loss function can be obtained from the equation by replacing the (unknown) solution with a parameterized quantum model representing the trial function and calculating the norm of the difference between the left and right sides of the equation.

[0168] The first-order ordinary differential equation (ODE) and Dirichlet boundary condition (Dirichlet BC) are:

[0169] [Mathematical Formula 24]

[0170]

[0171] [Mathematical Formula 25]

[0172]

[0173] The first-order ordinary differential equation (ODE) and Neumann boundary condition (Neumann BC) are:

[0174] [Mathematical Formula 26]

[0175]

[0176] [Mathematical Formula 27]

[0177]

[0178] Second-order ordinary differential equation (excluding first-order terms), Cauchy boundary conditions (Cauchy BC):

[0179] [Mathematical Formula 28]

[0180]

[0181] [Mathematical Formula 29]

[0182]

[0183] Linear second-order ordinary differential equation, Cauchy boundary conditions:

[0184] [Mathematical Formula 30]

[0185]

[0186] [Mathematical Formula 31]

[0187]

[0188] Second-order ordinary differential equation (with comprehensive first-order terms), Cauchy boundary conditions:

[0189] [Mathematical Formula 32]

[0190]

[0191] [Mathematical Formula 33]

[0192]

[0193] [Mathematical Formula 34]

[0194]

[0195] First-order partial differential equation:

[0196] [Mathematical Formula 35]

[0197]

[0198] [Mathematical Formula 36]

[0199]

[0200] In some cases, a differential problem can define more than one differential equation. For example, a differential problem can define partial differential equations with respect to multiple independent variables. As an additional example, higher-order differential equations These pairs of ordinary differential equations and It may be written as such. In such cases, while one of the differential equations may be solved as described herein, other differential equations may be solved using other methods, for example, - a differentiable quantum circuit (DQC) described in WO 2022 / 101483 A1, which is incorporated herein by reference in its entirety, or - amplitude-based methods described in pending application EP23186602.1, which is incorporated herein by reference in its entirety. Depending on the differential problem, it may be possible to solve all differential equations according to the embodiments described herein.

[0201] Below, a more general consideration is provided for deriving integration formulas based on differentiation problems. Boundary An open set (domain) having dependent variable u within A general differentiation problem of can be written as follows.

[0202] [Mathematical Formula 37]

[0203]

[0204] and are differential operators (potentially non-linear). Note that because this is an arbitrary differential operator, [Equation 37] can represent a rich class of differential equations, including partial differential equations (PDEs), ordinary differential equations (ODEs), and delayed differential equations (DDEs), to name just a few.

[0205] Neural network that has learned the solution to [Equation 37] It can be approximated by, represents a set of learnable parameters.

[0206] and Probability distributions for and Defines. Samples of these distributions are referred to as collocation points. The optimization of PINNs uses the following losses and optimization problems to weight (Here represents the solution to the differentiation problem) constitutes:

[0207] [Mathematical Formula 38]

[0208]

[0209]

[0210]

[0211] So far, Almost no restrictions were imposed on the architecture represented by [Equation 38]. Consequently, the optimization problem in [Equation 38] applies to quantum neural networks just as it does to classical neural networks. While the results of differential operators in classical neural networks can be evaluated at collocation points through automatic differentiation, the same can be achieved in quantum neural networks through the parameter-shift rule (PSR), which can be viewed as a quantum version of forward-mode automatic differentiation.

[0212] For (digital) quantum neural networks, data and network weights are typically encoded through rotations on a given qubit. The derivatives of these encodings typically involve two additional circuit configurations per encoding. N Data encodings, and d - For the derivatives, d The number of circuits required to evaluate the -th derivative is It increases in proportion to, and as a result, a significant number of circuit evaluations occur.

[0213] Furthermore, if boundary conditions are not properly considered, local optimization can lead to a waste of resources, which can result in (local) convergence to a 'false' local minimum.

[0214] Therefore, the embodiments in this disclosure are boundary from domain It uses a methodology for constructing a line integral that propagates boundary conditions inward. Invertible differential operator Regarding this, the suffix indicates its importance in boundary condition propagation, and [Equation 37] is as follows:

[0215] [Mathematical Formula 39]

[0216]

[0217] From [Equation 37] Keep in mind that, u neural network It is derived by replacing it with and organizing the terms.

[0218] boundary Domain from the points above Probability distribution of straight lines pointing inward Wow, the given line l Probability distribution of points lying along Introduces. This ensures that the following optimization problem is defined.

[0219] [Mathematical Formula 40]

[0220]

[0221] Here, is solving differential equations It is the parameterization of.

[0222] now from Operators that propagate boundary conditions up to It remains to define.

[0223] Dirichlet boundary conditions are boundary conditions on the target function. If the differentiation problem includes these boundary conditions, the point A boundary condition It can be selected from. Afterwards, pre This can be parameterized, and here And, v Is from It is the inner direction, and all 0 <s<S에 대해서 It is. At that time The choice is as follows:

[0224] [Mathematical Formula 41]

[0225]

[0226] Here x is a point on the line and the integral is from x until It represents the line integral along. Now, within the integrand u a parameterized function such as a neural network It can be parameterized, and then defines Volterra integral equations to be used within the optimization problem in [Equation 40]. The result can be written as follows.

[0227] [Mathematical Formula 42]

[0228]

[0229] Neumann boundary conditions are boundary conditions on the derivative of the target function. If the differentiation problem includes these boundary conditions, points and parameterized lines can be selected, as described above for Dirichlet boundary conditions. However, instead of Dirichlet boundary conditions, Neumann boundary conditions are It can take the form, and here silver It refers to the directional derivative toward. Subsequently The choice of yields the following:

[0230] [Mathematical Formula 43]

[0231]

[0232] Here, as before, is a point on the parameterized line and the integral is from x until It represents the line integral along . As before, within the integrand u a neural network It can be parameterized as. However, in this scenario, since the boundary condition is a Neumann condition, in order to avoid the resulting loss function being biased toward an arbitrary specific Dirichlet condition, in [Equation 43] this It is replaced by. Subsequently, the resulting formula defines the line integrals to be optimized in [Equation 40]. In this case, the resulting formula can be written as follows.

[0233] [Mathematical Formula 44]

[0234]

[0235] The formulation of boundary-propagated differential equation losses offers several advantages:

[0236] 1) Boundary conditions and internal terms—which are summed and weighted as separate terms in traditional PINN formulas—are combined into a single term.

[0237] 2) Boundary conditions through losses directly derived from the basis differential equations from It propagates internally. This contrasts with PINNs, which optimize points independently of each other.

[0238] 3) and Depending on the specific forms, a reduction in the order of automatic derivatives can be achieved in the optimization of [Equation 40] compared to [Equation 38]. For many ODEs, higher-order (automatic) derivatives are not required at all.

[0239] 4) When the evaluation is probabilistic, as in the measurement of a quantum neural network, the subsequent (quasi) Monte-Carlo estimation of the integral averages the noisy evaluations across the integration domain.

[0240] [Mathematical Equation 41] My integral is evaluated numerically. This is a line Sampling points along , through automatic differentiation This can be done by calculating and then numerically approximating the integral (e.g., through Monte Carlo integration, trapezoidal approximations, or quantum-enhanced Monte Carlo integration). extending from L Other lines N If evaluated at different points, the computational complexity of evaluating line integrals is am. NL At other points, the computational complexity is of the same order as the evaluation of standard physical-information loss. Finally, the integral within [Equation 43] represents a double line integral, as opposed to a surface integral. This can be obtained by repeating the integral estimation twice. Therefore, the evaluation of Neumann boundary integrals of the form within [Equation 43], even if nested, is of the order It remains as.

[0241] The formulation of loss functions for conferring Dirichlet and Neumann boundaries is not unique. For example, [Equation 43] is Includes Dirichlet boundaries for, as It can be used to normalize Dirichlet boundary conditions by replacing it.

[0242] While the methodologies described above apply to a broad class of differential equations, specific examples of those methodologies are specifically related to ordinary differential equations. Consider the ordinary differential equation defined by the following.

[0243] [Mathematical Formula 45]

[0244]

[0245] Here am. n The problem with the car's initial value n Note that this formulation includes arbitrary higher-order non-linear initial value problems because it can be rewritten as a set of combined first-order initial value problems. For example, These pairs of ordinary differential equations and It can be written as.

[0246] In this case, since the boundary condition includes a single line, the probability distribution within [Equation 40] includes a single line. We by and T It refers to the probability distribution of time between. Neural network By parameterizing the solution of [Equation 45] and combining it with [Equation 41], [Equation 40] subsequently derives the following optimization problem to solve the differential equation.

[0247] [Mathematical Formula 46]

[0248]

[0249] [Equation 38] Compared to the optimization problems used in the traditional PINN formulations described within, the input t About There are no derivatives of [Equation 46]. My optimization problem can be obtained directly from [Equation 45] by manually integrating both sides of [Equation 45] with respect to time. However, the general formula provided above applies to a large class of differential equations.

[0250] Compared to general PINNs, the elimination of higher-order neural network derivatives in optimization problems [Equation 46] is of particular importance in solving differential equations with quantum circuits. Combining the unnecessary nature of higher-order derivatives with gradient-free optimization allows for the selection of almost any quantum hardware to solve ordinary differential equations. N quantum neural networks with feature-encodings d Finding the second derivative is Because it requires evaluations, therefore this method (at least) of multiple quantum neural network shots per training step Provides reduction.

[0251] Quantum circuit inputs, which are weights or characteristics, are typically encoded as rotations on qubits. Because the circuit output is probabilistic, to achieve an estimation of the output, the quantum circuit is typically performed a certain number of times, so to speak. N It is evaluated as such. The gradients of quantum circuits with respect to their input characteristics can be calculated through parameter-shift rules, which are typically 2 E Requesting evaluation of circuits, E is the number of times the characteristic is encoded into the circuit. If W If represents the number of weights encoded into the circuit, then similarly 2 W Circuit evaluations are necessary to estimate the gradient with respect to the weights. If different collocation points C If evaluated at, it is a quantum physical information neural network (qPINN) per training epoch for gradient-based optimization , , , and Corresponding to the evaluations CN (1 + 2 E + 2 S + 4 E.W. )Requires circuit evaluations.

[0252] In contrast, the propagated learning objective function described in [Equation 46] is t It does not require the derivative with respect to, and therefore per epoch CN (1 + 2 W ) requires circuit evaluations. Therefore, in the context of quantum circuit learning for differential equations, the advantage of [Equation 46] compared to qPINNs is E It increases as (i.e., the number of feature encodings) increases.

[0253] Figures 6 through 10 present examples of homogeneous, non-homogeneous, and non-linear ordinary differential equations in applications within population modeling. The experimental setups are achievable on near-future (NISQ) quantum hardware. While the methods described above are applied to various types of differential equations and can be used to eliminate the need for higher-order automatic differentiation for some partial differential equations (PDEs), this section demonstrates the applicability of the methods to ordinary differential equations (ODEs).

[0254] In addition, the methodology is applied to almost any neural network architecture (including neural networks on classical computers), including multilayer perceptrons and spiking neural networks.

[0255] FIG. 5 is a quantum circuit diagram of the quantum circuit used in the experiments shown in FIG. 6 through 10. The top diagram (A) shows three qubits , , Shows a circuit diagram having. The qubits are initialized to a zero basis state. Unitary data encoding on qubits applying a Chebyshev-tower type feature map. is applied, and qubit number n Rotate Apply (shown in more detail in the middle column (B)). Next, variational Anzatz is applied, and this is three parameterized fecal unitaries It includes and leads to a triangular ring of entangling CNOT gates that implement a hardware-efficient ansatz (HEA). This is shown in more detail in the diagram (C) below. The circuit ends with a measurement.

[0256] This experimental setup is chosen to ensure the degree of achievability of current quantum hardware. A circuit of three qubits It can be manifested by, It encodes data through a Chebyshev-Tower type feature map, and represents a parameterized variational unitary comprising single-qubit rotations connected by a triangular loop of entangling CNOT gates, such as a hardware-efficient Ansatz. Input characteristics t linearly prior to Chebyshev-feature encoding It is scaled to. Variational parameters go It is initialized to be uniformly distributed within the range. Hamiltonian is single-qubit magnetization on the 0th qubit.

[0257] PennyLane and Nevergrad are used in all quantum simulation experiments.

[0258] Figures 6 through 10 show the results associated with models of population extinction, population growth in seasonal environments, and population evolution in environments with limited resources, given by their respective equations:

[0259] [Mathematical Formula 47]

[0260]

[0261] Starting at time t = 0, the maximum times are t = 2.0, t = 1.0, and t = 5.0, respectively, and the initial conditions are y = 1.5, y = 1.0, and y = 0.1, respectively.

[0262] FIGS. 6a through 6f are graphs showing experimental results comparing the method according to the embodiment with the method according to the prior art. In these experiments, a finite number of shots were used. The black lines (602, 612, 622, 632, 642, 652) represent the mean losses according to the qPINN method according to the prior art described above with reference to FIG. 1. The gray lines (604, 614, 624, 634, 644, 654) represent the mean losses according to the method according to the embodiment (indicated in the legend as 'propagated'). The shaded areas around the qPINN lines represent the 95% confidence intervals of the mean calculated through bootstrapping for neural networks trained with 20 random seeds. For the propagated lines, these confidence intervals are obscured by the line representing the mean.

[0263] The figures demonstrate lower Root Mean Square Error (RMSE) in terms of both the number of training epochs (Figs. 6a to 6c) and the number of training shots (Figs. 6d to 6f). All models were trained using an Adam optimizer with a learning rate of 0.01. Figs. 6a and 6d present the results for seasonal growth of populations according to environment, Figs. 6b and 6e show the logistic growth model, and Figs. 6c and 6f illustrate the population extinction model.

[0264] For the given circuit parameters, the number of collocation points and the number of repeated measurements were both set to 32 for the finite shot experiments, and the collocation points were arranged in an even grid. Numerical integration of the propagated losses was performed using a quasi-Monte Carlo method across the collocation points. Training is performed over 10,000 epochs per method. During training, finite shots are used, while the solution quality is calculated based on the expected value taken from an even grid of 128 points across the temporal domain. All results are for 20 different random seeds. For each system given in [Equation 38], the physical information solutions denoted by qPINN are compared with the propagated optimization scheme denoted as the 'propagated case' described in [Equation 46].

[0265] The experimental setup specifies the number of shots per training epoch depending on the optimization procedure used, which is summarized in Table 3.

[0266] Table 3: Number of quantum circuit shots required per training epoch for our proposed propagated methodology compared to existing qPINN methods. The approximately 6-fold difference between the two methods corresponds to two additional circuit evaluations required for each of the three data-encodings within our quantum circuits. NFor data encodings, the propagation method is the number of shots required per training epoch. Provides a reduction in the multiplier.

[0267] [Table 3]

[0268]

[0269] In FIGS. 6a through 6f, Adam is used as a gradient-based optimizer with a learning rate of 0.01. The results of FIGS. 7 through 9 are obtained using methods according to the embodiments described in FIGS. 6a through 6f, and again, FIGS. 7a, 8a, and 9a present the seasonal growth results of a population according to the environment, FIGS. 7b, 8b, and 9b show a logistic growth model, and FIGS. 7c, 8c, and 9c illustrate a population extinction model.

[0270] Figures 7a through 7c compare these results with the same gradient-based optimizer Adam with different learning rates. Curves 702, 712, and 722 show the results for a learning rate of 0.01, curves 704, 714, and 724 show the results for a learning rate of 0.001, and curves 706, 716, and 726 show the results for a learning rate of 0.0001. In general, a higher learning rate leads to faster convergence.

[0271] In contrast to the constant scaling of backpropagation in classical neural networks, automatic differentiation of learnable parameters in quantum neural networks entails computational overhead that scales linearly with the number of parameters; therefore, there is a stronger incentive to use gradient-free methods to train quantum neural networks. Consequently, gradient-based methods (with multiple learning rates) are compared with gradient-free methods.

[0272] Figures 8a through 8c compare these results with so-called gradient-free optimizers. For gradient-free optimization, the hyperparameters required to calculate the magnitude of the gradient, standard deviation c = 0.15, and a Simultaneous Perturbation Stochastic Approximation (SPSA) is used with = 2.0. Additionally, budget The One-Plus-One optimizer is used, and all other hyperparameters remain as set by Neverred. Curves 802, 812, and 822 once again show the results of the Adam optimizer with a learning rate of 0.01, curves 834, 844, and 854 show the results using the SPSA optimizer, and curves 836, 846, and 856 show the results of the One-Plus-One optimizer.

[0273] Investigations into the optimization of quantum neural networks in finite-shot learning are still in their early stages. Figures 9a through 9c show the results for essentially the same experiments as Figures 6a through 6c, respectively, but at the limit of an infinite number of shots rather than a (more realistic) finite number. For average qPINN losses of 902, 912, and 922 (black) versus neural networks trained over 20 random seeds, the propagated ('global') losses of 904, 914, and 924 (gray), with shaded regions representing the 95% confidence intervals of the bootstrapped mean, demonstrate a lower Root Mean Square Error (RMSE) in terms of the number of training epochs in the case of infinite shots. Both models were trained using an Adam optimizer with a learning rate of 0.01.

[0274] As shown in FIGS. 6 through 10, across the entire range of gradient-based optimizer settings and gradient-pre-optimizer settings, faster convergence is obtained in terms of both the number of shots (applicable) and the number of epochs. Additionally, lower end RMSEs are obtained in finite shot experiments and most infinite shot experiments.

[0275] Given the prevalence of infinite shot experiments, it is interesting to consider whether the performance of various algorithms at the infinite shot limit translates to the performance of finite shots.

[0276] Figure 10 shows a comparison of the final log root mean square error (log RMSE) values ​​achieved after 10,000 training epochs between the finite 1002, 1004, 1012, 1014, 1018, 1024, 1028 (light gray) and qPINN methods 1002, 1006, 1012, 1016, 1022, 1026 (left) and infinite 1006, 1008, 1016, 1018, 1026, 1028 (right) cases for the propagated methods 1004, 1008, 1014, 1018, 1024, 1028 (light gray) using an Adam optimizer with a learning rate of 0.01 (dark gray). The box plots show the loss distributions for 20 random seed experiments. Figure 10a shows the population extinction model, Figure 10b shows the logistic growth model, and Figure 10c shows the seasonal growth of the population according to the environment. Note that, particularly for the seasonal model, qPINN shows better RMSE than the propagated loss functions at the infinite shot limit, but worse RMSE at the finite shot limit.

[0277] Therefore, in some cases, algorithm performance within an infinite shot limit does not imply algorithm performance within a finite shot limit. The implications of this result are twofold.

[0278] 1. Shows that the infinite shot behavior of quantum neural networks does not necessarily explain the finite shot behavior; and

[0279] 2. The currently presented propagated loss formula has advantageous performance in a (physically feasible) finite shot system.

[0280] FIG. 11a shows a reference quantum machine learning model (quantum neural network, QNN-type model). The circuit (1100) is, for example, a cost function corresponding to a projection on the zero state. (1106), parameter Variational circuit parameterized by (1104) and Fourier feature map or Chebyshev feature map (1102)) is included. A measurement (1108) for ancilla may be included to select an appropriate sum term that may be required for some feature maps, such as Chebyshev feature maps.

[0281] Input value x One option for differentiating the circuit with respect to is to use the parameter shift rule, which is well known in the relevant art and illustrated in FIG. 11c. For example, this approach is valid in the learning phase where the cost function corresponds to a projection on the zero state, for example (including an ancillary qubit). The application of the parameter shift rule involves running the circuit twice for each gate that depends on the variable being differentiated, and once is x -dependent rotation angle Move by that amount, and the other time x -dependent rotation angle Shift by that amount, and sum the results. Multiple qubits applied to each qubit x - There may be dependent gates. For example, the Chebyshev feature map has two gates applied to each qubit xSince dependent gates are used, the total number of parameter shifts (and thus terms in the final sum) is 4n, where n is the number of encoding qubits. In the notation used in the diagram is gate l The rotation gate angle of It means moving by that amount, and similarly Is It refers to a gate angle shift of ). Therefore, for example, the gate

[0282] [Mathematical Formula 48]

[0283]

[0284] It moves as follows.

[0285] [Mathematical Formula 49]

[0286]

[0287] (and Similarly for), one gate at a time (essentially, in the current example, each pair Once for each. This leads to 4n modified circuits and is summed to obtain the derivative circuit. Other examples can use more circuits.

[0288] In cases where mid-circuit measurements are possible, alternative differentiation methods may be applied. In this case, controlled phase rotations can be decomposed using CNOT conjugation, and each parameter is individually re-uploaded x Two shifts can be applied. This is repeated twice for different bases of the measurement results. Thus, collectively, the Chebyshev feature map can be differentiated by 4n parameter shifts.

[0289] The above approaches are described in more detail in Kyriienko et al.’s ‘Generalized quantum circuit differentiation rules’ (Phys. Rev. A 104 (052417)) – incorporated herein by reference – in particular in Sections II.B and II.C.

[0290] Another option, illustrated in FIG. 11b, is to formally differentiate the feature map to obtain an effective generator It involves extracting and taking the derivative as a linear combination of the unitaries. To this end, from [Equation 11], the state (which does not necessarily need to be normalized), which is a Chebyshev polynomial with all amplitudes scaled is considered. A normalized version of this is prepared by the Chebyshev feature map. A model including x When taking the derivative with respect to, the resulting model It includes terms. This can be regarded as a state where each amplitude is a derivative of a Chebyshev polynomial scaled.

[0291] Chebyshev polynomials and their derivatives are related as follows.

[0292] [Mathematical Formula 50]

[0293]

[0294] Here Is k It refers to the second kind of Chebyshev polynomial of degree. This can be further extended as follows:

[0295] [Mathematical Formula 51]

[0296]

[0297] [Mathematical Formula 52]

[0298]

[0299] [Mathematical Formula 53]

[0300]

[0301] This means that the derivative of a Chebyshev polynomial is a linear combination of lower-order Chebyshev polynomials. Therefore, the valid operator can be extracted as follows:

[0302] [Mathematical Formula 54]

[0303]

[0304] operator It is non-unitary, but can be implemented using known techniques that implement non-unitary operators, such as linear combinations of units or embeddings in higher-order spaces.

[0305] For example, for the DQGM model, the following approach can be used. If the probability density function If is given as follows:

[0306] [Mathematical Formula 55]

[0307]

[0308] is normalized And, silver If it is a norm, the derivative (using the product rule) is as follows:

[0309] [Mathematical Formula 56]

[0310]

[0311]

[0312]

[0313] FIG. 12a is a hardware-level schematic diagram of operations realizing the logical operations shown in circuit diagrams such as FIG. 1 and FIG. 5. Unitary operators (e.g., encoding quantum kernel feature maps and their derivatives) can be decomposed into a series of logic gate operations. These logic gate operations are transformations within quantum Hilbert space across qubits encoding input parameters. To transform the internal states of these qubits, a classical control stack is used to send pulse information to a pulse controller that affects one or more qubits. The controller can send this series of pulses independently to each qubit in chronological order. An initialization pulse [initials] the qubits It is used to initialize to state (1202). Afterwards, for example, a series of single-qubit pulses are transmitted to the qubit array within (1204), which can apply a single-layer feature map. Additionally, a series of two-qubit pulses may be used to effectively entangle multiple qubits into a feature map (1206). The duration, type, intensity, and shape of these pulses determine the quantum logic operations to be realized. (1208) implies a 'break' in the indicated timeline, which means that a series of gates can be repeated in a similar manner in the direction of the time axis (1212). At the end of the series of pulses, one or more qubits are measured (1210).

[0314] FIG. 12b is a hardware-level schematic diagram of operations realizing the logical operations shown in circuit diagrams such as FIG. 1 and FIG. 5, specified for a photonic / optical quantum processor. Unitary operators (e.g., those used to encode quantum kernel feature maps and their derivatives) can be decomposed into a series of optical gate operations. These optical gate operations are transformations in quantum Hilbert space across optical modes. To transform the internal states of these modes, a classical control stack is used to send pulse information to a pulse controller affecting one or more modes. The controller can formulate programmable unitary transformations in a parameterized manner.

[0315] Initially, all modes (1214) are in a vacuum state (1216), which are subsequently compressed to create single-mode compressed vacuum states (1218). The duration, type, intensity, and shape of the control-optical gate transitions determine the quantum logical operations (1220) that are realized. At the end of the optical paths, one or more modes are measured by photon-number resolving, Fock basis measurement (1222), tomography, or threshold detectors.

[0316] FIG. 12c is a hardware-level schematic diagram of operations realizing the logical operations shown in circuit diagrams such as FIG. 1 and FIG. 5, specified for a Gaussian boson sampling device. Unitary operators (e.g., those used to encode quantum kernel feature maps and their derivatives) can be decomposed into a series of optical gate operations. These optical gate operations are transformations in quantum Hilbert space across optical modes. To transform the internal states of these modes, a classical control stack is used to send information to optical switches and delay lines. The controller can formulate programmable unitary transformations in a parameterized manner.

[0317] Initially, all modes (1226) are in a weak coherent state, which is mostly a vacuum state with a possibility of one or two photons and more than that count can be ignored. Subsequently, the photons pass through optical waveguides (1228), delay lines (1230), and two mode combiners (1232) that can be tuned to a classical control stack, which determine the quantum logical operations to be realized.

[0318] At the end of the optical paths, one or more modes are measured by photon-number separation (1234) or threshold detectors.

[0319] FIG. 12d is a hardware-level schematic diagram of operations realizing the logical operations shown in the circuit diagrams of FIG. 1 and FIG. 5, which can be executed on a neutral-atom-based quantum computer. In this type of hardware, unitary operators (e.g., those used to encode quantum feature maps and their derivatives) can be decomposed into two different types of operations: digital or analog. Both of these operations are transformations in quantum Hilbert space across atomic states.

[0320] Schematic diagram (a) of FIG. 12d illustrates a digital quantum circuit (1238), wherein local laser pulses can be used to individually address neutral atoms to realize transitions between atomic states, which effectively implements a set of standardized or 'digital' rotations on computational states. These digital gates may include any single-qubit rotations and controlled-Pauli-Z operations having any number of control qubits. Additionally, such digital gate operations may also include 2-qubit operations.

[0321] Schematic diagram (b) of FIG. 12d illustrates analog mode (1246) operation, where a global laser light pulse is applied simultaneously to a group of atoms or all atoms with specific characteristics such as detuning, Rabi frequencies, and Rydberg interactions to induce multi-qubit entanglement, thereby effectively deriving the evolution of the Hamiltonian (1244) of the atomic array in an analog manner. The combined quantum wave function evolves according to the Schrödinger equation, and in particular, unitary operators (Here Hamiltonian, t (referring to time) can be designed by pulse-shaping the parameterized coefficients of the Hamiltonian over time. In this way, a parameterized analog unitary block that entangles atoms and can operate as a fractional Anzatz, feature map, or other entanglement operation can be applied.

[0322] Digital and analog modes can be combined or switched, and a combination of the effects of each mode is derived. Schematic diagram (c) of FIG. 12d shows an example of such a digital-analog quantum circuit comprising blocks (1246_1-3) of digital qubit operations (single or multiple-qubit) and analog blocks (1248_1-3).

[0323] It can be proven that any computation can always be decomposed into a finite set of digital gates, including at least one multi-qubit digital gate (universality of sets of digital gates). This includes the ability to simulate general analog Hamiltonian evolutions by using trotterization or other simulation methods. However, trotterization is expensive, and decomposing multi-qubit Hamiltonian evolutions into digital gates is costly in terms of the number of operations required.

[0324] Digital-analog circuits define circuits that can be decomposed into both explicitly-digital and explicitly-analog operations. Internally, all operations are implemented as evolutions over controlled system Hamiltonians, but digital operations form a small set of pre-complied operations that are typical but not exclusive to single-qubits, whereas analog operations are used to evolve systems over natural Hamiltonians, for example, to achieve complex entanglement dynamics.

[0325] It can be shown that complex multi-qubit analog operations can be reproduced or simulated using only a relatively large number of digital gates, thus providing advantages for devices that achieve superior control of both digital and analog operations, such as neutral atomic quantum computers. In particular, considering the finite connectivity of purely digital devices, entanglement can propagate faster in terms of the actual elapsed time of a single analog block compared to a series of digital gates.

[0326] Furthermore, digital-analog quantum circuits for neutral quantum processors based on the Rydberg type of Hamiltonians can be analytically differentiated and can be used in variational and / or quantum machine learning schematics including the differential quantum circuit (DQC) schematics described in this application.

[0327] To transform the internal states of these modes, a classical control stack is used to transmit information to the optical components and lasers. The controller can formulate programmable unitary transformations in a parameterized manner.

[0328] At the end of the unitary transformation, the state of one or more atoms can be read by applying measurement laser pulses and then using a camera to observe the brightness to find which atomic qubit has changed to 'on' or 'off', 1 or 0. This bit information across the array is further processed according to the embodiments.

[0329] FIG. 12e is a hardware-level schematic diagram of operations realizing the logic operations shown in FIG. 1 and FIG. 5, embodied in a photon / optical quantum processor. The quantum model can be decomposed into a series of optical gate operations. These optical gate operations are transformations of photons within the quantum Hilbert space. To transform the internal states of these photons, a classical control stack is used to transmit information to a universal multiport interferometer. The controller can formulate programmable unitary transformations in a parameterized manner.

[0330] Initially, the photons (1255) are in Fock states, weak coherent states, or coherent states. The duration, type, intensity, and shape of the controlled-optical gate transformations determine the quantum logic operations (1256) that are realized.

[0331] At the end of the optical path, the modes are measured by photon number decomposition, width basis measurement (1257), tomography, or threshold detectors.

[0332] The hardware implementation examples shown in FIGS. 12a to 12e can also be used to implement the quantum circuits shown in FIGS. 13 to 16, where applicable.

[0333] FIG. 13a illustrates a system illustrating a digital-analog implementation example of a quantum algorithm. In particular, FIG. 13a illustrates a digital-analog implementation example in which qubit interactions can be turned on and off and single-qubit addressability exists for the application of single-qubit gates, which is the process described in (1300). To transform the internal states of these qubits, a classical control stack may be used to transmit information to a controller that affects one or more qubits. The controller may send multiple information carriers (e.g., pulses) in time and independently for each qubit. An initialization protocol is used to initialize the qubits, and in this example Initialize to state (1302). Although the current example shows only 4 qubits, the same principles can be extended in a clear manner to systems with more (or fewer) qubits than 4.

[0334] Subsequently, the quantum feature map is applied to (1304) to encode quantum information (e.g., input variables) into the Hilbert space associated with the quantum processor.

[0335] Following the application of the feature map, a variational Anzatz implemented by a variational quantum circuit (1306) is applied. In the present example, the variational Anzatz has different rotation angles Three single-qubit gates having - - It includes. These single-qubit gates are typically sets of standardized or 'digital' rotations on computational states applied to other qubits having different rotation angles / parameters. These digital gates are of rotations It includes arbitrary single-qubit rotations according to the argument. Since a single rotation / single-qubit gate can only rotate about a single axis, it is not sufficient to perform arbitrary rotations regardless of the parameter / angle. To complete a general rotation block, three gates connected in series with different rotation angles / parameters may be used, and in this example, are represented by a block of single-qubit gates (1306).

[0336] Subsequently, the entanglement in this digital-analog method is a block It is generated by the wave function evolution (1308) described by. During this evolution, the qubits interact with each other and allow the system to evolve during the time in which it is embodied. This process generates the entanglement required by the system. The combined quantum wave function is the Schrödinger equation, and in particular the unitary operators It evolves according to, and here is the Hamiltonian of the system (e.g., for neutral atoms, the Hamiltonian governing the system is the state occupancy of the Rydberg atoms) having Im) and, is the evolution time. In this way, a parametric analog unitary block that can entangle atoms and operate as a variational anzatz can be applied.

[0337] After the wave function evolution (1308), blocks of gates - - Similar to the process described in (1306), a different set of single-qubit gates can be applied.

[0338] Subsequently, the wave function can evolve once more, and finally, as described in (1310), a measurement in the computational basis occurs.

[0339] Optionally, additional steps indicated by ellipses (e.g., before, after, or between the displayed blocks) may be included. For example, before the application of the feature map. An initial state different from that may be prepared. Additionally, the blocks may be executed in a different order, and, for example, in some embodiments, block (1308) may precede block (1306). One or more of blocks (1304) through (1308) or variations thereof may be repeated one or more times prior to the measurement (1310).

[0340] Fig. 13b shows a quantum register A typical quantum feature map (1312) is shown, which begins by preparing a quantum state (1302) and then applying single-qubit rotations (1304) to all qubits at different rotation angles (typically). In this example, (here, Single-qubit rotations (1304) selected by) operate individually on each qubit and variables x It is parameterized by. For non-linear feature map encodings such as Chebyshev feature maps, the variable x It can be used as a rotation angle.

[0341] Applying a single rotation operation to each qubit in a register (1304) can be referred to as a single layer of rotation operations. A type of encoding classical information into a quantum state is known as angle encoding, which means that data vectors are represented by the angles of the quantum state. Angle encoding can be found in many quantum machine learning algorithms, and the main advantages are, respectively N having characteristics M To encode the input dataset solely n = log NM It means that it requires bits. Therefore, nConsidering an algorithm that is a polynomial for, it has a logarithmic runtime dependency with respect to the data size.

[0342] FIG. 13c illustrates different pulses (potentially having different pulse durations and / or amplitudes / heights) that can be transmitted to a quantum circuit (different qubits) to encode classical information. The rotation angle described by defines the duration and geometric structure of the pulse.

[0343] FIG. 14a presents a quantum circuit based on a digital-analog implementation example (1400) that lacks single-qubit addressing capability and uses wave function evolution that can be turned on and off for entanglement generation. FIG. 14a differs from FIG. 13a in that it shows a technique known as data re-uploading that alters quantum feature maps (1404 and 1410). Loading classical data into a quantum system is not a trivial task, and uploading large amounts of data makes it a much more critical issue. However, when big data cases are not considered, techniques such as data re-uploading can be utilized. As the name suggests, this involves uploading classical data multiple times and encoding them into single-qubit rotations through rotation angle / parameter encoding across the entire quantum circuit. This data re-uploading technique starts by setting the quantum register to an initial state (e.g., a state where all are zero (1402)), followed by angle encoding based on a feature map (1404), which in this case is a tower feature map.

[0344] After that, three single-qubit gates connected in series are used to perform a general rotation block operating as an Anzatz (1406), as previously explained above with reference to (1306). In this example, the encoding is - - rotations It is performed through variables. However, here, the type ( , , Any type of combined rotation of ) can be applied. In this example, since there is no single-qubit addressing capability, each and Gates at the same rotation angle It is applied to all qubits simultaneously. Subsequently, the wave function is allowed to evolve during the materialized time (1408).

[0345] Subsequently, the classical data is re-uploaded via angle encoding within single-qubit rotations (1410). However, each time the (same) data is uploaded, the information is encoded using different angles than those used in the previous data uploading steps. For example, the amount of rotation can be doubled in each data (re)uploading step (1404, 1410) (i.e., the rotation angle can be increased as 1-2-4, etc.).

[0346] Once again, rotations - - For each of these single-qubit gates, it is applied simultaneously to all qubits having the same rotation angle, but is different from (1406). Classical information can be repeatedly encoded into a quantum feature map through a data re-uploading technique, changing the single-qubit rotation angle (1404, 1410, etc.) each time after the wave function evolution. The resulting quantum feature map can be as expressive as a tower feature map by performing a serial data re-uploading process with different levels of rotation angles each time re-uploading is performed.

[0347] FIG. 14b illustrates another (and more realistic at the current level of technology) form of a digital-analog implementation example. In this implementation, the system has single-qubit addressing capability, but the wave function evolution cannot be turned off and is therefore always on. This system requires less control over qubit interactions (since they are always on) and requires only single-qubit addressing capability.

[0348] Single-qubit gates are applied while wave function evolution (qubit interaction) is also taking place. In the example shown, the quantum circuit (1418) is It starts by encoding information within a quantum state (1402), followed by a quantum feature map (1404) (similar to (1304)).

[0349] Subsequently, wave function evolution ( Variational Anzatz is applied, including the simultaneous application of blocks (1416) (similar to block (1306)) of single-qubit gates. After multiple repetitions of blocks (1416) having different rotation angles each time in the single-qubit gates, the circuit ends with a measurement in the computational base (1414).

[0350] FIG. 15a illustrates a quantum circuit (1500) implementing a digital-analog embodiment having single-qubit addressing capability, in which wave function evolution (e.g., atomic interactions) can be turned on and off. This quantum circuit is equivalent to the quantum circuit of FIG. 14a, and all other operations are the same as those described in (1400), whereas the time of wave function evolution (1508) (In a dimensionless coordinate system). The same Anzatz is used, which includes the same technique as data re-uploading using tower feature maps (1504, 1510) and single-qubit gates (1506, 1512) followed by wave function evolution (1508) (all similar to (1400)).

[0351] The reason these specific time periods are chosen for wave function evolution (1508) is that such evolution leads to the application of Controlled-Z (CZ) gates between pairs of qubits. The advantage of describing wave function evolution as a set of CZ gates is that this digital-analog embodiment of the quantum circuit resembles the structure of an equivalent digital quantum circuit, where the digital quantum circuit performs a set of CNOT gates between pairs of qubits included in the quantum circuit instead of the operation (1508). This allows for a clearer comparison between the digital and analog embodiments of these feature maps and enables the evaluation of their relative performance.

[0352] FIG. 15b illustrates a quantum circuit having entanglement generated through CZ gates between qubits, which is similar to (1500) and implements a digital-analog embodiment having single-qubit addressing capability (1518), but resembles atomic interaction / entanglement generation.

[0353] Figure 16 presents a workflow for generalized circuit differentiation. In quantum machine learning (QML) algorithms, various quantum feature maps and parametric quantum circuits (PQCs) are typically utilized to represent circuits required to solve specific problems. These quantum circuits contain tunable parameters that use classical optimizers to find optimal values ​​of the parameters present in the formulation. The differentiation required by most parametric optimization methods can be achieved in various ways, the most common being the Parameter Shift Rule (PSR). Examples of this type of differentiation for quantum feature maps are presented in Figures 11a through 11c.

[0354] One way to yield the analytic derivatives of quantum circuits is to measure the superpositions between quantum states. Analytic derivatives (e.g., an arbitrary Hermitian generator) generated by It can be used to differentiate the unitaries presented in ) (1604). However, to provide a less resource-intensive method for differentiation, the Parameter Shift Rule (PSR) was proposed. The PSR algorithm can provide analytic gradient estimates through measurements of expectation values ​​while shifting gate parameters to different values. The PSR algorithm is more commonly used to perform differentiation for QML algorithms; however, this is valid only for specific types of generators, namely involutory or idempotent generators. This is because the entire analytic derivative requires only measurements of two expectation values ​​(two eigenvalues). Although this simplifies the differentiation protocol, it also restricts it to working only for specific types of generators.

[0355] Thus, as improvements and generalizations of the PSR algorithm, a schematic diagram of the generalized parameter shift rule (GPSR) algorithm is shown in (1600). Such an approach allows for the differentiation of comprehensive quantum circuits having unitaries generated by operators with rich spectra (i.e., not limited to idempotent generators). The GPSR algorithm is based on spectral decomposition and shows the role of eigenvalue differences (spectral gaps) during differentiation with respect to the generator spectrum. Such an approach works for multiple non-degenerate gaps, in contrast to PSR, which is valid only for idempotent and idempotent operators with a single eigenvalue gap.

[0356] A workflow for generalized circuit differentiation can be described as follows: a function similar to (1600) including single- and multi-qubit gates (1602). f Creates a quantum circuit that encodes. Some adjustable parameters x parameterized by Select a unitary operator like (1604) and its constructor G The spectrum of is studied. Subsequently, using unique and positive spectral gaps, a system of equations containing spectral information can be generated, which can be produced along function expectations measured with shifted parameters and calculated parameter shifts. The solution of such a system is an arbitrary general generator G An analytical derivative of can be provided, and thus GPSR can be performed. GPSR is described in more detail in Kyriienko et al.’s ‘Generalized quantum circuit differentiation rules’, Phys. Rev. A 104 (052417), incorporated herein by reference.

[0357] FIG. 17a illustrates a system for determining a solution to a computational problem using a hybrid quantum computer system according to an embodiment. The system (1702) may include a quantum computer system (1704) comprising one or more quantum processors (1708) (e.g., a gate-based qubit quantum processor), and a controller system (1710) comprising input / output (I / O) devices forming an interface between the quantum processors and the outside world (e.g., one or more classical processors of a classical computer (1706)). For example, the controller system may include a system for generating control signals to control quantum processing elements. The control signals may include, for example, a series of pulses used to manipulate qubits (e.g., microwave pulses, voltage pulses, and / or optical pulses). Additionally, the controller may include output devices for reading qubits (e.g., read circuits) and control signals for reading quantum processing elements (e.g., read pulses for reading qubits). In some embodiments, at least a portion of such read circuits may be located on or integrated into a chip containing qubits.

[0358] The system may further include (pure classical information) inputs (1712) and (pure classical information) outputs (1714). Data processor systems may be configured to use a quantum computer to solve optimization problems. Input data may include information regarding the optimization problem to be solved. This information may include learning data such as input-output pairs, differential equations, boundary conditions, initial values, normalized values, etc. Input data may be used to classically compute values ​​(e.g., a series of pulses) that can be used by the system to construct quantum circuits, in particular quantum feature maps and / or parameterized quantum circuits, and to initialize and control qubit operations according to the quantum circuits. To this end, the classical computer may include a quantum circuit generator (1707). Input data may be used to classically compute values ​​(e.g., parameter settings) that can be used by the system to initialize quantum circuits implemented on a quantum processor. Similarly, the output data may include loss function values, sampling results, correlation operator expectations, optimization convergence results, optimized quantum circuit parameters and hyperparameters, and other classical data.

[0359] Each of one or more quantum processors may include a set of controllable quantum processing elements (e.g., a set of controllable 2-level systems referred to as qubits). The two levels and And, N - The wave function of a qubit quantum processor is of these (distinct) ground states It can be regarded as a superposition of complex values. However, embodiments of the present application are not limited to qubits but may include any multi-level quantum processing elements (e.g., qutrits) suitable for performing quantum computation. Examples of such quantum processors include noisy intermediate-scale quantum (NISQ) computing devices and fault-tolerant quantum computing (FTQC) devices.

[0360] A quantum processor can be configured to execute a quantum algorithm based on gate operations of a quantum circuit. The quantum processor can be implemented as a gate-based qubit quantum device, which initializes qubits to an initial state, enables interactions between qubits by sequentially applying quantum gates between other qubits, and allows measuring subsequent states of the qubits. To this end, input devices can be configured to configure the quantum processor to an initial state and control the gates that realize interactions between qubits. Similarly, output devices may include a read network for reading qubits that can be used to implement measurements of energy associated with the expected value of the system's Hamiltonian taken for a prepared state.

[0361] Additionally, one or more quantum processors may include a set of systems of continuous variables, such as optical or photonic quantum computers. Additionally, the quantum processor may include the Gaussian boson sampling device described above with reference to FIG. 12c or any of the other hardware described above with reference to FIG. 12a through 12e.

[0362] In some embodiments, the first data processor system may be implemented as a software program for simulating a quantum computer system (1704) including a quantum processor system (1708). Accordingly, in that case, the software program may be a classical software program that drives a classical computer (1706) so that quantum algorithms can be developed, executed, and tested on the classical computer without requiring access to a hardware implementation of the quantum processor system.

[0363] FIG. 17b is a block diagram illustrating an exemplary classical data processing system, e.g., a classical computer (1706), as described in the present disclosure. The classical data processing system (1720) may include at least one processor (1722) coupled to memory elements (1724) via a system bus (1726). Thus, the data processing system may store program code within the memory elements (1724). Additionally, the processor (1722) may execute program code accessed from the memory elements (1724) via the system bus (1726). In one aspect, the data processing system may be implemented as a computer suitable for storing and / or executing program code. However, it should be understood that the data processing system (1720) may be implemented in the form of any system including a processor and memory capable of performing the functions described herein.

[0364] The memory elements (1724) may include one or more physical memory devices, such as, for example, local memory (1728) and one or more bulk storage devices (1730). Local memory may refer to random access memory or other non-volatile memory device(s) typically used during the actual execution of program code. The bulk storage device may be implemented as a hard drive or other volatile data storage device. The classical data processing system (1720) may also include one or more cache memories (not shown) that provide temporary storage of at least some program code to reduce the number of times program code must be retrieved from the bulk storage device (1730) during execution.

[0365] Input / output (I / O) devices, denoted as a primary device (1732) and an output device (1734), may optionally be coupled to the data processing system. Examples of primary devices may include, but are not limited to, pointing devices such as a keyboard or mouse. Examples of output devices may include, but are not limited to, monitors or displays, speakers, or similar devices. The primary device and / or output device may be coupled to the data processing system directly or through arbitration I / O controllers. A network adapter (1736) may also be coupled to the data processing system to enable it to be coupled to other systems, computer systems, remote network devices and / or remote storage devices through arbitration private or public networks. The network adapter may include a data receiver for receiving data transmitted as data by said systems, devices and / or networks, and a data transmitter for transmitting data to said systems, devices and / or networks. Operating modems, cable operating modems, and Ethernet cards are examples of other types of network adapters that may be used with the classic data processing system (1720).

[0366] As visualized in FIG. 17b, memory elements (1724) can store an application (1738). It should be understood that the classical data processing system (1720) may additionally run an operating system (not shown) capable of facilitating the execution of the application. An application implemented in the form of executable program code may be executed by the classical data processing system (1720) (e.g., processor (1722)). In response to the execution of the application, the data processing system may be configured to perform one or more operations, which will be described in more detail in this specification.

[0367] In one aspect, for example, the classical data processing system (1720) may represent a client data processing system. In that case, the application (1738) may represent a client application configured to enable the classical data processing system (1720) to perform various functions described herein by reference to "client" when executed. Examples of clients may include, but are not limited to, personal computers, portable computers, mobile phones, and similar devices.

[0368] In another aspect, the data processing system may represent a server. For example, the data processing system may represent an (HTTP) server, in which case the application (1738) may be configured to perform (HTTP) server operations when executed. In yet another aspect, the data processing system may represent a module, unit, or function referred to in this specification.

[0369] The terms used herein are for the sole purpose of describing specific embodiments and are not intended to limit the invention. As used herein, the singular forms “a,” “an,” and “the” are intended to include plural forms as well, unless the context clearly indicates otherwise. It will be further understood that, as used herein, the terms “comprises” and / or “comprising” specify the presence of the stated characteristics, integers, steps, operations, elements, and / or components, but do not exclude the presence or addition of one or more other characteristics, integers, steps, operations, elements, components, and / or groups thereof.

[0370] The corresponding structures, materials, operations, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or operation for performing the function in combination with other specifically claimed elements.

[0371] The description of the embodiments is presented for illustration and illustrative purposes only and is not intended to limit or exhaust the invention in the disclosed form. Many modifications and variations will be obvious to those skilled in the art without departing from the scope and essence of the invention. The embodiments are selected and described to best illustrate the principles of the invention and practical applications, and to enable those skilled in the art to understand the invention in various embodiments with various modifications suitable for the specific use being considered.

Claims

Claim 1 A method for solving a differential problem using a hybrid computer system including a classical computer system and a quantum processor, wherein the differential problem defines a differential equation in a domain and boundary conditions on the boundary of the domain, and the method comprises: receiving or determining information regarding one or more quantum circuits that define operations of a parameterized quantum model of an execution function parameterized by variational parameters; receiving or determining an integral formula of the differential problem, wherein the integral formula defines an equation having a differential order smaller for at least one independent variable than the differential order of the differential equation for at least one independent variable, wherein the integral formula includes at least some of the boundary conditions, and wherein the integral formula includes an integral having a first integral limit corresponding to a point on the boundary and a second integral limit corresponding to a point within the domain, preferably a path integral. The method includes the step of determining a solution to the differential problem based on the parameterized quantum model and the loss function based on the integral formula of the differential problem, wherein the evaluation of the loss function for the point in the domain includes the calculation of the path integral, and the calculation of the path integral is based on one or more collocation points between the point on the boundary and the point in the domain, and the step of determining the solution includes the step of varying the variational parameters to determine a set of optimal parameters defining the solution of the differential equation, and the step of varying the variational parameters includes: - translating the operations of the one or more quantum circuits into control signals for reading and controlling the quantum elements of the quantum processor; - applying the operations of the one or more quantum circuits to the quantum elements based on the control signals;- A step of determining classical measurement data by estimating one or more expected values ​​based on a plurality of measurements of the above quantum elements; - A step of calculating a loss value based on the loss function and the above classical measurement data; and - A step of adjusting the variation parameters based on the loss value; comprising a method. Claim 2 A method according to claim 1, wherein the step of changing the variation parameters comprises one or more of the steps of determining or estimating the slope of the loss value for the variation parameters and adjusting the variation parameters based on the determined or estimated slope, determining a group of initial sets of variation parameters based on the associated loss values, and determining adjusted sets of variation parameters based on the initial sets of the associated loss values ​​and variation parameters. Claim 3 A method according to claim 1 or 2, wherein the step of calculating the loss value comprises the step of estimating the integral using at least one of a general Monte Carlo-based method, a quasi-Monte Carlo-based method, a quantum Monte Carlo-based method, a midpoint rule-based method, or a quadrature-based method. Claim 4 In any one of claims 1 to 3, the parameterized quantum model comprises: a quantum neural network; a quantum-kernel-based model; and / or is one of an amplitude-encoding-based model, wherein the quantum neural network preferably includes a variational circuit and a feature map for encoding independent variables, and preferably the function value of the trial function is associated with an observable quantity (e.g., expectation value) of the one or more quantum circuits, or the function value is associated with a functional of the output state of the one or more quantum circuits (e.g., superposition with a zero state, real component, or imaginary component), and preferably the evaluation of the quantum kernel includes determining the superposition between a first quantum state associated with a first input variable and a second quantum state associated with a second input variable, and preferably the superposition is determined based on a Hadamard test or a swap test, and preferably the one or more quantum circuits define operations for encoding primitive elements of the differential equation (e.g., terms such as f(x), f(x) / dx, g(x), etc.) within a plurality of quantum states of the quantum elements of the quantum processor, and each quantum state of the plurality of quantum states corresponds to a set of coefficients associated with the basis functions. Claim 5 A method according to any one of claims 1 to 4, wherein the differential equation is an ordinary differential equation, a partial differential equation, a stochastic differential equation, or a delayed differential equation, and the differential problem is an initial value problem. Claim 6 In any one of claims 1 to 5, the differential problem defines an additional differential equation, and the method further comprises the step of solving the additional differential equation using an additional method for solving the differential problems, and preferably, the additional method is a method based on differential quantum circuits, based on quantum kernels, or based on amplitude-encoding of a set of coefficients associated with basis functions representing an execution function. Claim 7 A method according to any one of claims 1 to 6, wherein the step of executing a quantum circuit by the quantum processor comprises translating the quantum circuit into a series of signals and using the series of signals to compute quantum elements in the quantum register of the quantum processor; and the step of receiving hardware measurement data comprises applying a read signal to the quantum elements of the quantum processor and measuring the quantum hardware measurement data in response to the read signal. Claim 8 A method according to any one of claims 1 to 7, wherein the hybrid quantum computer comprises a gate-based qubit device, a digital / analog quantum device, a neutral atom-based quantum device, an optimal qubit device, and / or a Gaussian boson sampling device. Claim 9 A method for solving a differential problem, wherein the differential problem defines a differential equation in a domain and boundary conditions on the boundary of the domain, and the method comprises the steps of: receiving or determining information about a parameterized model of an execution function parameterized by variational parameters; and receiving or determining an integral formula of the differential equation, wherein the integral formula defines an equation having a differential order smaller for at least one independent variable than the differential order of the differential equation for at least one independent variable, wherein the integral formula includes at least some of the boundary conditions, and wherein the integral formula includes a path integral having a first integral limit corresponding to a point on the boundary and a second integral limit corresponding to a point within the domain. A method comprising the step of determining the solution to the differential problem based on the parameterized model and the loss function associated with the integral formula of the differential problem, wherein the evaluation of the loss function for the point in the domain comprises the calculation of the integral, and the calculation of the integral is based on one or more collocation points between the point on the boundary and the point in the domain, and the step of determining the solution comprises the step of training a neural network to determine a set of optimal variational parameters, and the trained neural network represents the solution to the differential equation. Claim 10 In claim 9, the step of training the neural network comprises the step of varyingly optimizing the variational parameters, and the training step comprises: - evaluating the parameterized model of the trial function using first values ​​for the variational parameters; - calculating a loss value based on the evaluated parameterized model and the loss function; and - adjusting the variational parameters based on the loss value. Claim 11 A system for solving a differential problem, wherein the differential problem defines a differential equation in a domain and boundary conditions on the boundary of the domain, and the system includes a classical computer system and a special purpose processor, and the system receives or determines information about a parameterized model of an execution function parameterized by variational parameters; and receives or determines an integral formula for the differential problem, wherein the integral formula defines an equation having a differential order smaller for at least one independent variable than the differential order of the differential equation for at least one independent variable, the integral formula includes at least some of the boundary conditions, and the integral formula includes a path integral having a first integral limit corresponding to a point on the boundary and a second integral limit corresponding to a point within the domain. A system configured to perform the step of determining the solution to the differential problem based on the parameterized model and the loss function associated with the integral formula of the differential problem, wherein the evaluation of the loss function for the point in the domain includes the calculation of the integral, and the calculation of the integral is based on one or more collocation points between the point on the boundary and the point in the domain, and the step of determining the solution includes the step of training a neural network to determine a set of optimal variational parameters, and the trained neural network represents the solution to the differential equation. Claim 12 In claim 11, the special purpose processor is a quantum processor, and the step of changing the variation parameters comprises: - translating the operations of one or more quantum circuits into control signals for reading and controlling quantum elements of the quantum processor; - applying the operations of one or more quantum circuits to the quantum elements based on the control signals; - determining classical measurement data by estimating an expected value based on a plurality of measurements of the quantum elements; - calculating a loss value based on the loss function and the classical measurement data; and - adjusting the variation parameters based on the loss value. Claim 13 A system further configured to perform any one of the steps of any of the above-mentioned method claimed in any one of claims 2 through 10, in claim 11 or 12. Claim 14 A method comprising software code portions configured to execute the steps of the method according to any one of claims 1 to 10 when a computer program product is executed within the memory of the computer. Claim 15 A non-transient computer-readable storage medium storing at least one portion of software code, wherein the portion of software code is configured to perform the method claimed in any one of claims 1 through 10 when executed or processed by a computer.

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