Newton-type method for phase factor determination in quantum signal processing

Abstract

Systems and techniques that facilitate phase factor determination in quantum signal processing are provided. Various embodiments described herein comprise a system, which can comprise: a memory that can store computer executable components; and a processor, operably coupled to the memory, that can execute at least one of the computer executable components that can receive a first target real-valued function and a second target real-valued function that represent a target transformation on a quantum state; determine a system of non-linear equations based on the first target real-valued function and the second target real-valued function, the system of non-linear equations comprising a number of phase factors that define parameters of quantum operations; determine the phase factors using a modified Newton method to iteratively solve the system of non-linear equations; and configure the quantum processor to apply the phase factors to a quantum circuit to implement the target transformation on the quantum state.

Description

NEWTON-TYPE METHOD FOR PHASE FACTOR DETERMINATION IN QUANTUM SIGNAL PROCESSING BACKGROUND

[0001] The subject disclosure relates to quantum signal processing, and morespecifically, to a Newton-type method for phase factor determination in quantum signalprocessing. SUMMARY

[0002] The following presents a summary to provide a basic understanding of one ormore embodiments of the invention. This summary is not intended to identify key or criticalelements, or delineate any scope of the particular embodiments or any scope of the claims.Its sole purpose is to present concepts in a simplified form as a prelude to the more detailed description that is presented later. In one or more embodiments described herein, systems,computer-implemented methods, and / or computer program products that facilitate aNewton-type method for phase factor determination in quantum signal processing areprovided.

[0003] According to an embodiment, a system can comprise a memory that storescomputer executable components. The system can further comprise a processor, operably coupled to the memory, that can execute at least one of the computer executable components that can receive a first target real-valued function and a second target real-valued functionthat represent a target transformation on a quantum state within a quantum processor. In oneor more embodiments, the at least one of the computer executable components can furtherdetermine a system of non-linear equations based on the first target real-valued function and the second target real-valued function, the system of non-linear equations comprising anumber of phase factors that define parameters of quantum operations. In one or moreembodiments, the at least one of the computer executable components can further determinethe phase factors to implement the target transformation by using a modified newton method to iteratively solve the system of non-linear equations. In one or more embodiments, the at least one of the computer executable components can further configure the quantum processor to apply the phase factors to a quantum circuit to implement the target transformation on the quantum state.

[0004] According to another embodiment, a computer-implemented method cancomprise receiving, by a system operatively coupled to a processor, a first target real-valuedfunction and a second target real-valued function that represent a target transformation on aquantum state within a quantum processor. In one or more embodiments, the computer-implemented method can further comprise determining, by the system, a system of non-linear equations based on the first target real-valued function and the second target real- valued function, the system of non-linear equations comprising a number of phase factors that define parameters of quantum operations. In one or more embodiments, the computer- implemented method can further comprise determining, by the system, the phase factors to implement the target transformation by using a modified newton method to iteratively solve the system of non-linear equations. In one or more embodiments, the computer-implemented method can further comprise configuring, by the system, the quantumprocessor to apply the phase factors to a quantum circuit to implement the target transformation on the quantum state.

[0005] According to another embodiment, a computer program product for phasefactor determination in quantum signal processing comprising a computer readable storage medium having program instructions embodied therewith, the program instructionsexecutable by a processor to cause the processor to receive a first target real-valued functionand a second target real-valued function that represent a target transformation on a quantumstate within a quantum processor. In one or more embodiments, the program instructions canbe further executable by the processor to cause the processor to determine a system of non- linear equations based on the first target real-valued function and the second target real- valued function, the system of non-linear equations comprising a number of phase factorsthat define parameters of quantum operations. In one or more embodiments, the programinstructions can be further executable by the processor to cause the processor to determinethe phase factors to implement the target transformation by using a modified newton methodto iteratively solve the system of non-linear equations. In one or more embodiments, theprogram instructions can be further executable by the processor to cause the processor to configure the quantum processor to apply the phase factors to a quantum circuit to implement the target transformation on the quantum state. DESCRIPTION OF THE DRAWINGS

[0006] FIG. 1 illustrates a block diagram of example, non-limiting system that canfacilitate a Newton-type method for phase factor determination in quantum signal processingin accordance with one or more embodiments described herein.

[0007] FIG. 2 illustrates a block diagram of example, non-limiting system includingan approximation component that can facilitate a Newton-type method for phase factordetermination in quantum signal processing in accordance with one or more embodimentsdescribed herein.

[0008] FIG. 3 illustrates an example, non-limiting block diagram of determining asystem of non-linear equations based on a first target real-valued function and a secondtarget real-valued function in accordance with one or more embodiments described herein.

[0009] FIG. 4 illustrates an example, non-limiting block diagram of modes forapproximating target real-valued functions in accordance with one or more embodimentsdescribed herein.

[0010] FIG. 5 illustrates an example, non-limiting block diagram of determiningphase factors based on an accuracy level and mode in accordance with one or moreembodiments described herein.

[0011] FIG. 6 illustrates an example, non-limiting diagram of performance results ofa Newton-type method for phase factor determination in accordance with one or moreembodiments described herein.

[0012] FIGS. 7 illustrates an example, non-limiting diagram of performance resultsof a Newton-type method for phase factor determination in accordance with one or more embodiments described herein.

[0013] FIG. 8 illustrates an example, non-limiting diagram of performance results ofa Newton-type method for phase factor determination in accordance with one or more embodiments described herein.

[0014] FIG. 9 illustrates a flow diagram of an example, non-limiting method that canfacilitate a Newton-type method for phase factor determination in quantum signal processing in accordance with one or more embodiments described herein.

[0015] FIG. 10 illustrates a flow diagram of an example, non-limiting method thatcan facilitate a Newton-type method for phase factor determination in quantum signal processing in accordance with one or more embodiments described herein.

[0016] FIG. 11 illustrates a block diagram of an example, non-limiting operatingenvironment in which one or more embodiments described herein can be facilitated. DETAILED DESCRIPTION

[0017] The following detailed description is merely illustrative and is not intended tolimit embodiments and / or application or uses of embodiments. Furthermore, there is nointention to be bound by any expressed or implied information presented in the preceding Background or Summary sections, or in the Detailed Description section.

[0018] According to an embodiment, a system can comprise a memory that storescomputer executable components. The system can further comprise a processor, operably coupled to the memory, that can execute at least one of the computer executable componentsthat can receive a first target real-valued function and a second target real-valued functionthat represent a target transformation on a quantum state within a quantum processor. In oneor more embodiments, the at least one of the computer executable components can further determine a system of non-linear equations based on the first target real-valued function and the second target real-valued function, the system of non-linear equations comprising anumber of phase factors that define parameters of quantum operations. In one or moreembodiments, the at least one of the computer executable components can further determine the phase factors to implement the target transformation by using a modified newton methodto iteratively solve the system of non-linear equations. . In one or more embodiments, the atleast one of the computer executable components can further configure the quantum processor to apply the phase factors to a quantum circuit to implement the target transformation on the quantum state. Such embodiments of the system can provide anumber of advantages, including improving processing efficiency of phase factordetermination, improving robustness of phase factor determination, and improving convergence performance.

[0019] In one or more embodiments of the aforementioned system, the at least one ofthe computer executable components can further: approximate, based on a mode, the firsttarget real-valued function in a real or imaginary part of a first complex polynomial function, wherein the first complex polynomial function is of a first degree; approximate, based on themode, the second target real-valued function in a real or imaginary part of a second complexpolynomial function, wherein the second complex polynomial function is of a second degree; and configure the quantum processor to apply the phase factors based on the mode used for approximating the first target real-valued function and the second target real-valuedfunction. Such embodiments of the system can provide a number of advantages, includingimproving processing efficiency of phase factor determination and improving robustness of phase factor determination.

[0020] In one or more embodiments of the aforementioned system, the first targetreal-valued function or the second target real-valued function can correspond to polynomialinterpolations at Chebyshev points. Such embodiments of the system can provide a numberof advantages, including improving processing efficiency of phase factor determination.

[0021] In one or more embodiments of the aforementioned system, the phase factorscan represent the first target real-valued function and the second target real-valued function as the first complex polynomial function and the second complex polynomial function respectively if the first target function and the second target function are polynomial functions. Such embodiments of the system can provide a number of advantages, including improving processing efficiency of phase factor determination.

[0022] In one or more embodiments of the aforementioned system, the at least one ofthe computer executable components can further set a level of accuracy of the phase factorsfor solving the system of non-linear equations to implement the target transformation on thequantum state at the level of accuracy. Such embodiments of the system can provide anumber of advantages, including improving accuracy of phase factor determination andimproving convergence performance.

[0023] In one or more embodiments of the aforementioned system, the number ofphase factors can correspond to a maximum degree between the first degree and the second degree. Such embodiments of the system can provide a number of advantages, including improving processing efficiency of phase factor determination.

[0024] In one or more embodiments of the aforementioned system, iterativelysolving the system of non-linear equations to determine the phase factors to implement thetarget transformation on the quantum state can comprise using a quasi-Newton method.Such embodiments of the system can provide a number of advantages, including increasing processing efficiency of phase factor determination and improving convergence performance.

[0025] In one or more embodiments of the aforementioned system, iterativelysolving the system of non-linear equations to determine the phase factors to implement thetarget transformation on the quantum state can comprise using Newton’s method. Suchembodiments of the system can provide a number of advantages, including increasing processing efficiency of phase factor determination.

[0026] In one or more embodiments of the aforementioned system, the modifiedNewton method can comprise selecting an initial value of the phase factors, wherein a Jacobian matrix of the system of non-linear equations is orthogonal at the initial value. Such embodiments of the system can provide a number of advantages, including increasing processing efficiency of phase factor determination.

[0027] In one or more embodiments of the aforementioned system, the at least one ofthe computer executable components can further accelerate convergence using Aitken’s acceleration technique, wherein Aitken’s acceleration technique comprises: modifying a learning rate of iteratively solving the system of non-linear equations based on satisfaction of a criterion; and examining the criterion based on the level of accuracy. Such embodimentsof the system can provide a number of advantages, including improving convergenceperformance and increasing processing efficiency.

[0028] According to various embodiments, the above-described system can beimplemented as a computer-implemented method or as a computer program product.

[0029] One or more embodiments are now described with reference to the drawings,where like referenced numerals are used to refer to like elements throughout. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a more thorough understanding of the one or more embodiments. It is evident, however, in various cases, that the one or more embodiments can be practiced without these specific details.

[0030] Quantum Signal Processing (QSP) is a technique used in quantum computingfor applying polynomial transformations to quantum states, allowing approximation of atarget function with a complex polynomial. QSP has diverse applications in quantumcomputing such as fault tolerant quantum algorithms, quantum simulation, quantum machinelearning, and solving differential equations. QSP enables the efficient implementation of a desired transformation or function on quantum data by decomposing the operation into asequence of simpler quantum gates. In particular, QSP represents the complex polynomialsof a degree d by employing a product of 2x2 unitary matrices that are characterized by a setof (^+1) real numbers called phase factors. Phase factors, denoted by Φ, are criticalparameters that define specific operations performed by the quantum gates and are used to construct the unitary matrices that collectively implement the desired polynomial transformation on a quantum state.

[0031] However, despite a mathematical guarantee of the existence of the phasefactors Φ, efficiently and accurately determining the phase factors becomes increasinglydifficult as the degree ^ of the complex polynomial increases, complicating implementationof QSP in practical applications. That is, as the degree d of the complex polynomialincreases, the complexity of determining the phase factors grows exponentially, making the problem computationally intensive. The inherent nonlinearity and requirement for highprecision further exacerbates the difficulty of determining the phase factors, often leading tounstable or inaccurate results with existing methods. Existing techniques and algorithmsexhibit susceptibility to numerical instability, high computational costs, and poor scalabilityfor high-degree polynomials. These limitations can hinder practical implementations of QSPin quantum computing applications. Consequently, a more robust, efficient, and accuratemethod for phase factor determination can be desirable.

[0032] Furthermore, existing methods for phase factor determination in QSP, such asusing recursive matrix diagonalization algorithms, leverage linear algebra techniques.Although such existing methods offer a direct mathematical approach for phase factor determination, they can be highly numerically unstable, particularly for large polynomials orsystems sensitive to small perturbations (e.g., amplified numerical errors due to a recursivenature of the algorithm, leading to significant inaccuracies in the phase factors). Otherexisting methods, such as gradient descent methods, that address issues of numericalinstability are very limited in the context of phase factor determination in QSP due to thenature of optimization landscapes. More specifically, gradient descent methods are limited in phase factor determination due to the highly non-convex optimization landscape, which contains many local minima, saddle points, and flat regions that can trap the algorithm in suboptimal solutions. Moreover, such methods are sensitive to choices of hyperparameters, such as a learning rate, and can suffer from slow convergence or oscillation. Such limitations make gradient descent methods less reliable for determining phase factors in complex or high-degree polynomial cases where the optimization landscape is more challenging. Therefore, a numerically stable, robust, and accurate method for phase factor determination can be desirable.

[0033] In view of the problems discussed above, in relation to phase factordetermination in QSP, the present disclosure can be implemented to produce a solution toone or more of these problems by determining the phase factors using a modified Newton method. Newton’s method is an iterative numerical technique used to find successively better approximations of roots (or zeros) of a real-valued function by using the real-valuedfunction's derivative. The modified Newton method is a variant of Newton’s method thatrelies solely on a Jacobian matrix evaluated at an initial value for iteratively performing updates. By using the modified Newton method, the number of evaluations of the Jacobianmatrix can be reduced to one evaluation at the initial value, thereby improving processingefficiency for determining the phase factors. Therefore, determining the phase factors using amodified Newton method can improve the efficiency, robustness, and accuracy ofdetermining the phase factors in QSP.

[0034] The embodiments depicted in one or more figures described herein are forillustration only, and as such, the architecture of embodiments is not limited to the systems, devices and / or components depicted therein, nor to any particular order, connection and / or coupling of systems, devices and / or components depicted therein. For example, in one or more embodiments, the non-limiting systems described herein, such as non-limiting system 100 (e.g., system 100) as illustrated at FIG.1, and / or systems thereof, can further comprise, be associated with and / or be coupled to one or more computer and / or computing-based elements described herein with reference to an operating environment, such as the operating environment 1100 illustrated at FIG.11. For example, system 100 can be associated with, such as accessible via, a computing environment 1100 described below with reference to FIG.11, such that aspects of processing can be distributed between system 100 and the computing environment 1100. In one or more described embodiments, computer and / or computing-based elements can be used in connection with implementing one or more of the systems, devices, components and / or computer-implemented operations shown and / or described in connection with FIG.1 and / or with other figures described herein.

[0035] FIG. 1 illustrates block diagram of an example, non-limiting system 100 thatcan facilitate a Newton-type method for phase factor determination in quantum signalprocessing in accordance with one or more embodiments described herein. Aspects ofsystems (e.g., Newton method for phase factor determination system 102 and the like),apparatuses or processes in various embodiments of the present invention, can constitute oneor more machine-executable components embodied within one or more machines (e.g., embodied in one or more computer readable mediums (or media) associated with one or more machines). Such components, when executed by the one or more machines (e.g.,computers, computing devices, virtual machines, etc.), can cause the machines to performthe operations described.

[0036] Newton method for phase factor determination system 102 can compriseprocessor 104, memory 106, and phase factor determination component 101, the phase factordetermination component 101 comprising input component 110, determination component112, and / or evaluation component 114.

[0037] System 100 and / or the components of system 100 can be employed to usehardware and / or software to solve problems that are highly technical in nature (e.g., related to quantum computing, QSP, phase factor determination, etc.), that are not abstract and that cannot be performed as a set of mental acts by a human. Further, some of the processes performed may be performed by specialized computers for carrying out defined tasks relatedto phase factor determination in QSP. The system 100 and / or components of the system can be employed to solve new problems that arise through advancements in technologies mentioned above, quantum computing, and / or the like. The system 100 can provide technical improvements in terms of improving robustness and processing efficiency of phasefactor determination in QSP by using a modified Newton method, etc.

[0038] Discussion turns briefly to processor 104, memory 106 and bus 108 of system100. For example, in one or more embodiments, the system 100 can comprise processor 104(e.g., computer processing unit, microprocessor, classical processor, and / or like processor).In one or more embodiments, a component associated with system 100, as described herein with or without reference to the one or more figures of the one or more embodiments, can comprise one or more computer and / or machine readable, writable and / or executablecomponents and / or instructions that can be executed by processor 104 to enable performanceof one or more processes defined by such component(s) and / or instruction(s).

[0039] In one or more embodiments, system 100 can comprise a computer-readablememory (e.g., memory 106) that can be operably connected to the processor 104. Memory 106 can store computer-executable instructions that, upon execution by processor 104, cancause processor 104 and / or one or more other components of system 100 (e.g., phase factordetermination component 101, input component 110, determination component 112, evaluation component 114) to perform one or more actions. In one or more embodiments, memory 106 can store computer-executable components (e.g., phase factor determination component 101, input component 110, determination component 112, evaluation component 114).

[0040] System 100 and / or a component thereof as described herein, can becommunicatively, electrically, operatively, optically and / or otherwise coupled to one another via bus 108. Bus 108 can comprise one or more of a memory bus, memory controller, peripheral bus, external bus, local bus, and / or another type of bus that can employ one or more bus architectures. One or more of these examples of bus 108 can be employed. In one or more embodiments, system 100 can be coupled (e.g., communicatively, electrically, operatively, optically and / or like function) to one or more external systems (e.g., a non- illustrated electrical output production system, one or more output targets, an output target controller and / or the like), sources and / or devices (e.g., classical computing devices, communication devices and / or like devices), such as via a network. In one or more embodiments, one or more of the components of system 100 can reside in the cloud, and / or can reside locally in a local computing environment (e.g., at a specified location(s)).

[0041] As described above, in addition to the processor 104 and / or memory 106described above, system 100 can comprise one or more computer and / or machine readable, writable and / or executable components and / or instructions that, when executed by processor104, can enable performance of one or more operations defined by such component(s) and / orinstruction(s).

[0042] In various embodiments, input component 110 can receive a target real-valued function 116 and a target real-valued function 118, denoted by f and g respectively. Areal-valued function is a function that takes one or more real number inputs and maps themto a single real number as output. In various aspects, the target real-valued function 116 andthe target real-valued function 118 can represent target transformations or operations toimplement in QSP. Therefore, it can be desirable to approximate the target real-valuedfunction 116 and the target real-valued function 118 to achieve the target transformations oroperations in quantum computations.

[0043] In various cases, the target real-valued function 116can be an even or an oddfunction, and the target real-valued function 118 can be a corresponding odd or evenfunction. For example, if the target real-valued function 116is even, the target real-valuedfunction 118 can be odd. Conversely, if the target real-valued function 116is odd, the targetreal-valued function 118 can be even. In various instances, the target real-valued function116and the target real-valued function 118 can be any suitable electronic data (e.g., one or more scalars, one or more vectors, one or more matrices, one or more tensors, one or more character strings, or any suitable combination thereof) that indicates, specifies, or otherwise conveys a first target function and a second target function.

[0044] In various embodiments, the input component 110 can receive a degree 122,denoted by d. In various aspects, the degree 122 can represent a number of phase factors 124in QSP.

[0045] In various embodiments, the determination component 112 can, as describedherein, determine a system of non-linear equations based on the target real-valued function 116 and the target real-valued function 118. In various aspects, the system of non-linearequations can comprise a first complex polynomial function of a first degree and a secondpolynomial function of a second degree used to approximate the target real-valued function116 and the target real-valued function 118 respectively. The degree 122 can correspond toa maximum degree of the first complex polynomial function or the second complexpolynomial function.

[0046] QSP can refer to a matrix-valued function (e.g., a function whose output is amatrix) defined by the following equation:

[0048] The matrix-valued function can also be represented as the following equation:

[0049] denotes the firstcomplex polynomial function with parity-(d mod 2) and Q denotes the second complexpolynomial function with parity-(d-1 mod 2). In any case, it can be desirable to determinethe phase factors 124, denoted by Φ, to approximate the target real-valued function 116 and the target real-valued function 118.

[0050] In various embodiments, input component 110 can receive a mode 120. Invarious aspects, the evaluation component 114 can determine phase factors 124 based on themode 120. In various cases, the mode 120 can be any suitable electronic data (e.g., one or more scalars, one or more vectors, one or more matrices, one or more tensors, one or more character strings, or any suitable combination thereof). In various aspects, the mode 120 can specify how to combine the real or imaginary components of the first complex polynomial function and the second complex polynomial function within the system of non-linear equations to approximate the target real-valued function 116 and the target real-valued function 118.

[0051] In various instances, the system of non-linear equations can comprise a (d+1)-dimensional non-linear equation ^ = 0 that can be defined by the following equation:

[0052] ^, ^0, … , ^ where ^^^^^ = {^^} ^^^^ denotes a set of Chebyshev nodes (e.g., specific points usedin numerical interpolation, defined as the roots of Chebyshev polynomials, which are distributed to minimize interpolation error by clustering more densely at the interval'sendpoints). A solution to the system of non-linear equations satisfies the equations ^^[^] =^(^) and ^^[^] = ^(^).

[0053] Note that, as described herein, in the non-linear equation F, the target real-valued function 116 and the target real-valued function 118 are approximated in a real partof the first complex polynomial function P and in a real part of the second complexpolynomial function Q. However, the target real-valued function 116 and the target real- valued function 118 can be approximated in a real (e.g., denoted by Re) or imaginary part(e.g., denoted by Im) of the first complex polynomial function P and in a real part of thesecond complex polynomial function Q, as specified by the mode 120. Various aspects are described with respect to FIG.4. In any case

[0054] In various embodiments, the evaluation component 114 can, as describedherein, iteratively solve the system of non-linear equations to determine the phase factors124 using a modified Newton method (e.g., simplified Newton’s method). In variousinstances, the iteration process based on the modified Newton method can be defined by

[0055] In various cases, the evaluation component 114 can select an initial value ofthe phase factors 224 (e.g., phase factors ^(^)) in the modified Newton’s method.Specifically, the evaluation component 114 can select the initial value of the phase factors224 such that an inverse of a Jacobian matrix is also a transpose of the Jacobian matrix at theinitial value. In other words, at the initial value, the Jacobian matrix is orthogonal. Suchinitial value can improve convergence efficiency for determining the phase factors 124 byreducing bottlenecks of the modified Newton’s method.

[0056] In various aspects, the evaluation component 114 can select the initial valueof the phase factors 224 to be ^(^) = ^^ ^, 0, … , 0^. By selecting this initial value of the phasefactors 224, it can be shown that the Jacobian matrix ^^ ≔ ^′(^^) possesses a propertydefined by:

[0059] In various embodiments, the evaluation component 114 can compute amatrix-vector multiplication ^^^ using Fast Cosine Transform (FCT) and Fast SineTransform (FST). By using FCT and FST to compute vector multiplications, acomputational cost of ^(^ log(^)) can be achieved. Conversely, existing methods typicallyhave a computational cost of ^(^^), caused by bottlenecks that arise during matrix inversion. Thus, by selecting the initial value of the phase factors 124 such that the Jacobian matrix is orthogonal at the initial value, and by using FCT and FST for computation,efficiency of computation can be increased and significantly reduce bottlenecks of ModifiedNewton’s method.

[0060] In any instance, the evaluation component 114 can iteratively solve thesystem of non-linear equations using the modified Newton method with the initial value ofthe phase factors 124 to determine the phase factors 124 based on the mode 120.

[0061] In various embodiments, the evaluation component 114 can alternativelyutilize a quasi-Newton method (Broyden’s method) or the exact Newton method to determine phase factors 124. The quasi-Newton method is an approximate version of the Newton method. This means that the accurate Jacobian matrix at the initial step, which is crucial to the convergence of Broyden’s method, can be efficiently prepared with the selection of the initial value as described supra. By employing such accurately computedJacobian matrix and applying Broyden's update rule, a higher convergence performance canbe attained through the iterative process, which achieves ^(^^) complexity. In otherinstances, the exact Newton method can be used, wherein the exact Newton methodevaluates the Jacobian matrix with a complexity of ^(^^) at each iteration.

[0062] FIG. 2 illustrates a block diagram of example, non-limiting system 200including a modification component that can facilitate a Newton-type method for phasefactor determination in quantum signal processing in accordance with one or moreembodiments described herein. As shown, the system 200 can, in some cases, comprise the same components as the system 100, and can further comprise a modification component 202.

[0063] In various embodiments, the modification component 202 can employAitken’s acceleration technique (e.g., Aitken's delta-squared process) in the modifiedNewton’s method to determine the phase factors 124. Aitken’s acceleration technique is aseries acceleration method used for accelerating a rate of convergence of a sequence. Byemploying Aitken’s acceleration technique in the modified Newton’s method, improvedconvergence performance can be achieved. In particular, the modification component 202can employ Aitken’s acceleration technique to modify a learning rate of iteratively solvingthe system of non-linear equations based on satisfaction of a criterion. Given a sequence ^^, Aitken's acceleration technique generates a new sequence ^^^, which converges faster thanthe sequence ^^. Aitken’s acceleration technique uses the following formula to compute thenew sequence:

[0065] To achieve acceleration of convergence in phase factor determination in QSP,the formula can be adapted to the following equation:

[0067] In various aspects, the modification component 202 can examine a criterion ateach iteration in the iteration process based on the modified Newton method for determining the phase factors 124. In response to satisfaction of the criterion, the modificationcomponent 202 can modify the learning rate. In various embodiments, the modificationcomponent 202 can modify the learning rate by adding a gradient scaled by a constant to thecurrent phase factors (e.g., phase factors ^(^)). The learning rate can control a rate ofdescending in the energy landscape. Thus, the modification can effectively adjust thelearning rate to allow quicker convergence.

[0068] FIG. 3 illustrates an example, non-limiting block diagram 300 of determininga system of non-linear equations based on a first target real-valued function and a second target real-valued function in accordance with one or more embodiments described herein.

[0069] In various embodiments, the determination component 112 can receive thetarget real-valued function 116, the target real-valued function 118, and the degree 122. Invarious aspects, the determination component 112 can determine a system on non-linearequations based on the target real-valued function 116, the target real-valued function 118,and the degree 122. Specifically, the system of non-linear equations can comprise complexpolynomial function 308 and complex polynomial function 310, denoted by P and Qrespectively. In various aspects, the complex polynomial function 308 can approximate the target real-valued function 116 in a real or imaginary part of the complex polynomial function 308. Similarly, the complex polynomial function 310 can approximate the target real-valued function 118 in a real or imaginary part of the complex polynomial function 310.As previously described, the solution to the non-linear equation ^ = 0 satisfies the equations^^[^] = P(^) and ^^[^] = P(^) (e.g., or ^^[^] = P(^) ^^^ ^^[^] = P(^), ^^[^] =P(^) and ^^[^] = P(^), or ^^[^] = P(^) and ^^[^] = P(^) based on the mode 120). Inany case, the projection, denoted by P, represents a polynomial interpolation at ^^^^^, mapping from a space of functions on [0,1] to a space of polynomials with degrees less than d.

[0070] FIG. 4 illustrates an example, non-limiting block diagram 400 of modes forapproximating target real-valued functions in accordance with one or more embodiments described herein.

[0071] In various embodiments, the input component 110 can receive the mode 120.In various instances, the mode 120 can specify how to combine the real or imaginarycomponents of the first complex polynomial function and the second complex polynomial function within the system of non-linear equations to approximate the target real-valued function 116 and the target real-valued function 118.

[0072] For example, the mode 120 can be defined by mode 120(1). Mode 120(1)approximates the target real-valued function 116 in a real part of the first complexpolynomial function P and approximates the target real-valued function 118 in a real part ofthe second complex polynomial function Q. As another example, the mode 120 can be defined by mode 120(2). Mode 120(2) approximates the target real-valued function 116 in areal part of the first complex polynomial function P and approximates the target real-valuedfunction 118 in an imaginary part of the second complex polynomial function Q. As still another example, the mode 120 can be defined by mode 120(3). Mode 120(3) approximates the target real-valued function 116 in an imaginary part of the first complex polynomialfunction P and approximates the target real-valued function 118 in a real part of the secondcomplex polynomial function Q. As even another example, the mode 120 can be defined bymode 120(4). Mode 120(4) approximates the target real-valued function 116 in an imaginarypart of the first complex polynomial function P and approximates the target real-valuedfunction 118 in an imaginary part of the second complex polynomial function Q.

[0073] FIG. 5 illustrates an example, non-limiting block diagram 500 of determiningphase factors based on an accuracy level and mode in accordance with one or more embodiments described herein.

[0074] In various embodiments, the evaluation component 114 can receive the mode120 and an accuracy level 502. In various aspects, the evaluation component 114 candetermine the phase factors 124 based on the mode 120 and the accuracy level 502. That is,based on the accuracy level 502, the evaluation component 114 can engage the modificationcomponent 202 to modify the learning rate to accelerate convergence. Modification of thelearning rate can be executed based on satisfaction of a criterion. In various instances, thecriterion can be based on the accuracy level 502, denoted by ^. Specifically, the criterion canbe defined by ∥∥< ^ to increase convergence efficiencywithin the accuracy level 502. In response to satisfaction of the criterion, the modificationcomponent 202 can modify the current phase factors (e.g., phase factorsas ^(^)= +^(^)^^(^). In various aspects, the modification component 202 can evaluate thecriterion at each iteration in the iteration process of the modified Newton’s method. As aresult, the evaluation component 114 can receive the modified phase factors for the nextiteration. In other instances, in response to not satisfying the criterion, the modificationcomponent 202 can refrain from modifying the current phase factors. As a result, the evaluation component 114 can utilize the unmodified phase factors for the next iteration.

[0075] In various embodiments, after convergence has been achieved, the evaluationcomponent 114 can determine the phase factors 124 corresponding to the mode 120. For instance, the phase factors 124 can be denoted by. In various cases, ifthe mode 120 is defined by mode 120(1), the evaluation component 114 can output the phasefactors 124 as ^ = ^(^). In other cases, if the mode 120 is defined by mode 120(2), theevaluation component 114 can output the phase factors 124 as ^ = [^^ +−^^]. In still other cases, if the mode 120 is defined by mode 120(3), the evaluation component 114 can output the phase factors 124 as ^ = [^^ +^ ^, ^^, … , ^^^^, ^^ +^ ^]. In yet other cases, if the mode 120 is defined by mode 120(4), the evaluation component 114 can output the phase factors 124 as ^ = [^^ +^ ^, ^^^^, … , ^^].

[0076] FIG. 6 illustrates an example, non-limiting diagram 600 of performanceresults of a Newton-type method for phase factor determination in accordance with one or more embodiments described herein.

[0077] Depicted in non-limiting diagram 600 are performance results of phase factordetermination in QSP using the modified Newton’s method against existing methods. As shown, the modified Newton’s method exhibits a significant improvement in processingefficiency to determine the phase factors 124 for target function 602, defined by ^(^) =cos (^^). In particular, the modified Newton’s method performs 100 times faster than theexisting methods. Furthermore, the modified Newton’s method also achieves an accuracy of ^= 10^^^.

[0078] FIGS. 7 illustrates an example, non-limiting diagram 700 of performanceresults of a Newton-type method for phase factor determination in accordance with one or more embodiments described herein.

[0079] Depicted in non-limiting diagram 600 are performance results of phase factordetermination in QSP using the modified Newton’s method against existing methods. As shown, the modified Newton’s method exhibits a significant improvement in processing efficiency to determine the phase factors 124 for target function 602, defined by ^(^) =particular, the modified Newton’s method performs 100 times fasterthe existing methods. Furthermore, the modified Newton’s method also achieves an accuracyof ^ = 10^^^.

[0080] FIG. 8 illustrates an example, non-limiting diagram 800 of performanceresults of a Newton-type method for phase factor determination in accordance with one or more embodiments described herein.

[0081] Depicted in non-limiting diagram 600 are performance results of phase factordetermination in QSP using the modified Newton’s method against existing methods. As shown, the modified Newton’s method exhibits a significant improvement in processingefficiency to determine the phase factors 124 for target function 602, defined by ^(^) =^In particular, the modified Newton’s method performs 100 times faster than the existingmethods. Furthermore, the modified Newton’s method also achieves an accuracy of ^ =10^^^.

[0082] FIG. 9 illustrates a flow diagram of an example, non-limiting, computerimplemented method 900 that facilitates a Newton-type method for phase factordetermination in quantum signal processing in accordance with one or more embodimentsdescribed herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity.

[0083] At 902, non-limiting method 900 can include receiving, by a system (e.g.,Newton method for phase factor determination system 102 and / or input component 110)operatively coupled to a processor (e.g., 104), a first target real-valued function and a secondtarget real-valued function that represent a target transformation on a quantum state within a quantum processor.

[0084] At 904, non-limiting method 900 can include determining, by the system(e.g., determination component 112), a system of non-linear equations based on the firsttarget real-valued function and the second target real-valued function, the system of non-linear equations comprising a number of phase factors that define parameters of quantumoperations.

[0085] At 906, non-limiting method 900 can include determining, by the system(e.g., evaluation component 114), the phase factors to implement the target transformation by using a modified newton method to iteratively solve the system of non-linear equations.

[0086] At 908, non-limiting method 900 can include configuring, by the system, thequantum processor to apply the phase factors to a quantum circuit to implement the target transformation on the quantum state.

[0087] FIG. 10 illustrates a flow diagram of an example, non-limiting, computerimplemented method 1000 that facilitates a Newton-type method for phase factordetermination in quantum signal processing in accordance with one or more embodimentsdescribed herein. Repetitive description of like elements employed in other embodiments described herein is omitted for sake of brevity.

[0088] At 1002, non-limiting method 1000 can include selecting, by a system (e.g.,Newton method for phase factor determination system 102 and / or determination component112), an initial value of the phase factors, wherein a Jacobian matrix of the system of non- linear equations is orthogonal at the initial value.

[0089] At 1004, non-limiting method 1000 can include examining, by the system(e.g., modification component 202), a criterion based on a level of accuracy.

[0090] At 1006, non-limiting method 1000 can include determining, by the system(e.g., modification component 202), if the criterion is satisfied. If yes (e.g., the criterion is satisfied), the non-limiting method 1000 can proceed to 1010. If no (e.g., the criterion is not satisfied), the non-limiting method 1000 can proceed to 1008.

[0091] At 1008, non-limiting method 1000 can include modifying, by the system(e.g., modification component 202), the phase factors.

[0092] At 1010, non-limiting method 1000 can include determining, by the system(e.g., evaluation component 114), if convergence has been achieved. If yes (e.g., convergence has been achieved), the non-limiting method 1000 can proceed to 1014. If no (e.g., convergence has not been achieved), the non-limiting method 1000 can proceed to 1012.

[0093] At 1012, non-limiting method 1000 can include executing, by the system(e.g., evaluation component 114), a next iteration in solving the system of non-linear equations.

[0094] At 1014, non-limiting method 1000 can include outputting, by the system(e.g., evaluation component 114), the phase factors.

[0095] The Newton method for phase factor determination system 102 can providetechnical improvements to a processing unit associated with Newton method for phase factordetermination system 102. For example, by utilizing a Aitken’s acceleration technique tomodify a learning rate can exhibit improved convergence performance and increasedprocessing efficiency, thereby reducing the workload of a processing unit (e.g., processor 104). In this example, by reducing the workload of such a processing unit (e.g., processor104), Newton method for phase factor determination system 102 can thereby facilitateimproved performance, improved efficiency, and / or reduced computational cost associated with such a processing unit. Further, by selecting an initial state of the phase factors suchthat the Jacobian matrix is orthogonal, the amount of computation resources utilized byNewton method for phase factor determination system 102 is reduced by decreasing thenumber of evaluations in the modified Newton method, thereby reducing or removing the additional workload the processing unit. Newton method for phase factor determinationsystem 102 can thereby facilitate improved performance, improved efficiency, and / orreduced computational cost associated with phase factor determination in QSP.

[0096] A practical application of the Newton method for phase factor determinationsystem 102 is that it allows for determination of phase factors in QSP with increasedefficiency, accuracy, and robustness by utilizing a reduced amount of computing resources,in comparison to other methods. For example, phase factor determination is non-trivial, andexisting methods are susceptible to numerical instability, high computational costs, and poorscalability for high-degree polynomials. By using a modified Newton method for phasefactor determination, Newton method for phase factor determination system 102 can enablephase factor determination in QSP with improved scalability, decreased computationrequirements, and improved processing efficiency. Therefore, Newton method for phasefactor determination system 102 can enable phase factor determination in QSP that can beoperated with reduced hardware requirements and computation resources, thus promotingefficient quantum computation.

[0097] It is to be appreciated that the Newton method for phase factor determinationsystem 102 can utilize various combination of electrical components, mechanicalcomponents, and circuity that cannot be replicated in the mind of a human or performed by a human as the various operations that can be executed by Newton method for phase factordetermination system 102 and / or components thereof as described herein are operations thatare greater than the capability of a human mind. For instance, the amount of data processed, the speed of processing such data, or the types of data processed by Newton method forphase factor determination system 102 over a certain period of time can be greater, faster, ordifferent than the amount, speed, or data type that can be processed by a human mind over the same period of time. According to several embodiments, Newton method for phasefactor determination system 102 can also be fully operational towards performing one ormore other functions (e.g., fully powered on, fully executed, and / or another function) while also performing the various operations described herein. It should be appreciated that such simultaneous multi-operational execution is beyond the capability of a human mind. Itshould be appreciated that Newton method for phase factor determination system 102 caninclude information that is impossible to obtain manually by an entity, such as a human user. For example, the type, amount, and / or variety of information included in Newton method forphase factor determination system 102 can be more complex than information obtainedmanually by an entity, such as a human user.

[0098] FIG. 11 illustrates a block diagram of an example, non-limiting operatingenvironment 1100 in which one or more embodiments described herein can be facilitated.FIG. 11 and the following discussion are intended to provide a general description of asuitable operating environment 1100 in which one or more embodiments described herein atFIGs. 1-10 can be implemented.

[0099] Various aspects of the present disclosure are described by narrative text,flowcharts, block diagrams of computer systems and / or block diagrams of the machine logic included in computer program product (CPP) embodiments. With respect to any flowcharts, depending upon the technology involved, the operations can be performed in a different order than what is shown in a given flowchart. For example, again depending upon the technology involved, two operations shown in successive flowchart blocks may be performed in reverse order, as a single integrated step, concurrently, or in a manner at least partially overlapping in time.

[0100] A computer program product embodiment ("CPP embodiment" or “CPP”) is aterm used in the present disclosure to describe any set of one, or more, storage media (also called "mediums") collectively included in a set of one, or more, storage devices that collectively include machine readable code corresponding to instructions and / or data for performing computer operations specified in a given CPP claim. A "storage device" is any tangible device that can retain and store instructions for use by a computer processor. Without limitation, the computer readable storage medium may be an electronic storage medium, a magnetic storage medium, an optical storage medium, an electromagnetic storagemedium, a semiconductor storage medium, a mechanical storage medium, or any suitablecombination of the foregoing. Some known types of storage devices that include these mediums include: diskette, hard disk, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or Flash memory), static random access memory (SRAM), compact disc read-only memory (CD-ROM), digital versatile disk (DVD), memory stick, floppy disk, mechanically encoded device (such as punch cards or pits / lands formed in a major surface of a disc) or any suitable combination of the foregoing. A computer readable storage medium, as that term is used in the presentdisclosure, is not to be construed as storage in the form of transitory signals per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic wavespropagating through a waveguide, light pulses passing through a fiber optic cable, electricalsignals communicated through a wire, and / or other transmission media. As will be understood by those of skill in the art, data is typically moved at some occasional points in time during normal operations of a storage device, such as during access, de-fragmentation or garbage collection, but this does not render the storage device as transitory because the data is not transitory while it is stored.

[0101] Computing environment 1100 contains an example of an environment for theexecution of at least some of the computer code involved in performing the inventivemethods, such as modified newton method for phase factor determination code 1145. Inaddition to block 1145, computing environment 1100 includes, for example, computer 1101, wide area network (WAN) 1102, end user device (EUD) 1103, remote server 1104, public cloud 1105, and private cloud 1106. In this embodiment, computer 1101 includes processor set 1110 (including processing circuitry 1120 and cache 1121), communication fabric 1111, volatile memory 1112, persistent storage 1113 (including operating system 1122 and block 1145, as identified above), peripheral device set 1114 (including user interface (UI), device set 1125, storage 1124, and Internet of Things (IoT) sensor set 1125), and network module 1115. Remote server 1104 includes remote database 1130. Public cloud 1105 includes gateway 1140, cloud orchestration module 1141, host physical machine set 1142, virtual machine set 1143, and container set 1144.

[0102] COMPUTER 1101 may take the form of a desktop computer, laptopcomputer, tablet computer, smart phone, smart watch or other wearable computer, mainframe computer, quantum computer or any other form of computer or mobile device now known or to be developed in the future that is capable of running a program, accessing a network or querying a database, such as remote database 1130. As is well understood in the art of computer technology, and depending upon the technology, performance of a computer- implemented method may be distributed among multiple computers and / or between multiple locations. On the other hand, in this presentation of computing environment 1100, detailed discussion is focused on a single computer, specifically computer 1101, to keep the presentation as simple as possible. Computer 1101 may be located in a cloud, even though it is not shown in a cloud in Figure 11. On the other hand, computer 1101 is not required to be in a cloud except to any extent as may be affirmatively indicated.

[0103] PROCESSOR SET 1110 includes one, or more, computer processors of anytype now known or to be developed in the future. Processing circuitry 1120 may be distributed over multiple packages, for example, multiple, coordinated integrated circuit chips. Processing circuitry 1120 may implement multiple processor threads and / or multiple processor cores. Cache 1121 is memory that is located in the processor chip package(s) and is typically used for data or code that should be available for rapid access by the threads or cores running on processor set 1110. Cache memories are typically organized into multiple levels depending upon relative proximity to the processing circuitry. Alternatively, some, or all, of the cache for the processor set may be located “off chip.” In some computing environments, processor set 1110 may be designed for working with qubits and performing quantum computing.

[0104] Computer readable program instructions are typically loaded onto computer1101 to cause a series of operational steps to be performed by processor set 1110 of computer 1101 and thereby effect a computer-implemented method, such that the instructions thus executed will instantiate the methods specified in flowcharts and / or narrative descriptions of computer-implemented methods included in this document (collectively referred to as “the inventive methods”). These computer readable program instructions are stored in various types of computer readable storage media, such as cache 1121 and the other storage media discussed below. The program instructions, and associated data, are accessed by processor set 1110 to control and direct performance of the inventive methods. In computing environment 1100, at least some of the instructions for performingthe inventive methods may be stored in block 1145 in persistent storage 1113.

[0105] COMMUNICATION FABRIC 1111 is the signal conduction paths that allowthe various components of computer 1101 to communicate with each other. Typically, this fabric is made of switches and electrically conductive paths, such as the switches and electrically conductive paths that make up busses, bridges, physical input / output ports and the like. Other types of signal communication paths may be used, such as fiber optic communication paths and / or wireless communication paths.

[0106] VOLATILE MEMORY 1112 is any type of volatile memory now known orto be developed in the future. Examples include dynamic type random access memory (RAM) or static type RAM. Typically, the volatile memory is characterized by random access, but this is not required unless affirmatively indicated. In computer 1101, the volatile memory 1112 is located in a single package and is internal to computer 1101, but,alternatively or additionally, the volatile memory may be distributed over multiple packages and / or located externally with respect to computer 1101.

[0107] PERSISTENT STORAGE 1113 is any form of non-volatile storage forcomputers that is now known or to be developed in the future. The non-volatility of this storage means that the stored data is maintained regardless of whether power is being supplied to computer 1101 and / or directly to persistent storage 1113. Persistent storage 1113 may be a read only memory (ROM), but typically at least a portion of the persistent storage allows writing of data, deletion of data and re-writing of data. Some familiar forms of persistent storage include magnetic disks and solid state storage devices. Operating system 1122 may take several forms, such as various known proprietary operating systems or open source Portable Operating System Interface type operating systems that employ akernel. The code included in block 1145 typically includes at least some of the computercode involved in performing the inventive methods.

[0108] PERIPHERAL DEVICE SET 1114 includes the set of peripheral devices ofcomputer 1101. Data communication connections between the peripheral devices and the other components of computer 1101 may be implemented in various ways, such as Bluetooth connections, Near-Field Communication (NFC) connections, connections made by cables (such as universal serial bus (USB) type cables), insertion type connections (for example, secure digital (SD) card), connections made though local area communication networks and even connections made through wide area networks such as the internet. In various embodiments, UI device set 1125 may include components such as a display screen, speaker, microphone, wearable devices (such as goggles and smart watches), keyboard, mouse, printer, touchpad, game controllers, and haptic devices. Storage 1124 is external storage, such as an external hard drive, or insertable storage, such as an SD card. Storage 1124 may be persistent and / or volatile. In some embodiments, storage 1124 may take the form of a quantum computing storage device for storing data in the form of qubits. In embodiments where computer 1101 is required to have a large amount of storage (for example, where computer 1101 locally stores and manages a large database) then this storage may be provided by peripheral storage devices designed for storing very large amounts of data, such as a storage area network (SAN) that is shared by multiple, geographically distributed computers. IoT sensor set 1125 is made up of sensors that can be used in Internet of Things applications. For example, one sensor may be a thermometer and another sensor may be a motion detector.

[0109] NETWORK MODULE 1115 is the collection of computer software,hardware, and firmware that allows computer 1101 to communicate with other computers through WAN 1102. Network module 1115 may include hardware, such as modems or Wi- Fi signal transceivers, software for packetizing and / or de-packetizing data for communication network transmission, and / or web browser software for communicating data over the internet. In some embodiments, network control functions and network forwarding functions of network module 1115 are performed on the same physical hardware device. In other embodiments (for example, embodiments that utilize software-defined networking (SDN)), the control functions and the forwarding functions of network module 1115 are performed on physically separate devices, such that the control functions manage several different network hardware devices. Computer readable program instructions for performing the inventive methods can typically be downloaded to computer 1101 from an external computer or external storage device through a network adapter card or network interface included in network module 1115.

[0110] WAN 1102 is any wide area network (for example, the internet) capable ofcommunicating computer data over non-local distances by any technology for communicating computer data, now known or to be developed in the future. In some embodiments, the WAN may be replaced and / or supplemented by local area networks (LANs) designed to communicate data between devices located in a local area, such as a Wi- Fi network. The WAN and / or LANs typically include computer hardware such as copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers and edge servers.

[0111] END USER DEVICE (EUD) 1103 is any computer system that is used andcontrolled by an end user (for example, a customer of an enterprise that operates computer 1101), and may take any of the forms discussed above in connection with computer 1101. EUD 1103 typically receives helpful and useful data from the operations of computer 1101. For example, in a hypothetical case where computer 1101 is designed to provide a recommendation to an end user, this recommendation would typically be communicated from network module 1115 of computer 1101 through WAN 1102 to EUD 1103. In this way, EUD 1103 can display, or otherwise present, the recommendation to an end user. In some embodiments, EUD 1103 may be a client device, such as thin client, heavy client, mainframe computer, desktop computer and so on.

[0112] REMOTE SERVER 1104 is any computer system that serves at least somedata and / or functionality to computer 1101. Remote server 1104 may be controlled and usedby the same entity that operates computer 1101. Remote server 1104 represents the machine(s) that collect and store helpful and useful data for use by other computers, such as computer 1101. For example, in a hypothetical case where computer 1101 is designed and programmed to provide a recommendation based on historical data, then this historical data may be provided to computer 1101 from remote database 1130 of remote server 1104.

[0113] PUBLIC CLOUD 1105 is any computer system available for use by multipleentities that provides on-demand availability of computer system resources and / or other computer capabilities, especially data storage (cloud storage) and computing power, without direct active management by the user. Cloud computing typically leverages sharing of resources to achieve coherence and economies of scale. The direct and active management of the computing resources of public cloud 1105 is performed by the computer hardware and / or software of cloud orchestration module 1141. The computing resources provided by public cloud 1105 are typically implemented by virtual computing environments that run on various computers making up the computers of host physical machine set 1142, which is the universe of physical computers in and / or available to public cloud 1105. The virtual computing environments (VCEs) typically take the form of virtual machines from virtual machine set 1143 and / or containers from container set 1144. It is understood that these VCEs may be stored as images and may be transferred among and between the various physical machine hosts, either as images or after instantiation of the VCE. Cloud orchestration module 1141 manages the transfer and storage of images, deploys new instantiations of VCEs and manages active instantiations of VCE deployments. Gateway 1140 is the collection of computer software, hardware, and firmware that allows public cloud 1105 to communicate through WAN 1102.

[0114] Some further explanation of virtualized computing environments (VCEs) willnow be provided. VCEs can be stored as “images.” A new active instance of the VCE can be instantiated from the image. Two familiar types of VCEs are virtual machines and containers. A container is a VCE that uses operating-system-level virtualization. This refers to an operating system feature in which the kernel allows the existence of multiple isolated user-space instances, called containers. These isolated user-space instances typically behave as real computers from the point of view of programs running in them. A computer program running on an ordinary operating system can utilize all resources of that computer, such as connected devices, files and folders, network shares, CPU power, and quantifiable hardware capabilities. However, programs running inside a container can only use the contents of thecontainer and devices assigned to the container, a feature which is known as containerization.

[0115] PRIVATE CLOUD 1106 is similar to public cloud 1105, except that thecomputing resources are only available for use by a single enterprise. While private cloud 1106 is depicted as being in communication with WAN 1102, in other embodiments a private cloud may be disconnected from the internet entirely and only accessible through a local / private network. A hybrid cloud is a composition of multiple clouds of different types (for example, private, community or public cloud types), often respectively implemented by different vendors. Each of the multiple clouds remains a separate and discrete entity, but the larger hybrid cloud architecture is bound together by standardized or proprietary technology that enables orchestration, management, and / or data / application portability between the multiple constituent clouds. In this embodiment, public cloud 1105 and private cloud 1106 are both part of a larger hybrid cloud.

[0116] The embodiments described herein can be directed to one or more of asystem, a method, an apparatus and / or a computer program product at any possible technical detail level of integration. The computer program product can include a computer readable storage medium (or media) having computer readable program instructions thereon for causing a processor to carry out aspects of the one or more embodiments described herein. The computer readable storage medium can be a tangible device that can retain and store instructions for use by an instruction execution device. The computer readable storage medium can be, for example, but is not limited to, an electronic storage device, a magneticstorage device, an optical storage device, an electromagnetic storage device, asuperconducting storage device and / or any suitable combination of the foregoing. A non- exhaustive list of more specific examples of the computer readable storage medium can also include the following: a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), a static random access memory (SRAM), a portable compact disc read-only memory (CD-ROM), a digital versatile disk (DVD), a memory stick, a floppy disk, a mechanically encoded device such as punch-cards or raised structures in a groove having instructions recorded thereon and / or any suitable combination of the foregoing. A computer readable storage medium, as used herein, is not to be construed as being transitory signals per se, such as radio waves and / or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide and / or other transmission media(e.g., light pulses passing through a fiber-optic cable), and / or electrical signals transmitted through a wire.

[0117] Computer readable program instructions described herein can be downloadedto respective computing / processing devices from a computer readable storage medium and / or to an external computer or external storage device via a network, for example, theInternet, a local area network, a wide area network and / or a wireless network. The networkcan comprise copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers and / or edge servers. A network adapter card or network interface in each computing / processing device receives computer readable program instructions from the network and forwards the computer readable program instructions for storage in a computer readable storage medium within the respective computing / processing device. Computer readable program instructions for carrying out operations of the one or more embodiments described herein can be assembler instructions, instruction-set-architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state-setting data, configuration data for integrated circuitry, and / or source code and / or object code written in any combination of one or more programming languages, including an object oriented programming language such as Smalltalk, C++ or the like, and / or procedural programming languages, such as the "C" programming language and / or similar programming languages. The computer readable program instructions can execute entirely on a computer, partly on a computer, as a stand- alone software package, partly on a computer and / or partly on a remote computer or entirely on the remote computer and / or server. In the latter scenario, the remote computer can be connected to a computer through any type of network, including a local area network (LAN) and / or a wide area network (WAN), and / or the connection can be made to an external computer (for example, through the Internet using an Internet Service Provider). In one or more embodiments, electronic circuitry including, for example, programmable logic circuitry, field-programmable gate arrays (FPGA) and / or programmable logic arrays (PLA) can execute the computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry, in order to perform aspects of the one or more embodiments described herein.

[0118] Aspects of the one or more embodiments described herein are described withreference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to one or more embodiments described herein. It will be understood that each block of the flowchart illustrations and / or block diagrams, andcombinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer readable program instructions. These computer readable program instructions can be provided to a processor of a general-purpose computer, special purpose computer and / or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, can create means for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks. These computer readable program instructions can also be stored in a computer readable storage medium that can direct a computer, a programmable data processing apparatus and / or other devices to function in a particular manner, such that the computer readable storage medium having instructions stored therein can comprise an article of manufacture including instructions which canimplement aspects of the function / act specified in the flowchart and / or block diagram blockor blocks. The computer readable program instructions can also be loaded onto a computer, other programmable data processing apparatus and / or other device to cause a series of operational acts to be performed on the computer, other programmable apparatus and / or other device to produce a computer implemented process, such that the instructions which execute on the computer, other programmable apparatus and / or other device implement the functions / acts specified in the flowchart and / or block diagram block or blocks.

[0119] The flowcharts and block diagrams in the figures illustrate the architecture,functionality and / or operation of possible implementations of systems, computer- implementable methods and / or computer program products according to one or more embodiments described herein. In this regard, each block in the flowchart or block diagrams can represent a module, segment and / or portion of instructions, which comprises one or more executable instructions for implementing the specified logical function. In one or more alternative implementations, the functions noted in the blocks can occur out of the order noted in the Figures. For example, two blocks shown in succession can be executedsubstantially concurrently, and / or the blocks can sometimes be executed in the reverse order,depending upon the functionality involved. It will also be noted that each block of the block diagrams and / or flowchart illustration, and / or combinations of blocks in the block diagrams and / or flowchart illustration, can be implemented by special purpose hardware-based systems that can perform the specified functions and / or acts and / or carry out one or more combinations of special purpose hardware and / or computer instructions.

[0120] While the subject matter has been described above in the general context ofcomputer-executable instructions of a computer program product that runs on a computerand / or computers, those skilled in the art will recognize that the one or more embodiments herein also can be implemented at least partially in parallel with one or more other program modules. Generally, program modules include routines, programs, components and / or data structures that perform particular tasks and / or implement particular abstract data types. Moreover, the aforedescribed computer-implemented methods can be practiced with othercomputer system configurations, including single-processor and / or multiprocessor computersystems, mini-computing devices, mainframe computers, as well as computers, hand-held computing devices (e.g., PDA, phone), and / or microprocessor-based or programmableconsumer and / or industrial electronics. The illustrated aspects can also be practiced indistributed computing environments in which tasks are performed by remote processing devices that are linked through a communications network. However, one or more, if not all aspects of the one or more embodiments described herein can be practiced on stand-alone computers. In a distributed computing environment, program modules can be located in both local and remote memory storage devices.

[0121] As used in this application, the terms “component,” “system,” “platform”and / or “interface” can refer to and / or can include a computer-related entity or an entity related to an operational machine with one or more specific functionalities. The entities described herein can be either hardware, a combination of hardware and software, software, or software in execution. For example, a component can be, but is not limited to being, a process running on a processor, a processor, an object, an executable, a thread of execution, a program and / or a computer. By way of illustration, both an application running on a server and the server can be a component. One or more components can reside within a process and / or thread of execution and a component can be localized on one computer and / or distributed between two or more computers. In another example, respective components can execute from various computer readable media having various data structures stored thereon. The components can communicate via local and / or remote processes such as in accordance with a signal having one or more data packets (e.g., data from one component interacting with another component in a local system, distributed system and / or across a network such as the Internet with other systems via the signal). As another example, a component can be an apparatus with specific functionality provided by mechanical parts operated by electric or electronic circuitry, which is operated by a software and / or firmware application executed bya processor. In such a case, the processor can be internal and / or external to the apparatus andcan execute at least a part of the software and / or firmware application. As yet another example, a component can be an apparatus that provides specific functionality throughelectronic components without mechanical parts, where the electronic components can include a processor and / or other means to execute software and / or firmware that confers at least in part the functionality of the electronic components. In an aspect, a component can emulate an electronic component via a virtual machine, e.g., within a cloud computing system.

[0122] In addition, the term “or” is intended to mean an inclusive “or” rather than anexclusive “or.” That is, unless specified otherwise, or clear from context, “X employs A or B” is intended to mean any of the natural inclusive permutations. That is, if X employs A; X employs B; or X employs both A and B, then “X employs A or B” is satisfied under any of the foregoing instances. Moreover, articles “a” and “an” as used in the subject specification and annexed drawings should generally be construed to mean “one or more” unless specified otherwise or clear from context to be directed to a singular form. As used herein, the terms “example” and / or “exemplary” are utilized to mean serving as an example, instance, or illustration. For the avoidance of doubt, the subject matter described herein is not limited by such examples. In addition, any aspect or design described herein as an “example” and / or “exemplary” is not necessarily to be construed as preferred or advantageous over other aspects or designs, nor is it meant to preclude equivalent exemplary structures and techniques known to those of ordinary skill in the art.

[0123] As it is employed in the subject specification, the term “processor” can referto substantially any computing processing unit and / or device comprising, but not limited to, single-core processors; single-processors with software multithread execution capability; multi-core processors; multi-core processors with software multithread execution capability; multi-core processors with hardware multithread technology; parallel platforms; and / or parallel platforms with distributed shared memory. Additionally, a processor can refer to an integrated circuit, an application specific integrated circuit (ASIC), a digital signal processor (DSP), a field programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), a discrete gate or transistor logic, discrete hardware components, and / or any combination thereof designed to perform the functions described herein. Further, processors can exploit nano-scale architectures such as, but notlimited to, molecular and quantum-dot based transistors, switches and / or gates, in order tooptimize space usage and / or to enhance performance of related equipment. A processor can be implemented as a combination of computing processing units.

[0124] Herein, terms such as “store,” “storage,” “data store,” data storage,”“database,” and substantially any other information storage component relevant to operationand functionality of a component are utilized to refer to “memory components,” entities embodied in a “memory,” or components comprising a memory. Memory and / or memory components described herein can be either volatile memory or nonvolatile memory or can include both volatile and nonvolatile memory. By way of illustration, and not limitation, nonvolatile memory can include read only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable ROM (EEPROM), flashmemory and / or nonvolatile random-access memory (RAM) (e.g., ferroelectric RAM(FeRAM). Volatile memory can include RAM, which can act as external cache memory, for example. By way of illustration and not limitation, RAM can be available in many forms such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM) and / or Rambus dynamic RAM (RDRAM). Additionally, the described memory components of systems and / or computer-implemented methods herein are intended to include, without being limited to including, these and / or any other suitable types of memory.

[0125] What has been described above includes mere examples of systems andcomputer-implemented methods. It is, of course, not possible to describe every conceivable combination of components and / or computer-implemented methods for purposes of describing the one or more embodiments, but one of ordinary skill in the art can recognize that many further combinations and / or permutations of the one or more embodiments are possible. Furthermore, to the extent that the terms “includes,” “has,” “possesses,” and the like are used in the detailed description, claims, appendices and / or drawings such terms are intended to be inclusive in a manner similar to the term “comprising” as “comprising” is interpreted when employed as a transitional word in a claim.

[0126] The descriptions of the various embodiments have been presented forpurposes of illustration but are not intended to be exhaustive or limited to the embodiments described herein. Many modifications and variations will be apparent to those of ordinaryskill in the art without departing from the scope and spirit of the described embodiments.The terminology used herein was chosen to best explain the principles of the embodiments, the practical application and / or technical improvement over technologies found in the marketplace, and / or to enable others of ordinary skill in the art to understand the embodiments described herein.

Claims

CLAIMS1. A system, comprising:a memory that stores computer executable components; and a processor, operably coupled to the memory, that executes at least one of the computer executable components that: receives a first target real-valued function and a second target real-valued function that represent a target transformation on a quantum state within a quantumprocessor; determines a system of non-linear equations based on the first target real- valued function and the second target real-valued function, the system of non-linear equations comprising a number of phase factors that define parameters of quantumoperations; determines the phase factors to implement the target transformation by usinga modified Newton method to iteratively solve the system of non-linear equations;and configures the quantum processor to apply the phase factors to a quantumcircuit to implement the target transformation on the quantum state.

2. The system of claim 1, wherein the at least one of the computer executablecomponents further: approximates, based on a mode, the first target real-valued function in a real orimaginary part of a first complex polynomial function, wherein the first complex polynomial function is of a first degree; and approximates, based on the mode, the second target real-valued function in a real orimaginary part of a second complex polynomial function, wherein the second complex polynomial function is of a second degree; and configures the quantum processor to apply the phase factors based on the mode used for approximating the first target real-valued function and the second target real-valued function.

3. The system of any of the preceding claims, wherein the first target real-valuedfunction or the second target real-valued function correspond to polynomial interpolations at Chebyshev points.

4. The system of any of claims 2 to 3, wherein the phase factors represent the first targetreal-valued function and the second target real-valued function as the first complex polynomial function and the second complex polynomial function respectively if the first target real-valued function and the second target real-valued function are polynomial functions.

5. The system of any of the preceding claims, wherein the at least one of the computerexecutable components further: sets a level of accuracy of the phase factors for solving the system of non-linearequations to implement the target transformation on the quantum state at the level ofaccuracy.

6. The system of any of claims 2 to 5, wherein the number of phase factors correspondsto a maximum degree between the first degree and the second degree.

7. The system of any of the preceding claims, wherein iteratively solving the system ofnon-linear equations to determine the phase factors to implement the target transformationon the quantum state comprises using a quasi-Newton method.

8. The system of any of claims 1 to 6, wherein iteratively solving the system of non-linear equations to determine the phase factors to implement the target transformation on thequantum state comprises using Newton’s method.

9. The system of any of the preceding claims, wherein the modified Newton methodcomprises: selecting an initial value of the phase factors, wherein a Jacobian matrix of the system of non-linear equations is orthogonal at the initial value.

10. The system of any of claims 5 to 9, wherein the at least one of the computerexecutable components further: accelerates convergence using Aitken’s acceleration technique, wherein Aitken’s acceleration technique comprises: modifying a learning rate of iteratively solving the system of non-linear equations based on satisfaction of a criterion; andexamining the criterion based on the level of accuracy.

11. A computer-implemented method, comprising:receiving, by a system operatively coupled to a processor, a first target real-valuedfunction and a second target real-valued function that represent a target transformation on aquantum state within a quantum processor;determining, by the system, a system of non-linear equations based on the first target real-valued function and the second target real-valued function, the system of non-linearequations comprising a number of phase factors that define parameters of quantumoperations; determining, by the system, the phase factors to implement the target transformationby using a modified Newton method to iteratively solve the system of non-linear equations;and configuring, by the system, the quantum processor to apply the phase factors to a quantum circuit to implement the target transformation on the quantum state.

12. The computer-implemented method of claim 11, further comprising:approximating, by the system, and based on a mode, the first target real-valued function in a real or imaginary part of a first complex polynomial function, wherein the first complex polynomial function is of a first degree; and approximating, by the system and based on the mode, the second target real-valuedfunction in a real or imaginary part of a second complex polynomial function, wherein the second complex polynomial function is of a second degree; and configures the quantum processor to apply the phase factors based on the mode used for approximating the first target real-valued function and the second target real-valued function.

13. The computer-implemented method of any of claims 11 to 12, wherein the first targetreal-valued function or the second target real-valued function correspond to polynomial interpolations at Chebyshev points.

14. The computer-implemented method of any of claims 12 to 13, wherein the phasefactors represent the first target real-valued function and the second target real-valued function as the first complex polynomial function and the second complex polynomialfunction respectively if the first target real-valued function and the second target real-valued function are polynomial functions.

15. The computer-implemented method of any of claims 11 to 14, further comprising:setting, by the system, a level of accuracy of the phase factors for solving the system of non-linear equations to implement the target transformation on the quantum state at the level of accuracy.

16. The computer-implemented method of any of claims 11 to 15, wherein iterativelysolving the system of non-linear equations to determine the phase factors to implement thetarget transformation on the quantum state comprises using a quasi-Newton method orNewton’s method.

17. The computer-implemented method of any of claims 11 to 16, further comprising:selecting, by the system, an initial value of the phase factors, wherein a Jacobian matrix of the system of non-linear equations is orthogonal at the initial value.

18. The computer-implemented method of any of claims 15 to 17, further comprising:accelerating, by the system, convergence using Aitken’s acceleration technique, wherein Aitken’s acceleration technique comprises: modifying a learning rate of iteratively solving the system of non-linear equations based on satisfaction of a criterion; and examining the criterion based on the level of accuracy.

19. A computer program product facilitating phase factor determination in quantumsignal processing, the computer program product comprising a computer readable storagemedium having program instructions embodied therewith, the program instructionsexecutable by a processor to cause the processor to: receive, by the processor, a first target real-valued function and a second target real-valued function that represent a target transformation on a quantum state within a quantumprocessor; determine, by the processor, a system of non-linear equations based on the first target real-valued function and the second target real-valued function, the system of non-linearequations comprising a number of phase factors that define parameters of quantumoperations; determine the phase factors to implement the target transformation by using amodified Newton method to iteratively solve the system of non-linear equations; andconfigure the quantum processor to apply the phase factors to a quantum circuit to implement the target transformation on the quantum state.

20. The computer program product of claim 19, wherein the program instructions arefurther executable by the processor to cause the processor to: approximate, based on a mode, the first target real-valued function in a real orimaginary part of a first complex polynomial function, wherein the first complex polynomial function is of a first degree; and approximate, based on the mode, the second target real-valued function in a real orimaginary part of a second complex polynomial function, wherein the second complex polynomial function is of a second degree; and configure the quantum processor to apply the phase factors based on the mode used for approximating the first target real-valued function and the second target real-valued function.

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