Quantum resource estimation using reparameterization methods

Abstract

A system, computer-implemented method, and computer program product are provided that facilitate estimation of quantum resources for calculating an expectation value of a stochastic process using a reparameterization method. According to one aspect, the system may include a processor that executes computer-executable components stored in a memory. The computer-executable components may include a reparameterization component that applies a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution. The computer-executable components may further include an estimation component that estimates at least one defined criterion of a quantum computer that will be used to calculate the expectation value of the stochastic process associated with the target probability distribution.

Description

[Technical field]

[0001] The subject disclosure relates to estimating quantum resources for calculating the expectation value of a stochastic process, and more particularly, to estimating quantum resources for calculating the expectation value of a stochastic process using reparameterization methods. Summary of the Invention

[0002] The following provides a summary to provide a basic understanding of one or more aspects of the present invention. This summary is not intended to identify key or critical elements or to clarify the scope of particular aspects or claims. Its sole purpose is to present concepts in a simplified form as a prelude to the more detailed description presented later. In one or more aspects described herein, a system, device, computer-implemented method, and / or computer program product are described that can facilitate the estimation of quantum resources for calculating the expectation value of a stochastic process using a re-parameterization method.

[0003] According to an aspect, a system may include a processor that executes computer-executable components stored in a memory. The computer-executable components may include a reparameterization component that applies a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution. The computer-executable components may further include an estimation component that estimates at least one defined criterion for a quantum computer used to calculate an expectation value of a stochastic process associated with the target probability distribution.

[0004] According to another aspect, the computer-implemented method can include applying, by a system operatively coupled to the processor, a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution. The computer-implemented method can further include estimating, by the system, at least one defined criterion for a quantum computer used to calculate an expectation value of a stochastic process associated with the target probability distribution.

[0005] According to another aspect, a computer program product includes a computer-readable storage medium having program instructions embodied therein, the program instructions executable by a processor to cause the processor to apply a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution, the program instructions being further executable by the processor to cause the processor to estimate at least one defined criterion of a quantum computer used to calculate an expectation value of a stochastic process associated with the target probability distribution. [Brief description of the drawings]

[0006] [Figure 1] FIG. 1 illustrates a block diagram of an example, non-limiting system that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Diagram 2] FIG. 1 illustrates a block diagram of an example, non-limiting system that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Diagram 3]FIG. 1 illustrates a block diagram of an example, non-limiting system that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 4] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Diagram 5] FIG. 1 illustrates an example, non-limiting circuit that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 6A] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 6B] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 6C] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 7A] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 7B] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 8A] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 8B] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 9A] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 9B] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 9C] FIG. 1 illustrates an example, non-limiting graph that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 10] FIG. 1 illustrates a flow diagram of an example, non-limiting computer-implemented method that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. [Figure 11] FIG. 1 illustrates a block diagram of an example non-limiting operating environment that can facilitate one or more aspects described herein. [Figure 12] FIG. 1 illustrates a block diagram of an exemplary non-limiting cloud computing environment in accordance with one or more aspects of the subject disclosure. [Figure 13]FIG. 1 illustrates a block diagram of example non-limiting abstraction model layers in accordance with one or more aspects of the subject disclosure. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0007] The following detailed description is merely illustrative and is not intended to limit the embodiments and / or the application or uses of the embodiments, and is further not intended to be bound by any express or implied information presented in the Background or Summary sections above or in the Detailed Description section.

[0008] One or more aspects will now be described with reference to the drawings, wherein like reference 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 one or more aspects. It will be apparent, however, that in various instances, one or more aspects may be practiced without these specific details.

[0009] As referred to herein, a "derivative" and / or a "derivative asset" is a contract between an issuer and a holder, valid until its expiration date. As referred to herein, an "entity" can include a human, a client, a user, a computing device, a software application, an agent, a machine learning (ML) model, an artificial intelligence (AI), and / or another entity. It will be understood that when an element is referred to as being "coupled" to another element herein, one or more different types of bonds can be described, including, but not limited to, a chemical bond, a communicative bond, an electrical bond, an electromagnetic bond, a operative bond, an optical bond, a physical bond, a thermal bond, and / or another type of bond.

[0010] 1, 2, and 3 illustrate block diagrams of exemplary non-limiting systems 100, 200, and 300, respectively, that can facilitate quantum resource estimation for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Each of the systems 100, 200, and 300 can comprise a quantum resource estimation system 102. The quantum resource estimation system 102 of the system 100 illustrated in FIG. 1 can comprise a memory 104, a processor 106, a reparameterization component 108, an estimation component 110, and / or a bus 112. The quantum resource estimation system 102 of the system 200 illustrated in FIG. 2 can further comprise a variational component 202. The quantum resource estimation system 102 of the system 300 depicted in FIG. 3 can further comprise an error analysis component 302.

[0011] Although some aspects of the subject disclosure describe an example application of the quantum resource estimation system 102 to estimate quantum resources to calculate the expected value of a stochastic process, such as, for example, a derivative asset, it should be understood that the subject disclosure is not so limited. For example, the quantum resource estimation system 102 can estimate quantum resources to calculate the expected value of another stochastic process (e.g., any type of stochastic process).

[0012] It should be understood that the aspects of the subject disclosure illustrated in the various figures disclosed herein are for illustration purposes only, and thus the architecture of such aspects is not limited to the systems, devices, and / or components shown therein. For example, in some aspects, system 100, system 200, system 300, and / or quantum resource estimation system 102 may further comprise various computer and / or computing-based elements described herein with reference to operating environment 1100 and FIG. 11. In some aspects, such computer and / or computing-based elements may be used in connection with implementing one or more of the systems, devices, components, and / or computer-implemented operations shown and described in connection with FIG. 1, FIG. 2, FIG. 3, and / or other figures disclosed herein.

[0013] Memory 104 may store one or more computer- and / or machine-readable, writeable, and / or executable components and / or instructions that, when executed by processor 106 (e.g., a classical processor, a quantum processor, and / or another type of processor), can facilitate performance of operations defined by the executable components and / or instructions. For example, memory 104 may store computer- and / or machine-readable, writeable, and / or executable components and / or instructions that, when executed by processor 106, can facilitate performance of various functions described herein relating to quantum resource estimation system 102, reparameterization component 108, estimation component 110, variation component 202, error analysis component 302, and / or another component associated with quantum resource estimation system 102 as described herein with or without reference to various figures of the subject disclosure.

[0014] The memory 104 may include volatile memory (e.g., random access memory (RAM), static RAM (SRAM), dynamic RAM (DRAM), and / or another type of volatile memory) and / or non-volatile memory (e.g., read only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), and / or another type of non-volatile memory) that may utilize one or more memory architectures. Further examples of memory 104 are described below with reference to system memory 1116 and FIG. 11. Such examples of memory 104 may be utilized to implement any aspect of the subject disclosure.

[0015] The processor 106 may include one or more types of processors and / or electronic circuitry (e.g., classical processors, quantum processors, and / or other types of processors and / or electronic circuitry) that may implement one or more computer- and / or machine-readable, writable, and / or executable components and / or instructions that may be stored in the memory 104. For example, the processor 106 may perform various operations that may be specified by such computer- and / or machine-readable, writable, and / or executable components and / or instructions, including, but not limited to, logic, control, input / output (I / O), and / or arithmetic, etc. In some aspects, the processor 106 may include one or more central processing units, multi-core processors, microprocessors, dual microprocessors, microcontrollers, system-on-a-chip (SOC), array processors, vector processors, quantum processors, and / or other types of processors. Further examples of the processor 106 are described below with reference to the processing unit 1114 and FIG. 11. Any such example of the processor 106 may be utilized to implement any aspect of the subject disclosure.

[0016] The quantum resource estimation system 102, memory 104, processor 106, reparameterization component 108, estimation component 110, variation component 202, error analysis component 302, and / or other components of the quantum resource estimation system 102 as described herein may be communicatively, electrically, operatively, and / or optically coupled to one another via a bus 112 to perform the functions of the system 100, the system 200, the system 300, the quantum resource estimation system 102, and / or any components coupled thereto. The bus 112 may include one or more memory buses, memory controllers, peripheral buses, external buses, local buses, quantum buses, and / or other types of buses that may utilize various bus architectures. Further examples of the bus 112 are described below with reference to the system bus 1118 and FIG. 11. Such examples of the bus 112 may be utilized to implement any aspect of the subject disclosure.

[0017] The quantum resource estimation system 102 may include any type of component, machine, device, facility, apparatus, and / or equipment that includes a processor and / or is capable of effective and / or operable communication with wired and / or wireless networks. All such aspects are contemplated. For example, the quantum resource estimation system 102 may include a server device, a computing device, a general purpose computer, a special purpose computer, a quantum computing device (e.g., a quantum computer), a tablet computing device, a handheld device, a server class computing machine and / or database, a laptop computer, a notebook computer, a desktop computer, a mobile phone, a smartphone, a consumer electronics product and / or equipment, an industrial and / or commercial device, a digital assistant, a multimedia Internet-enabled phone, a multimedia player, and / or another type of device.

[0018] The quantum resource estimation system 102 can be coupled (e.g., communicatively, electrically, operatively, optically, and / or via another type of coupling) to one or more external systems, sources, and / or devices (e.g., classical and / or quantum computing devices, communication devices, and / or another type of external system, source, and / or device) using wires and / or cables. For example, the quantum resource estimation system 102 can be coupled (e.g., communicatively, electrically, operatively, optically, and / or via another type of coupling) to one or more external systems, sources, and / or devices (e.g., classical and / or quantum computing devices, communication devices, and / or another type of external system, source, and / or device) using data cables, including but not limited to high-definition multimedia interface (HDMI) cables, recommended standard (RS) 232 cables, Ethernet cables, and / or other data cables.

[0019] In some aspects, the quantum resource estimation system 102 may be coupled (e.g., communicatively, electrically, operatively, optically, and / or via another type of coupling) to one or more external systems, sources, and / or devices (e.g., classical and / or quantum computing devices, communications devices, and / or another type of external system, source, and / or device) via a network. For example, such a network may include wired and / or wireless networks, including but not limited to a cellular network, a wide area network (WAN) (e.g., the Internet), a local area network (LAN), and / or another network. The quantum resource estimation system 102 may communicate with one or more external systems, sources, and / or devices, e.g., computing devices, using substantially any desired wired and / or wireless technology, including but not limited to Wireless Fidelity (Wi-Fi), Global System for Mobile Communications (GSM), Universal Mobile Telecommunications System (UMTS), Worldwide Interoperability for Microwave Access (WiMAX), Enhanced General Packet Radio Service (Enhanced GPRS), Third Generation Partnership Project (3GPP), Long Term Evolution (LTE), and other related technologies. ), 3rd Generation Partnership Project 2 (3GPP2), Ultra Mobile Broadband (UMB), High Speed ​​Packet Access (HSPA), Zigbee and other 802.XX wireless technologies and / or legacy communication technologies, BLUETOOTH®, Session Initiation Protocol (SIP), ZIGBEE®, RF4CE protocol, WirelessHART protocol, 6LoWPAN (IPv6 over Low Power Wireless Area Network), Z-Wave, ANT, Ultra Wideband (UWB) standard protocols, and / or other proprietary and non-proprietary communication protocols.Thus, in some aspects, the quantum resource estimation system 102 may include hardware (e.g., a central processing unit (CPU), a transceiver, a decoder, quantum hardware, a quantum processor, and / or other hardware), software (e.g., a set of threads, a set of processes, running software, quantum pulse schedules, quantum circuits, quantum gates, and / or other software), or a combination of hardware and software that can facilitate communication of information between the quantum resource estimation system 102 and external systems, sources, and / or devices (e.g., computing devices, communications devices, and / or another type of external system, source, and / or device).

[0020] Quantum resource estimation system 102 may include one or more computer- and / or machine-readable, writeable, and / or executable components and / or instructions that, when executed by processor 106 (e.g., a classical processor, a quantum processor, and / or another type of processor), can facilitate performance of operations defined by such components and / or instructions. Further, in many aspects, any component associated with quantum resource estimation system 102 as described herein with or without reference to various figures of the subject disclosure can include one or more computer- and / or machine-readable, writeable, and / or executable components and / or instructions that, when executed by processor 106, can facilitate performance of operations defined by such components and / or instructions. For example, the reparameterization component 108, the estimation component 110, the variational component 202, the error analysis component 302, and / or any other components associated with (e.g., communicatively, electronically, operatively, and / or optically coupled to, and / or utilized by, the quantum resource estimation system 102 as disclosed herein may include such computer and / or machine readable, writeable, and / or executable components and / or instructions.As a result, according to various aspects, the quantum resource estimation system 102 and / or any components associated therewith as disclosed herein may utilize the processor 106 to execute such computer and / or machine readable, writable, and / or executable components and / or instructions to facilitate performance of one or more operations described herein with reference to the quantum resource estimation system 102 and / or any such components associated therewith.

[0021] The quantum resource estimation system 102 may facilitate (e.g., via the processor 106) the execution of operations performed by and / or associated with the reparameterization component 108, the estimation component 110, the variational component 202, the error analysis component 302, and / or another component associated with the quantum resource estimation system 102 as disclosed herein. For example, as described in more detail below, the quantum resource estimation system 102 may facilitate (e.g., via the processor 106) applying a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution and / or estimating at least one defined criterion for a quantum computer used to calculate an expectation value of a stochastic process associated with the target probability distribution.

[0022] In another example, as described in more detail below, the quantum resource estimation system 102 may further facilitate (e.g., via the processor 106) applying a transformation operation to the variationally prepared quantum state to generate a quantum state corresponding to a target probability distribution having a defined mean of the target probability distribution, a defined standard deviation of the target probability distribution, and / or one or more explicit parameters specifying the target probability distribution; applying a quantum fault-tolerant operation to the variationally prepared quantum state to prepare the quantum state as a superposition for possible paths of a discrete-time multivariate stochastic process; training a variational quantum circuit to prepare the variationally prepared quantum state and to reduce the computational cost of quantum arithmetic operations performed by the quantum computer to calculate the expectation value of the stochastic process associated with the target probability distribution; training the variational quantum circuit using a Hamiltonian operator to generate a basis state corresponding to the target probability distribution; and calculating one or more errors associated with at least one of the application of a quantum fault-tolerant operation to the variationally prepared quantum state to generate a quantum state, the estimation of the at least one defined criterion, or the calculation of the expectation value of the stochastic process associated with the target probability distribution.

[0023] In the above examples, the at least one defined criterion may include attributes, conditions, properties, parameters, and / or configurations of the quantum computer that enable the quantum computer to achieve a defined quantum superiority in calculating the expectation value of the stochastic process associated with the target probability distribution. In the above examples, the probability distribution may include a standard normal probability distribution and the target probability distribution may include a normal probability distribution.

[0024] According to various aspects of the subject disclosure described herein, to perform one or more of the operations described above, the quantum resource estimation system 102, the reparameterization component 108, the estimation component 110, the variational component 202, and / or the error analysis component 302 may define and / or implement one or more of the algorithms (e.g., algorithms 2.1, 3.1, 3.2, 4.1, and / or 4.2) and / or one or more of the equations (e.g., equations (1)-(109)) described below with reference to Sections 1.0-11.0. For example, with reference to Sections 3.2 and 10.0 described below, the reparameterization component 108 may apply a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution (e.g., a standard normal probability distribution) to generate a quantum state corresponding to a target probability distribution (e.g., a normal probability distribution). In this example, the reparameterization component 108 can apply a quantum fault-tolerant operation to the variationally prepared quantum state to prepare the quantum state as a superposition for possible paths of a discrete-time multivariate stochastic process. In this example, the reparameterization component 108 can apply a transformation operation to the variationally prepared quantum state to generate a quantum state corresponding to a target probability distribution having at least one of a defined mean of the target probability distribution, a defined standard deviation of the target probability distribution, or one or more explicit parameters that specify the target probability distribution.

[0025] In another example, with reference to sections 3.1.2 and 3.2.3 described below, the estimation component 110 can estimate at least one defined criterion of a quantum computer used to calculate an expected value of a stochastic process associated with a target probability distribution. For example, the estimation component 110 can estimate at least one defined criterion of a quantum computer used to calculate the value of an asset, such as, for example, a derivative asset, associated with a target probability distribution. For example, the estimation component 110 can estimate at least one defined criterion, including, but not limited to, attributes, conditions, properties, parameters, configurations, and / or another criterion of the quantum computer that enable the quantum computer to achieve a defined quantum superiority in calculating the expected value of a stochastic process associated with the target probability distribution (e.g., the value of a derivative asset).

[0026] In another example, see Sections 3.2.1 and 11.0 described below, the variational component 202 can train a variational quantum circuit to prepare a variationally prepared quantum state. For example, the variational component 202 can train a variational quantum circuit to prepare a variationally prepared quantum state by training the variational quantum circuit using a Hamiltonian operator to generate a basis state corresponding to a target probability distribution. It should be appreciated that the variational component 202 can train a variational quantum circuit to prepare a variationally prepared quantum state to reduce the computational cost of quantum arithmetic operations performed by a quantum computer to calculate the expected value of a stochastic process associated with a target probability distribution (e.g., the value of a derivative asset).

[0027] In another example, referring to Section 3.2.2 below, the error analysis component 302 can calculate one or more errors associated with at least one of applying a quantum fault-tolerant operation to a variationally prepared quantum state to generate a quantum state, estimating at least one defined criterion, or calculating an expected value of a stochastic process associated with a target probability distribution (e.g., the value of a derivative asset).

[0028] In accordance with various aspects, as described in the following sections, the quantum resource estimation system 102 can determine an upper bound on the resources (e.g., quantum computing resources) involved in providing a useful quantum advantage in pricing derivatives. To that end, the quantum resource estimation system 102 can provide a first complete resource estimate for pricing useful quantum derivatives using auto-callable options and target accrual redemption forwards (TARF) derivatives as an example benchmark use case. The quantum resource estimation system 102 can provide a new method (reparameterization method) for pricing quantum derivatives that overcomes and avoids obstacles in known approaches. It should be appreciated that the reparameterization method, which can be defined and / or implemented by the quantum resource estimation system 102 as described in the following sections, hybridizes a pre-trained variational circuit with fault-tolerant quantum computing, significantly reducing the resources (e.g., quantum computing resources) involved in estimating the value of a derivative. As described below, in accordance with various aspects of the subject disclosure, the quantum resource estimation system 102 may determine that autocallable options and TARF derivatives as exemplary benchmark use cases require, for example, approximately eight thousand (8k) logical qubits, an estimated T-depth of approximately 50 million, and an estimated logical clock speed of approximately 10 megahertz (MHz) to achieve a defined quantum supremacy.

[0029] 1.0 Derivatives Pricing

[0030] Pricing derivatives contracts using Monte Carlo methods is computationally intensive in the financial sector, making quantum supremacy in this application highly valuable. In accordance with various aspects, as described in the following sections, the quantum resource estimation system 102 can provide the first detailed resource estimates of the conditions involved in achieving quantum supremacy in derivatives pricing. To accomplish this, the quantum resource estimation system 102 can define and / or implement novel methods, described below in accordance with one or more aspects of the subject disclosure, for loading stochastic processes into a quantum computer.

[0031] As defined above and referred to herein, a "derivative" and / or "derivative asset" is a contract between an issuer and a holder that is in effect until its maturity date. Examples of these derivative assets include, but are not limited to, forward contracts, options, autocallable options, target accrual redemption notes (TARNs), TARFs, and / or another derivative asset. Each derivative defines a payoff that defines what the holder gets. The payoff depends on the value of one or more underlying assets during the contract period. Examples of these underlying assets include, but are not limited to, stocks, currencies, commodities, and / or another underlying asset. Derivative contracts are widespread in the financial realm with a variety of uses ranging from risk hedging to speculation. The objective of derivative pricing is to determine the value of entering into a derivative contract today, given the uncertainty regarding the future value of the underlying asset and therefore the payoff.

[0032] The underlying asset is typically modeled as a stochastic process under assumptions such as no arbitrage, which means that a particular asset is not priced differently in different markets and therefore it is not possible to buy an asset from one market and immediately sell it in another market at a profit. A common model is that the underlying asset evolves under geometric Brownian motion.

[0033]

number

[0034] Let be a vector of values ​​of d underlying assets at time t.

[0035]

number

[0036] Let be the path of the discrete-time multivariate stochastic process that describes the value of those assets. Both notations are used for the paths in this document. The corresponding probability density function is

[0037]

number

[0038] As shown in the figure.

[0039]

number

[0040] Let be the payoffs of derivatives on these assets. Equation (1), defined below, can be used to price the derivatives.

[0041]

number

[0042] If the underlying stochastic processes are modeled with geometric Brownian motion, they have the following transition probabilities:

[0043]

number

[0044] Where:

[0045]

number

[0046] Equation (2) defined above at time t is

[0047]

number

[0048] In

[0049]

number

[0050] Note that the parameter r and σ depend on the asset vector at time t-1 via j are the risk-free rate and volatility of the jth asset, respectively, Δt is the duration between steps of the stochastic process, and Σ is the d × d positive definite covariance matrix of the d underlying assets.

[0051]

number

[0052] where -1≦ρ ij ≦1 is the correlation between assets i and j. For any particular path

[0053]

number

[0054] The probability of

[0055]

number

[0056] It is.

[0057] The risk-free rate referred to above is the rate of return on an investment in a risk-free asset. Although such assets are purely theoretical, Treasury bonds are commonly used to represent such assets, and we approximate r as the yield on Treasury bonds minus the current inflation rate.

[0058] Classically, some simple derivatives in this model are easy to price, for example European call options, which can be analytically priced using the Black-Scholes equation. Easy-to-price derivatives are often path-independent and their payoffs are exercise time.

[0059]

number

[0060] This is in contrast to path-dependent derivatives, which are more difficult to price and are often priced in practice using classical Monte Carlo techniques.

[0061] Using classical Monte Carlo methods, the accuracy of derivatives pricing is

[0062]

number

[0063] It converges as O(1 / M), where M is the number of samples. In general, quantum algorithms based on amplitude estimation can be used to improve this to O(1 / M). Recent work has explored how to specialize this advantage to option pricing and risk analysis. This is only a quadratic speedup, and the subject disclosure focuses on derivatives complex enough to have a large M in practice. In accordance with various aspects, as described in the following sections, the quantum resource estimation system 102 can provide end-to-end quantum resource estimation of autocallable options and TARFs, two examples of such derivatives, both of which are computationally expensive path-dependent derivatives. In doing so, the quantum resource estimation system 102 can refine and optimize (e.g., via the reparameterization component 108, the estimation component 110, and / or the variational component 202) the loading of the underlying distributions over the asset paths into the quantum state. This loading step remains open (e.g., unsolved) in the prior art, and the quantum resource estimation system 102 (e.g., via the estimation component 110) can provide an initial description of the resources required to accomplish such a loading step.

[0064] In addition to estimating the resources that can be used for path loading, the quantum resource estimation system 102 can also provide several optimizations, including a deliberate shift in the calculation from price space to return space, and new reparameterization methods that result in significant resource reductions and are summarized in Table 1.

[0065] [Table 1]

[0066] Table 1 shown above shows the 2×10 -31 shows resources estimated by implementing a quantum resource estimation system 102 according to one or more aspects of the subject disclosure to price derivatives using different methods for a target error of . In such a quantum resource estimation system 102 implementation, there is a basket of autocallable (auto) options with knock-in options and 5 autocall days, and a TARF with one underlying asset and 26 simulation days. In such a quantum resource estimation system 102 implementation, it was determined that the Grover-Rudolph method cannot be applied in practice and that the Riemann summation method can include a normalization assumption to avoid exponentially growing errors in T. As shown in detail in the Riemann Sum (no-norm) row of Table 1, even when those normalization issues are avoided, the reparameterization method that can be defined and / or implemented by the quantum resource estimation system 102 according to one or more aspects of the subject disclosure performs best. Section 3.1 below describes regularization of Riemann summations, and detailed resource estimations that may be defined and / or implemented by estimation component 110 are described later in Sections 3.1.2 and 3.2.3.

[0067] 1.1 Discrete Derivatives Pricing

[0068] To map the derivatives pricing problem to quantum states,

[0069]

number

[0070] is discretized. Classically, this is less important due to the high precision available, but to study the minimum qubit criteria, discretization is explicitly considered.

[0071] Each value

[0072]

number

[0073] Suppose that x is discretized into distinct n-qubit registers, i.e., mapped to a regular lattice. Then, the discrete space of paths can be defined as ω∈Ω. Now, the price expectation is the sum

[0074]

number

[0075] It is.

[0076] Here, the probability p(ω) can be defined in several ways. For example, the midpoint of a lattice cell is

[0077]

number

[0078] It is possible to obtain that

[0079] Where:

[0080]

number

[0081] is restricted to discrete midpoints. Alternatively, p(ω) can be defined as an integral over discrete cells. These expressions are the same in the fine grid limit, and in accordance with one or more aspects of the subject disclosure, the midpoint method is used.

[0082] 1.2 Price Space vs. Return Space

[0083] As mentioned above, geometric Brownian motion can be used to model the price of an underlying asset. As referred to herein, this is called a price space description of the underlying stochastic process. In the price space, the transition probabilities are given by a multivariate log-normal distribution.

[0084] In an alternative but equivalent representation, we consider a stochastic process for the log returns of the underlying assets and perform all calculations in the return space. When asset prices follow a log-normal distribution, the log returns are made normally distributed. The vector of underlying log returns for d assets at time t is

[0085]

number

[0086] The transition probabilities can then be given by a multivariate normal distribution.

[0087]

number

[0088] Where:

[0089] μ = (μ1, μ2, … μ d ) T , (9)

[0090]

number

[0091] where σ, Δt, Σ, and r are the same Brownian motion parameters as in price space. Note that they are no longer conditioned on their values ​​at previous time steps. In fact, the path distribution in return space contains dT independent Gaussians.

[0092] Note the overloaded notation from the price space formulation, as these representations are interchangeable. At any time t', the price of asset j is

[0093]

number

[0094] can be calculated from the return space using:

[0095] The calculation just described is used when the stochastic process is modeled in a return space, but the payoffs are defined in terms of asset prices. In various aspects of the subject disclosure described herein, it will be made clear from the context in which space such aspects are operating.

[0096] Switching between price and return space changes us from log-normal to normal loadings. In general, normal loadings are easier because their underlying stochastic evolution is independent of the price at the previous time step, which can be seen by comparing equations (3) and (10) defined above. Thus, the probability distribution over all T time steps of the stochastic process is

[0097]

number

[0098] teeth,

[0099]

number

[0100] These can be calculated simultaneously by:

[0101] This advantage can complement the quantum arithmetic used to evaluate the exponent in equation (11) defined above. In various aspects of the subject disclosure, the quantum resource estimation system 102 can take advantage of this advantage by using the reparameterization methods described herein. Furthermore, when dealing with derivatives where the payoff is directly defined in terms of log returns and is independent of individual asset prices, this is another reason why the quantum resource estimation system 102 can operate in the return space.

[0102] 2.0 Core Approach

[0103] The targeted disclosure approach to derivatives pricing extends the quantum mean estimation method. We define the normalized payoff of any path as

[0104]

number

[0105] Let it be given by.

[0106] Algorithm 2.1, defined below, can proceed in four phases. First, probability distributions are loaded in the form of a superposition over all possible paths. Second, payoffs for all possible paths are computed in quantum parallel. Third, expected payoffs are stored in the amplitudes of marked states. Fourth, the amplitude estimates are used to retrieve the amplitudes using an O(1 / ε) query for a given target precision ε>0.

[0107] Algorithm 2.1 – The Core Approach to Derivatives Pricing

[0108] Use n, d, and T, all of which are positive integers.

[0109] We find an operator P that loads a superposition of probabilistically weighted paths into a register of ndT qubits.

[0110] 1. Apply operator P to the quantum state

[0111]

number

[0112] Prepare the following.

[0113] 2.

[0114]

number

[0115] The quantum register

[0116]

number

[0117] Insert this into the calculation.

[0118] 3. Introducing Ancilla qubits

[0119]

number

[0120] Rotate the value in the register to that amplitude.

[0121]

number

[0122] 4. Use amplitude estimation to extract the probability that the ancillary is |1〉, which is the (e.g. discretized) expected payoff

[0123]

number

[0124] This can be rescaled to

[0125]

number

[0126] get.

[0127] Note that steps 1-3 in Algorithm 2.1 load the exact answer after one run. If it were possible to read out the amplitudes directly, the quantum resource estimation system 102 could compute the expectation of all paths in a fixed number of queries. This is unfortunately not possible, so amplitude estimation introduces a linear overhead to extract an answer of a given precision. This may be an important conceptual difference from classical Monte Carlo, where samples are taken from paths. In Algorithm 2.1, the quantum resource estimation system 102 can compute (e.g., all) possible paths and take samples (where the amplitudes are estimated) of the expected payoffs.

[0128] Another distinguishing feature of the quantum approach is that the quantum resource estimation system 102 can normalize the payoff and store it in the amplitude of the state. This normalization can be rescaled at the end, but the error is also scaled up, which can affect the error scaling. In the Riemann summation method described in Section 3.1, this version of normalization rescaling can cause errors to accumulate quickly.

[0129] 2.1 Amplitude Estimation of Derivatives Pricing

[0130] Typically, path-dependent derivatives such as autocallables and TARFs are priced using Monte Carlo methods.

[0131]

number

[0132] is derived by modeling the underlying stochastic process, and the expected payoff is then calculated using the estimator.

[0133]

number

[0134] This estimator has an error of ε=O(M -1 / 2 ) converges to the true expected value.

[0135] This convergence is achieved by using the quantum amplitude estimation for Monte Carlo: ε = O(M -1 ) the amplitude estimation can be quadratically accelerated to

[0136]

number

[0137] It takes as input a unitary operator A for n+1 qubits such that:

[0138] Here, the parameter a is unknown. Here, the last qubit acts as a label that distinguishes |Ψ0〉 states from |Ψ1〉 states.

[0139] The amplitude estimation is performed using the operator (often called the Grover operator)

[0140]

number

[0141] a is determined by repeated application of

[0142]

number

[0143] and

[0144]

number

[0145] is a reflection operator. By using phase estimation and the quantum Fourier transform, a can be reduced to O(M -1 ) accuracy. Unfortunately, the depth of the resulting quantum circuit is O(1 / ε), which requires the use of resource-expensive quantum Fourier transforms. Recent developments have introduced other approaches that aim to reduce the resources required to perform amplitude estimation and can eliminate the quantum phase estimation.

[0146] The most efficient variant of amplitude estimation known to date is Iterative Quantum Amplitude Estimation (IQAE). It has been empirically shown that IQAE outperforms other known variants. It omits quantum phase estimation, but achieves a performance four times better than the canonical phase estimation approach. Furthermore, for practical considerations, the following bounds are shown to hold:

[0147]

number

[0148] Where:

[0149]

number

[0150] denotes the worst-case number of oracle invocations, i.e., applications of Q, to achieve an estimation error of ε>0 with confidence 1-α,α∈(0,1).

[0151] 2.2 Route distribution loading

[0152] In order for Algorithm 2.1 to achieve practical quantum dominance, the path loadings and the resources to compute the payoffs are taken into account. In some cases, analytical forms exist that allow the path loadings to be simplified. For example, for path-independent derivatives, the distribution on the path is not relevant. What is relevant is the distribution of the final underlying price S, such as the log-normal distribution given by the Black-Scholes model. T For quantum states, the only distributions are for x, y, y, and z. This means that the distributions can be analytically calculated and variationally or explicitly loaded into the quantum state. Unfortunately, for quantum dots, the analytical form of this distribution means that these derivatives are typically classically simple. Thus, in accordance with various aspects of the subject disclosure, the quantum resource estimation system 102 may focus on path-dependent derivatives where superpositions over paths are calculated.

[0153] Loading of general distributions becomes exponentially difficult, but several methods have been proposed. If the distribution is efficiently integrable, then there exists the Grobel-Rudolph method as an efficient quantum algorithm for loading. However, this algorithm has limited applicability in practice to derivatives pricing, since the associated probability distributions are quantum, but still involve Monte Carlo integration which can be just as easily avoided using amplitude estimation. The drawbacks of this method are explained in more detail in Section 7.0.

[0154] Another approach to loading the path distribution involves the use of quantum generative adversarial networks (qGANs), which are attractive because they reduce the loading overhead, but it is not yet clear how to predict what the training overhead of a given qGAN will be in practice.

[0155] 2.3 Error analysis

[0156] In this section, we consider the various factors that contribute to the overall error in the quantum approach to option pricing. There are three main factors that contribute to the error in the approach of Algorithm 2.1: δ =f max -f min Let us assume that.

[0157] Truncation Error: The price of a derivative is determined by an integral over all possible values ​​of the underlying price or return. It is infeasible to calculate the integral over an infinite domain, and therefore the quantum resource estimation system 102 may limit the domain of the integral as follows: the price and / or logarithmic return are determined to be within the range [B l ,B u ]. This restriction of the domain excludes the probability mass of α. The density function for each step is P max There is an upper limit of f δ If there is an upper limit of

[0158]

number

[0159] There may be a truncation error, expressed as:

[0160] Discretization error: This error (ε disc The error (denoted as n) arises from the use of a Riemann sum over a finite lattice of points to approximate the integral. This error can be reduced by increasing the number of qubits (n) to approximate the sum.

[0161] Amplitude estimation error: In amplitude estimation, the error is 1 / ε in the state preparation procedure and price calculation. amp When iterations are used, ε amp An error occurs.

[0162] Truncation and discretization errors are discussed in more detail below.

[0163] 2.3.1 Truncation Error

[0164] In this section, we present the truncation error in the return space and then extend it directly to the price space. max Using Chernoff tail bounds on the Gaussian, the log return of asset i falls within the interval [μ i -wσ max ,μ i +wσ max The probability of being outside of

[0165]

number

[0166] The union bound allows us to determine whether any logarithmic return (for example, for d assets over T time steps) falls within the interval

[0167]

number

[0168] The probability of being outside is

[0169]

number

[0170] The initial asset price is bounded by

[0171]

number

[0172] Then, the corresponding interval in the price space is

[0173]

number

[0174] is given by:

[0175] The quantum resource estimation system 102 can then define a truncated window of dT distinct n-qubit register values ​​that are w standard deviations around the mean value for each time step. The truncation error can then be expressed as:

[0176]

number

[0177] is given by:

[0178] 2.3.2 Discretization error

[0179] The final output of the amplitude estimation algorithm represents a Riemann sum that approximates a truncated multidimensional integral. The integral is over dT variables corresponding to d assets at T time steps. Each underlying asset and / or return is scaled over the interval [B l ,B u To calculate the discretization error, we apply a multidimensional variation of the midpoint formula as follows: Let there be n qubits used to represent each underlying asset, and the domain is 2 ndT cells, and for each value in the register we associate the value of the integrand at the midpoint of the corresponding cell. Assume that β provides an upper bound on the second derivative of the integrand (e.g., this means that the deviation from linearity over a range of length l is βl 2 / 2).

[0180] Consider the error from the discretization accumulated over a single cell. Each cell has side lengths (B u -B l ) / 2 nand is a hypercube of dimension l. Note that by symmetry, the linear component of the deviation from the value at the cell center integrates to zero over the cell. Thus, the error at each cell is the length of the side centered at the origin, l = (B u -B l ) / 2 n The term βx on the dT hypercube 2 It is possible for the integral to be bounded by / 2.

[0181]

number

[0182] Aggregating the errors across all cells is

[0183]

number

[0184] to provide.

[0185] The number of standard deviations used in discretization and the maximum eigenvalue σ of the covariance matrix max With respect to, the total discretization error is

[0186]

number

[0187] It is bounded by.

[0188] For the target discretization error, (23) further becomes:

[0189]

number

[0190] gives the total number of qubits that can be used to represent d assets for T time steps, given by

[0191] Truncation and discretization errors apply generally to the methods introduced here, but each method has further method-specific sources of error, which are discussed separately for each approach.

[0192] 3.0 Advantages of Methods in Pricing Quantum Derivatives

[0193] The following sections describe two approaches that may work effectively to practically price quantum derivatives: Riemann summation and the reparameterization method of the subject disclosure. Riemann summation has been introduced previously, and a first resource analysis, the application of which achieves quantum supremacy, is described herein in accordance with one or more aspects of the subject disclosure. This analysis reveals the limitations of error scaling due to normalization. As described below, the new reparameterization method, which may be defined and / or implemented by the quantum resource estimation system 102 in accordance with one or more aspects of the subject disclosure, avoids the shortcomings of other methods and in fact provides the first end-to-end path to quantum supremacy.

[0194] 3.1 Riemann sum

[0195] The Riemann summation method provides an approach to construct the P-path loading operator in Algorithm 2.1. ndT Let be the size of the Hilbert space that contains all possible paths.

[0196]

number

[0197] is the maximum value of the d asset multivariate transition probability in equation (2). Then,

[0198]

number

[0199] teeth,

[0200]

number

[0201] and

[0202]

number

[0203] The normalized transition probability for all choices of . The asset price of each asset at each time step is in the interval [0,S max The steps of the method summarized in Algorithm 3.1 defined below consist of the normalization coefficients

[0204]

number

[0205] Using

[0206]

number

[0207] We calculate the derivative price as . Note that the normalization factor in the last step scales exponentially with T. P max < 1, no normalization is involved and this factor is redundant. However, P max >1, the error grows exponentially, making this approach impractical.

[0208] Algorithm 3.1 – Riemann sum pricing

[0209] Use n, d, and T, all of which are positive integers.

[0210]

number

[0211] The operator W applies the transition probability of the stochastic process to the ancillary process via t ,t=1,…,T.

[0212] 1. Apply Hadamard to the ndT qubit to prepare an equal superposition of all paths.

[0213] 2. Initial Price

[0214]

number

[0215] Load into the 0th nd qubit register.

[0216] 3.T transition operators W t By applying each of

[0217]

number

[0218] Configure.

[0219] where N=2 ndT It is.

[0220] 4.

[0221]

number

[0222] is inserted into the quantum register and calculated,

[0223]

number

[0224] Get the.

[0225] 5. Introducing Ancilla qubits

[0226]

number

[0227] Rotate the value in the register to that amplitude.

[0228]

number

[0229] 6. Use amplitude estimation to extract the probability that the ancillary is |1〉, which is the (e.g. discretized) expected payoff

[0230]

number

[0231] This can be rescaled to

[0232]

number

[0233] get.

[0234] Normalization factor P max is more tractable in return space, where the probability density function is given by equation (8). The log return is given by the asset volatility σ j When it is discretized to ±w times of

[0235]

number

[0236] It is.

[0237] If the d assets are uncorrelated, then:

[0238]

number

[0239] Therefore, P max If <= 1,

[0240]

number

[0241] However, choosing a small discretization window w increases the truncation error described in Section 2.3.1,

[0242]

number

[0243] in the case of,

[0244]

number

[0245] which grows linearly with the number of assets and time steps in the computation.

[0246] 3.1.1 Riemann summation error analysis

[0247] In addition to the truncation and discretization errors of Section 2.3, the Riemann summation approach contains errors due to scaling issues and quantum arithmetic.

[0248] When operating in the return space, one transition operator is used to compute (12) and

[0249]

number

[0250] The transition operator performs amplitude coding of the maximum summation error ε dens The payoff operator for calculating equations (27) and (28) is the payoff error ε f Assuming that we introduce

[0251]

number

[0252] Neglecting the quadratic error term, we obtain:

[0253]

number

[0254] Here, the logarithmic return for each asset and each time step is given by the domain [-wσ max ,wσ max ] is constructed to discretize

[0255] Probability density error ε dens is given by equation (12)

[0256]

number

[0257] have

[0258]

number

[0259] and arises from the calculation of the ancillar rotation in equation (26). The term in the exponential function in equation (12) is

[0260]

number

[0261] It can be described as:

[0262] Where:

[0263]

number

[0264] and C ij are classical variables containing volatility and correlation parameters from the correlation matrix Σ. In equation (33),

[0265]

number

[0266] For each calculation of ε A There is an error of d+d2)T times in total.

[0267]

number

[0268] The term, by construction, is |w|σ max where each quantum register representing the logarithmic return R is bounded by a window [-wσ max ,wσ max Using the error analysis for addition and multiplication in Section 8.2, the total error in computing Equation (33) is

[0269]

number

[0270] It is.

[0271] Then, using the error propagation analysis of Section 8.2, we compute the exponential, square root, arcsine, and sine functions on a quantum register that already contains arithmetic errors, and we get ε dens teeth,

[0272]

number

[0273] It is bounded by.

[0274] Each rescaling performed on the input variables introduces a corresponding rescaling error. max In addition to the rescaling, the quantum resource estimation system 102 scales the payoffs to 1 / f δ The final answer is,

[0275]

number

[0276] These rescalings can be taken into account by multiplying with , and thus the error in the estimate of the truncated integral with the amplitude estimate is

[0277]

number

[0278] The quantum resource estimation system 102 can then bound the error with a Riemann summation approach.

[0279]

number

[0280] Here, ε trunc , ε disc, and ε amp is defined as in Section 2.3.

[0281] 3.1.2 Resource Estimation

[0282] As an example, consider a basket autocallable with 5 autocall days and parameters T=20, d=3, and ε total / f δ ≦2×10 -3 The target error is set to be 1. Select w~5 so that the truncation error of equation (20) is within the total target error, and equation (30) is P max ≒4 3 This gives a very large scaling factor,

[0283]

number

[0284] However, there may be some ways to address this regularization problem, such as the method inspired by importance sampling and discussed in Section 9.2. Therefore, let us assume that some method has been invented to address regularization, and P max =1.

[0285] Then, using the resource calculations discussed in Section 9.1, we set ε arith ≦2×10 -3 where p is the integer part of the fixed-point representation defined in (63). In this case, the Q operator uses 21k qubits and has a T-depth of 23k, including the resources to compute the price from the logarithmic return using (11). Δt=1 / 20 and σ min For the selection of =0.1, we calculate that β≒17. max = 0.4 and w = 5. Therefore, for the selection of n, ε disc ≒f δ 10 -5 and

[0286] ε trunc ≦f δ 10 -4 (37) It is.

[0287] Amplitude estimation goal ε amp to 10 -3 , the target reliability is α=10 -2 Then,

[0288]

number

[0289] This means that the total T depth is about 1.9×10 8 This means that.

[0290] Using the same analysis, for a TARF contract with d=1, T=26, and Δt=1 / 26 (see, for example, Section 4.2), assuming annualized volatility of the underlying asset σ=0.4, ε total / f δ ≦2×10 -3 The target error is 1.7×10 8 A total T depth of 100 k qubits can be achieved with 10 k qubits.

[0291] 3.2 Reparameterization Method

[0292] The limitations of normalization in Riemann summation motivate the development and implementation of new methods of loading stochastic processes. In the reparameterization method, the quantum resource estimation system 102 can move to modeling assets in return space. As described in Section 1.2, in return space, the underlying assets include uncorrelated normal distributions. It should be recognized that these different distributions can be loaded by preparing many standard normals (e.g., in parallel) and then applying an affine transformation to obtain the mean and standard deviation. In this approach, one extracts a specific subroutine (loading the standard normal into the quantum state) and uses it as a resource to load the full distribution of the underlying paths. The normal loading subroutine itself can be pre-computed and optimized using variational methods. This is an advantageous combination of fault-tolerant quantum computing and variational compilation, which will be discussed in Section 3.2.1. Overall, the reparameterization method avoids the normalization problem in Riemann summation and reduces computational costs and / or resources. The steps in reparameterizing pricing are described in Algorithm 3.2 defined below. In this context, a path ω R ∈Ω R is a series of logarithmic returns

[0293]

number

[0294] Note that the reparameterization method refers to P max Remove the problematic dependency on operator G j can be implemented with relatively few resources using variationally learned circuits, as described in the following section.

[0295] Algorithm 3.2 - Reparameterization method pricing

[0296] Use n, d, and T, all of which are positive integers.

[0297] Standard Gaussian distribution

[0298]

number

[0299] We gain access to an operator G that loads g into an n-qubit register. i , 2 n Let be the value of the probability mass function of a standard Gaussian distribution discretized into bins.

[0300] 1. Apply dT Gaussian operators G to ndT qubits. This constitutes:

[0301]

number

[0302] Where:

[0303]

number

[0304] is the square of this multivariate standard Gaussian ndT Over all distinct realizations of

[0305]

number

[0306] denotes the corresponding probability.

[0307] 2.Σ=LL T Let be the Cholesky decomposition of the covariance matrix. Affine transformation

[0308]

number

[0309] We adjust the center and volatility of each Gaussian by performing R and p(ω R ) is used.

[0310] 3. If the payoff can be calculated directly from the log return,

[0311]

number

[0312] The quantum register

[0313]

number

[0314] and calculate directly.

[0315] First, if the payoff is defined in terms of price and not the correct log return,

[0316]

number

[0317] Calculate the price space path ω for each asset using: The calculation just described can be done in parallel for each asset (e.g., each derivative asset).

[0318] 4. Introducing an ancilla qubit and f(ω R ) register value to its amplitude as follows:

[0319]

number

[0320] 5. Use amplitude estimation to extract the probability that the ancillary is |1〉, which is the (e.g. discretized) expected payoff

[0321]

number

[0322] This can be rescaled to

[0323]

number

[0324] get.

[0325] 3.2.1 Variationally Trained Gaussian Loader

[0326] The standard Gaussian loading operator G can be precomputed since it is problem independent in the reparameterization method. This section describes an approach to variationally optimize this operator. _i =-w+iΔx, where i=0,…2 n -1, and Δx=2w / 2 n The standard normal distribution g(x _i ) in the example below. The domain has been fixed at w=5, so the total range of values ​​considered is 2w=10. This choice leaves about 5×10 outside the domain. -7 Finally, we take into account the different metrics used to normalize the function in real space and the wave function in the quantum register, which is normalized so that the sum of its squared elements is 1, and thus we aim to load the quantum register with the following distribution (for example the target distribution):

[0327] g(x i )×Δx, Σ i g(x i )×Δx=1. (41)

[0328] Note that due to the finite truncation region, the target distribution is normalized to 1-α. In principle, the distribution could be renormalized to 1 in the chosen interval of width 2w. In any case, this choice provides a negligible difference when compared to the error observed in training.

[0329] The variational ansatz chosen is represented by the so-called Ry-controlled NOT (Ry-CNOT) ansatz, which has linear connectivity (see, e.g., Section 11.0). Various aspects of the subject disclosure provide a novel strategy for optimizing circuits in this context, which relies on energy-based methods and is also detailed in Section 11.0. In essence, the target cost function is the energy of the associated quantum harmonic oscillator problem, whose ground state is naturally Gaussian. Note that the discretized quantum harmonic oscillator solution (e.g., the squared coefficient of the solution) can only match a normal distribution in the limit of Δx→0. To correct for this, we perform the following reoptimization, which directly targets the infinity norm between the two distributions.

[0330]

number

[0331] Here, the quantum state encoded in the register is the coefficient

[0332]

number

[0333] is defined as follows:

[0334] Note that directly learning Equation (42) as the cost function is inefficient, so pre-training using an energy-based approach is used. ∞We observe how the cost function exhibits a much more corrugated landscape in the circuit parameter space compared to the energy of the associated quantum harmonic oscillator problem.

[0335] Note that for different choices of register size n, the circuits that encode these Gaussian states can be pre-trained in advance, so that training is not included as part of the run time for any given derivatives pricing problem. Results for different register sizes and circuit Ansatz depths are shown in Figure 4. More details are provided in Section 11.0.

[0336] 4 illustrates an exemplary non-limiting graph 400 that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repeated descriptions of similar elements and / or processes utilized in each embodiment are omitted for the sake of brevity.

[0337] Graph 400 illustrates the L1 ∞ The numerical study shows that the states that can be variationally prepared approach the target exponentially faster in the depth and therefore the number of gate operations. This observation is in good agreement with the behavior expected from the Solovay-Kitaev theorem, which gives an upper bound on the number of gates that can be used to achieve a desired accuracy of the cost function. Indeed, for any target operation U∈SU(2 n ), SU(2 n A sequence of operators in a dense subset of

[0338]

number

[0339] exists, and the error ε in the energy is the depth D = O(log c (1 / ε)). The entanglement block in the circuit causes the SU(2 n ) operations are a subset of SU(2 n ), it can be numerically observed that the exponential reduction in error with the number of gates still holds.

[0340] At the end of this section, we investigate the portability of these results in a fault-tolerant regime, which could allow the application of the entire derivatives pricing algorithm. The numerical results provide evidence of a fairly efficient Gaussian state preparation in terms of the circuit depth of Ry-CNOT, but an additional step is taken in terms of a fault-tolerant implementation of such circuits. In this new framework, the continuous rotation Ry gate is extended as a finite product of discrete operations. Following again the Solov'i-Kitaev theorem or a more specialized result

[27] , it is also possible to have an efficient representation of any SU(2) operator in terms of a sequence of Clifford+T gates that scales logarithmically with the threshold error ε. We investigate how it is possible to transfer the previously obtained results in this regime, where the angle of rotation can only take discretized values. Thus, for each parameter

[0341]

number

[0342] is in the format j*2π / M digit where j is an integer. In Section 11.0 we will assume that the error introduced by such a digitization can be expressed only by 1 / M digit We numerically show how the σ decreases systematically with mesh size.

[0343] 3.2.2 Error analysis

[0344] The total error in the reparameterization approach is:

[0345]

number

[0346] Here, ε trunc , ε disc , and ε amp are the truncation, discretization, and amplitude estimation errors bounded in Section 2.3. Here, the term ε arith arise from the individual errors introduced during the preparation of the Gaussians and the calculation of the payoffs. i ) is L ∞ Error ε dens and the payoff calculation is ε f Assuming that the maximum error is

[0347]

number

[0348] Here, x=(x1,x2,…,x dT ) Expanding the integrand and keeping only the linear error term gives:

[0349] ε arith ≦2wdTf δ ε dens +ε f , (45)

[0350] Where:

[0351]

number

[0352] is used by truncation of the probability mass function.

[0353] 3.2.3 Resource Estimation

[0354] Calculate the resources that can be required for the same basket autocallable as in section 3.1.2, where d=3, T=20, Δt=1 / 20, σ max = 0.4, σ min = 0.1, and w = 5, and the contract has 5 autocall days. Further assume that each Gaussian is prepared with n = 5 qubits, ε dens =2×10 -6 , ε amp =ε f =10 -4 So that, ε total / f δ ≒2×10 -3 From Figure 4, we use 5 qubits and a circuit depth of 6 to obtain the total error of L ∞ ~2×10 -6 It can be seen that a Gaussian state can be prepared with α = 10, requiring 7 layers of Ry gates. With these inputs and the resource calculations described in Section 10.0, constructing the Q operator using reparameterization requires 7.5k qubits and has a T-depth of 5.7k, which includes the calculation of the price from the logarithmic return, i.e., equation (11). -2 At the target confidence level of , the total T depth is 4.6 × 10 7 Using the reparameterization method, total / f δ ≒2×10 -3 To price the TARF of section 3.1.2 with d=1, T=26, Δt=1 / 26 and σ=0.4 with an accuracy of 6×10 7 with a total T-depth of 100 and 9.5k qubits.

[0355] 4.0 Payoff

[0356] 4.1 Autocallable Contracts

[0357] Typically, an autocallable contract is defined in terms of an asset return compared to a predefined reference level and includes a notional amount that is used to calculate the dollar value of the contract. In the case of a single underlying asset, an autocallable can include:

[0358] Strike (rights exercise) K i , exercise time i , fixed payoff f i A set of binary options {(K i ,t i ,f i )} i=0…m-1 These are t i <t i+1 Assume that the data are sorted such that

[0359] Strike K o and short knock input with barrier b, and

[0360] · If any binary option pays off, all subsequent options and puts are knocked out.

[0361] The strike and barrier parameters are defined in terms of the return of the underlying asset price S(t) compared to a reference level, which, without loss of generality, can be taken as the initial spot price S0 of the underlying asset. The payoff f of a binary option i is similarly defined as a dimensionless parameter that indicates the return. In a return space whose basis vectors represent the logarithmic returns of the underlying assets (see Section 1.2), checking whether the underlying asset has crossed a strike or barrier K given a logarithmic return value R is a simple function of e R It may include checking whether ≧K.

[0362]

number

[0363] Let be the normalized payoff given by:

[0364]

number

[0365] Algorithm 4.1 – Autocallable Payoff Implementation

[0366] Parameter {(K i ,t i ,f i )} i=0…m-1 ,K o Find an autocallable with a and b.

[0367] 1. For each time step t=1...T, the cumulative return calculated by the selected path distribution loading method

[0368]

number

[0369] Assume access to

[0370] 2.Each t i A parallel comparator is applied to

[0371]

number

[0372] get.

[0373] 3.

[0374]

number

[0375] Successively, for each bit in the strike register, we select θ i We apply a controlled rotation of x to the accumulator qubit, which is shown in Figure 5. This introduces an m-qubit ancillary register a.

[0376] 4. Cumulative returns

[0377]

number

[0378] By applying a comparator in parallel to

[0379]

number

[0380] to indicate whether the put option was knocked in and whether it is in the money. These bits are then ORed together to obtain |put〉1, which holds whether the payoff from the put option is being taken into account.

[0381] 5.

[0382]

number

[0383] and normalize it using equation (46) to obtain the put option payoff |f p 〉.

[0384]

number

[0385] Calculate.

[0386] 6. Then, the controlled R y Using rotation to control both |put〉 and a[m-1]

[0387]

number

[0388] Rotate bitwise to the target qubit.

[0389] 5 illustrates an exemplary non-limiting circuit 500 that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repeated descriptions of similar elements and / or processes utilized in each embodiment are omitted for the sake of brevity.

[0390] Circuit 500 comprises an exemplary non-limiting circuit that can be used to accumulate binary option payoffs in an autocallable having five binary options, where qubits s0,...,s5 represent five strike values ​​K i Represents a Boolean comparison on the payoff f i (e.g., given by a particular phase of the R--R rotation) occurs only if no payoff occurred previously. The entire payoff is loaded into the amplitude of the qubit e0.

[0391] The amplitude estimate is the expected return of the contract.

[0392]

number

[0393] The dollar value of

[0394]

number

[0395] where N is the notional amount specified in the contract. Autocallables can be defined not only on one asset but also on baskets of assets. Typical examples include BestOf and WorstOf, where the contract's return is based on the return of the best or worst performing asset in the basket. These may be treated similarly to the single asset case, where the return of each asset is

[0396]

number

[0397] are first compared to find the maximum or minimum (eg, as appropriate).

[0398] Steps 2 and 4 of Algorithm 4.1 defined above can be implemented using logic circuits (e.g., comparators, ANDs, ORs) that do not introduce errors, while steps 3 and 6 involve the use of a controlled Ry rotation whose decomposition into T gates is a function of the summation error ε, which can be selected depending on the desired accuracy of the computation. Step 5 is the most resource-heavy component of the payoff circuit, which is the quantum register

[0399]

number

[0400] This can include computing,the,division of that register by a classical constant in the,denominator of (46), and computing the square root and arcsine,of the register.,The resource criteria of all the above circuit components are,detailed in Section 8.1, and the corresponding arithmetic and,gate synthesis errors are discussed in Section 8.2.

[0401] Consider again the autocallable contract in Section 3.1.2 and Section 3.2.3, which has 5 autocall days, is defined on d = 3 assets, and is simulated using T = 32 time steps. The total additive payoff error ε f Target ε f =10 -4 For,,assuming that the computation can be parallelized as much as possible, the,circuit to compute the autocallable payoff contains 1.6k qubits,,and a,T,depth of 2k.

[0402] 4.2 TARF

[0403] In this section, we describe the TARF implementation for a single underlying asset in the price space.

[0404] The TARF is as follows:

[0405] Forward price F, payment dates t1,…,t m , two strike prices K upper and K. lower ≦F, knockout price K o , and accrual cap

[0406]

number

[0407] The date is t i ≦t i+1 Assume that the data are sorted such that

[0408] ·Each time t i In, TARF has the payoff as follows:

[0409]

number

[0410] Any t i At price K o > F, all subsequent payoffs are knocked out.

[0411] · An accrual cap condition that if the total profits accumulated by any payment date exceed C, the contract holder will receive only those payoffs that bring the total profits equal to C, and the remaining forward contracts will be knocked out.

[0412]

number

[0413] Let be the normalized payoff given by:

[0414]

number

[0415] Algorithm 4.2 – TARF Payoff Implementation

[0416] Parameters (F, t1, …, t m ,K upper ,K lower ,K o , C) is obtained.

[0417] 1. Knockout and Accrual Cap Qubits |0〉 o , |0〉 c Start from.

[0418] 2. All time t i We apply three comparators in parallel to

[0419]

number

[0420] get.

[0421] 3. Each time t i About

[0422] (a)

[0423]

number

[0424] | o and onto the new qubit, which is | 〉 o It is possible to relabel

[0425] (b) Compute a payoff (e.g., conditional on different strike qubits) and add the payoff to a register that records the total payoff;

[0426]

number

[0427] get.

[0428] (c) Applying a comparator,

[0429]

number

[0430] Calculate whether the accrual cap has been reached to get

[0431] (d)

[0432]

number

[0433] Calculate the quantity that makes the sum equal to C to get

[0434] (e) Normalized payoffs when the accrual cap is met and not met

[0435]

number

[0436] , and then calculate the AND of each condition, (for example, a new) knockout qubit |·〉 o NOT, the accrual cap qubit |·〉 c Add the appropriate one controlled by NOT of

[0437]

number

[0438] get.

[0439] (f)

[0440]

number

[0441] and the cap qubit | 〉 c and store the result in a new qubit, which is | 〉 c It is possible to relabel

[0442] 4. |θ〉

[0443]

number

[0444] Calculate

[0445] 5. The final encoded qubit |0〉e Above, the controlled angle 2 for each quantum bit i in |θ〉 -i A series of controlled R y Apply rotation.

[0446] 5.0 Discussion

[0447] Various aspects of the subject disclosure described herein provide a complete resource and error analysis for pricing financial derivatives using quantum computers. In particular, these various aspects use autocallables and TARFs, two types of path-dependent options that are relevant in practice and difficult to price classically, as illustrative case studies. To this extent, various aspects of the subject disclosure provide a new method of loading stochastic processes that overcomes limitations of existing approaches. Although these various aspects include geometric Brownian motion, the subject disclosure is not so limited, and the approaches described herein can be easily extended to stochastic or local volatility methods, for example, by loading multiple independent stochastic processes and having conditional or non-stationary reparameterizations.

[0448] The resource estimates presented herein provide a target performance threshold for quantum computers to be able to take advantage of derivatives pricing. Assuming a one second target for pricing autocallable options, a 10 1 The code distance that can support one logical operation will have a logical clock rate of 10 MHz.

[0449] Current estimates of logical clock rates are around 10 kilohertz (kHz), i.e., roughly three orders of magnitude below, but future research into algorithms, circuit optimization, error correction, and hardware will continue to improve resource criteria and execution times. For example, in the case of Shor's algorithm, careful analysis across several publications has reduced the estimated resource criteria by nearly three orders of magnitude. The subject disclosures described herein represent a first milestone in the journey toward quantum advantage for pricing financial derivatives.

[0450] 6.0 Background on Derivatives

[0451] 6.1 Forward

[0452] An example of a derivative is a forward contract (often simply called a forward). In a forward, the holder promises to buy or sell a particular asset to the issuer at a fixed price F, known as the forward price, on a specified date in the future. A simple path-independent example might be when the holder promises to buy a quantity x of an asset m months from now for $F per asset. Forwards are typically settled in cash, meaning that rather than an exchange of money for an asset at maturity, there is only an exchange of money determined by the payoff, based on the value of the asset. For example, if the price of an asset at maturity date T is S, then T If so, the payoff is f(S T )=x(S T -F), and f(S T )>0, the contract holder makes a profit (the issuer makes a loss), and f(S T )<0, the opposite is true.

[0453] 6.2 Options

[0454] Another example of a derivative is an option. An option can be considered a conditional forward. Unlike a forward, where the issuer is obligated to buy or sell an asset, in an option contract the holder has the option to buy or sell a specific asset from the issuer at a predetermined price at a future date. If the holder chooses to buy or sell the asset, the holder has chosen to exercise the option. Like forward contracts, option contracts are usually settled in cash based on the value of the asset on the exercise date. An example of a path-independent option with a single underlying asset is the European call option, where the issuer has the option to buy the asset at the expiration date at a strike price K. The payoff at expiration is f(S T )=max(S T -K,0). In a European put option, the issuer has the option to sell an asset at the expiration date at a strike price K, and f(S T )=max(KS T ,0). Another example of a path-independent option is a binary option with a fixed payoff if the underlying asset is above or below the strike at time T.

[0455] 6.3 Path Dependence and Discounted Payoffs

[0456] An example of a path-dependent derivative is the knock-out European call option. It is the same as a European call option but with an additional knock-out price π. If the underlying asset rises above this value at any time between 0 and T, the contract becomes worthless. This path-dependent payoff function has the following form:

[0457]

number

[0458] Including the value of the underlying asset at times other than T introduces path dependence. Another example is a knock-input option with a payoff such as:

[0459]

number

[0460] Here, the contract is a knock-in because it has a nonzero payoff only if the asset falls below a value π, the amount by which it falls below the strike price K.

[0461] The examples discussed so far have only one payment date at time T, when the exchange occurs between the contract issuer and holder. Some path-dependent options can have several payment dates, where multiple payments are made at different times throughout the life of the contract.

[0462] We now introduce the concept of discounted payoff. As expected, the current price of a derivative is related to its expected payoff in the future. However, a time lag of the payoff can also be considered to account for the opportunity cost of investing in a risk-free asset with interest rate r. A contract is a contract that is traded at a time t from the present. i Payoff at f i , the discounted payoff can be defined as:

[0463]

number

[0464] The price of a derivative contract is given by the expected value of the discounted payoff in the stochastic process of the underlying asset. In practice, path-dependent derivatives are much more difficult to price computationally and are often priced using Monte Carlo simulations of paths. This contrasts with models of path-independent derivatives that may have analytical solutions, such as the Black-Scholes model of European call options. Path-dependent options offer an opportunity to take advantage of quantum speedups to Monte Carlo.

[0465] 6.4 Autocallable Options

[0466] A typical example of an autocallable option is a collection of binary options, each of which pays a different fixed payout on a different payment date, and which knocks out the entire instrument (i.e., negates all future payoffs) if it pays out on any of the payment dates. More formally, (K,t i ,f i ) at time t i If the underlying asset value is above the strike price K, the payoff is f i where the autocallables are the set:

[0467] {(K,t1,f1),(K,t2,f2),…,(K,T,f T )}, (52)

[0468] Here, {t i} and {f i} typically grows linearly. Binary options (K,t i ,f i ) pays out (in the money), all subsequent options {(K,t j ,f j )} j>i is knocked out (e.g., disabled).

[0469] In practice, these binary options are often bundled with short knock-in options, i.e., knock-in options granted to the issuer by the holder, thereby reducing the risk for the issuer and lowering the price for the client. As with the set of binary options, this put option can also be bundled with a binary option (K,t j ,f j ) will be knocked out if any of them pay out.

[0470] As is common with many options, autocallables can be extended to have multiple underlying assets. In this case, it is typical to tie the entire option payoff to the best or worst performing asset, where performance is defined in terms of returns, and the knock-in option's payoff is proportional to the return of the best or worst performing asset if that asset falls below the strike price. The strikes for the underlying j are often the same in the return space, i.e., for each asset K j The strike price is

[0471]

number

[0472] Note that k is the same constant for all underlyings. Thus, if the worst performing asset is in-the-money (e.g., above the strike price), then all other assets are. Conversely, if the best performing asset is below the strike price (or out-of-the-money), then all other assets are. In principle, different underlyings could have independent strike prices, but this is not common, so all strike prices are

[0473]

number

[0474] An assumption may be made that is defined as:

[0475] Contingent payoffs, and knock-inputs, mean that autocallables have strongly path-dependent payoffs, which means that they are computationally expensive in practice, taking sometimes 5 to 10 seconds using classical Monte Carlo methods with at least 40,000 paths.

[0476] 6.5 Target Accrual Redemption Forward

[0477] A target accrual redemption note (TARN) is a derivative with a payoff capped at a specific target amount. Historically, this term referred only to bills, which is where it got its name, but now it has come to include any derivative with an accrual cap. Various aspects of the disclosure being targeted use a commonly used TARN called a target accrual redemption forward (TARF) as an illustrative case for implementing the quantum resource estimation system 102. A TARF is a forward set with several knockout conditions. Specifically, it is a derivative with a single underlying asset having several (e.g., 20 - 60) payment dates and a forward price F. Throughout the contract, K upper =F and there can be two fixed strike prices K lower <F. At each payment date t i there are several possible payoffs as follows:

[0478]

Number

[0479] where

[0480]

Number

[0481] is the price of the underlying asset at payment date t i and α is a positive constant.

[0482]

Number

[0483] Note that when , the payoff is negative and the holder of the derivative incurs a loss. The constant α makes this loss asymmetric, and when it does occur it is often 1 or 2.

[0484] In addition, TARF has two knock-out conditions based on the knock-out threshold π and the accrual cap C. The first condition states that if on any payment date the price of the underlying asset is equal to or greater than π, the derivative contract will be knocked out immediately (e.g., without any payment on that day). The second condition states that on any payment date t i The total payoff for holders is payoff f i would exceed the accrual cap C, the contract holders would instead receive only the amount that would bring their total payout to C, and the contract would be knocked out.

[0485] 7.0 Disadvantages of Global Rudolph Loading

[0486] The Groenberg-Rudolph algorithm is often cited as an efficient way to create quantum superpositions that correspond to classical distributions. Given a probability distribution {p i}, the algorithm creates a quantum superposition of the form:

[0487]

number

[0488] The algorithm is inductive in nature. First, we define a probability distribution over some number of vectors in the domain of interest. m We start by dividing the space into regions and assuming that there is a way to create the following states:

[0489]

number

[0490] Where:

[0491]

number

[0492] is the probability that the random variable is in region i. Then, add one qubit to the state of equation (55) and m The region is further divided into two probability distributions with the following evolution form: m+1 We aim to discretize:

[0493]

number

[0494] Here, α i (β i ) is the probability that the random variable is in the left or right half of region i.

[0495]

number

[0496] and

[0497]

number

[0498] Let be the left and right boundaries of region i, then the function

[0499]

number

[0500] is the probability that x is also in the left half of region i, given that it is in region i. A circuit can be constructed that can perform the following calculations:

[0501]

number

[0502] Where:

[0503]

number

[0504] Then, the angle θ at the m+1th quantum bit i A controlled rotation of the

[0505]

number

[0506] |θ i After uncompute

[0507]

number

[0508] which is the extension of the state of (55) to one additional qubit. After performing this iteration n=log2N times, we obtain a discretization of the distribution over a total of N points spanning n qubits.

[0509] In practice, the efficiency of the Groebel-Rudolph method depends on the ability to perform the integration of equation (57) in superposition. The argument in the original formulation is that any probability distribution that can be efficiently integrated classically using stochastic methods (e.g., using Monte Carlo) can equivalently be efficiently integrated quantum-wise. However, since the ultimate goal of quantum derivatives pricing is to provide a faster alternative to Monte Carlo integration for probability distributions, performing this integration as part of the initial state preparation without a corresponding quantum speedup negates the advantages offered by amplitude estimation as an alternative to Monte Carlo. Although efficient from a complexity perspective, this means that Groebel-Rudolph is an insufficient method for quantum advantage in derivatives pricing.

[0510] More recently, an approximate method for implementing the Grover-Rudolph algorithm for the standard normal probability distribution has been presented, and the authors propose the formula in equation (57) written as follows:

[0511]

number

[0512] This can be approximated as follows:

[0513]

number

[0514] The authors emphasize that for small δ, the approximation in (62) becomes sufficiently accurate for m ≥ 7 as the δ parameter decreases with each iteration of the Groenberg-Rudolph algorithm that adds qubits to the discretization. However, because the Groenberg-Rudolph construction is iterative, it is possible that terms for m < 7 can be calculated before the above approximation is possible. Thus, the integrals in (57) are calculated classically and loaded into the corresponding quantum registers. Although this approximation allows a simplification of the general Groenberg-Rudolph algorithm for the standard normal distribution after a certain point in the iteration, it does not change the fact that it may involve computing integrals over the entire domain of the probability distribution, and thus becomes practically infeasible for the same reasons as the original Groenberg-Rudolph method.

[0515] 8.0 Fixed-Point Quantum Arithmetic Resources

[0516] This section describes general quantum arithmetic operations and provisions for the composition of arbitrary rotations. These operations are used for resource estimation and error analysis. Quantum arithmetic can be included in the path loading using the Riemann sum method (Section 3.1), the reparameterization method (Section 3.2), and the payoff calculation described in Section 4.0. For the Riemann sum method, all arithmetic operations included in Eq. (12) can be performed, as well as the computation of the arcsine and square root of the quantum registers for the payoff calculation in Eq. (15). Algorithms for efficient performance of individual arithmetic operations have been identified, and resources are typically reported as a number of Toffoli gates or T-gates. Where arithmetic algorithms are implemented from prior work in the literature, gate costs are reported herein for the gate set reported by the authors.

[0517] In a fault-tolerant setting, an estimate of the T-depth of the circuit in the Clifford + T-gate set decomposition can be made, and an assumption can be made that a Toffoli gate can be constructed with a T-depth of 1 using ancilla qubits. For each operation, an assumption can be made that the resulting circuit can be parallelized as much as possible.

[0518] 8.1 Resource Estimation

[0519] All calculations are done in fixed-point arithmetic. The n-bit representation of a number x is:

[0520]

number

[0521] where x i ∈0,1 denotes the i-th bit of the binary representation of x, and p denotes the number of bits to the left of the binary point. The selection of n and p controls the error that can be tolerated in each calculation, and the resources that can be used to perform arithmetic operations on registers. Once the values ​​of (n,p) are chosen such that the total arithmetic error is acceptable for the problem under consideration, keep them constant throughout the analysis. It is possible to adjust these values ​​for different components of the circuit to reduce the total resources that can be used, but for simplicity of explanation of the subject disclosure, this potential optimization can be ignored.

[0522] TF f and T f Let denote the number of Toffoli gates and T-depth used to compute an arithmetic function or logical operation f. The estimate of the operation is a function of the fixed-point register size (n,p) used to represent the underlying quantum states involved in the computation.

[0523] Addition and / or Subtraction

[0524] It performs addition of two given n-qubit registers with a Toffoli cost of 2n-1. Subtraction is given by ab = ~(~a + b) and can therefore be implemented as addition with 2n additional X gates, which does not change the Toffoli count.

[0525] Given the T-depth cost of controlled and uncontrolled addition, an addition circuit can be T-depth regardless of register size. add = 10, and controlled addition can be constructed with a T-depth cost of O(n) for registers of size n.

[0526] Multiplication

[0527] Multiplication uses a controlled adder circuit and the following Toffoli counts:

[0528]

number

[0529] This method may also be used for division of a quantum register by a classical value, which can be done by inverting the classical value and utilizing a multiplication algorithm.

[0530] Fixed-point multiplication involves n controlled additions and is therefore O(n 2 ) for T-depth cost. These methods can use ancillary qubits proportional to the register size, but because the circuit involves inversely computing the ancillary, the ancillary can be reused for each subsequent addition that is not done in parallel. Because the computation can be parallelized over d assets and T time steps, include an additional T*d*n qubits when counting the sum to account for these potential ancillary qubits.

[0531] Furthermore, by considering the registers of one factor as z≧1 independent registers of size n / z, we can parallelize each multiplication circuit and perform each controlled addition in parallel on z sub-registers. This allows the z sub-results to be accumulated into a final result using n·(z-1) additional qubits, and z-1 additions. z=1 indicates that no additional parallelization is employed. If the pairwise accumulate additions can also be parallelized, then the total T-depth cost of a parallelized fixed-point multiplication is given by:

[0532]

number

[0533] square root

[0534] We exploit the square root algorithm, which can be extended for quantum registers in fixed-point representation. For a number x of size (n,p), we compute the square root by treating x as an n-digit integer and shifting the result to the right (np) / 2 times.

[0535]

number

[0536] This amounts to doing the following:

[0537]

number

[0538] The Toffoli count for this square root algorithm is:

[0539]

number

[0540] As reported by the authors, the T-depth of this algorithm is T sqSince (n)=5n+3, 2n+1 quantum bits can be used.

[0541] Logical Operations

[0542] For comparisons between quantum registers, or between a quantum register and a constant,

[0543]

number

[0544] We use a logarithmic comparator with a Toffoli / T depth of 1, which includes an intermediate ancillary. A logical OR operation on a two-qubit input can be performed with a Toffoli / T depth of 1.

[0545] Exponential Function

[0546] A general quantum algorithm can be used to compute smooth classical functions using parallel piecewise polynomial approximations. It is applied to estimate the resources to compute exponential functions. The algorithm takes parameters k and M, which control the polynomial order and the number of domain subintervals selected for the piecewise approximation, respectively. The Toffoli sum is given by:

[0547]

number

[0548] The algorithm can also be used to calculate the inverse sine function, which involves k iterations of multiplication and addition, using a k-th degree polynomial for the approximation. Furthermore, for the selected M subintervals, it includes M comparison circuits between an n-qubit input register and a classical value.

[0549]

number

[0550] Using a comparator with a T-depth of , the T-depth of the parallel polynomial evaluator is:

[0551]

number

[0552] where z is the optional parallelization factor introduced in the above resource estimation for the multiplication circuit.

[0553] The number of qubits for the parallel polynomial evaluation scheme for a choice of polynomial order k and number of subintervals M is:

[0554]

number

[0555] arcsine

[0556] To calculate the arcsine, we use the general quantum algorithm described above, just as can be done for exponential functions. However, the derivative of:

[0557]

number

[0558] Since diverges around ±1, the authors of the general quantum algorithm propose the following transformation:

[0559]

number

[0560] to handle the interval x ∈ [0.5,1]. Since the computation of the inverse sine may involve a conditional square root evaluation of the argument, we also compute the square root whenever the inverse sine is computed (e.g., equation (15)) and use the following transformation:

[0561]

number

[0562] Resource estimation considerations become similar to those in the prior art approach. We obtain:

[0563] · indicates whether the above transformations should be applied,

[0564]

number

[0565] Assuming the values ​​in the quantum register are normalized, it is possible to use two Toffoli gates.

[0566] ·

[0567]

number

[0568] or

[0569]

number

[0570] Conditional subtraction and conditional copy depending on the value of the comparator above to prepare either TF add +n Toffoli gates can be used.

[0571] ·TF for square root calculation sq Toffoli Gate.

[0572] Toffoli gate used for polynomial evaluation to calculate the arcsine

[0573] For x<0.25

[0574]

number

[0575] , otherwise

[0576]

number

[0577] A conditional copy and conditional subtraction that again depends on the comparator result from the first step to get

[0578] Using these considerations and the Toffoli count of the polynomial approximation of arcsin(x), we obtain

[0579]

number

[0580] The total toffoli count to calculate is:

[0581]

number

[0582] of a number x represented in a register of size (n,p), calculated similarly to the exponential function

[0583]

number

[0584] The T depth for computing is:

[0585] T arcsq (n,p,z)=T sq (n)+T pp (n,z)+8n+6, (75)

[0586] Here, T sq (n)=5n+3 is the T-depth for the square root algorithm.

[0587] This operation is called q arcsq qubits, where the qubit reference for the arcsine may be given by equation (70) for choices of k and M and 2n+1 for the square root operation as follows:

[0588] q arcsq (n,k,M)=q pp (n,k,M)+2n+1 (76)

[0589] R y

[0590] R in the variational preparation of the Gaussian discussed in section 3.2.1 y (θ) rotation, and the controlled R for encoding the payoffs into the ancillary amplitudes in Eq. (16). y Rotations, as well as transition probabilities in the Riemann summation method are used. Using the Repeat-Until-Success method, any single-qubit unitary can be performed to within accuracy ε with a T-depth of approximately 1.15log2(1 / ε) using one ancilla qubit and measurements.

[0591] When the rotation angle θ is stored in a separate register |θ〉, a series of R y (θ k ) rotations, each controlled by the kth qubit in |θ〉, where

[0592]

number

[0593] It is.

[0594] A single controlled R n is a single R n depth, three Rs n count, and a single ancilla qubit, can be implemented using decomposition. However, since each rotation contributes an error ε, if |θ〉 is an n-qubit register (e.g., having p bits to the left of the binary point), the end-to-end operation can be implemented with a precision of ε with a maximum T-depth of 1.15n log2(n / ε). θ k A controlled R with <arcsin(ε) n Note that the amplitude increase due to the rotation is less than ε and is thus unnecessary, and reduce this depth slightly. Thus, using this observation and Equation (77),

[0595]

Number

[0596] calculate the total number of rotations that result in. This gives the final T-depth of the controlled Ry(θ) operation as follows:

[0597]

Number

[0598] where

[0599]

Number

[0600] is.

[0601] 8.2 Error analysis

[0602] Given the fixed-point representation of equation (63), each arithmetic operation requiring registers will result in some approximation error depending on the particular method used. Here we outline the arithmetic errors associated with each operation described in the previous section.

[0603] Addition and / or multiplication

[0604] It uses fixed-point addition and multiplication and adds two (n,p) sized registers.

[0605]

number

[0606] If multiplication yields an error bounded by: then the error associated with multiplication is, at most,

[0607]

number

[0608] For registers X and Y of size (n,p), each register has an additive error ε X , ε Y If we already include the factor b and each factor X, Y is upper bounded by b, the error in the calculation of X, Y is given by:

[0609] ε mul =b*(ε X +ε Y )+ε X ε Y +ε M (n,p) (80)

[0610] Exponential Function

[0611] Parallel polynomial evaluation methods are used to estimate the resources and associated error in computing exponential functions. The error associated with the algorithm depends on the degree of the selected polynomial approximation and the choice of the number of subintervals, but the authors of such algorithms have been able to estimate the error in the order of 10 -5 From 10 -9 We provide explicit error estimates and corresponding resources for errors in the range . We use these for the overall error estimate. We compute the exponential function of the register, which itself has arithmetic error ξ. We compute the error ε in computing the exponential function of the register. exp Showing that the total arithmetic error in computing the exponential function of the register can be approximated to first order in the Taylor expansion of exp(-x+ξ) as follows:

[0612]

number

[0613] square root

[0614] As discussed in the previous section, for the square root calculation, consider the square root algorithm extended to a quantum register in a fixed-point representation. The mapping in (66) can result in a maximum error of

[0615]

number

[0616] When computing the square root of a register x that already contains a (say positive) additive error ξ, the total additive error due to the square root operation is

[0617]

number

[0618] This is the estimated

[0619]

number

[0620] This can be easily understood by observing that using the square root algorithm gives:

[0621]

number

[0622] and then

[0623]

number

[0624]

number

[0625] where the first inequality is, for positive x and ξ,

[0626]

number

[0627] is derived from.

[0628] arcsine

[0629] To calculate the arcsine, we again used the polynomial estimation method, and the sample resource estimate of the error rate was 10 -5 From 10 -90.5. We first bound the error from computing the arcsine on a register, including the arithmetic error ξ. As discussed in Section 8.1, we compute arcsin(x) for x≦0.5. Furthermore, when we compute the function arcsin(x) in the algorithms of the subject disclosure described herein, we do so only for x≧0. This gives us a domain for the arcsin(x) error computation, 0≦x≦0.5. Note that given this domain, the slope of arcsin(x) is always monotonically increasing, with a maximum at x=0.5. Thus, computing the error at x=0.5 gives us the following upper bound:

[0630]

number

[0631] Here, ε arcsinは , the error due to computing the arcsine given the choice of polynomial order and number of subintervals.

[0632] sine

[0633] As discussed in the previous section, we compute the sin(θ) function with a series of controlled Ry rotations controlled by qubits from a register containing the angle θ. We bound the error from computing sin(θ) when the register assumed to represent θ actually represents θ+ξ due to arithmetic error. To quantify the upper bound, we note that in the region 0≦θ≦π / 2, the slope of sin(θ) decreases monotonically and thus has its maximum slope at θ=0. Thus, computing the error when θ=0 gives us the following upper bound:

[0634]

number

[0635] Here, the inequality sin(a+b)≦sin(a)+b for b≧0 is used, and ε sinis the error arising from the gate decomposition of the Ry operator described in Section 8.1.

[0636] 9.0 Riemann sum

[0637] 9.1 Riemann summation path loading resource estimation

[0638] In this section, we consider the T depth and number of qubits to compute equation (12) in the quantum register and encode the value into the amplitude of the ancillary qubits as described in Algorithm 3.1. The computation is done in log-return space (see Section 1.2) and includes resource estimates for the operations described in Section 8.1.

[0639] T f and q f Let denote the T-depth and number of qubits of operation f, respectively. Assuming that the computation over d assets and T time steps can be parallelized as much as possible, we obtain

[0640]

number

[0641] have

[0642]

number

[0643] The resource contribution to compute

[0644] T to calculate the term (R-μ) that can be done in parallel for d assets and T time steps add Here, T*d*n qubits are used to hold the logarithmic return R for s assets and time steps.

[0645] For all R in the expansion of (33) (e.g., in parallel for all d and T), which obtains and / or uses T*d*n additional qubits 2 T for the term mul .

[0646] All R in the expansion of (33) (e.g., parallel with respect to d,T) i R j T for the term mul *d2 / (d / 2) and T*d2*n qubits.

[0647] To sum all the terms in equation (33) in parallel

[0648]

number

[0649] The qubits from the previous step can be reused here.

[0650] · The choice of parameter values ​​is determined by the desired accuracy of approximation, given by equation (70) q exp With q exp T to calculate the exponential function in equation (8) using an additional qubit exp .

[0651] The qubit resource is given by (76),

[0652]

number

[0653] T to calculate the arcsine and square root of arcsq .

[0654] · All sums for t'∈[2,T] in equation (11)

[0655]

number

[0656] To calculate T add *(T-1) and (T-1)*d*n qubits

[0657] ·q exp T to compute the prices over all assets and all time steps in equation (11) in parallel using *d*T additional qubits. exp .

[0658] ·

[0659]

number

[0660] A T-depth of 1.15nlog2(n / ε) for an ancillary rotation to precision ε, controlled on the register on which R is computed. This is done by y It contains n ancilla qubits using decomposition.

[0661] Additionally, when using the parallel multiplication scheme described in Section 8.1 to compute prices across assets and time steps, additional registers of size T*d*n are included to implement a summation circuit with constant T-depth and (z-1)*T*d additional qubits, where z≧1 is an arbitrary parallelization factor that is chosen. The number of additional qubits to compute the (R-μ) terms and sums in Equation (33) is calculated for each R i Note that these are not included because they can be done in place using existing registers to hold R after computing the sum and exponential functions in equation (11) (which can be done, for example, before computing the sum). i This is possible because the value of is not used again.

[0662] Then the total T-depth of the Riemann summation path loading process of precision ε for d assets and T time steps using registers of size (n,p) is

[0663] T RS (n,p,d,T,ε)=n 2 +2n 2 d2 / d+10(d2+d)+10T+9n+5+1.15nlog2(n / ε)+2T exp (n,p,ε)+T arcsin (n,p,ε), (85) It is.

[0664] Here, T for ε exp and T arcsin The dependence of θ indicates that the polynomial approximation parameters k and M in equation (69) for each function depend on the target precision of the process. The total number of qubits involved is

[0665] q RS (n,p,d,T,ε)=Tn(4d+d2)+3n+1+q exp (n,p,ε)(1+dT)+q arcsin (n,p,ε). (86) It is.

[0666] 9.2 Importance Sampling for Regularization in Riemann Summation

[0667] In this section, we introduce a technique closely related to classical importance sampling to overcome the exponential scaling problem presented in Algorithm 3.1. The main idea is to approximate the target distribution by another distribution that can be efficiently loaded, and use quantum arithmetic only to adjust the (e.g. multiplicative) errors.

[0668] P>1, and

[0669]

number

[0670] Assume a univariate probability density function f:[0,1]→[0,P] where P is the probability density function f(x) / P, and a payoff function g:[0,1]→[0,1]. In the considered context, g is applied only once. It can therefore be assumed that it takes values ​​in [0,1] without changing the overall complexity of the approach. As introduced earlier, consider a scaled function f(x) / P and corresponding operator F, as well as a corresponding operator G, to prepare a state with n+2 qubits given by:

[0671]

number

[0672] where x i = i / N. Then the probability of measuring |11〉 for the last two qubits is

[0673]

number

[0674] and when multiplied by P, if X~f,

[0675]

number

[0676] corresponds to a Riemann sum that approximates

[0677] Furthermore, the probability distribution h(x i ) ∈ [0,1]. That is, it is known that we can efficiently construct the quantum operator H as follows:

[0678]

number

[0679] Assume that there exists h such that for all x, f(x) / (h(x)N)∈[0,1], and then we define a new operator F h Configure:

[0680]

number

[0681] H and F h Combining these gives us:

[0682]

number

[0683] This means that the probability of measuring |11〉 for the last two qubits is given by:

[0684]

number

[0685] That is, when X~f

[0686]

number

[0687] Therefore, if we have such a probability distribution h, we can multiply it by P and rescale it without any rescaling.

[0688]

number

[0689] We construct a state that corresponds directly to P. It can be easily seen that if P≦1, we can set h(x)=1 / N and recover the original approach without importance sampling.

[0690] For multivariate probability density functions, three cases are distinguished. First, for t = 0, …, T, we have a univariate function f t In this case, the univariate approach can be applied directly, and each f t The corresponding h t Second, P>1 and

[0691]

number

[0692] A non-separable multivariate probability density function f:[0,1] where d →[0,P]. Each dimension is discretized using n qubits, i.e., N d Then, for all x, f(x) / (h(x)N d )∈[0,1], and the analysis is similar to the univariate case. Finally, consider the case of a multivariate probability density function obtained from a stochastic process and given by:

[0693]

number

[0694] where x t ∈[0,1] d , and for t = 0, …, T, f0(x0), f t (x t |x t-1 )∈[0,P]. We assume the following separable probability distributions:

[0695]

number

[0696] This can be efficiently loaded and the corresponding decomposition is

[0697]

number

[0698] where:

[0699]

number

[0700] At this time,

[0701]

number

[0702] It is.

[0703] Therefore, h t If h can be found such that it can be loaded efficiently,

[0704]

number

[0705]

number

[0706] and then the exponential scaling overhead P T+1 Efficiently load stochastic processes without using

[0707]

number

[0708] is set,

[0709]

number

[0710] and it can be easily seen that the original approach can be recovered without importance sampling.

[0711] Note that even if it is not possible to find an h that satisfies all criteria, the approach just described can still help reduce the overhead resulting from scaling.

[0712] 10.0 Reparameterized Pathway Loading Resource Estimation

[0713] To prepare a standard normal distribution that can be used by the quantum resource estimation system 102 in implementing the reparameterization loading approach of the subject disclosure, the quantum resource estimation system 102 (e.g., via the variational component 202) can utilize the variational techniques described in Section 3.2.1 and corresponding gate and / or qubit costs, depending on the desired accuracy of the approximation. In addition, as described in Algorithm 3.2, the affine transformation

[0714]

number

[0715] Note that the affine transformation is used to derive asset prices from log returns, which for asset j at time t' is:

[0716]

number

[0717] Where:

[0718]

number

[0719] is sum

[0720]

number

[0721] is the i-th component of Equation (98). One complication in Equation (98) is that the asset prices cannot be calculated completely in parallel among the l assets because the log returns of any correlated assets contribute to the calculation of each other's price. If all assets are pairwise correlated, the quantum resource estimation system 102 can calculate the contribution to the price of each asset from the log returns of all l assets at that time step, and use a total of d 2 However, the quantum resource estimation system 102 can perform d additions in parallel, where the contribution of the return of asset j to the price of asset (j+i)%d is calculated for a choice of i∈[0,d−1], since all d such operations have distinct source and target registers. Then, by d additions, the term

[0722]

number

[0723] is calculated and the quantum resource estimation system 102 calculates in a separate register for each t′

[0724]

number

[0725] When computing , the above computation can be parallelized across all time steps.

[0726] The arithmetic error in computing Equation (98) can be reduced (e.g., minimized) by increasing the qubit register size to the largest possible value for the sum over time steps T and assets d. If each Gaussian prepared in Equation (38) is discretized using n qubits, then:

[0727]

number

[0728] The qubits are expressed as

[0729]

number

[0730] It is sufficient to hold the maximum value of

[0731]

number

[0732] The qubit has coefficients for all i,j.

[0733]

number

[0734] Assuming that,

[0735]

number

[0736] This condition achieves the same for (e.g., the definition |ρ ij |≦1) covariance matrix Σ ij =Δtρ ij σ i σ jIt is not difficult to meet typical situations of practical interest, which can be argued by looking at the elements of σ. Typically, the annualized volatility is less than 100% (i.e., σ i <1), the time step typically satisfies Δt<1, which means that the price of the underlying asset can be sampled more frequently than once a year. However, if neither condition is met, the quantum resource estimation system 102 may increase the number of time steps in the calculation by |Σ ij A smaller Δt can be chosen to ensure |<1.

[0737] In the reparameterization approach for derivatives defined on d assets and T time steps, the contributions to the T-depth and number of qubits for loading the paths and computing the asset prices are:

[0738] To load the Gaussian state in equation (38) using the variational method of section 3.2.1

[0739]

number

[0740] T-depth, where each Gaussian is prepared in parallel and the variational Ansatz has depth L. This step involves T*d*n qubits, where n qubits are used to prepare each Gaussian state.

[0741] Additional

[0742]

number

[0743] In equation (98), which contains qubits, for t'∈[2,T], all the sums

[0744]

number

[0745] To calculate T add *(T-1).

[0746] In equation (98),

[0747]

number

[0748] and

[0749]

number

[0750] T to compute all the contributions for the additional qubits add *d.

[0751] in equation (98) over assets and time steps

[0752]

number

[0753] T for calculating the contribution add .

[0754] · T for calculating the exponential function in equation (98) over assets and time steps exp , and q given by Eq. (70) exp With q exp *d*T additional qubits.

[0755] Overall, the total T depth for path loading using the reparameterization method with precision ε for d assets and T time steps is

[0756] T RP(n,d,T,L,ε)=1.15nlog2(n / ε)(L+1)+10(d+T)+T exp (n,ε), (99)

[0757] where the number of qubits is

[0758]

number

[0759] where:

[0760]

number

[0761] It is.

[0762] 11.0 Methods for Gaussian Loader Training

[0763] This section describes an approximate method for initializing quantum registers using a Variational Quantum Eigensolver (VQE) approach. The algorithm features a parameterized circuit that generates a parameterized state |ψ({θ})〉 that approximately represents a goal state |φ0〉, and updates its parameters {θ} to optimize the expectation of a suitable cost function. We show that the choice of the cost function to optimize is critical to successful training.

[0764] Energy-Based Training

[0765] As a first method, the quantum resource estimation system 102 may employ a physics-based approach and define an operator H to assume its lowest possible value E when its expected E value is evaluated at the target state, as follows:

[0766] E0=〈Ψ0|H|Ψ0〉 (101)

[0767] In physics, the operator H is usually called the Hamiltonian, E the energy, and |φ0〉 the ground state.

[0768]

number

[0769] It is well known that is the ground state of the quantum harmonic oscillator Hamiltonian:

[0770] H=P 2 2m+m(X-x0) 2 2, (103)

[0771] where X is the position operator in real space, P=-iddx is the momentum operator, m is a parameter that determines the variance of the desired Gaussian distribution, and x0 is the center of the Gaussian distribution.

[0772]

number

[0773] To find the state φ(x) such that 2 ) can be adjusted.

[0774] Note that it is always possible to find the generating Hamiltonian function such that the ground states are the square root of the smooth distribution function to be loaded (eg, the target probability distribution).

[0775] To translate such considerations into an operational workflow, the quantum resource estimation system 102 can define how to compute the expectation value of equation (103) using a quantum computer. To this end, the operator X 2can be observed to be diagonal in the computational basis, which is the number of bit string histograms N count (j) can be measured directly from the operator P 2 is diagonal in the momentum basis. This implies the addition of a centered quantum Fourier transform (QFT) circuit after the state preparation block. The quantum resource estimation system 102 can use the centered Fourier transform to allow negative momenta. As mentioned above, the quantum resource estimation system 102 calculates the i =-w+iΔx, where i=0,…2 n -1, and Δx=2w / 2 n Without loss of generality, the quantum resource estimation system 102 may select a region centered around zero.

[0776]

number

[0777] can be calculated as follows:

[0778]

number

[0779]

number

[0780] Here, N shots is the total number of circuit iterations in the space and momentum basis. N counts (j) (where 0≦N counts (j)≦N shots ,Σ j N counts (j)=N shots) is the number of measurements collapsed to the qubit basis state corresponding to the binary representation of the integer j. This strategy avoids the use of the Pauli representation of (103), which would involve an exponentially growing Pauli string scaled by the qubit register size.

[0781] The first step of the program is to numerically verify the possibility of preparing states that systematically converge to equation (102) using quantum circuits. By adopting a variational approach, we avoid expensive quantum arithmetic operations at the expense of introducing a source of error that is always present in numerical variational approaches. The most trivial one concerns the possibility of being trapped in a local minimum during the (classical) optimization procedure. The second, and more serious one, is connected to the expressiveness of the trial states generated by (e.g. shallow) quantum circuits.

[0782] The main option for Ansatz is the so-called R y -CNOT circuit. The quantum resource estimation system 102

[0783]

number

[0784] The initial state defined on the n-qubit register can be set to unitary.

[0785]

number

[0786] is expanded under the action of

[0787]

number

[0788] Gives.

[0789] This circuit uses an entangled U that spans the length of the qubit register. ENT followed by a single qubit rotation

[0790]

number

[0791] In the experimental case study described above, the quantum resource estimation system 102 estimates the quantum bit q i is the quantum bit q i-1 and the control qubit q i+1 Select a ladder of CNOT gates with linear connectivity so that i = 1, ..., n-2 are the targets of U. R One additional layer of gates is applied, making the number of variational parameters n×(L+1).

[0792] Since single qubit rotations are all local operations,

[0793]

number

[0794] can be written as a tensor product of rotations of a single qubit as follows:

[0795]

number

[0796]

number

[0797] is the quantum bit q i on the Bloch sphere of , k=1,…,L+1. The complete unitary circuit operation is described by:

[0798]

number

[0799] And the parameterized state is:

[0800]

number

[0801] Unitary

[0802]

number

[0803] Note that although describes the complete circuit, it does not describe pre-measurement changes in the basis that can be included to collapse the wavefunction in momentum space as explained above.

[0804] For each n and L parameter, the quantum resource estimation system 102 may repeat the optimization run, for example, eight times, to gather sufficient statistics, since the optimization may get stuck in a suboptimal local minimum. Since the quantum resource estimation system 102 may use a classical emulation of the quantum circuit, the only source of error in the optimization comes from the classical optimizer. In the experimental runs performed by the quantum resource estimation system 102 to implement the case study described above, the quantum resource estimation system 102 first performed a warm-up run with the COBYLA optimizer, followed by a longer run using the BFGS optimizer. To increase the efficiency of the optimization, the starting point of the VQE run at depth L uses the optimal parameters found in the previous optimization at depths L-2 or L-1, if available. It is noted that the parts of the algorithm that involve classical optimization feedback can be significantly improved, for example, using gradient-based methods or imaginary-time inspired update schemes.

[0805] L ∞ training refinement

[0806] As described above, the quantum resource estimation system 102 uses L as a cost function. ∞ In the first embodiment, the pre-optimized circuit obtained using the energy optimization method can be used as a starting guess for reoptimization according to the following: ∞ It is shown how optimization does not always provide acceptable results.

[0807] 6A, 6B, and 6C show example non-limiting graphs 600a, 600b, 600, respectively, that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repeated descriptions of similar elements and / or processes utilized in each embodiment are omitted for the sake of brevity.

[0808] In each of the graphs 600a, 600b, and 600c shown in Figures 6A, 6B, and 6C, respectively, an optimization run obtained using the energy-based method is represented by a plot 602. In each of the graphs 600a, 600b, and 600c shown in Figures 6A, 6B, and 6C, respectively, the direct L ∞ The optimization is represented by plot 604. In each of the graphs 600a, 600b, and 600c shown in Figures 6A, 6B, and 6C, respectively, L ∞ A mixed strategy in which the energy-based optimization is further refined using optimization is represented by plot 606. To obtain the data plotted in each of graphs 600a, 600b, and 600c, the quantum resource estimation system 102 may perform eight independent runs with the given parameters n qubits and L.

[0809] The complete optimization results are shown in Figures 7A and 7B.

[0810] 7A and 7B show example non-limiting graphs 700a and 700b, respectively, that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repeated descriptions of similar elements and / or processes utilized in each embodiment are omitted for the sake of brevity.

[0811] Graph 700a illustrates the L between the prepared distribution and the target distribution. ∞ The norm difference is shown as a function of circuit depth L for different qubit register sizes n. The best of eight independent optimizations for each parameter is plotted in graph 700a. Plot 702 in graph 700a corresponds to an optimization performed using the energy of the quantum harmonic oscillator as a cost function. Plot 704 in graph 700a corresponds to an optimization performed using L as a cost function. ∞ Supports sophisticated optimizations using

[0812] Graph 700b shows the difference in energy of the associated quantum harmonic oscillator model as a function of circuit depth L for different qubit register sizes n. The best of eight independent optimizations for each parameter are plotted in graph 700b. Plot 702 in graph 700b corresponds to an optimization performed using the energy of the quantum harmonic oscillator as a cost function. Plot 704 in graph 700b corresponds to an optimization performed using the energy of the quantum harmonic oscillator as a cost function. ∞ As expected, L ∞ Refinement targeting does not improve this quantity. The results of the numerical study, shown in graphs 700a and 700b, show that the convergence to the exact ground state is exponential in depth and therefore in the number of gate operations.

[0813] L ∞ Norm Direct optimization failure

[0814] Here, an empirical explanation for the observed failure of direct norm optimization techniques is provided. To this end, the quantum resource estimation system 102 combines energy-based methods and direct L ∞ The quantum resource estimation system 102 can explore the cost function landscape for both optimization and optimization methods.

[0815]

number

[0816] Starting from , we can make cuts in the parameter space according to the following prescription:

[0817]

number

[0818] Where:

[0819]

number

[0820] is a vector containing uniformly distributed random numbers in the range [-1, 1], and λ ∈ [-π, π] is a scalar that parameterizes the deformation from the optimal solution.

[0821] 8A and 8B show example non-limiting graphs 800a and 800b, respectively, that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repetitive descriptions of similar elements and / or processes utilized in each embodiment are omitted for the sake of brevity.

[0822] Graphs 800a and 800b shown in FIGS. 8A and 8B, respectively, show three different cut directions (e.g., vector

[0823]

number

[0824] 800a and 800b show probing of the cost function landscape for three different realizations of the parameter space. The plots shown at the top of the graphs 800a and 800b represent the cost function landscape against energy E, and the plots shown at the bottom of the graphs 800a and 800b represent the cost function landscape against L, calculated for two different settings n=5, 7 and depth L=6, 10, respectively, using three different cuts along the parameter space. ∞ 4 represents the cost function landscape over the norm.

[0825] From graphs 800a and 800b, L ∞ We observe that the cost function defined by the norm is much more corrugated than the one defined by the energy E of the related quantum mechanical toy blob, which instead exhibits a smooth surface. ∞ The basins of attraction of the cost functions overlap because they are close to the Gaussian function for which the ground state of the physics problem is achieved, and therefore L ∞ The second optimization with the norm will not get stuck in high-cost local minima outside such basins.

[0826] Variational Parameter Digitization

[0827] The numerical results presented above provide evidence of a rather efficient Gaussian state preparation, in terms of circuit depth, for parameterized circuits, but additional steps need to be taken in terms of a fault-tolerant implementation of such circuits. In this new framework, the continuous rotation RY gate can be extended as a finite product of discrete operations. Following again the Solov'i-Kitaev theorem or more specialized results, it is also possible to have an efficient representation of any SU(2) operator in terms of a sequence of Clifford+T gates that scales logarithmically with the threshold error ε. It is examined how the previously obtained results can be propagated in this regime, where the angle of rotation can only take discretized values. Thus, for each parameter

[0828]

number

[0829] is of the form i*2π / M digit It can be assumed that the vectors can only be expressed by: where i is an integer.

[0830] The quantum resource estimation system 102 can employ a protocol to optimize parameters on a lattice. First, the quantum resource estimation system 102 can project the original continuous-valued parameters onto a lattice that takes the closest lattice point for each parameter. Then, the quantum resource estimation system 102 optimizes the L ∞ A local search can be performed on the lattice to find a better combination of digitized parameters that reduces (e.g., minimizes) the norm difference. The quantum resource estimation system 102 can numerically show that the error introduced by such digitization systematically decreases with mesh size. Considering the L∞ norm difference error introduced by such digitization, the quantum resource estimation system 102 can show that it is 1 / M digitIn all cases, the quantum resource estimation system 102 can obtain values ​​that are compatible with a continuous-valued solution or the mesh size is reduced as M digit ~10 5 When it reaches this, it becomes 2π / M digit Better values ​​can be obtained that are equivalent to discretizing the space at ≈0.0001 radians (rad).

[0831] 9A, 9B, and 9C show example non-limiting graphs 900a, 900b, and 900c, respectively, that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more embodiments described herein. Repeated descriptions of similar elements and / or processes utilized in each embodiment are omitted for the sake of brevity.

[0832] Graph 900a shown in FIG. 9A illustrates the L between the prepared distribution and the target distribution. ∞ The norm difference is calculated for two different circuit depths L with n = 4 qubits and a digitized mesh size M digit As a function of each M digit For L, the quantum resource estimation system 102 can "digitize" the eight parameter set obtained by a previous independent optimization (e.g., performed considering a continuous range of rotation angle values). The empty (e.g., blank) square and circle symbols represent with respect to the entire data set, while the non-empty symbols represent the minimum value in the set. The horizontal line 902 indicates the best value obtained in the previous optimization performed considering a continuous range for rotation angle values ​​for each L parameter. In some instances, the digitization helps to escape the local minimum and achieve a slightly better solution. The diagonal line 904 is a visual guide and represents the best value obtained by the previous optimization performed considering a continuous range for rotation angle values ​​for each L parameter. digit and 0.1 / M digit Represents.

[0833] Graph 900b shown in FIG. 9B includes an example, non-limiting, alternative embodiment of graph 900a, in which graph 900b illustrates the L∞ The norm difference is calculated for two different circuit depths L with n = 5 qubits and a digitized mesh size M digit Graph 900c shown in FIG. 9C includes an example, non-limiting alternative embodiment of graph 900a, in which graph 900c illustrates the relationship between the prepared distribution and the target distribution. ∞ The norm difference is calculated for two different circuit depths L with n = 6 qubits and a digitized mesh size M digit is shown as a function of .

[0834] 10 illustrates a flow diagram of an exemplary non-limiting computer-implemented method 1000 that can facilitate estimation of quantum resources for calculating the expectation value of a stochastic process using a reparameterization method according to one or more aspects described herein. Repeated descriptions of similar elements and / or processes utilized in each aspect have been omitted for the sake of brevity.

[0835] At 1002, the computer-implemented method 1000 may include applying, by a system operably coupled to a processor (e.g., processor 106) (e.g., via the quantum resource estimation system 102 and / or the reparameterization component 108), a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution.

[0836] At 1004, the computer-implemented method 1000 may include estimating, by the system (e.g., via the quantum resource estimation system 102 and / or the estimation component 110), at least one defined criterion for a quantum computer to be used to calculate an expected value of a stochastic process associated with the target probability distribution (e.g., the value of a derivative asset).

[0837] The quantum resource estimation system 102 may be associated with various technologies, for example, the quantum resource estimation system 102 may be associated with quantum computing technologies, quantum hardware and / or software technologies, quantum algorithm technologies, machine learning technologies, artificial intelligence technologies, cloud computing technologies, and / or other technologies.

[0838] The quantum resource estimation system 102 may provide technical improvements to systems, devices, components, operational steps, and / or process steps associated with the various techniques identified above. For example, the quantum resource estimation system 102 may apply a quantum fault-tolerant operation to a variationally prepared quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution and / or estimate at least one defined criterion of a quantum computer used to calculate an expected value of a stochastic process associated with the target probability distribution (e.g., the value of a derivative asset). In this example, the at least one defined criterion may include an attribute, condition, property, parameter, or configuration of the quantum computer that enables the quantum computer to achieve a defined quantum advantage in calculating the expected value of a stochastic process associated with the target probability distribution (e.g., the value of a derivative asset). In this example, the quantum resource estimation system 102 may thus be implemented to identify a quantum resource (e.g., a quantum computer, a quantum processor, and / or another quantum resource) that can leverage the advantages of quantum computing to calculate the expected value of a stochastic process (e.g., the value of a derivative asset such as an option contract), while incurring a minimal amount of computational costs compared to other quantum resources.

[0839] The quantum resource estimation system 102 can provide technical improvements to a processing unit (e.g., the processor 106, a quantum processor, and / or another processor) associated with the quantum resource estimation system 102. For example, as described above, the quantum resource estimation system 102 can also estimate and thus determine at least one defined criterion that can enable the quantum computer to exploit the advantages of quantum computing to calculate the expected value of a stochastic process (e.g., the value of a derivative asset such as an option contract), while incurring a minimal amount of computational costs compared to other quantum resources. In this example, a quantum processor in a quantum computer can be developed (e.g., planned, designed, and / or manufactured) and / or modified to include at least one defined criterion that can be estimated and / or identified by the quantum resource estimation system 102 to enable the quantum computer to calculate the expected value of a stochastic process (e.g., the value of a derivative asset such as an option contract), while incurring a minimal amount of computational costs.

[0840] A practical application of the quantum resource estimation system 102 is that it can be implemented using a classical computing device (e.g., a classical computer) to estimate at least one defined criterion that can enable a quantum computer to take advantage of quantum computing to compute one or more solutions (e.g., heuristics) to complex various problems (e.g., estimation problems, optimization problems, and / or other problems) in various domains (e.g., finance, chemistry, medicine, and / or other domains). For example, a practical application of the quantum resource estimation system 102 is that it can be implemented using a classical computing device (e.g., a classical computer) to estimate at least one defined criterion that can enable a quantum computer to take advantage of quantum computing to compute one or more solutions (e.g., heuristics) to estimation problems and / or optimization problems in the domains of chemistry, medicine, and / or finance, such solutions can be used, for example, to engineer new chemical compounds, new drugs, and / or new option premiums.

[0841] It should be appreciated that the quantum resource estimation system 102 provides a new approach driven by relatively new quantum computing techniques. For example, the quantum resource estimation system 102 provides a new approach to estimating at least one defined criterion that can enable a quantum computer to exploit the advantages of quantum computing to calculate the expected value of a stochastic process (e.g., the value of a derivative asset such as an options contract), while incurring a minimal amount of computational cost compared to other quantum resources.

[0842] The quantum resource estimation system 102 may utilize hardware or software to solve problems that are highly technical in nature, not abstract, and cannot be performed as a set of mental acts by a human. In some aspects, one or more of the processes described herein may be executed by one or more specialized computers (e.g., specialized processing units, specialized classical computers, specialized quantum computers, and / or another type of specialized computer) to perform defined tasks associated with the various technologies identified above. The quantum resource estimation system 102 and / or components thereof may be utilized to solve new problems that arise through the use of advances in the technologies described above, quantum computing systems, cloud computing systems, computer architectures, and / or other technologies.

[0843] It should be appreciated that the quantum resource estimation system 102 may utilize various combinations of electrical components, mechanical components, and circuitry that cannot be replicated by a human mind or performed by a human being because various operations capable of being performed by the quantum resource estimation system 102 and / or its components described herein are operations that are beyond human intellectual capabilities. For example, the amount of data processed by the quantum resource estimation system 102 in a particular period of time, the speed at which such data is processed, or the types of data may be greater in amount, faster in speed, or different types of data than a human mind can process in the same period of time.

[0844] According to some aspects, the quantum resource estimation system 102 may also be fully operable to perform one or more other functions (e.g., fully powered on, fully running, and / or another function) while performing various operations described herein. It should be understood that performing such simultaneous multiple operations exceeds human intellectual capabilities. It should also be understood that the quantum resource estimation system 102 may include information that an entity such as a human user cannot manually obtain. For example, the type, amount, and / or variety of information included in the quantum resource estimation system 102, reparameterization component 108, estimation component 110, variation component 202, and / or error analysis component 302 may be more complex than information manually obtained by a human user.

[0845] In some aspects, the quantum resource estimation system 102 may be associated with a cloud computing environment. For example, the quantum resource estimation system 102 may be associated with a cloud computing environment 1250, described below with reference to FIG. 12, and / or one or more functional abstraction layers (e.g., hardware and software layer 1360, virtualization layer 1370, management layer 1380, and / or workload layer 1390), described below with reference to FIG.

[0846] The quantum resource estimation system 102 and / or its components (e.g., the reparameterization component 108, the estimation component 110, the variational component 202, the error analysis component 302, and / or another component) may utilize one or more computing resources of a cloud computing environment 1250, described below with reference to FIG. 12, and / or one or more functional abstraction layers (e.g., quantum software), described below with reference to FIG. 13, to perform one or more operations in accordance with one or more aspects of the subject disclosure described herein. For example, the cloud computing environment 1250 and / or such one or more functional abstraction layers may include one or more classical computing devices (e.g., a classical computer, a classical processor, a virtual machine, a server, and / or another classical computing device), quantum hardware and / or quantum software (e.g., a quantum computing device, a quantum computer, a quantum processor, a quantum circuit simulation software, a superconducting circuit, and / or other quantum hardware and / or quantum software) that may be utilized by the quantum resource estimation system 102 and / or its components to perform one or more operations in accordance with one or more aspects of the subject disclosure described herein. For example, the quantum resource estimation system 102 and / or components thereof may utilize such one or more classical and / or quantum computing resources to perform one or more classical and / or quantum: mathematical functions, calculations, and / or equations; computing and / or processing scripts; algorithms; models (e.g., artificial intelligence (AI) models, machine learning (ML) models, and / or other types of models); and / or other operations in accordance with one or more aspects of the subject disclosure described herein.

[0847] Although the subject disclosure includes detailed descriptions of cloud computing, it should be understood that implementation of the teachings described herein is not limited to a cloud computing environment. Rather, aspects of the invention can be implemented in conjunction with any other type of computing environment now known or later developed.

[0848] Cloud computing is a model of service delivery for enabling convenient, on-demand network access to a shared pool of configurable computing resources (e.g., networks, network bandwidth, servers, processing, memory, storage, applications, virtual machines, and services) that can be rapidly provisioned and released with minimal administrative effort or interaction with the provider of the service. The cloud model just described may include at least five characteristics, at least three service models, and at least four deployment models.

[0849] Its features are as follows:

[0850] On-Demand Self-Service: Cloud consumers can automatically and unilaterally provision computing capacity, such as server time or network storage, as needed, without the need for human interaction with the service provider.

[0851] Broad network access: Functionality is available over the network and accessed through standard mechanisms that facilitate use by heterogeneous thin or thick client platforms (eg, mobile phones, laptops, and PDAs).

[0852] Resource Pooling: To serve multiple consumers using a multi-tenant model, provider computing resources are pooled, with different physical and virtual resources dynamically allocated and reallocated on demand. Consumers generally have no control or knowledge of the exact location of the provided resources, but there is a sense of location independence in the sense that they may be able to specify the location at a higher level of abstraction (e.g. country, state, or data center).

[0853] Rapid Elasticity: Capabilities can be rapidly provisioned and rapidly scaled out, sometimes automatically, and rapidly released and rapidly scaled in. To the consumer, the capabilities available for provisioning often appear unlimited and can be purchased in any quantity at any time.

[0854] Measured Services: Cloud systems leverage metering capabilities to automatically control and optimize resource usage at several levels of abstraction appropriate to the type of service (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, and reported, providing transparency to both providers and consumers of the services being used.

[0855] The service model is as follows:

[0856] Software as a Service (SaaS): The functionality provided to the consumer is the use of the provider's applications running on a cloud infrastructure. The applications are accessible from a variety of client devices through thin-client interfaces such as web browsers (e.g., web-based email). The consumer does not manage or control the underlying cloud infrastructure, including networks, servers, operating systems, storage, or individual application functionality, with the possible exception of limited user-specific application configuration settings.

[0857] Platform as a Service (PaaS): The capability offered to a consumer is to deploy consumer-created or acquired applications, written using programming languages ​​and tools supported by the provider, onto a cloud infrastructure. The consumer does not manage or control the underlying cloud infrastructure, including networks, servers, operating systems, or storage, but does control the deployed applications and, potentially, the application hosting environment configuration.

[0858] Infrastructure as a Service (IaaS): The functionality provided to the consumer is to provision processing, storage, network, and other basic computing resources, and the consumer can deploy and run any software, which may include operating systems and applications. The consumer does not manage or control the underlying cloud infrastructure, but does control the operating systems, storage, and deployed applications, and possibly limited control over selected networking components (e.g., host firewalls).

[0859] The deployment models are as follows:

[0860] Private Cloud: The cloud infrastructure is operated solely for the organization. It may be managed by the organization or a third party and may reside on-premise or off-premise.

[0861] Community Cloud: Cloud infrastructure is shared across multiple organizations to support a specific community with a shared concern (mission, security requirements, policy, compliance considerations). It may be managed by the organization or a third party and may reside on-premise or off-premise.

[0862] Public Cloud: The cloud infrastructure is made available to the public or large industry organizations and is owned by an organization that sells cloud services.

[0863] Hybrid Cloud: A cloud infrastructure is a composition of two or more clouds (private, community, or public) bound together while remaining a unique entity by standardized or proprietary technologies that allow for the portability of data and applications (e.g., cloud bursting for load balancing between clouds).

[0864] Cloud computing environments are service-oriented with an emphasis on statelessness, low coupling, modularity, and semantic interoperability. At the heart of cloud computing is an infrastructure that includes a network of interconnected nodes.

[0865] For ease of explanation, the computer-implemented methodologies are depicted and described as a series of acts. It should be understood and appreciated that the subject innovation is not limited by the depicted acts and / or the order of acts, e.g., acts can occur in various orders and / or simultaneously, and with other acts not presented or described herein. Moreover, not all depicted acts are necessarily required to implement a computer-implemented methodology in accordance with the disclosed subject matter. In addition, those skilled in the art will understand and appreciate that a computer-implemented methodology can alternatively be represented as a series of interrelated states, via a state diagram or events. Moreover, it should be further appreciated that the computer-implemented methodologies disclosed hereinafter and throughout this specification can be stored in an article of manufacture to facilitate the transportation and transfer of such computer-implemented methodologies to a computer. The term article of manufacture, as used herein, is intended to encompass a computer program accessible from any computer-readable device or storage medium.

[0866] To provide a context for the various aspects of the disclosed subject matter, Figure 11 and the following description are intended to provide a general description of a suitable environment in which the various aspects of the disclosed subject matter can be implemented. Figure 11 illustrates a block diagram of an exemplary non-limiting operating environment in which one or more aspects described herein can be facilitated. Repeated descriptions of similar elements utilized in other aspects described herein have been omitted for the sake of brevity.

[0867] 11, a suitable operating environment 1100 for implementing various aspects of the subject disclosure may also include a computer 1112. The computer 1112 may also include a processing unit 1114, a system memory 1116, and a system bus 1118. The system bus 1118 couples system components including, but not limited to, the system memory 1116 to the processing unit 1114. The processing unit 1114 may be any of a variety of available processors. Dual microprocessors and other multi-processor architectures may also be utilized as the processing unit 1114. The system bus 1118 may be any of several types of bus structures, including a memory bus or memory controller, a peripheral bus or external bus, and / or a local bus using any of a variety of available bus architectures, including, but not limited to, Industry Standard Architecture (ISA), Micro Channel Architecture (MSA), Enhanced ISA (EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB), Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus (USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and Small Computer System Interface (SCSI).

[0868] The system memory 1116 may also include volatile memory 1120 and non-volatile memory 1122. A basic input / output system (BIOS), containing the basic routines for transferring information between elements within the computer 1112, such as during start-up, is stored in the non-volatile memory 1122. The computer 1112 may also include removable / non-removable, volatile / non-volatile computer storage media. FIG. 11 illustrates, for example, disk storage 1124. The disk storage 1124 may also include devices such as, but not limited to, magnetic disk drives, floppy disk drives, tape drives, Jaz drives, Zip drives, LS-100 drives, flash memory cards, memory sticks, and the like. The disk storage 1124 may also include storage media, either separately or in combination with other storage media. To facilitate connection of the disk storage 1124 to the system bus 1118, a removable or non-removable interface, such as interface 1126, is typically used. 11 also illustrates software that acts as an intermediary between users and the basic computer resources described in suitable operating environment 1100. Such software may include, for example, an operating system 1128. The operating system 1128, which may be stored on disk storage 1124, acts to control and allocate resources of the computer 1112.

[0869] System applications 1130 take advantage of the management of resources by the operating system 1128 through program modules 1132 and program data 1134 stored, for example, in either the system memory 1116 or the disk storage 1124. It is to be appreciated that the subject disclosure may be implemented with various operating systems or combinations of operating systems. A user enters commands or information into the computer 1112 through input devices 1136. The input devices 1136 include, but are not limited to, pointing devices such as a mouse, trackball, stylus, touch pad, keyboard, microphone, joystick, game pad, satellite dish, scanner, TV tuner card, digital camera, digital video camera, web camera, and the like. These and other input devices are connected to the processing unit 1114 through the system bus 1118 via interface ports 1138. The interface ports 1138 include, for example, serial ports, parallel ports, game ports, and universal serial bus (USB). The output devices 1140 use some of the same types of ports as the input devices 1136. Thus, for example, a USB port may be used to provide input to the computer 1112 and to output information from the computer 1112 to an output device 1140. Output adapter 1142 is provided to illustrate that there are some output devices 1140 such as monitors, speakers, printers, among other output devices 1140, that may require special adapters. Output adapters 1142 include, by way of example and not limitation, video cards and sound cards that provide a means of connection between the output device 1140 and the system bus 1118. It should be noted that other devices and / or systems of devices provide both input and output capabilities, such as a remote computer 1144.

[0870] The computer 1112 can operate in a networked environment using logical connections to one or more remote computers, such as a remote computer 1144. The remote computer 1144 can be a computer, a server, a router, a network PC, a workstation, a microprocessor-based device, a peer device or other common network node, and can typically include many or all of the elements described relative to the computer 1112. For simplicity, only a memory storage device 1146 is shown with the remote computer 1144. The remote computer 1144 is logically connected to the computer 1112 through a network interface 1148 and physically connected through a communication connection 1150. The network interface 1148 encompasses wired and / or wireless communication networks, such as a local area network (LAN), a wide area network (WAN), a cellular network, and / or another wired and / or wireless communication network. LAN technologies include Fiber Distributed Data Interface (FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ring. WAN technologies include, but are not limited to, point-to-point links, integrated services digital networks (ISDN) and variations thereon, packet switched networks, and circuit switched networks such as digital subscriber lines (DSL). The communications connection(s) 1150 refer to the hardware / software utilized to connect the network interface 1148 to the system bus 1118. The communications connection(s) 1150 are shown internal to the computer 1112 for clarity, but may be external to the computer 1112. The hardware / software for connecting to the network interface 1148 may also include, for purposes of illustration only, internal and external technologies such as modems, including ordinary telephone grade modems, cable modems, and DSL modems, ISDN adapters, and Ethernet cards.

[0871] Referring now to FIG. 12, an exemplary cloud computing environment 1250 is illustrated. As illustrated, the cloud computing environment 1250 includes one or more cloud computing nodes 1210 with which local computing devices used by cloud consumers, such as, for example, a personal digital assistant (PDA) or mobile phone 1254A, a desktop computer 1254B, a laptop computer 1254C, and / or an automobile computer system 1254N, may communicate. Although not illustrated in FIG. 12, the cloud computing node 1210 may further include a quantum platform (e.g., a quantum computer, quantum hardware, quantum software, and / or another quantum platform) with which local computing devices used by the cloud consumers may communicate. The nodes 1210 may communicate with each other. They may be grouped (not illustrated) in one or more networks, such as private, community, public, hybrid clouds, or combinations thereof, either physically or virtually, as described herein above. This allows the cloud computing environment 1250 to provide infrastructure, platform, and / or software as a service without the need for cloud consumers to maintain resources on their local computing devices. It will be understood that the types of computing devices 1254A-N shown in FIG. 12 are intended as examples only, and that the computing node 1210 and cloud computing environment 1250 can communicate with any type of computerized device via any type of network and / or network-addressable connection (e.g., using a web browser).

[0872] Referring now to Figure 13, a set of functional abstraction layers provided by cloud computing environment 1250 (Figure 12) is shown. It should be understood in advance that the components, layers, and functions shown in Figure 13 are intended to be illustrative only, and aspects of the present invention are not limited thereto. As shown, the following layers and corresponding functions are provided:

[0873] Hardware and software layer 1360 includes hardware and software components. Examples of hardware components include mainframe 1361, RISC (reduced instruction set computer) architecture based servers 1362, servers 1363, blade servers 1364, storage devices 1365, and networks and network components 1366. In some aspects, software components include network application server software 1367, database software 1368, quantum platform routing software (not shown in FIG. 13), and / or quantum software (not shown in FIG. 13).

[0874] The virtualization layer 1370 provides an abstraction layer in which the following examples of virtual entities may be provided: virtual servers 1371, virtual storage 1372, virtual networks including virtual private networks 1373, virtual applications and operating systems 1374, and virtual clients 1375.

[0875] In one example, management layer 1380 may provide the functions described below. Resource provisioning 1381 provides dynamic procurement of computing and other resources utilized to execute tasks within the cloud computing environment. Metering and pricing 1382 provides cost tracking as resources are utilized within the cloud computing environment and charging or billing for the consumption of those resources. In one example, these resources may include application software licenses. Security provides identity verification of cloud consumers and tasks, and protection of data and other resources. User portal 1383 provides consumers and system administrators with access to the cloud computing environment. Service level management 1384 provides allocation and management of cloud computing resources such that required service levels are met. Service level agreement (SLA) planning and fulfillment 1385 provides proactive coordination and procurement of cloud computing resources in anticipation of future demand in accordance with SLAs.

[0876] The workload layer 1390 provides examples of functionality for which a cloud computing environment may be utilized. Non-limiting examples of workloads and functions that may be provided from this layer include mapping and navigation 1391, software development and lifecycle management 1392, virtual classroom instructional delivery 1393, data analytics processing 1394, transaction processing 1395, and quantum resource estimation software 1396.

[0877] The present invention may be a system, a method, an apparatus, and / or a computer program product in any possible level of technical detail integration. The computer program product may include a computer readable storage medium (or media) having computer readable program instructions for causing a processor to execute aspects of the present invention. The computer readable storage medium may be a tangible device that can hold and store instructions used by an instruction execution device. The computer readable storage medium may be, for example, but not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the above. A non-exhaustive list of more specific examples of computer readable storage media may include portable computer diskettes, hard disks, random access memories (RAM), read-only memories (ROM), erasable programmable read-only memories (EPROMs or flash memories), static random access memories (SRAMs), portable compact disk read-only memories (CD-ROMs), digital versatile disks (DVDs), memory sticks, floppy disks, punch cards or mechanically encoded devices such as raised structures in grooves with instructions recorded, and any suitable combination of the above. As used herein, a computer-readable storage medium should not be construed as a transitory signal per se, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through a waveguide or other transmission medium (e.g., light pulses passing through a fiber optic cable), or electrical signals transmitted over wires.

[0878] The computer readable program instructions described herein can be downloaded to each computing / processing device from a computer readable storage medium or to an external computer or external storage device via a network, such as the Internet, a local area network, a wide area network, and / or a wireless network. The network can include copper transmission cables, optical transmission fibers, wireless transmissions, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface of each computing / processing device receives the 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. The computer readable program instructions for performing the operations of the present invention can be assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine dependent instructions, microcode, firmware instructions, state setting data, configuration data for integrated circuits, or source or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk or C++, and procedural programming languages ​​such as the "C" programming language or similar programming languages. The computer readable program instructions may be executed entirely on the user's computer as a stand-alone software package, may be executed partially on the user's computer, may be executed partially on the user's computer and partially on a remote computer, or may be executed entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer via any type of network, including a local area network (LAN) or a wide area network (WAN), or may be connected to an external computer (e.g., via the Internet using an Internet Service Provider).In some embodiments, electronic circuitry including, for example, a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA) can execute computer readable program instructions by utilizing state information of the computer readable program instructions to personalize the electronic circuitry to perform aspects of the invention.

[0879] Aspects of the present invention are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the present invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations 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, or other programmable data processing apparatus generating machine, such that the instructions executed via the processor of the computer or other programmable data processing apparatus can create means for implementing the functions / acts specified in one or more blocks of the flowcharts and / or block diagrams. These computer-readable program instructions, which can instruct a computer, programmable data processing apparatus, and / or other device to function in a particular manner, can also be stored in a computer-readable storage medium, such that the computer-readable storage medium having instructions stored thereon comprises an article of manufacture containing instructions for implementing aspects of the functions / acts specified in one or more blocks of the flowcharts and / or block diagrams. Also, the computer readable program instructions may be loaded into a computer, other programmable data processing apparatus, or other device to cause a series of operational acts to be performed on the computer, other programmable apparatus, or other device to generate a computer-implemented process, such that the instructions executing on the computer, other programmable apparatus, or other device implement the functions / acts specified in one or more blocks of the flowcharts and / or block diagrams.

[0880] The flowcharts and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various aspects of the present invention. In this regard, each block in the flowcharts or block diagrams may represent a module, segment, or part of an instruction, which includes one or more executable instructions for implementing the specified logical function. In some alternative implementations, the functions described in the blocks may occur out of the order described in the figures. For example, two blocks shown in succession may in fact be executed substantially simultaneously, or the blocks may be executed in the reverse order depending on the functionality involved. It should also be noted that each block of the block diagrams and / or flowchart diagrams, and combinations of blocks in the block diagrams and / or flowchart diagrams, may be implemented by a special-purpose hardware-based system that performs the specified functions or acts, or executes a combination of special-purpose hardware and computer instructions.

[0881] Although the subject matter has been described above in the general context of computer-executable instructions for a computer program product executing on one computer and / or multiple computers, those skilled in the art will recognize that the subject disclosure may also be implemented or practiced in combination with other program modules. Generally, program modules include routines, programs, components, data structures, and / or other program modules that perform particular tasks and / or implement particular abstract data types. Moreover, those skilled in the art will appreciate that the computer-implemented methods of the present invention may be practiced using other computer system configurations, including single-processor or multi-processor computer systems, minicomputer devices, mainframe computers, as well as computers, handheld computer devices (e.g., PDAs, phones), microprocessor-based or programmable consumer or industrial electronic devices, and the like. The illustrated aspects may also be practiced in distributed computing environments where tasks are performed by remote processing devices linked through a communications network. However, some, if not all, aspects of the subject disclosure may be practiced in stand-alone computers. In a distributed computing environment, program modules may be located in both local and remote memory storage devices. For example, in one or more aspects, the computer-executable components can be executed from a memory that can include or consist of one or more distributed memory units. As used herein, the terms "memory" and "memory unit" are interchangeable. Furthermore, one or more aspects described herein can execute code of the computer-executable components in a distributed manner, e.g., multiple processors coupled or cooperating to execute code from one or more distributed memory units. As used herein, the term "memory" can encompass a single memory or memory unit in one location, or a memory or memory unit in one or more locations.

[0882] As used in this application, terms such as "component," "system," "platform," "interface," and the like, can refer to and / or include computer-related entities or entities associated with an operating machine having one or more specific functionalities. The entities disclosed 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, a process running on a processor, a processor, an object, an executable file, a thread of execution, a program, and / or a computer. As an example, both an application running on a server and a 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 among two or more computers. In another example, each component 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 by following a signal having one or more data packets (e.g., data from one component interacting with another component via a signal having one or more data packets, in a local system, a distributed system, and / or across a network such as the Internet with other systems). As another example, a component may be a device having particular functionality provided by mechanical parts operated by electrical or electronic circuits, which in turn are operated by a software or firmware application executed by a processor, in which case the processor may be internal or external to the device and may execute at least a portion of the software or firmware application.As yet another example, a component may be a device that provides certain functionality without mechanical parts through electronic components, which may include a processor or other means for executing software or firmware that at least partially provides the functionality of the electronic component. In an aspect, a component may emulate an electronic component via, for example, a virtual machine in a cloud computing system.

[0883] Furthermore, the term "or" is intended to mean an inclusive "or" rather than an exclusive "or". That is, unless otherwise specified or clear from the context, "X uses A or B" is intended to mean any of the natural inclusive permutations. That is, if X uses A, if X uses B, or if X uses both A and B, then "X uses A or B" is satisfied in any of the foregoing cases. Furthermore, the articles "a" and "an" as used in this specification and the accompanying drawings should generally be construed to mean "one or more" unless otherwise specified or clear from the context to mean the 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 disclosed herein is not limited by such examples. Additionally, any aspect or design described herein as "example" and / or "exemplary" is not necessarily to be construed as preferred or advantageous over other aspects or designs, and is not intended to exclude equivalent exemplary structures and techniques known to those skilled in the art.

[0884] As used herein, the term "processor" may refer to substantially any computing processing unit or device, including, but not limited to, a single-core processor, a single processor with software multithreading execution capability, a multi-core processor, a multi-core processor with software multithreading execution capability, a multi-core processor with hardware multithreading technology, a parallel platform, and a parallel platform with distributed shared memory. In addition, a processor may 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), discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. Furthermore, a processor may utilize nanoscale architectures, such as, but not limited to, molecular and quantum dot-based transistors, switches, gates, to optimize space usage or improve the performance of user equipment. A processor may also be implemented as a combination of computing processing units. In the subject disclosure, terms such as "store," "storage," "data store," "data storage," "database," and substantially any other information storage component associated with the operation and functionality of a component are utilized to refer to a "memory component," an entity embodied in a "memory," or a component that includes a memory. It is to be understood that the memory and / or memory components described herein may be either volatile or non-volatile memory, or may include both volatile and non-volatile memory.By way of example, and not limitation, non-volatile memory may include read only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable ROM (EEPROM), flash memory, or non-volatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). Volatile memory may include RAM, which may act as external cache memory, for example. By way of example, and not limitation, RAM is available in various forms, such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), SyncLink DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), Rambus dynamic RAM (RDRAM), etc. Additionally, the disclosed memory components of the systems or computer-implemented methods herein are intended to include, without being limited to, these and any other suitable types of memory.

[0885] What has been described above includes merely exemplary systems and computer-implemented methods. Of course, it is not possible to describe every conceivable combination of components or computer-implemented methods for purposes of describing the subject disclosure, but one of ordinary skill in the art will recognize that many additional combinations and permutations of the subject disclosure are possible. Moreover, to the extent that terms such as "including," "having," and "holding" are used in the detailed description, claims, appendices, and drawings, such terms are intended to be inclusive in the same manner as "comprising," as "comprising" is interpreted when "comprising" is employed as a transitional term in the claims.

[0886] The description of the various embodiments is provided for illustrative purposes, but is not intended to be exhaustive or limiting of the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terms used in this specification are selected to best explain the principles of the embodiments, practical applications or technical improvements to the technology found in the market, or to allow those skilled in the art to understand the embodiments disclosed herein.

Claims

1. A computer-implemented method comprising: applying, by a system operatively connected to a processor, a quantum operation to a first quantum state corresponding to a probability distribution to generate a second quantum state corresponding to a target probability distribution, said applying step including: Applying dT Gaussian operators G to ndT quantum bits of a quantum computer, [0010] and configuring said first quantum state given by: [0025] is the path of a discrete-time multivariate stochastic process running over the 2 ndT realizations of this multivariate standard Gaussian, the Gaussian operator G loading a Gaussian distribution onto an n-qubit register, [0030] denotes the corresponding probability, n, d, and T being positive integers; adjusting each Gaussian distribution of the first quantum state by performing an affine transformation, such that the resulting second quantum state has a return path ω R and a probability p(ω R ) corresponding to the target probability distribution; Including steps The method comprises: Quantum register of said quantum computer [0045] To [0050] A step of calculating by inserting Rotate the register value to the amplitude of the ancilla qubit [006] generating a second quantum state given by: performing amplitude estimation in the second quantum state to determine the probability that the ancilla qubit is |1〉; The computer-implemented method further comprising:

2. The computer-implemented method of claim 1, characterized in that performing the affine transformation generates the second quantum state corresponding to the target probability distribution having at least one of a defined mean of the target probability distribution, a defined standard deviation of the target probability distribution, or one or more explicit parameters that specify the target probability distribution.

3. The computer-implemented method of claim 1, wherein the first quantum state is a superposition over possible paths of the discrete-time multivariate stochastic process.

4. The computer-implemented method of claim 1, wherein the first quantum state is a variationally prepared quantum state prepared by executing a trained variational quantum circuit. training, by the system, the variational quantum circuit to prepare the first quantum state and to reduce computational cost of quantum arithmetic operations performed by the quantum computer to calculate an expectation of a stochastic process associated with the target probability distribution.

5. The computer-implemented method of claim 4, further comprising: training, by the system, the variational quantum circuit to prepare the variationally prepared quantum state, the step including using a Hamiltonian operator to generate a ground state corresponding to the target probability distribution.

5. The computer-implemented method of claim 4, further comprising:

7. The computer-implemented method of claim 1, further comprising the step of estimating at least one defined criterion of a quantum computer that will be used by the system to calculate an expectation of a stochastic process associated with the target probability distribution.

8. The method of claim 7, further comprising: calculating, by the system, one or more errors associated with at least one of applying the quantum operations to the variationally prepared quantum state to generate the quantum state, estimating the at least one defined criterion, or calculating the expectation of the stochastic process associated with the target probability distribution.

8. The computer-implemented method of claim 7, further comprising:

9. The computer-implemented method of claim 7, wherein the at least one defined criterion is selected from a group consisting of attributes, conditions, properties, parameters, or configurations of the quantum computer that enable the quantum computer to achieve a defined quantum supremacy when calculating the expected value of the stochastic process associated with the target probability distribution, the probability distribution comprising a standard normal probability distribution, and the target probability distribution comprising a normal probability distribution.

10. The computer-implemented method of claim 1, wherein the quantum operation is a fault-tolerant operation.

11. A system arranged to perform a method according to a computer-implemented method as claimed in any one of claims 1 to 10.

12. A computer-readable medium comprising instructions which, when executed by a computer, cause the computer to perform a method according to any one of the computer-implemented methods described in claims 1 to 10.

Images