Quantum resource estimation using reparametrization method

Abstract

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

Description

Background Technology

[0001] This subject matter discloses methods for estimating quantum resources to compute the expectation of a stochastic process, and more specifically, methods for estimating quantum resources using reparameterization to compute the expectation of a stochastic process. Summary of the Invention

[0002] The following summary provides a basic understanding of one or more embodiments of the invention. This summary is not intended to identify key or essential elements or to define any scope of a particular embodiment or any scope of the claims. Its sole purpose is to present the concepts in a simplified form as a prelude to the more detailed description that follows. In one or more embodiments described herein, systems, apparatuses, computer-implemented methods, and / or computer program products are described that can use reparameterization methods to facilitate the estimation of quantum resources to compute the expected value of a stochastic process.

[0003] According to an embodiment, the system may include a processor that executes computer-executable components stored in memory. The computer-executable components may include a reparameterization component that applies quantum fault-tolerant operations to variationally prepared quantum states corresponding to a probability distribution to produce quantum states corresponding to a target probability distribution. The computer-executable components may also include an estimation component that estimates at least one defined criterion for the quantum computer to compute the expected value of a stochastic process associated with the target probability distribution.

[0004] According to another embodiment, the computer-implemented method may include applying quantum fault-tolerant computation to variationally prepared quantum states corresponding to a probability distribution via a system operatively coupled to a processor to generate quantum states corresponding to a target probability distribution. The computer-implemented method may further include estimating at least one defined criterion of the quantum computer via the system for calculating the expected value of a stochastic process associated with the target probability distribution.

[0005] According to another embodiment, the computer program product includes a computer-readable storage medium having program instructions contained therein, which are executable by a processor to cause the processor to apply quantum fault-tolerant operations to variationally prepared quantum states corresponding to a probability distribution to produce quantum states corresponding to a target probability distribution. The program instructions are also executable by the processor to cause the processor to estimate at least one defined criterion for the quantum computer to compute the expected value of a stochastic process associated with the target probability distribution. Attached Figure Description

[0006] Figure 1 , Figure 2 and Figure 3The diagram illustrates a block diagram of an example non-limiting system that can each use a reparameterization method to facilitate the estimation of quantum resources to compute the expected value of a stochastic process, according to one or more embodiments described herein.

[0007] Figure 4 , Figure 6A , Figure 6B , Figure 6C , Figure 7A , Figure 7B , Figure 8A , Figure 8B , Figure 9A , Figure 9B and Figure 9C The illustration shows an example non-limiting plot of one or more embodiments described herein, showing how reparameterization methods can be used to facilitate the estimation of quantum resources to compute the expected value of a stochastic process.

[0008] Figure 5 The illustration shows an example non-limiting circuit that uses a reparameterization method to facilitate the estimation of quantum resources in order to compute the expected value of a stochastic process, according to one or more embodiments described herein.

[0009] Figure 10 The illustration shows a flowchart of an example non-limiting computer implementation of a method that can use reparameterization methods to facilitate the estimation of quantum resources in accordance with one or more embodiments described herein, in order to compute the expected value of a stochastic process.

[0010] Figure 11 A block diagram illustrating an example non-limiting operating environment that may facilitate one or more embodiments described herein is shown.

[0011] Figure 12 The illustration shows a block diagram of an example non-limiting cloud computing environment according to one or more embodiments disclosed in this subject matter.

[0012] Figure 13 The diagram illustrates a block diagram of an example non-limiting abstract model layer according to one or more embodiments disclosed in this subject matter. Detailed Implementation

[0013] The following detailed description is merely illustrative and is not intended to limit the embodiments and / or their application or use. Furthermore, it is not intended to be construed as being bound by any express or implied information provided in the foregoing Background or Aspects or Detailed Description sections.

[0014] One or more embodiments will now be described with reference to the accompanying drawings, wherein the same reference numerals are used throughout to refer to the same elements. In the following description, numerous specific details are set forth for purposes of explanation, in order to provide a more thorough understanding of one or more embodiments. However, it will be apparent that, in various cases, one or more embodiments may be practiced without these specific details.

[0015] As described herein, “derivatives” and / or “derivative assets” are contracts between issuers and holders that are valid until a maturity date. As described herein, “entity” can include people, clients, users, computing devices, software applications, agents, machine learning (ML) models, artificial intelligence (AI) and / or other entities. It is understood that when an element is referred to herein as being “coupled” to another element, it can describe one or more different types of coupling, including but not limited to chemical coupling, communication coupling, electrical coupling, electromagnetic coupling, operational coupling, optical coupling, physical coupling, thermal coupling and / or other types of coupling.

[0016] Figure 1 , Figure 2 and Figure 3 Block diagrams of example non-limiting systems 100, 200, and 300, each according to one or more embodiments described herein, are illustrated, which can each employ reparameterization methods to facilitate the estimation of quantum resources to compute the expected value of a stochastic process. Systems 100, 200, and 300 may each include a quantum resource estimation system 102. Figure 1 The quantum resource estimation system 102 of the system 100 depicted may include a memory 104, a processor 106, a reparameterization component 108, an estimation component 110, and / or a bus 112. Figure 2 The quantum resource estimation system 102 of the system 200 depicted may also include a variational component 202. Figure 3 The quantum resource estimation system 102 of the system 300 depicted may also include an error analysis component 302.

[0017] While some embodiments disclosed in this subject matter describe example applications of the quantum resource estimation system 102 for estimating quantum resources to compute the expected value of stochastic processes such as, for example, derived assets, it should be understood that this subject matter disclosure is not limiting. For example, the quantum resource estimation system 102 can estimate quantum resources to compute the expected value of other stochastic processes (e.g., any type of stochastic process).

[0018] It should be understood that the embodiments of this subject matter disclosed in the various figures herein are merely illustrative, and therefore, the architecture of such embodiments is not limited to the systems, devices, and / or components described therein. For example, in some embodiments, system 100, system 200, system 300, and / or quantum resource estimation system 102 may also include the operating environment 1100 referenced herein and Figure 11 Various computers and / or computing-based components are described. In several embodiments, such computers and / or computing-based components can be combined to achieve integration. Figure 1 , Figure 2 , Figure 3 Used for one or more of the operations implemented by other systems, devices, components and / or computers illustrated and described herein.

[0019] Memory 104 may store one or more computer and / or machine-readable, writable, and / or executable components and / or instructions that, when executed by processor 106 (e.g., a classical processor, a quantum processor, and / or other types of processor), may facilitate the performance of the operations defined by the executable components and / or instructions. For example, memory 104 may store computer and / or machine-readable, writable, and / or executable components and / or instructions that, when executed by processor 106, may facilitate the execution of various functions of the components described herein in relation to quantum resource estimation system 102, reparameterization component 108, estimation component 110, variational component 202, error analysis component 302, and / or other components described herein with or without reference to the various diagrams disclosed herein related to quantum resource estimation system 102.

[0020] Memory 104 may include volatile memory (e.g., random access memory (RAM), static RAM (SRAM), dynamic RAM (DRAM), and / or other types 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 other types of non-volatile memory) that may employ one or more memory architectures. Further examples of memory 104 are provided in the following references. Figure 11 The system memory 1116 is used for description. Such an example of memory 104 can be used to implement any embodiment disclosed in this subject matter.

[0021] Processor 106 may include one or more types of processors and / or electronic circuit systems (e.g., classical processors, quantum processors, and / or other types of processors and / or electronic circuit systems) capable of implementing one or more computer and / or machine-readable, writable, and / or executable components and / or instructions that can be stored on memory 104. For example, processor 106 may perform various operations that can 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), arithmetic, etc.). In some embodiments, 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. Reference is made to processing unit 1114 and... Figure 11 To further illustrate processor 106, we will describe a further example. Such an example of processor 106 can be used to implement any embodiment disclosed in this subject matter.

[0022] The quantum resource estimation system 102, memory 104, processor 106, reparameterization component 108, estimation component 110, variational component 202, error analysis component 302, and / or another component of the quantum resource estimation system 102 described herein can be coupled to each other via bus 112 in a communicative, electrical, operational, and / or optical manner to perform the functions of system 100, system 200, system 300, quantum resource estimation system 102, and / or any component coupled thereto. 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 employ various bus architectures. Reference is made below to system bus 1118 and... Figure 11 To further illustrate bus 112, we will describe a bus 112. Such an example of bus 112 can be used to implement any embodiment disclosed in this subject matter.

[0023] The quantum resource estimation system 102 may include any type of component, machine, device, facility, apparatus, and / or instrument, including a processor and / or capable of effective and / or operative communication with wired and / or wireless networks. All such embodiments are contemplated. For example, the quantum resource estimation system 102 may include server equipment, computing devices, general-purpose computers, special-purpose computers, quantum computing devices (e.g., quantum computers), tablet computing devices, handheld devices, server-type computers and / or databases, laptop computers, notebook computers, desktop computers, mobile phones, smartphones, consumer appliances and / or instrumentation, industrial and / or commercial equipment, digital assistants, multimedia internet-enabled telephones, multimedia players, and / or other types of devices.

[0024] The quantum resource estimation system 102 can be coupled (e.g., communication ground, electrical ground, operative ground, optical ground, and / or coupled via another type of coupler) to one or more external systems, sources, and / or devices (e.g., classical and / or quantum computing devices, communication devices, and / or other types of external systems, sources, and / or devices) using wires and / or cables. For example, the quantum resource estimation system 102 can be coupled (e.g., communication ground, electrical ground, operative ground, optical ground, and / or coupled via another type of coupler) to one or more external systems, sources, and / or devices (e.g., classical and / or quantum computing devices, communication devices, and / or other types of external systems, sources, and / or devices) 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.

[0025] In some embodiments, the quantum resource estimation system 102 may be coupled (e.g., communicative ground, electrical ground, operative ground, optical ground, and / or via another type of coupler) to one or more external systems, sources, and / or devices (e.g., classical and / or quantum computing devices, communication devices, and / or other types of external systems, sources, and / or devices) via a network. For example, such a network may include wired and / or wireless networks, including but not limited to cellular networks, wide area networks (WANs) (e.g., the Internet), local area networks (LANs), and / or other networks. The quantum resource estimation system 102 can communicate with one or more external systems, sources, and / or devices, such as computing devices using virtually any desired wired and / or wireless technology, including but not limited to: Wi-Fi, GSM, UMTS, WiMAX, Enhanced General Packet Radio Service (Enhanced GPRS), 3GPP Long Term Evolution (LTE), 3GPP2 Ultra Mobile Broadband (UMB), High Speed ​​Packet Access (HSPA), Zigbee and other 802.XX wireless technologies and / or legacy telecommunications technologies, Bluetooth, and Session Initiation Protocol (SIP). RF4CE protocol, wireless HART protocol, 6LoWPAN (IPv6 over low-power wireless networks), Z-Wave, ANT, ultra-wideband (UWB) standard protocol and / or other proprietary and non-proprietary communication protocols. Therefore, in some embodiments, the quantum resource estimation system 102 may include hardware (e.g., a central processing unit (CPU), transceiver, decoder, quantum hardware, quantum processor and / or other hardware), software (e.g., a set of threads, a set of processes, executing software, a quantum pulse schedule, quantum circuits, quantum gates and / or other software), or a combination of hardware and software that can facilitate information communication between the quantum resource estimation system 102 and external systems, sources and / or devices (e.g., computing devices, communication devices and / or other types of external systems, sources and / or devices).

[0026] The quantum resource estimation system 102 may include one or more computer and / or machine-readable, writable, and / or executable components and / or instructions that, when executed by the processor 106 (e.g., a classical processor, a quantum processor, and / or other types of processors), can improve the performance of the operations defined by these components and / or instructions. Furthermore, in many embodiments, any component associated with the quantum resource estimation system 102 as described herein with or without reference to the various accompanying drawings disclosed in this subject matter may include one or more computer and / or machine-readable, writable, and / or executable components and / or instructions that, when executed by the processor 106, can improve the performance of the 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 component associated with the quantum resource estimation system 102 disclosed herein (e.g., communicatively, electrically, operatively, and / or optically coupled and / or employed by the quantum resource estimation system 102) may include such computer and / or machine-readable, writable, and / or executable (multiple) components and / or (multiple) instructions. Therefore, according to various embodiments, as disclosed herein, the quantum resource estimation system 102 and / or any component associated therewith may employ processor 106 to execute such computer and / or machine-readable, writable, and / or executable (multiple) components and / or (multiple) instructions to improve the performance of one or more operations described herein with reference to the quantum resource estimation system 102 and / or any such component associated therewith.

[0027] The quantum resource estimation system 102 can facilitate (e.g., via processor 106) the performance of operations performed and / or associated with the reparameterization component 108, estimation component 110, variational component 202, error analysis component 302, and / or with another component associated with the quantum resource estimation system 102 as described herein. For example, the quantum resource estimation system 102 can facilitate (e.g., via processor 106) the following: applying quantum fault-tolerant operations to variationally prepared quantum states corresponding to probability distributions to produce quantum states corresponding to target probability distributions; and / or estimating at least one defined criterion for a quantum computer to compute the expected value of a stochastic process associated with a target probability distribution.

[0028] In another example, as described below, the quantum resource estimation system 102 may further facilitate (e.g., via processor 106) the following: applying transformation operations to a variationally prepared quantum state to produce a quantum state corresponding to a target probability distribution having a defined mean, a defined standard deviation, and / or one or more explicit parameters specifying the target probability distribution; applying quantum fault-tolerant operations to the variationally prepared quantum state to prepare the quantum state as a superposition of possible paths of a discrete-time multivariate stochastic process; training a variational quantum circuit to prepare the variationally prepared quantum state and reducing the computational cost of quantum arithmetic operations performed by a quantum computer to compute the expectation of the stochastic process associated with the target probability distribution; using a Hamiltonian operator to train the variational quantum circuit to generate a ground state corresponding to the target probability distribution; compute one or more errors associated with at least one of the following: applying quantum fault-tolerant operations to the variationally prepared quantum state to produce the quantum state; estimating at least one defined standard; or computed the expectation of the stochastic process associated with the target probability distribution.

[0029] In the above examples, at least one defined criterion may include properties, conditions, characteristics, parameters, and / or configurations of the quantum computer that enable the quantum computer to achieve the defined quantum advantage when computing the expected value of a stochastic process associated with a 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.

[0030] According to various embodiments disclosed in the subject matter described herein, in order to perform one or more operations described above, the quantum resource estimation system 102, reparameterization component 108, estimation component 110, variational component 202, and / or error analysis component 302 may define and / or implement one or more algorithms (e.g., algorithms 2.1, 3.1, 3.2, 4.1, and / or 4.2) and / or one or more equations (e.g., equations (1)–(109)) as described below with reference to sections 1.0–11.0. For example, with reference to sections 3.2 and 10.0 described below, reparameterization component 108 may apply quantum fault-tolerant operations to a variationally prepared quantum state corresponding to a probability distribution (e.g., a standard normal probability distribution) to produce a quantum state corresponding to a target probability distribution (e.g., a normal probability distribution). In this example, reparameterization component 108 may apply quantum fault-tolerant operations to a variationally prepared quantum state to prepare the quantum state as a superposition of 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 using at least one of the following: the mean of the target probability distribution as defined; the standard deviation of the target probability distribution as defined; or one or more explicit parameters of the target probability distribution.

[0031] In another example, referring to sections 3.1.2 and 3.2.3 described below, estimation component 110 may estimate at least one defined criterion for the quantum computer to compute the expected value of a stochastic process associated with a target probability distribution. For example, estimation component 110 may estimate at least one defined criterion for the quantum computer to compute, for example, the value of a derived asset associated with the target probability distribution. For example, estimation component 110 may estimate at least one defined criterion, including but not limited to properties, conditions, characteristics, parameters, configurations, and / or another criterion of the quantum computer that enable it to achieve a defined quantum advantage when computing the expected value of a stochastic process associated with the target probability distribution (e.g., the value of a derived asset).

[0032] In another example, referring to sections 3.2.1 and 11.0 of the following description, variational component 202 can train a variational quantum circuit to prepare a variationally prepared quantum state. For example, variational component 202 can train the variational quantum circuit to prepare a variationally prepared quantum state by generating a ground state corresponding to a target probability distribution using a Hamiltonian operator. It should be understood that 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 derived asset).

[0033] In another example, referring to Section 3.2.2 described below, the error analysis component 302 can calculate one or more errors associated with at least one of the following: applying quantum fault-tolerant operations to a variationally prepared quantum state to generate a quantum state; estimating at least one defined criterion; or calculating the expected value of a stochastic process associated with a target probability distribution (e.g., the value of a derived asset).

[0034] According to various embodiments and as described in the following sections, the quantum resource estimation system 102 can determine an upper limit on the resources involved (e.g., quantum computing resources) to provide a valuable quantum advantage in derivatives pricing. To this end, the quantum resource estimation system 102 can use auto-callable options and target accrued redemption forward (TARF) derivatives as example benchmark use cases to provide a first complete resource estimate for useful quantum derivatives pricing. The quantum resource estimation system 102 can overcome the blocking challenges in known methods and provides a new approach to avoid these for quantum derivatives pricing—the reparameterization method. It should be understood that the reparameterization method, which can be defined and / or implemented by the quantum resource estimation system 102, as described in the following section, mixes pre-trained variational circuits with fault-tolerant quantum computing, thereby significantly reducing the resources (e.g., quantum computing resources) involved in estimating the value of derivatives. As described below and according to various embodiments disclosed in this subject matter, the quantum resource estimation system 102 can determine automatically redeemable options and TARF derivatives as example benchmark use cases involving, for example: approximately 8,000 (8k) logical qubits; approximately 50 million T depths; and an estimated logical clock speed of approximately 10 megahertz (MHz), thereby achieving the defined quantum advantage.

[0035] 1.0 Derivative Pricing

[0036] Pricing derivative contracts using the Monte Carlo method consumes a significant amount of computation in the financial sector, and quantum advantage would be highly valuable in this application. According to various embodiments and as described in the following sections, quantum resource estimation system 102 can provide a first detailed resource estimate of the conditions involved in achieving quantum advantage in derivative pricing. To achieve this, quantum resource estimation system 102 can define and / or implement the novel methods described below according to one or more embodiments disclosed in this subject matter, thereby loading stochastic processes into a quantum computer.

[0037] As defined above and in this document, a “derivative” and / or “derivative asset” is a contract between an issuer and a holder that is valid until a maturity date. Examples of such derivative assets include, but are not limited to, forward contracts, options, automatically callable options, target accrued notes (TARNs), TARFs, and / or other derivative assets. Each derivative defines a return, which defines the income the holder will receive. The return depends on the value of one or more underlying assets over the entire duration of the contract. Examples of such underlying assets include, but are not limited to, stocks, currencies, commodities, and / or other underlying assets. Derivative contracts are ubiquitous in the financial world, used for everything from hedging risk to speculation. Because the future value of the underlying asset and the resulting return are uncertain, the goal of derivative pricing is to determine the value of a derivative contract entered into today.

[0038] The underlying asset is typically modeled as a stochastic process under assumptions such as no arbitrage. No arbitrage assumes that no particular asset is priced differently in different markets, making it impossible to buy an asset in one market and immediately sell it in another to profit. A common model is the evolution of the underlying asset under geometric Brownian motion. Let... Let be a vector of d asset values ​​at time t. This describes the discrete-time multivariate stochastic process of these asset values. Both notations are used for paths in this paper. The corresponding probability density function is given by... To indicate. Command These are the returns of certain derivatives of these assets. The following equation (1) can be used to price derivatives.

[0039]

[0040] If the target stochastic process is modeled using geometric Brownian motion, then they have transition probabilities:

[0041]

[0042] in

[0043]

[0044]

[0045]

[0046] Note that equation (2) defined above at time t is passed through at time t-1. In And it relates to the asset vector. Parameters r and σ jLet be the risk-free interest rate and volatility of asset j, respectively; Δt be the duration between steps in the stochastic process; and Σ be the d×d positive definite covariance matrix of asset d.

[0047]

[0048] Where -1 ≤ ρ ij ≤1 represents the correlation between assets i and j. Then any specific path... The probability is:

[0049]

[0050] The risk-free rate mentioned above is the rate of return on investing in risk-free assets. Although such assets are purely theoretical, government bonds are often used to represent such assets, and r is approximated as the yield on government bonds minus the current inflation rate.

[0051] Traditionally, some simple derivatives under this model are easy to price, such as European call options, which can be analyzed and priced using the Black-Scholes equation. Easy-to-price derivatives are often path-independent, where the payoff is simply the time to exercise. The final price is a function of the price at that point. This contrasts sharply with path-dependent derivatives, which are more difficult to price. In practice, path-dependent derivatives are typically priced using the classic Monte Carlo method.

[0052] Using the classic Monte Carlo method, the accuracy of derivative pricing converges to... 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 considered how this advantage can be specifically used for option pricing and risk analysis. Since this is only a secondary acceleration, this topic focuses on derivatives that are complex enough to have a large M in practice. According to various embodiments and as described in the following sections, quantum resource estimation system 102 can provide end-to-end quantum resource estimation for two examples of such derivatives, such as automatically redeemable options and TARFs, which are computationally expensive path-dependent derivatives. In doing so, quantum resource estimation system 102 (e.g., via reparameterization component 108, estimation component 110, and / or variational component 202) can refine and optimize the quantum states of the underlying distribution loaded onto the asset path. This loading step remains open in the prior art (e.g., unsolved) and quantum resource estimation system 102 (e.g., via estimation component 110) can provide a first account of the resources involved in completing this loading step.

[0053] In addition to estimating resources that can be used for path loading, the quantum resource estimation system 102 offers several optimizations, including intentionally shifting computation from the price space to the return space and novel reparameterization methods. These methods significantly reduce resources and are summarized in Table 1.

[0054]

[0055] Table 1 described above illustrates resource estimation by implementing a quantum resource estimation system 102 according to one or more embodiments disclosed in this subject matter, using different methods to target an error of 2 × 10⁻⁶. -3 Pricing of derivatives. In such an implementation of the quantum resource estimation system 102, there are automatically redeemable options and knock-in put options with 5 automatically redeemable dates, and TARFs with one underlying date and 26 simulated dates. In such an implementation of the quantum resource estimation system 102, it is determined that the Grover-Rudolph method is not applicable in practice, and the Riemann summation method may involve normalizing the assumptions to avoid exponentially growing errors in T. Even if these normalization problems are avoided, as detailed in the Riemann summation (unnormalized) row of Table 1, the reparameterization method defined and / or implemented by the quantum resource estimation system 102 according to one or more embodiments disclosed in this subject matter performs best. Riemann summation normalization is described in Section 3.1 below, and detailed resource estimation that can be defined and / or implemented by the estimation component 110 is discussed in Sections 3.1.2 and 3.2.3 below.

[0056] 1.1 Discretized Derivative Pricing

[0057] To map the derivatives pricing problem to quantum states, the value Discretization. Traditionally, this is not so important because it allows for high precision, but discretization has been explicitly considered in order to study the minimum qubit standard.

[0058] Let each value Discretized to distinct n-qubit registers, i.e., mapped to a regular grid. The discrete space of the paths can then be defined as ω∈Ω. The price expectation is now the sum:

[0059]

[0060] The probability p(ω) can be defined in several ways. For example, the midpoint of the grid cell can be chosen such that...

[0061]

[0062] in This is limited to discrete midpoints. Alternatively, p(ω) can be defined as an integral over discrete elements. These representations are identical under the constraint of fine meshes and, according to one or more embodiments disclosed in this subject matter, use the midpoint method.

[0063] 1.2 Price Range and Return Range

[0064] As mentioned above, geometric Brownian motion can be used to model the price of an underlying asset. This is referred to as a price-space description of the underlying stochastic process, as described in this paper. In the price space, the transition probabilities are given by a multivariate log-normal distribution.

[0065] Another equivalent representation is to consider the stochastic process of the logarithmic return of the underlying asset and perform all calculations within the return space. Logarithmic returns are normally distributed when asset prices follow a log-normal distribution. The vector of the underlying logarithmic return of asset d at time t can be defined as... The transition probability can then be given by a multivariate normal distribution:

[0066]

[0067] in

[0068]

[0069]

[0070] Furthermore, σ, Δt, Σ, and r are the same Brownian motion parameters as in the price space. Note that this is no longer conditional on the value of the previous time step. In fact, the path distribution in the return space comprises Gaussians independent of dT.

[0071] Note the overloaded symbols in the price space formula, as these representations are interchangeable. At any time t', the price of asset j can be calculated from the return space using the following equation:

[0072]

[0073] This computation is used when a stochastic process is modeled in a return space, but the return is defined based on asset prices. In the various embodiments of the subject matter disclosed herein, it will be clearly indicated from the context in which such embodiments operate.

[0074] The switching between the price space and the return space changes from loading from a log-normal distribution to loading from a normal distribution. Generally, normal loading is easier because the random evolution of their targets is independent of the prices at previous time steps, as can be seen by comparing equations (3) and (10) defined above. Therefore, the probability distribution for all T time steps of the stochastic process... The following equations can be used to calculate them simultaneously:

[0075]

[0076] This advantage can compensate for the quantum arithmetic used to evaluate the exponent in equation (11) defined above. In the various embodiments disclosed in this subject matter, the quantum resource estimation system 102 can take advantage of this advantage by using the reparameterization method described herein. Additionally, this is another reason why the quantum resource estimation system 102 can operate in the return space when using derivatives that define returns directly from logarithmic returns and are independent of the price of a single asset.

[0077] 2.0 Core Methodology

[0078] The derivatives pricing method disclosed in this topic extends the quantum mean estimation method. Let the normalized return of any path be given by the following equation:

[0079]

[0080] Algorithm 2.1, as defined below, can be performed in four stages. First, the probability distributions are superimposed onto all possible paths. Second, the payoffs for all possible paths are calculated in quantum parallelism. Third, the expected payoffs are stored in the amplitudes of the labeled states. Fourth, the amplitude estimates, with a given target precision ε>0, are used to... Use a query to read the amplitude.

[0081] Algorithm 2.1 – The Core Method for Derivatives Pricing

[0082] Use parameters n, d, and T, all of which are positive integers.

[0083] Get Operator This is used to load the probability weighted superposition of the paths onto the register of the ndT qubit.

[0084] 1. Applying operators To prepare quantum states:

[0085]

[0086] 2. Calculations in quantum registers

[0087]

[0088] 3. Introduce auxiliary qubits and The value of the register is rotated to its amplitude:

[0089]

[0090] 4. Use amplitude estimation to extract the probability of the auxiliary parameter being |1〉, which is (e.g., the discretized) expected return. Rescale it to obtain

[0091] Note that steps 1-3 in Algorithm 2.1 load the accurate answer after a single execution. If the amplitude could be read directly, the quantum resource estimation system 102 could compute the expectation of all paths in a constant number of queries. Unfortunately, this is not possible, and therefore amplitude estimation introduces linear overhead to extract the answer with a given precision. This is perhaps a key conceptual difference from classical Monte Carlo methods that sample from paths. In Algorithm 2.1, the quantum resource estimation system 102 can compute (e.g., all) possible paths and obtain a sample of the expected gain (e.g., the estimated amplitude).

[0092] Another significant feature of the quantum method is that the quantum resource estimation system 102 can normalize the returns to store them in the amplitude of the state. This normalization can be rescaled at the end and can affect error scaling, as the error is also amplified. In the Riemann summation method discussed in Section 3.1, this rescaled version of the normalization can rapidly accumulate errors.

[0093] 2.1 Amplitude Estimation for Derivatives Pricing

[0094] Typically, path-dependent derivatives (such as automatically redeemable and TARF) are priced using the Monte Carlo method. It is generated by modeling the stochastic process of the target, and then an estimator is used to calculate the expected return:

[0095]

[0096] According to the central limit theorem, the estimator converges to the true expected value with an error of ∈ = O(M). -1 / 2 ).

[0097] This convergence can be quadratically accelerated to ∈=O(M) using Monte Carlo quantum amplitude estimation. -1 Amplitude estimation uses unitary operators on n+1 qubits. As input, make

[0098]

[0099] The parameter 'a' is unknown. Here, the final qubit acts as a label to distinguish between the |ψ0> state and the |ψ1> state.

[0100] Amplitude estimation is achieved by repeatedly applying an operator (often called the Grover operator). To determine a, where and It is a reflection operator. By using phase estimation and quantum Fourier transform, a can be obtained with an accuracy of O(M). -1 The result is determined under the condition that the depth of the quantum circuit is O(1 / ∈). Unfortunately, the depth scaling of the resulting quantum circuit is O(1 / ∈) and involves the use of resource-intensive quantum Fourier transforms. Recent developments have introduced alternative methods aimed at reducing the resources required to perform amplitude estimation and eliminating quantum phase estimation altogether.

[0101] The most efficient amplitude estimation variant known to date is the Iterative Quantum Amplitude Estimation (IQAE). Empirical evidence shows that IQAE outperforms other known variants. Although it omits quantum phase estimation, its performance is four times higher than that of gauge phase estimation methods. Furthermore, it has been shown that the following bounds hold for practical considerations:

[0102]

[0103] in This represents the number of Oracle calls in the worst-case scenario, i.e. The application of this method aims to achieve an estimation error of ε>0 at a confidence level of 1-α, where α∈(0,1).

[0104] 2.2 Path Distribution Loading

[0105] To enable Algorithm 2.1 to achieve a practical quantum advantage, the resources used for performing path loading and payoff calculations are considered. In some cases, there are analytical forms that simplify path loading. For example, in the case of path-independent derivatives, the distribution along the path is not considered; only the final underlying asset price S is involved. T The distribution of quantum resources, such as the log-normal distribution given by the Black-Scholes model, means that the distribution can be computed analytically and then loaded into the quantum state in a variational or explicit manner. Unfortunately, for quantum advantage, the analytical form for this distribution means that these derivatives are generally easy to find classically. Therefore, according to the various embodiments disclosed in this subject matter, the quantum resource estimation system 102 can focus on the path-dependent derivatives superimposed on the paths to be computed.

[0106] While loading the overall distribution is exponentially difficult, several methods have been proposed. If the distribution is efficiently integrable, efficient quantum loading algorithms do exist, namely the Grover-Rudolf method. However, this algorithm has limited practical applicability to derivatives pricing because the relevant probability distribution still involves Monte Carlo integration, albeit quantum integration, which is precisely what can be avoided using amplitude estimation. More details on the limitations of this method are described in detail in Section 7.0.

[0107] Alternative approaches for loading path distributions could involve using quantum generative adversarial networks (qGANs). While this is attractive for reducing overhead, it remains unclear how to predict the cost of training a given qGAN in practice.

[0108] 2.3 Error Analysis

[0109] This section examines various factors that contribute to the overall error in quantum methods for option pricing. Three main components introduce error in the method of Algorithm 2.1. Let f δ =f max -f min .

[0110] Truncation error: The price of a derivative is determined by the integral of all possible values ​​of the underlying price or return. Computing the integral over an infinite field is infeasible; therefore, the quantum resource estimation system 102 can restrict the integration domain as follows: the price and / or logarithmic return are restricted to a range [B]. i B u Within the domain, this restriction omits the probabilistic quality of α. Given an upper bound of P for the density function at each step... max The maximum profit is f. δ This may result in truncation error, which is caused by... To express.

[0111] Discretization error: This error (from ∈ disc The error (denoted as ) arises from the use of Riemann summation to approximate the integral over a finite point grid. This error can be reduced by increasing the number of qubits (n) used to approximate the summation.

[0112] Amplitude estimation error: when using 1 / ∈ amp During the repetitive state preparation process and price calculation, amplitude estimation is generated ∈ amp The error.

[0113] The truncation error and discretization error will be described in more detail below.

[0114] 2.3.1 Truncation Error

[0115] This section introduces the truncation error in the return space, as it is subsequently extended directly to the price space. Using σ max Let be the largest eigenvalue of the covariance matrix Σ. Using the Chernov tail boundary on a Gaussian, the logarithmic return for asset i lies in the interval [μ]. i -wσ max μ i +wσ max The upper limit of the probability outside of ] is By using the union bound, any logarithmic return (e.g., asset d above time step T) lies within the interval. The upper limit of the external probability is Let the initial asset price be at Within the range. Then the corresponding interval in the price space is determined by... Provided.

[0116] The quantum resource estimation system 102 can then define a cutoff window for the values ​​of the n-qubit registers for different dT, which is w standard deviations around the average value for each time step. The cutoff error is given by the following equation:

[0117]

[0118] 2.3.2 Discretization Error

[0119] The final output of the amplitude estimation algorithm represents the Riemann sum of an approximately truncated multidimensional integral. The integral is performed over a dT variable corresponding to d assets at a time step T. It is assumed that each underlying asset and / or return is constrained to an interval [B]. l B u [Within]. To calculate the discretization error, a multidimensional change of the point rule is applied, as follows: Let n qubits be used to represent each target asset, and the domain is divided into 2... ndT There are units, and each value in the register is associated with the integral value at the midpoint of the corresponding unit. Assume β provides an upper limit to the second derivative of the integral (e.g., this can be restated as a deviation from linearity over a length l by βl). 2 (with 2 as the boundary).

[0120] Consider the discretization error accumulated on a single cell. The side length of each cell is (B u -B l ) / 2 n Furthermore, it is a hypercube of dimension l. Note that, by virtue of symmetry, the linear component of the deviation from the value at the center of the element integrates to zero over the element. Therefore, the error in each element can be expressed by the term βx. 2 / 2 With the origin as the center and a side length of l = (B) u -B l ) / 2 n The integral over the dT hypercube is used to define it.

[0121]

[0122] By aggregating the errors across all units, the following is provided:

[0123]

[0124] Based on the number of standard deviations used in discretization and the largest eigenvalue σ of the covariance matrix max The total discretization error is defined by the following formula:

[0125]

[0126] For the target discretization error, equation (23) also gives the total number of qubits that can be used to represent the d assets for time step T, given by the following equation:

[0127]

[0128] Truncation error and discretization error are generally applicable to the methods described in this paper, although each method has additional method-specific error sources, which will be discussed separately for each method.

[0129] Advantages of Quantum Derivatives Pricing Methods (3.0)

[0130] The following sections describe two methods that can be effectively implemented in practice for pricing quantum derivatives: Riemann summation and the reparameterization method disclosed in this subject matter. Riemann summation has been previously introduced, and the first resource analysis described according to one or more embodiments disclosed in this subject matter is for applications aiming to achieve quantum advantage. This analysis reveals limitations due to error scaling caused by normalization. The novel reparameterization method, which can be defined and / or implemented by the quantum resource estimation system 102 according to one or more embodiments disclosed below, avoids the drawbacks of other methods and provides a first end-to-end path to achieving quantum advantage in practice.

[0131] 3.1 Riemann Summation

[0132] The Riemann summation method is described in Algorithm 2.1, which constructs... Methods for path loading operators. Let N=2 ndT It is the size of the Hilbert space containing all possible paths. Let... It is the maximum value of the multivariate transition probability of asset d in equation (2). yes and Normalized transition probabilities for all choices. Let the asset price at each time step be in the interval [0, S]. max Discretization is performed in the following algorithm 3.1. The steps summarized in the algorithm are normalized using a normalization factor. To calculate the price of derivatives, where Note that the normalization factor in the final step size scales exponentially in T. If P max If P < 1, then the factor is redundant as normalization is not involved. However, if P maxIf the value is greater than 1, the error increases exponentially, making this method impractical.

[0133] Algorithm 3.1 – Riemann Summation Pricing

[0134] Use parameters n, d, and T, all of which are positive integers.

[0135] Get the pair operator The access rights, which apply the transition probabilities of the stochastic process to the auxiliary process via the following terms:

[0136]

[0137] 1. Applying Hadamards to ndT qubits to prepare equal superpositions of all paths.

[0138] 2. Set the initial price Load it into the 0th nd qubit register.

[0139] 3. Apply T transition operators To construct each of them

[0140]

[0141] Where N = 2 ndT .

[0142] 4. The calculation is input into the quantum register to obtain

[0143]

[0144] 5. Introduce auxiliary qubits and The value of the register is rotated to its amplitude:

[0145]

[0146] 6. Use amplitude estimation to extract the probability of the auxiliary parameter being |1〉, i.e., (e.g., discretized) expected return. Rescale it to obtain

[0147] Normalization factor P max The probability density function is more easily handled in the return space given by equation (8). If the logarithmic return of each asset at each time step is discretized as the asset volatility σ... j ±w times, then

[0148]

[0149] When asset d is unrelated, then

[0150]

[0151] Therefore, for P max ≤1, select However, choosing a smaller discretization window w increases the truncation error described in Section 2.3.1, and for but This increases proportionally to the number of assets and the time step in the calculation.

[0152] 3.1.1 Riemann Summation Error Analysis

[0153] In addition to the truncation error and discretization error described in Section 2.3, the Riemann summation method also includes errors due to scaling considerations and quantum arithmetic.

[0154] When working in the return space, a transition operator is used to compute equation (12) and perform the operation in equation (26). The amplitude encoding. Assume the transfer operator introduces a maximum addition error ∈ dens The revenue operator used to calculate equations (27) and (28) introduces a revenue error ∈ f Then the total arithmetic error of the quantity estimated by amplitude estimation is:

[0155]

[0156] If we ignore the quadratic error term, then

[0157]

[0158] The logarithmic return for each asset and each time step has been constructed as a domain [-wσ] max wσ max Discretization.

[0159] probability density error ∈ dens Using the equation (12) calculate And it is generated by the auxiliary rotation in equation (26). The terms within the exponent in equation (12) can be written as:

[0160]

[0161] in and C ij These are classical variables, containing volatility and correlation parameters derived from the correlation matrix Σ. In equation (33), each calculation... Therefore, an error is generated. A And there are a total of (d+d2)·T multiplications. Each The term is constructed by using |w|σ maxBounded by a boundary, where each quantum register representing the logarithmic return R is constructed as a representation window [-wσ]. max wσ max The value in ] is used. Using the error analysis for addition and multiplication in Section 8.2, the total error in equation (33) is calculated as:

[0162]

[0163] Then, the error propagation analysis from Section 8.2 is used to compute the exponential, square root, arcsine, and sine functions on the quantum register, which already include arithmetic errors ∈ dens ,∈ dens It can be defined by the following:

[0164]

[0165] Each rescaling of the input variable introduces a corresponding rescaling error. In addition to P described in the previous section... max In addition to rescaling, the quantum resource estimation system 102 can also scale the revenue by 1 / f δ Scale to fit within [0, 1]. The final answer can be multiplied by To explain these rescalings, and therefore, the error in estimating the truncation integral via amplitude estimation is... Scaling. The quantum resource estimation system 102 can then define the error in the Riemann summation method:

[0166]

[0167] Where ∈ trunc ,∈ disc and ∈ amp As defined in Section 2.3.

[0168] 3.1.2 Resource Estimation

[0169] For example, consider an automatically redeemable basket with 5 automatic redeemable dates and parameters T=20, d=3, and a target error ∈ total / f δ ≤2×10 -3 Choosing w to be 5 ensures that the truncation error in equation (20) is within the total target error. Equation (30) gives P. max ≈4 3 This makes the scaling factor... It is very large. However, there may be some methods to handle this normalization problem, such as those inspired by importance sampling and discussed in Section 9.2. Therefore, suppose some methods are invented to handle the normalization, and let P... max =1.

[0170] Then, using the resource computation discussed in Section 9.1, we define ∈ in the case of n = 34 and p = 2. arith ≤2×10 -3 Here, p is the integer part of the fixed-point representation as defined in equation (63). In this case, The operator uses 21k qubits, and the T depth is 23k, which includes the resources used to calculate the price from the logarithmic return using equation (11). For Δt = 1 / 20 and σ min Choosing σ = 0.1, we calculate β ≈ 17. max =0.4 and w=5.

[0171] Therefore, for the choice of n, ∈ disc ≈f δ 10 -5 and

[0172] ∈ trunc ≤f δ ·10 -4 (37)

[0173] If the target is selected ∈ amp Perform 10 -3 Amplitude estimation with a target confidence level of α = 10 -2 Then you can obtain This means the total T depth is approximately 1.9 × 10⁻⁶. 8 .

[0174] Using the same analysis, for TARF contracts with d=1, T=26, and Δt=1 / 26 (e.g., refer to Section 4.2), assuming the annualized volatility of the underlying asset is σ=0.4, then the total T depth is 1.7×10 8 With 15k qubits, it is possible to achieve ∈ total / f δ ≤2×10 -3 The target error.

[0175] 3.2 Reparameterization Method

[0176] The limitations of normalization in Riemann summation have spurred the development and implementation of novel methods for loading stochastic processes. In the reparameterization approach, the quantum resource estimation system 102 can be shifted to modeling assets in the return space. As described in Section 1.2, the target assets in the return space comprise uncorrelated normal distributions. Recognizing that these disparate distributions can be loaded by preparing (e.g., in parallel) many standard normals, and then applying affine transformations to obtain the mean and standard deviation, this method extracts specific subroutines that load the standard normals into the quantum state and use them as resources for the complete distribution of the target path. The normal distribution of the loading subroutines themselves can then be pre-computed and optimized using variational methods. This is a favorable combination of fault-tolerant quantum computing and variational compilation, and will be discussed in Section 3.2.1. Overall, the reparameterization approach avoids the normalization problem in Riemann summation and reduces computational costs and / or resources. The steps of reparameterized pricing are described in Algorithm 3.2 as defined below. Note that in this context, the path ω... R ∈Ω R Refers to a series of logarithmic returns The reparameterization method eliminates the dependence on P max This avoids dependency issues and allows operators to be implemented using variationally trained circuits with relatively few resources. As described in the following sections.

[0177] Algorithm 3.2 – Pricing via Reparameterization

[0178] Use parameters n, d, and T, all of which are positive integers.

[0179] Get the pair operator Access permissions, operators Standard Gaussian distribution Load it into the n-qubit register. Let g i It is discretized into 2 n -bins are the probability mass function values ​​of the standard Gaussian distribution.

[0180] 1. Apply the dT Gaussian operator Applied to ndT qubits. This constructs:

[0181]

[0182] in All 2 in the multivariate standard Gaussian distribution ndT It runs on top of different implementations, and This represents the corresponding probability.

[0183] 2. Order This is the Cholesky decomposition of the covariance matrix. Perform an affine transformation. To adjust the center and volatility of each Gaussian. This is determined by ω. R and p(ω) R ) to represent the corresponding reward path and probability.

[0184] 3. If the return can be directly calculated from the logarithmic return, then... The calculations are directly processed in the quantum register.

[0185]

[0186] If returns are defined based on price rather than just logarithmic returns, then first use To calculate the price space path ω for each asset. This calculation can be performed in parallel for each asset (e.g., each derivative asset).

[0187] 4. Introduce auxiliary qubits and set f(ω) R The value of the register is rotated to its amplitude:

[0188]

[0189] 5. Use amplitude estimation to extract the probability of the auxiliary parameter being |1〉, i.e., the expected return (e.g., discretized). Rescale it to obtain

[0190] 3.2.1 Variationally trained Gaussian loader

[0191] Standard Gaussian loading operator It can be pre-computed because, in the reparameterization method, it is problem-independent. This section describes a method for variational optimization of this operator. Consider preparing a standard normal distribution g(x) i Define point x i = -w+iΔx for the discrete grid case, where i = 0, ... 2 n -1 and Δx = 2w / 2 n In the following example, the domain is fixed at w = 5, such that the full range of values ​​considered is 2w = 10. This choice leaves ~5 × 10 outside the domain. -7 The probability quality. Finally, consider the different indices used to normalize the function in real space and the WAFE function in the quantum register, which are normalized such that the sum of their squared elements is 1. Therefore, the aim is to load the following distribution (e.g., the target distribution) into the quantum register:

[0192] g(x i )×Δx,∑ i g(x i )×Δx=1, (41)

[0193] Note that the target distribution is normalized to 1-α due to the finite cutoff region. In principle, the distribution could be renormalized to 1 over a selected interval of width 2w. Either way, this choice provides negligible differences compared to the error observed during training.

[0194] The chosen variational fit (ansatz) is represented by the so-called Ry-Controlled NOT (Ry-CNOT) fit (ansatz), which has linear connectivity (e.g., refer to Section 11.0). Various embodiments disclosed in this subject provide novel strategies for optimizing circuits in this context, relying on energy-based methods, which are also described in detail in Section 11.0. In short, the objective cost function is the energy of the relevant quantum harmonic oscillator problem, whose ground state is naturally Gaussian. Note that the solution to the discretized quantum harmonic oscillator (e.g., the squared modulus of the solution) may coincide with a normal distribution only in the limit Δx→0. To solve this problem, subsequent re-optimization is performed directly with respect to the infinite norm between the two distributions.

[0195]

[0196] The quantum state encoded in the register is determined by coefficients. To define.

[0197] Please note that directly using equation (42) as the cost function for training is inefficient, and an energy-based method is used for pre-training. Observing the energy of L compared to the relevant quantum harmonic oscillator problem, ∞ How the cost function can exhibit more ripples in the circuit parameter space.

[0198] Please note that the circuits encoding these Gaussian states for different choices of register size n can be pre-trained, and therefore the training is not included as part of the runtime of any given derivative pricing problem. Figure 4 The results shown are for different register sizes and circuit design (ansatz) depths. More details are provided in Section 11.0.

[0199] Figure 4 An example non-limiting graph 400 illustrates how reparameterization methods can be used to facilitate the estimation of quantum resources to compute the expected value of a stochastic process according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements and / or processes employed in the respective embodiments have been omitted.

[0200] Figure 400 illustrates the variational Ry-CNOT circuit trained (e.g., via quantum resource estimation system 102 and / or variational component 202) to approximate different register sizes n. L∞ Error. This numerical study shows that the states that can be prepared approach the target exponentially in depth, and therefore in the number of gate operations. This observation is very consistent with the expected behavior of the Solovy-Kitaev theorem, which provides an upper limit on the number of gates that can be used to achieve the desired accuracy of the cost function. In fact, for any objective operation U∈SU(2 n ), in the dense subset SU(2 n A sequence containing operators This causes the error in energy ε to vary with depth. It decreases exponentially. Although the SU(2) generated by the entanglement block in the circuit... n A subset of operations will not generate SU(2) that are arbitrarily close to the exact unitary U. n Dense subsets (e.g., generators of target states) can be used, but it can be observed numerically that the error still holds true as the number of gates decreases exponentially.

[0201] Finally, this section examines the portability of these results in a fault-tolerant regime, which could enable the application of the entire derivative pricing algorithm. While the numerical results for Ry-CNOT provide evidence for fairly efficient Gaussian state preparation in terms of circuit depth, additional steps are taken to account for the fault-tolerant implementation of such circuits. In this new framework, the continuously rotating Ry gate is extended to a finite product of discrete operations. Again following the Solovy-Kitayev theorem or more specialized results

[27] , any SU(2) operator can also be efficiently represented using a sequence of Clifford+T gates logarithmically scaled with a threshold error ε. The study investigates how to obtain the results before transmission when the rotation angle can only take discretized values. Therefore, it is assumed that each parameter Only available in the format j*2π / M digit Let be the denoting factor, where j is an integer. Section 11.0 numerically illustrates how the error introduced by such digitization changes with the grid size 1 / M. digit And the systemic reduction.

[0202] 3.2.2 Error Analysis

[0203] The total error in the reparameterization method is:

[0204]

[0205] Where ∈ trunc ,∈ disc and ∈ amp These are the truncation error, discretization error, and amplitude estimation error defined in Section 2.3. Here, the term ∈ arith This stems from a single error introduced during the preparation of the Gaussian and the calculation of the gain. Assume that each Gaussian prepared has a value of g(x).i ) has L ∞ Error ∈ dens Furthermore, the profit calculation introduces the maximum error ∈ f The total error will then be:

[0206]

[0207] Where x = (x1, x2, ..., x...) dT Extending the integral and retaining only the linear error term, we give...

[0208] ∈ arith ≤2wdTf δ ∈ dens +∈ f (45)

[0209] in It is used because the probability mass function is truncated.

[0210] 3.2.3 Resource Estimation

[0211] Calculate the resources that may be involved in the automatic redemption of the same basket 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 automatic redemption dates. Further assume that each Gaussian is prepared using n=5 qubits, such that ∈ dens =2×10 -6 ,∈ amp =∈ f =10 -4 This gives the total error ∈ total / f δ ≈2×10 -3 .from Figure 4 It can be observed that L can be fabricated using 5 qubits and a circuit depth of 6. ∞ ~2×10 -6 The Gaussian state requires 7 layers of Ry gates. Using these inputs and computed using the resources described in Section 10.0, reparameterization is used to construct... The operator involves 7.5k qubits and a depth of T of 5.7k, which includes calculating the price according to the logarithmic return equation (11). For α = 10 -2 The target confidence level is 4.6 × 10⁻⁶, and the total T depth is 4.6 × 10⁻⁶. 7 Using the reparameterization method, with d=1, T=26, Δt=1 / 26 and σ=0.4, a total T depth of 6×10⁻⁶ is used. 7 With 9.5k qubits, the TARF pricing in Section 3.1.2 will achieve an accuracy of ∈ total / f δ ≈2×10 -3 .

[0212] 4.0 Revenue

[0213] 4.1 Automatic Redeemable Contracts

[0214] Automatic redeemable contracts are typically defined based on asset returns relative to a predefined reference level and include a notional value used to calculate the contract's dollar value. For a single underlying asset, automatic redemption may include:

[0215] Binary option set {(K i , t i f i )} i＝0...m-1 Each option has a strike price K. i Exercise time t i and fixed income f i Assume these are sorted such that t i <t i+1 .

[0216] • A short knock-in put option with strike price Ko and barrier B, and

[0217] The condition is that if any binary option pays out, all subsequent follow-on options and put options are eliminated.

[0218] The strike price and barrier parameter are defined based on the payoff of the underlying asset price S(t) relative to a reference level, and can be considered, without loss of generality, as the initial spot price of the underlying asset S0. The payoff f of a binary option... i Similarly defined, it is a dimensionless parameter representing the return. In the return space, the basis vectors represent the logarithmic return of the underlying asset (see Section 1.2). Given a logarithmic return value R, checking whether the underlying asset has exceeded the strike price or barrier K may involve checking whether e R ≥K. Let The normalized return is given by the following formula:

[0219]

[0220] Algorithm 4.1 – Implementation of Automatic Redeemable Returns

[0221] Using parameter {(K i , t i f i )} i＝0...m-1 K o Use b to obtain automatic redeemability.

[0222] 1. For each time step t = 1...T, assume that the cumulative return is obtained. It is calculated using the selected path distribution loading method.

[0223] 2. For each t i Parallel application comparators are used to obtain the exercise price register.

[0224] 3. Order Serially, for each bit of the exercise price register, θ is set while all previous bits are zero. i Controlled rotation is applied to accumulator qubits. This is in Figure 5 The diagram in the middle illustrates this. This introduces an m-qubit auxiliary register a.

[0225] 4. For each cumulative return Parallel application comparators to obtain registers This indicates whether the put option has been knocked in and whether it is in-the-money. These bits are then ORed together to obtain the payoff for holding |put>1 if |put>1 is considered.

[0226] 5. Calculation Then normalize it using equation (46) to obtain the put option payoff |f p >. Calculation

[0227] 6. Then control |put> and a[m-1] to use controlled R y Rotate, Rotate bit by bit into the target qubit.

[0228] Figure 5 An example non-limiting circuit 500 is illustrated, which uses a reparameterization method to facilitate the estimation of quantum resources to compute the expected value of a stochastic process according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements and / or processes employed in the various embodiments are omitted.

[0229] Circuit 500 includes an example, non-restricted circuit that can be used to accumulate binary option payouts in an automatically redeemable binary option with five binary options. Here, qubits s0, ..., s5 represent the value K of the five strike prices. i Boolean comparison. Return f i (For example, given by a specific stage in the RY rotation) This only occurs if no gain has occurred previously. The total gain is loaded into the amplitude of the qubit e0.

[0230] Amplitude estimation allows for the calculation of contracts. The expected return, and its dollar value will be determined by Given, where N is the notional value specified in the contract. Automatic redemption can also be defined on a basket of assets, not just a single asset. Typical examples include BestOf and WorstOf, where the contract's return is based on the returns of the best-performing or worst-performing assets in the basket, respectively. These can be handled in a manner similar to a single-asset case, where each asset... The returns are first compared to find the largest or smallest (e.g., as needed).

[0231] Steps 2 and 4 in Algorithm 4.1 defined above can be executed using logic operation circuits (e.g., comparators, AND, OR), which do not introduce errors. Steps 3 and 6 involve the use of controlled R rotations, which decompose into T gates that are functions of the addition error ε, which can be selected based on the required computational precision. Step 5 is the most resource-rich component in the gain circuit, which can involve quantum registers. The calculation involves dividing the register by the classical constant in the denominator of equation (46) and calculating the square root and arcsine of the register. Section 8.1 details the resource standards for all the circuit components described above, and the corresponding arithmetic and gate synthesis errors in Section 8.2.

[0232] Reconsider the automatically redeemable contract from Sections 3.1.2 and 3.2.3, which has 5 automatically redeemable dates, defined on an asset with d=3 and simulated using a time step of T=32. The total summation error ∈ f To achieve this, the operations distributed across steps 3, 5, and 6 of Algorithm 4.1 determine the resources available to each component. For ∈ f =10 -4 Assuming the computation can be parallelized as much as possible, the circuitry for calculating the automatic redeemable returns involves 1.6k qubits and a T depth of 2k.

[0233] 4.2TARF

[0234] This section introduces the TARF implementation for a single asset within the price space.

[0235] TARF is:

[0236] Forward price F, payment dates t1, ..., t m Two execution prices K upper and K lower ≤F, knock out price K o and accrual cap Assuming the dates are sorted, such that t i ≤t i+1 .

[0237] • At each time ti TARF offers rewards:

[0238]

[0239] The condition for triggering the action is: if at any t i The price is higher than K. o If the value is greater than F, then all subsequent gains are knocked out.

[0240] • Accrual cap condition, such that if the total accumulated revenue on any payment date exceeds C, the contract holder receives only the revenue, such that the total revenue equals C and the remaining forward contracts are knocked out.

[0241] make The normalized return is given by the following formula:

[0242]

[0243] Algorithm 4.2 - TARF Profit Implementation

[0244] The parameters are (F, t1, ..., t). m K upper K lower K o ,C) of TARF.

[0245] 1. From knock-out and acknowledgment upper limit qubits |0> o |0> c start.

[0246] 2. For all times t i Three comparators are applied in parallel and some of their results are combined to obtain the register.

[0247] 3. For each t i :

[0248] (a) will and |·> o OR operation until a new label can be created |·> o On the new quantum bits;

[0249] (b) Calculate the payoff (e.g., conditioned on different exercise price qubits) and add the payoff to a register that tracks the total payoff to obtain

[0250] (c) Apply a comparator to calculate whether the accrual limit has been reached, in order to obtain...

[0251] (d) Calculate the quantity that makes the total equal to C, to obtain

[0252] (e) Calculate the normalized revenue in cases where the accrual cap is reached and not reached in parallel. And under the corresponding conditions, AND (e.g., new) knock-out qubits |·> o NOT and upper limit of qubits |·> c Add controlled, appropriate gains to the non-(NOT) to obtain

[0253] (f) will and upper limit of qubits |·> c Perform an AND operation and store the result in a variable that can be relabeled. c In the new qubits.

[0254] 4. The calculation is |θ>.

[0255] 5. In the final encoded qubit |θ> e Above, the application performs a series of angles 2 for each qubit i of |θ>. -i Controlled R y Rotate.

[0256] 5.0 Discussion

[0257] The various embodiments of the subject matter disclosed herein provide comprehensive resource and error analysis for pricing financial derivatives using quantum computers. Specifically, these different embodiments use automatically redeemable and TARF as example case studies, which are two path-dependent option types that are relevant in practice and difficult to price classically. To this extent, the various embodiments disclosed herein provide new methods for overcoming the limitations of existing approaches to load stochastic processes. Although these different embodiments involve geometric Brownian motion, this disclosure is not limiting, as the methods described herein can be readily extended, for example, to stochastic or local volatility methods by loading multiple independent stochastic processes and having conditional or non-stationary reparameterization.

[0258] The resource estimates presented in this paper give the target performance threshold for quantum computers, which can demonstrate advantages in derivatives pricing. Assuming a pricing target of 1 second for automatically redeemable options, the quantum processor, with a logic clock rate of 10 MHz at the code distance, can support 10... 1 One logical operation.

[0259] Although current estimates of logic clock rates are around 10 kHz, roughly three orders of magnitude lower, future work in algorithms, circuit optimization, error correction, and hardware will continue to improve resource metric and runtime. For example, in the case of Shor's algorithm, a careful analysis of several publications has reduced the estimated resource metric by nearly three orders of magnitude. The publication of the subject described in this paper marks the first milestone in the journey toward quantum advantage in pricing financial derivatives.

[0260] 6.0 Background on Derivatives

[0261] 6.1 Forward Contracts

[0262] An example of a derivative is a forward contract, often simply called a forward. With a forward contract, the holder undertakes to buy or sell an asset from the issuer at a fixed price F (called the forward price) on a specified future date. A simple, path-independent example is that the holder undertakes to buy x quantities of an asset at a price of $F per asset m months from now. Forward contracts are typically cash-settled, meaning that at maturity, it is not an exchange of currency and asset, but rather a transaction based on the value of the asset, and only a currency exchange occurs determined by that transaction. For example, if the price of the asset at maturity T is S... T Then the profit is determined by f(S) T )=x(S T -F) is given, where if f(S) T If f(S) > 0, then the contract holder profits (the issuer loses) and if f(S) > 0, then the contract holder profits (the issuer loses). T If ) < 0, then the opposite is true.

[0263] 6.2 Options

[0264] Another example of derivatives is options. Options can be viewed as conditional forward contracts. Through an option contract, the holder has the option to buy or sell an asset from the issuer at a predetermined price on a future date, unlike a forward contract where the issuer is obligated to buy or sell the asset. If the holder chooses to buy or sell the asset, they have chosen to exercise the option. Similar to forward contracts, option contracts are typically settled in cash based on the asset's value on the exercise date. A European call option is an example of a path-independent option with a single underlying asset, where the issuer has the option to buy the asset at the strike price K on the expiration date. The payout on expiration can be written as f(S). T ) = max(S T -K, 0). A European put option allows the issuer to sell an asset at the strike price K on the expiration date, with a payout of f(S). T ) = max(KS TAnother example of path-independent options is binary options, which have a fixed payoff if the underlying asset is above or below the strike price at time T.

[0265] 6.3 Path Dependency and Discounted Returns

[0266] An example of a path-dependent derivative is a knock-out European call option. This is the same as a European call option, but with an additional knock-out price π. If the underlying asset is above that value at any time from 0 to T, the contract becomes worthless. The path-dependent payoff function has the following form:

[0267]

[0268] Including the underlying asset value at times other than T is the reason for introducing path dependency. Another example is a knock-in put option with a reward:

[0269]

[0270] Here, the contract is knocked in because it only has a non-zero return when the asset is below a certain value π, which is the amount by which the asset is below the strike price K.

[0271] In the examples discussed so far, there is only one payment date, where the exchange occurs between the contract issuer and the holder at time T. Some path-dependent options can have multiple payment dates, with payments made multiple times at different points throughout the contract's duration.

[0272] Now let's introduce the concept of discounted returns. As expected, the price of any derivative today is related to its expected future return. However, we can also consider the time delay of returns to account for the opportunity cost of investing in a risk-free asset with an interest rate of r. If the contract is due at time t from today... i There is a profit at f i The discounted return can then be defined as:

[0273]

[0274] The price of a derivative contract is given by the expected value of the discounted return of the underlying asset under a stochastic process. In practice, path-dependent derivatives are computationally more difficult to price and are typically priced using Monte Carlo simulations of the path. This contrasts sharply with models for some path-independent derivatives, which can even have analytical solutions, such as the Black-Scholes model for European call options. Path-dependent options offer Monte Carlo simulations an opportunity to leverage quantum speedup.

[0275] 6.4 Automatic Redeemable Options

[0276] A typical example of automatically callable options is a binary option set, where each binary option pays a different fixed payout on different payout dates, and if the product is knocked out on any payout date (e.g., all future payouts are cancelled), the entire product is cancelled. More formally, let (K, t) i f i () is a binary option, if the value of the underlying asset changes at time t i If the price is higher than the execution price K, then there is a profit f. i Automatic redeemability is a set:

[0277] {(K,t1,f1),(K,t2,f2),...,(K,T,f T )}, (52)

[0278] Where {t i} and {f i} typically increases linearly. If any binary option (K, t) i f i If the payment is made (in-the-money), then all subsequent options {(K, t)} will be subject to the following conditions: j f j ) j＞i It was knocked out (e.g., canceled).

[0279] In practice, these binary options are often bundled with short knock-in put options (i.e., knock-in put options given by the holder to the issuer) to reduce the issuer's risk and lower the price for the client. Similar to the binary option set, if any binary option (K, t) i f i If payment is made, the put option will also be knocked out.

[0280] Like many options, automatic redeemability can be extended to have multiple underlying assets. In this case, the overall option payout is typically linked to the best-performing or worst-performing asset, where performance is defined by return, and if that asset falls below the strike price, the payout of the put option is proportional to the return of the best-performing or worst-performing asset. Note that the strike price against underlying asset j is typically the same throughout the payout range; that is, for each asset K... j The exercise price can be written as Here, k is the same constant for all underlying assets. Therefore, if the worst-performing asset is in-the-money (e.g., above the strike price), then all other assets are also in-the-money. Conversely, if the best-performing asset is below the strike price (or out-of-the-money), then all other assets are also in-the-money. In principle, different underlying assets can have independent strike prices, but since this is uncommon, it can be assumed that all strike prices are defined as...

[0281] Contingent payouts and knock-in put options imply automatically redeemable payouts with strong path dependence. This means that their pricing computations in practice are very costly, sometimes taking five to ten seconds using the classic Monte Carlo method with at least 40,000 paths.

[0282] 6.5 Target Accrued Redemption Forward

[0283] A Target Accrued Redeemable Note (TARN) is any derivative whose yield cap is a specified target amount. Historically, the term referred only to notes, hence the name, but now it encompasses any derivative with an accrual cap. Various embodiments disclosed in this subject use a common TARN called a Target Accrued Redeemable Forward (TARF) as an example case study to implement Quantum Resource Estimation System 102. A TARF is a set of forwards with several knock-out conditions. Specifically, it is a derivative with a single underlying asset having multiple (e.g., 20-60) payment dates and a forward price F. Throughout the contract, there can be two fixed strike prices K. upper =F and K lower <F. On each payment day t i There are several possible returns:

[0284]

[0285] in The subject matter is on the payment date t i The price, and α is a positive constant. Note that when At this point, the return is negative, and therefore the holder of the derivative will suffer a loss. The constant α makes this loss asymmetric, and if it occurs, it is usually one or two.

[0286] Additionally, TARF will have two knock-out conditions based on a knock-out threshold π and an accrual cap C. The first condition stipulates that if the underlying asset price is higher than or equal to π on any payment date, the derivative contract will be immediately knocked out (e.g., if there is no payment on that date). The second condition is that if on any payment date t... i The total return for the holder is due to return f i If the accrued amount exceeds the cap C, the contract holder will only receive the sum of their total earnings, which is C, and then the contract will be knocked out.

[0287] 7.0 Grover-Rudolph underload

[0288] The Grover-Rudolf algorithm is generally considered an efficient method for creating quantum superpositions that correspond to classical distributions. For a given probability distribution {p} of a random variable x. i The algorithm creates quantum superpositions of the following form:

[0289]

[0290] This algorithm is essentially inductive. It first assumes that there exists a method to partition the probability distribution into a certain 2-1 within the region of interest. m Each region and its status:

[0291]

[0292] in It is the probability that the random variable is located in region i. Then, its goal is to add a qubit to the state of equation (55) to further utilize the evolution of the following form, 2 m Each region is subdivided into 2 probability distributions. m+1 Discretization:

[0293]

[0294] Where α i (β i Let be the probability that the random variable lies in the left or right half of region i. and The function represents the left and right boundaries of region i:

[0295]

[0296] Let x be the probability that x lies in region i, and it also lies in the left half of the region. If a circuit can be constructed that performs the following calculation:

[0297]

[0298] in Then angle θ i Controlled rotation on the (m+1)th qubit produces:

[0299]

[0300] Cancel the calculation of |θ i After that,

[0301]

[0302] This is the extension of the state in equation (55) to an additional qubit. Performing this iteration n = log2N times gives a discretization of the distribution of N points on the n qubits.

[0303] In practice, the efficiency of the Grover-Rudolf method depends on the ability to perform the integral in equation (57) superimposed. The argument in the original formula is that a probability distribution that can be classically efficiently integrated using probabilistic methods (e.g., Monte Carlo methods) can be quantum efficiently integrated equivalently. However, since the ultimate goal of quantum derivative pricing is to provide a faster alternative to Monte Carlo integration over probability distributions, performing this integration as part of initial state preparation without any corresponding quantum acceleration negates the advantage of amplitude estimation as a Monte Carlo alternative. While efficient from a complexity standpoint, this means that Grover-Rudolf is insufficient as a method for quantum advantage in derivative pricing.

[0304] Recently, an approximation method for the Grover-Rudolf algorithm was proposed for the standard normal probability distribution, in which the authors suggested writing the expression in equation (57) as:

[0305]

[0306] It can be approximated as:

[0307]

[0308] For small δ, the δ parameter decreases with each iteration of the Grover-Rudolf algorithm adding qubits to the discretization, and the authors emphasize that for m ≥ 7, the approximation in equation (62) becomes sufficiently accurate. However, since the Grover-Rudolf construction is iterative, terms for m < 7 can be computed before the above approximation becomes possible. Therefore, the integral in equation (57) is classically computed and then loaded into the corresponding quantum register. While this approximation allows for simplification of the general Grover-Rudolf algorithm for the standard normal distribution after a certain point in the iteration, it does not change the fact that it can involve computing integrals over the entire domain of the probability distribution, and is therefore practically infeasible for the same reasons as the original Grover-Rudolf method.

[0309] 8.0 Fixed-Point Quantum Computing Resources

[0310] This section introduces the basics of common quantum arithmetic operations and arbitrary rotation synthesis. These operations are used for resource estimation and error analysis. Quantum arithmetic can involve path loading using the Riemann summation method (Section 3.1) and the reparameterization method (Section 3.2), as well as the payoff calculations described in Section 4.0. For the Riemann summation method, all arithmetic operations involved in Equation (12) can be performed, and for the payoff calculations in Equation (15), the arcsine and square root of the quantum register can be computed. Algorithms that efficiently perform individual arithmetic operations have been identified, where resources are typically reported as multiple Toffoli gates or T gates. In cases where arithmetic algorithms are performed from previous work in the literature, the gate costs of the gate sets reported by the authors are reported.

[0311] In the fault-tolerant setting, the depth T of the circuit in the Clifford+T gate set decomposition can be estimated, and it can be assumed that the Toffoli gate can be constructed using auxiliary qubits with a depth T of 1. For each operation, it can be assumed that the resulting circuit can be parallelized in any possible way.

[0312] 8.1 Resource Estimation

[0313] All calculations are performed using fixed-point arithmetic. The n-bit representation of the number x is:

[0314]

[0315] Where x i ∈ 0, 1 represents the i-th bit of the binary representation of x, and p represents the number of bits to the left of the decimal point. The choice of n and p controls the allowable error in each calculation and the resources available to perform arithmetic on the register. Once the values ​​of (n, p) are chosen such that the overall arithmetic error acceptable for the problem under consideration, they are kept constant throughout the analysis. These values ​​can be customized for different components of the circuit and the overall resources available can be reduced, but for the sake of simplicity in the subject matter disclosure, this potential optimization can be ignored.

[0316] Let TF f and T f This represents the number of Toffoli gates and the T-depth used to compute the arithmetic function or logical operation f. The estimated value of the operation is a function of the fixed-point register size (n, p), which will be used to represent the target quantum state involved in the computation.

[0317] Addition and / or subtraction

[0318] Using Tooffoli, addition of two n-bit registers is performed with a cost of 2n-1. Note that subtraction is given by ab = ~(~a+b), and can therefore be implemented as addition of 2n additional X gates, which does not change the Tooffoli count.

[0319] Consider the T-depth cost of controlled and uncontrolled addition, where the adder circuit can operate at a depth T independent of register size. add Constructed when =10, and controlled addition for a register of size n, with a depth cost of T. Constructed under the condition of...

[0320] multiplication

[0321] For multiplication, a controlled adder circuit and a Toffoli counter are used:

[0322]

[0323] This method can also be used to divide a quantum register by its classical value, which can be done by inverting the classical value and using a multiplication algorithm.

[0324] Fixed-point multiplication involves n controlled additions, therefore its depth T cost is These methods can use auxiliary qubits proportional to the register size, but the circuitry includes decomputing the auxiliary qubits, meaning they can be reused for each subsequent addition that wasn't performed in parallel. Since the computation can be parallelized across d assets and T time steps, an additional T*d*n qubits are included when calculating the total to account for these potential auxiliary qubits.

[0325] Additionally, each multiplication circuit is parallelized by treating the register of a factor as z≥1 independent registers of size n / z, and each controlled addition can occur in parallel for z sub-registers. This can be achieved by accumulating z sub-results into the final result using n·(z-1) additional qubits and z-1 additions. z=1 indicates that no additional parallelization is used. If pairwise accumulation can also be parallelized, the total T-depth cost of the parallelized fixed-point multiplication can be given by:

[0326]

[0327] square root

[0328] The square root algorithm is employed, which can be extended to quantum registers in fixed-point representations. For a number x of size (n, p), it is calculated by treating x as an n-bit integer and then shifting the result to the right by (np) / 2 times. This is equivalent to execution.

[0329]

[0330] The Toffoli count for this square root algorithm is:

[0331]

[0332] The authors reported that the depth T of the algorithm is T. sq (n) = 5n + 3 and 2n + 1 qubits can be used.

[0333] Logical operations

[0334] To perform comparisons between quantum registers or between a quantum register and a constant, a Tooffoli / T depth of [value missing] is used. The logarithmic comparator includes a computational aid to eliminate intermediate steps. A logical OR operation on a 2-qubit input can be performed with a Tofoli / T depth of 1.

[0335] index

[0336] A general quantum algorithm can be used to compute smooth classical functions using parallel piecewise polynomial approximations. It is applied to estimate the resources required to compute the exponent. The algorithm takes parameters k and M, which control the polynomial degree and the number of subintervals chosen for the piecewise approximation, respectively. The total number of Toffolis is given by the following formula:

[0337]

[0338] This algorithm can also be used to compute the arcsine function, involving k iterations of multiplication and addition, where a k-th degree polynomial is used for approximation. Additionally, for M selected subintervals, it involves M comparison circuits between an n-qubit input register and a classical value. Using a depth of T... The comparator, the parallel polynomial evaluation circuit has a depth of T as follows:

[0339]

[0340] Where z is an optional parallelization factor introduced in the resource estimation of the above multiplication circuit.

[0341] The number of qubits in a parallel polynomial evaluation scheme that selects polynomial degree k and subinterval number M is:

[0342]

[0343] Arbitrary string

[0344] To calculate the arcsine, similar to that used for the exponent, the aforementioned general quantum algorithm is employed. However, due to the reciprocal...

[0345]

[0346] Diverging around ±1, the authors of the universal quantum algorithm used a transformation:

[0347]

[0348] To handle the interval x∈[0.5,1]. Since the calculation of the arcsine may involve the calculation of the conditional square root of the parameter, and whenever the arcsine is calculated, the square root is also calculated (e.g., equation (15)), and a transformation is used.

[0349]

[0350] Then, resource estimation is considered in a manner similar to that in existing technical methods. The results are as follows:

[0351] • Comparator used to check if x < 0.25 To indicate whether the above transformation should be applied, assuming the value in the quantum register is normalized, the comparator can use two Toffoli gates.

[0352] • Based on the above comparator values, conditional subtraction and conditional replication are performed for preparation. or Conditional copying can use n Tofoli, and conditional subtraction can use TF. add +n Tofoli gates.

[0353] TF for square root calculation sq A Toffoli door.

[0354] • The Toffoli gate used for polynomial computation to compute arcsines.

[0355] Conditional copying and conditional subtraction again depend on the comparator result from the first step; for x < 0.25, we get... Or in other cases, get

[0356] Based on the above considerations and the Toffoli counts for the arcsin(x) polynomial approximation, the calculation is performed... The total Tofoli count is:

[0357]

[0358] Used to calculate the number x represented in a register of size (n, p). The depth T is calculated in a manner similar to an exponential method:

[0359] T arcsq (n, p, z) = T sq (n)+T pp (n, z) + 8n + 6, (75)

[0360] Where T sq (n) = 5n + 3 is the T-depth of the square root algorithm.

[0361] This operation will involve q arcsq qubits, where the qubit standard for arcsine is given by equation (70), for choosing k and M, and 2n+1 for the square root operation:

[0362] q arcsq (n, k, M) = q pp (n, k, M) + 2n + 1 (76)

[0363] R y

[0364] The Gaussian variational preparation discussed in Section 3.2.1 uses R y (θ) Rotation and Controlled R y The rotation encodes the gain as the auxiliary amplitude in equation (16) and the transition probability in the Riemann summation method. Using the repetition-until-success method, arbitrary single-qubit unitary operations can be performed with an auxiliary qubit and measurement at a depth of approximately 1.15log2(1 / ∈) in the case of T.

[0365] When the rotation angle θ is stored in a separate register |θ>, it is obtained and / or a series of R values ​​are used. y (θ k ) rotation, each rotation is controlled on the k-th qubit of |θ>, where

[0366]

[0367] Single controlled R n Decomposition can be used in R n Depth 1, R n This is performed with a count of 3 and a single auxiliary qubit. However, each rotation introduces an error ε, so if |θ> is an n-qubit register (e.g., p bits to the left of the binary point), the end-to-end operation can be performed with precision ε with a depth of at most 1.15nlog2(n / ∈). This is achieved by noting that the amplitude is due to any controlled R n The depth increases with rotation, and decreases slightly with a slight decrease, where θ k<arcsin(∈) is less than ε and is therefore unnecessary. Therefore, using this observation and equation (77), the total number of rotations is calculated as This gives the final T depth of the controlled Ry(θ) operation:

[0368]

[0369] in

[0370] 8.2 Error Analysis

[0371] Given a fixed-point representation of equation (63), each arithmetic operation involving registers introduces some approximation error, depending on the specific method used. The arithmetic errors associated with each operation described in the previous section are outlined here.

[0372] Addition and / or multiplication

[0373] Using fixed-point addition and multiplication methods, where the addition of two (n, p)-sized registers is introduced by... The defined error, and the error associated with multiplication, is at most:

[0374]

[0375] For registers X and Y of size (n, p), where each register already contains the addition error ∈ X ,∈ Y Furthermore, since each factor X and Y is defined by b, the error in the calculation of X·Y is given by the following formula:

[0376] ∈ mul =b*(∈ X +∈ Y )+∈ X ∈ Y +∈ M (n, p) (80)

[0377] index

[0378] A parallel polynomial evaluation method is employed to estimate the resources and associated errors in computing the exponent. The error associated with the algorithm depends on the choice of polynomial approximation and the number of subintervals selected, but the algorithm's authors are listed as 10. -5 Up to 10 -9 The error provides an explicit error estimate and corresponding resources. These are used in overall error estimation. The exponent of the register itself contains an arithmetic error ξ. The error when calculating the register exponent is represented as ∈ exp The total arithmetic error in calculating the register exponent can be approximated as the first order ξ in the Taylor expansion of exp(-x+ξ):

[0379]

[0380] square root

[0381] As described in the previous section, for square root calculation, we consider the square root algorithm, which is an extension for quantum registers in fixed-point representations. The mapping in equation (66) may introduce the maximum error:

[0382]

[0383] When calculating the square root of register x, which already contains (e.g., a positive) addition error ξ, the total addition error of the square root operation is determined by... The definition. This is easily understood by observing using the square root algorithm, which provides an estimate. in but:

[0384]

[0385] The first inequality is... Therefore, for x and ξ...

[0386] Arbitrary string

[0387] For the arcsine calculation, the polynomial evaluation method is used again, where the error rate is estimated to be within a range of 10 for the sample resource. -5 Up to 10 -9 The error in calculating the arcsine is delimited to a register containing the arithmetic error ξ. As described in Section 8.1, arcsin(x) is calculated for x ≤ 0.5. Additionally, when calculating the function arcsin(x) in the algorithm disclosed in this subject matter, it is only performed if x ≥ 0. This gives a domain of 0 ≤ x ≤ 0.5 for the calculation of the arcsin(x) error. Given this domain, note that the slope of arcsin(x) is always monotonically increasing, reaching its maximum at x = 0.5. Therefore, when x = 0.5, the calculation error is given an upper limit:

[0388]

[0389] Where ∈ arcsin The error in calculating the arcsine is determined by the choice of the polynomial degree and the number of subintervals.

[0390] sine

[0391] As described in the previous section, a series of controlled Ry rotations are used to compute the sin(θ) function, these rotations being controlled by qubits in a register containing the angle θ. The error in the sin(θ) computation is defined when the register that should represent θ actually represents θ+ξ due to arithmetic error. To quantify the upper limit, note that in the domain 0 ≤ θ ≤ π / 2, the slope of sin(θ) is monotonically decreasing, and therefore has a maximum slope at θ = 0. Therefore, the error in computing θ = 0 gives an upper limit:

[0392]

[0393] For b≥0, the inequality sin(a+b)≤sin(a)+b is used, and where ∈ sin This is the error caused by the gate decomposition of the Ry operator described in Section 8.1.

[0394] 9.0 Riemann Summation

[0395] 9.1 Riemann Summation Path Loading Resource Estimation

[0396] This section investigates the computation of the T depth and qubit count of equation (12) in a quantum register, and the encoding of this value as the amplitude of the auxiliary qubits as described in Algorithm 3.1. The computation is performed in logarithmic return space (see Section 1.2), and it involves resource estimation of the operations described in Section 8.1.

[0397] Let T f and q f Let T represent the depth of operation f and the number of qubits, respectively. Assuming that computations across assets and time steps T can be parallelized as much as possible, then for computations using the values ​​given by equation (12)... To calculate The contribution of resources is as follows:

[0398] • T used to calculate the term (R-μ) add It can be done in parallel for d assets and T time steps, where T*d*n qubits are used to store the logarithmic return R for all assets and time steps.

[0399] • For all R values ​​in the expansion of equation (33) 2 Item T mul (For example, in parallel for all d and T), obtain and / or use T*d*n additional qubits.

[0400] • For all R values ​​in the expansion of equation (33) i R j Terms (e.g., in parallel in d, T) and T*d²*n qubits in T mul *d2 / (d / 2).

[0401] · This is used to sum all terms in equation (33) in parallel. The qubits from the previous step can be reused here.

[0402] ·T exp , used for using q exp An additional qubit is used to compute the exponent in equation (8), where q exp Equation (70) provides the parameter values ​​to be selected based on the required approximate accuracy.

[0403] ·T arcsq This is used to calculate arcsine sums using the qubit resources given by equation (76). The square root of.

[0404] ·T add The *(T-1) and (T-1)*d*n qubits are used to calculate all sums in equation (11) for t′∈[2,T].

[0405] ·T exp , used to use ratio q exp *d*T more qubits, to compute the prices of all assets and all time steps in equation (11) in parallel.

[0406] A depth of T of 1.15nlog2(n / ∈) is used to perform auxiliary rotations with accuracy ε, in the calculation Controlled by registers. This involves using controlled R. y The decomposition consists of n auxiliary qubits.

[0407] Furthermore, if the parallel multiplication scheme described in Section 8.1 is used to compute prices across assets and time steps, an additional register of size T*d*n is included to implement an adder circuit with a constant depth T and (z-1)*T*d extra qubits, where z≥1 is the chosen optional parallelization factor. It should be noted that the count of the extra qubits used to compute the (R-μ) term and the sum in equation (33) is not yet included, as these can be stored in existing registers for each R. i This is possible because R is calculated after the sum and exponent in equation (11) (e.g., it can be done before calculating the sum). i The value is no longer used.

[0408] Then, using a register of size (n, p), for asset d and time step T, the total depth of the Riemann summation path loading process for accuracy ε is:

[0409] TRS (n, p, d, T, ∈) = n 2 +2n 2 d² / d + 10(d² + d) + 10T + 9n + 5 +

[0410] 1.15nlog2(n / ∈)+2T exp (n, p, ∈) + T arcsin (n, p, ∈) (85)

[0412] Among them, T exp and T arcsin The dependence on ε means that the polynomial approximation parameters k and M for each function in equation (69) will depend on the target precision of the process. The total number of qubits involved is:

[0413] q RS (n,p,d,T,∈)=Tn(4d+d2)+3n+1+q exp (n, p, ∈ (1+dT)+q arcsin (n, p, ∈) (86)

[0415] 9.2 The Importance of Sampling for Normalization in Riemann Summation

[0416] This section introduces a technique closely related to classical importance sampling to overcome the exponential scaling problem shown in Algorithm 3.1. The main idea is to approximate the target distribution with another distribution that can be efficiently loaded, and then use only quantum arithmetic to adjust for errors (e.g., multiplication).

[0417] Suppose a univariate probability density function f: [0, 1] → [0, P], where P > 1 and And the payoff function g: [0, 1] → [0, 1]. In the context under consideration, g will be applied only once. Therefore, it can be assumed to take values ​​in [0, 1] without changing the overall complexity of the method. As mentioned earlier, consider the scaling function f(x) / P and the corresponding operator. and the corresponding operators To prepare the state given by the following equation on n+2 qubits:

[0418]

[0419] Where x i = i / N. Then, the probability of measuring |11> in the last two qubits is given by the following formula:

[0420]

[0421] Furthermore, when multiplied by P, it corresponds to the Riemann sum approximation:

[0422] For X ~ f,

[0423] Furthermore, consider the probability distribution h(x) that can be efficiently loaded into the quantum state. i )∈[0,1], that is, we know how to efficiently construct a quantum operator H such that:

[0424]

[0425] Suppose there exists h such that for all x, f(x) / (h(x)N)∈[0,1], then construct a new operator defined as follows.

[0426]

[0427] Will and Combinations lead to:

[0428]

[0429] This means that in the last two qubits, the probability of measuring |11> is given by the following equation:

[0430]

[0431] That is, for X ~ f, the Riemannian and approximation Therefore, if such a probability distribution h can be found, then constructing a distribution with P can be achieved without needing to rescale by multiplying by P. The directly corresponding state. It is easy to see that for P≤1, h(x)=1 / N can be set to recover the original method without importance sampling.

[0432] In the case of multivariate probability density functions, three cases are distinguished. First, for t = 0, ..., T, the separable function can be written as a univariate function f. t The product of f. In this case, a univariate approach can be applied directly, and it can be applied to each f. t Find the corresponding h t Secondly, the inseparable multivariate probability density function f: [0, 1] d →[0, P], where P>1 and Assuming each dimension is discretized using n qubits, i.e., a total of N qubits are used. d There are N grid points. Then, find the probability distribution h such that for all x, f(x) / (h(x)N) dThe analysis is similar to that of a univariate case, where the probability density function ∈ [0, 1]. Finally, we consider the case of a multivariate probability density function from a stochastic process, given by the following equation:

[0433]

[0434] Where x t ∈[0,1] d And for t=0,...,T,f t (x t |x t-1 )∈[0, P]. Assume the separable probability distribution is:

[0435]

[0436] It can efficiently load and decompose accordingly. in but:

[0437]

[0438] Therefore, if it is found that h alone makes h t It can be loaded efficiently and

[0439]

[0440]

[0441] Then, without exponential scaling of the overhead P T+1 In this case, a random process can be effectively loaded. Similarly, it is easy to see that for P≤1, It can be set and This allows for the restoration of the original method without requiring importance sampling.

[0442] Note that even though it may not always be possible to find an h that satisfies all the conditions, this method still helps to reduce the overhead of scaling.

[0443] 10.0 Reparameterized Path Loading Resource Estimation

[0444] To prepare a standard normal distribution that the quantum resource estimation system 102 can use when implementing the reparameterized loading method disclosed in this subject matter, the quantum resource estimation system 102 (e.g., via variational component 202) can employ the variational method described in Section 3.2.1 and the corresponding gate and / or qubit costs, depending on the required approximation accuracy. In addition, the affine transformation described in Algorithm 3.2 is computed. Costs will also be incurred. Note that the affine transformation is used to calculate the asset price based on the logarithmic return; for asset j at time t', the price is:

[0445]

[0446] in It is the sum The i-th component. One complexity in equation (98) is that the price of each asset cannot be calculated in full parallel across d assets, because the logarithmic return of any related asset will contribute to the calculation of each other's prices. In the case that all assets are paired and correlated, the quantum resource estimation system 102 can calculate the contribution to each asset price from the logarithmic returns of all d assets at that time step, requiring a total of d 2 The addition is performed d times to calculate the prices of all assets at each time step. 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+1)%d is calculated for the choice of i∈[0,d-1], since all d such operations have different source and destination registers. Then, d rounds of addition are performed to calculate the item for all assets. Furthermore, if the quantum resource estimation system 102 calculates for each t' in a separate register... The above calculations can also be parallelized across all time steps.

[0447] The arithmetic error in calculating equation (98) can be reduced (e.g., minimized) by increasing the size of the qubit register to accommodate as much as possible the maximum sum of time step T and asset d. If each Gaussian prepared in equation (38) is discretized using n qubits, then One qubit will be sufficient to store the expression. The maximum value. Assume that for all i, j, the coefficient... against Additional This can also be achieved with a single qubit. For typical cases with practical significance, this condition is not difficult to satisfy; this can be achieved by examining the covariance matrix ∑. ij =Δtρ ij σ i σ j (For example, where according to the definition |ρ ij The argument is based on elements with a volatility of |≤1). Typically, annualized volatility is less than 100% (i.e., σ...). i <1) and the time step typically satisfies Δt < 1, meaning the price of the underlying asset can be sampled more frequently, not just once a year. However, if none of these conditions are met, the quantum resource estimation system 102 can choose a smaller Δt to guarantee |∑ ij | < 1, but at the cost of increasing the number of time steps in the computation.

[0448] In the reparameterization method, for derivatives defined on asset d and time step T, the contributions of depth T and qubit count for loading paths and calculating asset prices are:

[0449] · The depth T of the Gaussian states is used to load in equation (38) using the variational method in Section 3.2.1, where each Gaussian is prepared in parallel and the variational assumption has a depth L. This step involves T*d*n qubits, where n qubits are used to prepare each Gaussian state.

[0450] ·T add *(T-1), used in equation (98) to calculate all sums for t′∈[2,T]. Involving additional One quantum bit.

[0451] ·T add *d, used to calculate the values ​​in equation (98) and All the contributions of more qubits.

[0452] ·T add Used to calculate the cross-asset and time step in equation (98) contribute.

[0453] ·T exp The exponents across assets and time steps in equation (98) are used to calculate q given by equation (70). exp q exp *d*T additional qubits.

[0454] In summary, for asset d and time step T, the total depth T of path loading using the reparameterization method for precise ε is:

[0455] T RP (n,d,T,L,∈)=1.15nlog2(n / ∈)(L+1)+10(d+T)+T exp (n, ∈), (99)

[0457] Using qubit counting

[0458]

[0459] in

[0460] Methods for training Gaussian loaders (11.0)

[0461] This section describes an approximate method for initializing a quantum register using a Variational Quantum Feature Solver (VQE) approach. The algorithm is characterized by a parameterized circuit that generates a parameterized state |ψ({θ})>, which approximates the target state |φ0> and updates its parameters {θ} to optimize the expected value of a suitable cost function. It is shown here that the choice of cost function for optimization is crucial for successful training.

[0462] Energy-based training

[0463] As a first method, the quantum resource estimation system 102 can employ a physics-based approach and define an operator H such that, when its expected value E is evaluated at the target state, it is assumed that its lowest possible value E0 is:

[0464] E0 = <ψ0|H|ψ0>, (101)

[0465] In physics applications, the operator H is often called the Hamiltonian, E is the energy, and ||φ0> represents the ground state. The Gaussian function is well-known:

[0466]

[0467] It is the ground state of the Hamiltonian of the quantum harmonic oscillator.

[0468] H = P 2 2m+m(X-x0) 2 2, (103)

[0469] Where X is the position operator in real space, and 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. In this case, in order to find the state φ0(x), such that... The quantum resource estimation system 102 can adjust m to m = 1 / (2σ) 2 ).

[0470] Note that it is always possible to find a generating Hamiltonian function such that its ground state is the smooth distribution function to be loaded (e.g., the square root of the target probability distribution).

[0471] To translate these considerations into an operational workflow, the quantum resource estimation system 102 can define a method for using a quantum computer to compute the expected value of equation (103). For this purpose, the operator X can be observed... 2 Since the computational basis is diagonal, N can be directly counted from the histogram of the bit string generated by the collapse of the repeating wavefunction. counts (j) Perform the measurement. Operator P 2In the momentum basis, it is diagonal. This means that after the state preparation module, a centered quantum Fourier transform (QFT) circuit is added. The quantum resource estimation system 102 can use the centered Fourier transform to allow negative momentum. As described above, the quantum resource estimation system 102 can estimate negative momentum in the discrete location space x. i = -w+iΔx, where i = 0, ... 2 n -1 and Δx = 2w / 2 n Without sacrificing generality, the quantum resource estimation system 102 can choose a zero-centered domain. Energy It can be calculated in the following way:

[0472]

[0473]

[0474] Where N shots N is the total number of circuit repetitions in space and momentum bases. counts (j)(where 0≤N) counts (j)≤N shots ,∑ j N counts (j)=N shots ) is the number of measurements that collapse to the qubit ground state, which corresponds to the binary representation of the integer j. This strategy bypasses the use of the Pauli representation in equation (103), which would include a Pauli string with the number of measurements using the size of the qubit register increasing exponentially.

[0475] The first step of the procedure is to use quantum circuits to numerically verify whether it is possible to prepare states that systematically converge to equation (102). Using a variational approach avoids expensive quantum arithmetic operations, but at the cost of introducing error sources inherent in numerical variational methods. The most trivial problem concerns the possibility of getting trapped in local minima during the (classical) optimization process. The second, and more profound, problem relates to the representational power of experimental states generated by (e.g., shallow) quantum circuits.

[0476] The main option for the proposed design is the so-called R. y -CNOT circuit. The initial state is defined on an n-qubit register, which the quantum resource estimation system 102 can set to... In You Under the influence of the wave function, a trial wave function is evolved.

[0477] The circuit consists of a series of L blocks, which are rotated from single qubits. The structure is constructed, followed by an entangled unit U spanning the length of a qubit register. ENT. In the above experimental case study, the quantum resource estimation system 102 selected a ladder of CNOT gates with linear connections such that the qubit q i is the qubit q i-1 and the control qubit q i+1 is the target, where i = 1,..., n - 2. Finally, an additional layer of U R gates is applied such that the number of variational parameters is n×(L + 1).

[0478] Since single-qubit rotations are all local operations, it can be written as a tensor product of single-qubit rotations:

[0479]

[0480] is a rotation about the Y-axis on the Bloch sphere of the qubit q i , and k = 1,..., L + 1. The complete description of the unitary circuit operation is as follows:

[0481]

[0482] And the parameterized state is

[0483]

[0484] Note that the unitary describes the complete circuit but not the basis change of the prediction. The basis change of the prediction involves the collapse of the wave function in momentum space as described above.

[0485] For each n and L parameter, the quantum resource estimation system 102 can repeat the optimization run, e.g., eight times, to collect sufficient statistics because the optimization can get stuck at a suboptimal minimum. Since the quantum resource estimation system 102 can use classical simulation of quantum circuits, 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 above case study, the quantum resource estimation system 102 first uses the COBYLA optimizer to perform a warm-up run and then uses the BFGS optimizer for a longer run. To improve efficiency, the starting point for VQE running at depth L uses the best parameters found at depth L - 2 or L - 1 during the previous optimization (if available). Note that the part of the algorithm involving classical optimization feedback can be greatly improved, e.g., using gradient-based methods or imaginary-time inspired update schemes.

[0486] L ∞ Training Refinement

[0487] As described above, the quantum resource estimation system 102 can use a pre-optimized circuit obtained using an energy optimization method as the cost function L∞ The initial guess was then re-optimized. Figure 6A , Figure 6B and Figure 6C The direct L is shown ∞ Optimization fails to deliver acceptable results.

[0488] Figure 6A , Figure 6B and Figure 6C Example non-limiting graphs 600a, 600b, and 600c, respectively, illustrating how reparameterization methods can be used to facilitate the estimation of quantum resources to compute the expected value of a stochastic process according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements and / or processes employed in the various embodiments have been omitted.

[0489] In separate Figure 6A , Figure 6B and Figure 6C In each of the graphs 600a, 600b, and 600c depicted in the diagram, curve 602 represents the optimized run obtained using the energy-based method. In each of these graphs... Figure 6A , Figure 6B and Figure 6C In each of the curves depicted in graphs 600a, 600b, and 600c, curve 604 represents the direct L. ∞ Optimization. In each of the following: Figure 6A , Figure 6B and Figure 6C In each of the curves depicted in graphs 600a, 600b, and 600c, curve 606 represents the use of L. ∞ The optimization further refines the hybrid strategy based on energy optimization. To obtain the data plotted in graphs 600a, 600b, and 600c, the quantum resource estimation system 102 can perform eight independent runs with given parameters n qubits and L.

[0490] The complete result of the optimization is as follows Figure 7A and Figure 7B As shown.

[0491] Figure 7A and Figure 7B Example non-limiting graphs 700a and 700b, respectively, illustrating how reparameterization methods can be used to facilitate the estimation of quantum resources to compute the expected value of a stochastic process according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements and / or processes employed in the various embodiments have been omitted.

[0492] Figure 700a illustrates the relationship between the prepared distribution and the target distribution for different qubit register sizes n. ∞The norm difference is a function of the circuit depth L. The curve in plot 700a shows the best of eight independent optimizations for each parameter. Curve 702 in plot 700a corresponds to the optimization performed using the energy of the quantum harmonic oscillator as the cost function. Curve 704 in plot 700a corresponds to the optimization performed using L... ∞ As a further refinement and optimization of the cost function.

[0493] Figure 700b illustrates the energy difference of the relevant quantum harmonic oscillator models as a function of circuit depth L for different qubit register sizes n. The curves plotted in Figure 700b represent the best of eight independent optimizations for each parameter. Curve 702 in Figure 700b corresponds to the optimization performed using the energy of the quantum harmonic oscillator as a cost function. Curve 704 in Figure 700b corresponds to the optimization performed using L... ∞ This is a further refinement and optimization of the cost function. As expected, for L... ∞ The refinement did not improve this number. The numerical results shown in plots 700a and 700b indicate that convergence to the exact ground state is exponential in depth, and therefore exponential in the number of gate operations.

[0494] L ∞ Failure of direct optimization of norm

[0495] This section provides an empirical explanation for the observed failures of direct norm optimization techniques. To this end, the quantum resource estimation system 102 can probe energy-based and direct L-norm optimization techniques. ∞ The cost function landscape of the two methods is optimized. The quantum resource estimation system 102 can optimize the parameter configuration. Begin by performing the cut in parameter space, following these guidelines.

[0496]

[0497] in It is a vector containing a uniformly distributed random number in the range [-1, 1], and λ∈[-π, π] is a scalar that parameterizes the deformation to form the optimal solution.

[0498] Figure 8A and Figure 8B Example non-limiting graphs 800a and 800b, respectively, illustrating how reparameterization methods can be used to facilitate the estimation of quantum resources to compute the expected value of a stochastic process according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements and / or processes employed in the various embodiments have been omitted.

[0499] Figure 8A and Figure 8BThe curves depicted in Figures 800a and 800b illustrate the cost function landscape detection (e.g., vector) for three different cutting directions. (Three different implementations). The curves shown at the top of graphs 800a and 800b represent the cost function landscape for energy E, while the curves shown at the bottom of graphs 800a and 800b represent L calculated using three different cuts along the parameter space. ∞ Norms and cost functions for landscapes, respectively, for two different settings n=5,7 and depth L=6,10.

[0500] As can be seen from curves 800a and 800b, L ∞ The cost function defined by the norm is more rippled than the cost function defined by the energy E of the relevant quantum mechanical toy problem; the latter, conversely, exhibits a smoother surface. The energy cost function and L... ∞ The attraction basins of the cost function overlap because the ground state of the physical problem is close to the Gaussian function to be achieved, so the second optimization with the L∞ norm will not stop at a high-cost local minimum outside the basin.

[0501] Variational parameter digitization

[0502] While the numerical results above provide fairly strong evidence for the preparation of Gaussian states in terms of the circuit depth of parameterized circuits, additional steps are necessary to address the fault-tolerant implementation of such circuits. In this new framework, continuously rotating RY gates can be extended to finite products of discrete operations. Again, following the Solovy-Kitayev theorem or more specialized results, any U(2) operator can also be efficiently represented using a sequence of Clifford+T gates logarithmically scaled with respect to the threshold error ∈. It is possible to investigate how the results can be obtained prior to the transformation in this state, where the angle of rotation can only take discretized values. Therefore, it can be assumed that each parameter... Only available in the format i*2π / M digit Let represent , where i is an integer.

[0503] The quantum resource estimation system 102 can employ a protocol to optimize parameters on a grid. First, the quantum resource estimation system 102 can project the original continuous-valued parameters onto the grid, taking the grid point closest to each parameter. Then, the quantum resource estimation system 102 can perform a local search on the grid to find a better combination of digitized parameters, reducing (e.g., minimizing) Lobstacles compared to the target distribution. ∞ Norm difference. The quantum resource estimation system 102 can numerically demonstrate that the error introduced by such digitization systematically decreases with increasing grid size. If the quantum resource estimation system 102 considers the L introduced by such digitization... ∞ The norm difference error then decreases to 1 / M.digit It can be observed that, in all cases, the quantum resource estimation system 102 can obtain values ​​compatible with, or even better than, continuous-value solutions, when the grid size reaches M. digit ~10 5 At that time, this is equivalent to using 2π / M digit The space is discretized using approximately 0.0001r radians (rad).

[0504] Figure 9A , Figure 9B and Figure 9C Example non-limiting graphs 900a, 900b, and 900c, respectively, illustrating how reparameterization methods can be used to facilitate the estimation of quantum resources to compute the expected value of a stochastic process according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements and / or processes employed in the various embodiments have been omitted.

[0505] Figure 9A The curves depicted in Figure 900a illustrate the relationship between the prepared distribution and the target distribution for two different circuit depths L and for qubits with n=4. ∞ Norm difference and digital grid size M digit The functional relationship. For each M digit The quantum resource estimation system 102 can "digitize" an eight-parameter set (e.g., a continuous domain considering rotation angle values) obtained through previous independent optimization. Empty (e.g., hollow) squares and circles represent the complete dataset, while solid symbols represent minimum values ​​in the set. Horizontal line 902 represents the best value obtained from the previous optimization, considering a continuous domain of rotation angle values ​​for each L parameter. In some examples, digitization helps to escape local minima and achieve a slightly better solution. Diagonal line 904 is a guide-to-the-eye and represents the function 1 / M. digit and 0.1 / M digit .

[0506] Figure 9B The curve 900b in the figure includes an example non-limiting alternative embodiment of curve 900a, wherein curve 900b illustrates the relationship between the prepared distribution and the target distribution for two different circuit depths L and for n=5 qubits. ∞ Norm difference as digital grid size M digit The function. Figure 9C The curve 900c depicted includes an example non-limiting alternative embodiment of curve 900a, wherein curve 900c illustrates the relationship between the prepared distribution and the target distribution for two different circuit depths L and for n=6 qubits. ∞Norm difference as digital grid size M digit The function.

[0507] Figure 10 The illustration shows a flowchart of an example non-limiting computer implementation of a method 1000 that uses a reparameterization approach to facilitate the estimation of quantum resources to compute the expected value of a stochastic process, according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements and / or processes employed in the various embodiments have been omitted.

[0508] At 1002, the computer-implemented method 1000 may include performing quantum fault-tolerant operations on a variationally prepared quantum state corresponding to a probability distribution via a system operatively coupled to a processor (e.g., processor 106) (e.g., via quantum resource estimation system 102 and / or reparameterization component 108) to produce a quantum state corresponding to a target probability distribution.

[0509] At 1004, the computer-implemented method 1000 may include at least one defined criterion for estimating a quantum computer by a system (e.g., via quantum resource estimation system 102 and / or estimation component 110), the at least one defined criterion being used to calculate the expected value of a stochastic process (e.g., the value of a derived asset) associated with a target probability distribution.

[0510] The quantum resource estimation system 102 can be associated with various technologies. For example, the quantum resource estimation system 102 can be associated with quantum computing technology, quantum hardware and / or software technology, quantum algorithm technology, machine learning technology, artificial intelligence technology, cloud computing technology and / or other technologies.

[0511] Quantum resource estimation system 102 can provide technical improvements to systems, devices, components, operating procedures, and / or processing steps associated with the various techniques identified above. For example, quantum resource estimation system 102 can: apply quantum fault-tolerant computation to variationally prepared quantum states corresponding to probability distributions to generate quantum states corresponding to a target probability distribution; and / or estimate at least one defined criterion for a quantum computer used to compute the expected value (e.g., the value of a derivative asset) of a stochastic process associated with the target probability distribution. In this example, the at least one defined criterion may include properties, conditions, characteristics, parameters, or configurations of the quantum computer that enable it to achieve a defined quantum advantage in computing the expected value (e.g., the value of a derivative asset) of a stochastic process associated with the target probability distribution. In this example, quantum resource estimation system 102 can thus be implemented to identify quantum resources that can leverage the advantage of quantum computing to compute the expected value of a stochastic process (e.g., the value of a derivative asset such as an option contract), while accumulating the minimum computational cost relative to other quantum resources.

[0512] The quantum resource estimation system 102 can provide technological improvements to the processing units associated with it (e.g., processor 106, a quantum processor, and / or another processor). For example, as described above, the quantum resource estimation system 102 can estimate and therefore also determine at least one defined criterion that enables a quantum computer to leverage the advantages of quantum computing to compute the expected value of a stochastic process (e.g., the value of a derivative asset such as an option contract) while accumulating minimal computational cost relative to other quantum resources. In this example, a quantum processor in a quantum computer can be developed (e.g., engineered, 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, enabling the quantum computer to compute the expected value of a stochastic process (e.g., the value of a derivative asset such as an option contract) while accumulating minimal computational cost.

[0513] A practical application of the quantum resource estimation system 102 is that it can use classical computing devices (e.g., classical computers) to estimate at least one defined criterion. This criterion allows the quantum computer to leverage the advantages of quantum computing to compute one or more solutions (e.g., heuristics) to various problems (e.g., estimation problems, optimization problems, and / or other problems) of varying complexity in various fields (e.g., finance, chemistry, medicine, and / or other fields). For example, a practical application of the quantum resource estimation system 102 is that it can use classical computing devices (e.g., classical computers) to estimate at least one defined criterion. This criterion allows the quantum computer to leverage the advantages of quantum computing to compute one or more solutions (e.g., heuristics) to estimation problems and / or optimization problems in the fields of chemistry, medicine, and / or finance, where such solutions can be used to design, for example, new compounds, new drugs, and / or new option premiums.

[0514] It should be understood that the quantum resource estimation system 102 provides a new approach driven by relatively new quantum computing technologies. For example, the quantum resource estimation system 102 provides a new method for estimating at least one defined criterion that enables quantum computers to leverage the advantages of quantum computing to compute the expected value of stochastic processes (e.g., the value of derivative assets such as option contracts) while accumulating the minimum computational cost relative to other quantum resources.

[0515] The quantum resource estimation system 102 can employ hardware or software to solve problems that are inherently highly technical, not abstract, and cannot be performed as a set of human mental actions. In some embodiments, one or more processes described herein can be executed by one or more dedicated computers (e.g., dedicated processing units, dedicated classical computers, dedicated quantum computers, and / or other types of dedicated computers) to perform the tasks defined in relation to the various technologies identified above. The quantum resource estimation system 102 and / or its components can be used to solve new problems arising from advancements in the aforementioned technologies, quantum computing systems, cloud computing systems, computer architectures, and / or the use of other technologies.

[0516] It should be understood that the quantum resource estimation system 102 can utilize various combinations of electrical components, mechanical components, and circuits that cannot be replicated in the human mind or performed by humans, because the various operations that can be performed by the quantum resource estimation system 102 and / or its components, as described herein, are operations greater than the capabilities of human thought. For example, the amount of data processed by the quantum resource estimation system 102 within a given time period, the speed at which such data is processed, or the type of such data can be different from the quantity, speed, or data type that human thought can process within the same time period.

[0517] According to several embodiments, the quantum resource estimation system 102 can also fully operate to perform one or more other functions (e.g., full power-on, full execution, and / or another function) while simultaneously performing the various operations described herein. It should be understood that such simultaneous multi-operation execution is beyond the capabilities of human thought. It should also be understood that the quantum resource estimation system 102 may include information that cannot be manually obtained by an entity (such as a human user). For example, the type, quantity, and / or kind of information included in 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 be more complex than information manually obtained by a human user.

[0518] In some embodiments, 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 the following references Figure 12 The cloud computing environment described is referenced in section 1250 and / or below. Figure 13 One or more functional abstraction layers (e.g., hardware and software layer 1360, virtualization layer 1370, management layer 1380 and / or workload layer 1390) are described in relation to each other.

[0519] The quantum resource estimation system 102 and / or its components (e.g., reparameterization component 108, estimation component 110, variational component 202, error analysis component 302, and / or another component) may employ the following references Figure 12 The cloud computing environment described is referenced in section 1250 and / or below. Figure 13 One or more computing resources of one or more functional abstraction layers (e.g., quantum software) described herein are used to perform one or more operations of one or more embodiments of the subject matter disclosed herein. For example, cloud computing environment 1250 and / or such one or more functional abstraction layers may include one or more classical computing devices (e.g., classical computers, classical processors, virtual machines, servers, and / or other classical computing devices), quantum hardware and / or quantum software (e.g., quantum computing devices, quantum computers, quantum processors, quantum circuit simulation software, superconducting circuits, and / or other quantum hardware and / or quantum software) that can be employed by quantum resource estimation system 102 and / or its components to perform one or more operations of one or more embodiments of the subject matter disclosed herein. For example, quantum resource estimation system 102 and / or its components may employ one or more such classical and / or quantum computing resources to perform one or more classical and / or quantum mathematical functions, calculations and / or equations; calculation 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 of one or more embodiments of the subject matter disclosed herein.

[0520] It should be understood that although this disclosure includes a detailed description of cloud computing, the implementation of the teachings recorded herein is not limited to a cloud computing environment. Rather, embodiments of the invention can be implemented in conjunction with any other type of computing environment now known or developed hereafter.

[0521] Cloud computing is a service delivery model that enables convenient, on-demand network access to a shared pool of configurable computing resources (e.g., networks, network bandwidth, servers, processing, storage, applications, virtual machines, and services) that can be rapidly configured and deployed with minimal management effort or interaction with service providers. This cloud model may include at least five features, at least three service models, and at least four deployment models.

[0522] Its characteristics are as follows:

[0523] On-demand self-service: Cloud consumers can automatically and unilaterally provide computing power, such as server time and network storage, as needed without requiring manual interaction with service providers.

[0524] Extensive network access: Functionality is available via the network and accessed through standard mechanisms that facilitate use by heterogeneous thin or thick client platforms (e.g., mobile phones, laptops, and PDAs).

[0525] Resource pooling: The provider's computing resources are pooled to serve multiple consumers using a multi-tenant model, where different physical and virtual resources are dynamically allocated and reallocated based on demand. Location independence exists because consumers typically cannot control or know the exact location of the resources provided, but may be able to specify a location at a higher level of abstraction (e.g., country, state, or data center).

[0526] Rapid elasticity: Capacity can be configured quickly and elastically, and in some cases automatically, to scale horizontally rapidly and scale down quickly with rapid deployments. For consumers, the available capacity often appears unlimited and any amount can be purchased at any time.

[0527] Measurement services: Cloud systems automatically control and optimize resource usage by leveraging metering capabilities at some level of abstraction appropriate to service types (e.g., storage, processing, bandwidth, and active user accounts). Resource usage can be monitored, controlled, and reported, providing transparency to both service providers and consumers.

[0528] The service model is as follows:

[0529] Software as a Service (SaaS): The functionality provided to consumers is the use of an application from a provider running on cloud infrastructure. The application can be accessed from various client devices via a thin client interface such as a web browser (e.g., web-based email). Consumers do not manage or control the underlying cloud infrastructure, including the network, servers, operating system, storage devices, or even individual application functionality, except perhaps for limited user-specific application configuration settings.

[0530] Platform as a Service (PaaS): This provides consumers with the ability to deploy applications created by consumers or acquired using programming languages ​​and tools supported by the provider onto cloud infrastructure. Consumers do not manage or control the underlying cloud infrastructure, including networks, servers, operating systems, or storage devices, but they can control the deployed applications and, if any, the application configuration within the hosting environment.

[0531] Infrastructure as a Service (IaaS): This provides consumers with processing, storage, networking, and other basic computing resources on which they can deploy and run any software, including operating systems and applications. Consumers do not manage or control the underlying cloud infrastructure, but they can control the operating system, storage devices, deployed applications, and have limited control over selected networking components (e.g., host firewalls).

[0532] The deployment model is as follows:

[0533] Private cloud: Cloud infrastructure for organization operations only. It can be managed by the organization or a third party and can reside on-premises or externally.

[0534] Community cloud: A cloud infrastructure shared by multiple organizations to support a specific community with common concerns (e.g., mission, security requirements, policies, and compliance considerations). It can be managed by an organization or a third party and can reside on-premises or externally.

[0535] Public cloud: Cloud infrastructure available to the public or large industry groups and owned by the organization that sells cloud services.

[0536] Hybrid cloud: A cloud infrastructure consisting of two or more clouds (private, community, or public) that remain a single entity but are bound together by standardization or proprietary technologies to enable data and application portability (e.g., cloud bursting for load balancing between clouds).

[0537] Cloud computing environments are service-oriented, emphasizing statelessness, loose coupling, modularity, and semantic interoperability. At the heart of cloud computing is the infrastructure that includes a network of interconnected nodes.

[0538] For simplicity of explanation, the computer-implemented method is depicted and described as a series of actions. It should be understood that the subject matter innovation is not limited to the actions shown and / or the order of actions; for example, actions may occur in various orders and / or simultaneously, and with other actions not presented or described herein. Furthermore, not all shown actions are necessary for implementing the computer-implemented method according to the disclosed subject matter. Moreover, those skilled in the art will understand and recognize that the computer-implemented method may alternatively be represented as a series of interrelated states via state diagrams or events. Furthermore, it should be understood that the computer-implemented methods disclosed below and throughout this specification can be stored on an article of art to facilitate the transfer and assignment of such computer-implemented methods to a computer. The term "article of art" as used herein is intended to encompass any computer program accessible from any computer-readable device or storage medium.

[0539] In order to provide context for all aspects of the disclosed topic, Figure 11 The following discussion is intended to provide a general description of the appropriate environment in which the various aspects of the disclosed topics can be realized. Figure 11 The diagram illustrates an example non-limiting operating environment that may facilitate one or more embodiments described herein. For the sake of brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted.

[0540] refer to Figure 11 The suitable operating environment 1100 for implementing various aspects of this disclosure may further include a computer 1112. The computer 1112 may further include a processing unit 1114, system memory 1116, and a system bus 1118. The system bus 1118 couples system components (including, but not limited to, 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 multiprocessor architectures may also be used as the processing unit 1114. The system bus 1118 may be any of several types of bus architectures, including memory buses or memory controllers, peripheral buses or external buses, and / or local buses using any kind of available bus architecture, including but not limited to Industry Standard Architecture (ISA), Micro Channel Architecture (MSA), Extended 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).

[0541] System memory 1116 may also include volatile memory 1120 and non-volatile memory 1122. A basic input / output system (BIOS) containing basic routines such as those for transferring information between components within the computer 1112 during startup is stored in non-volatile memory 1122. Computer 1112 may also include removable / non-removable, volatile / non-volatile computer storage media. Figure 11 The illustration shows, for example, a disk storage device 1124. The disk storage device 1124 may also include, but is not limited to, devices such as disk drives, floppy disk drives, tape drives, Jaz drives, Zip drives, LS-100 drives, flash memory cards, or memory sticks. The disk storage device 1124 may also comprise a storage medium alone or in combination with other storage media. To facilitate connection of the disk storage device 1124 to the system bus 1118, a removable or non-removable interface, such as interface 1126, is typically used. Figure 11 Software that acts as an intermediary between the user and basic computer resources, as described in a suitable operating environment 1100, is also depicted. Such software may also include, for example, an operating system 1128. The operating system 1128, which may be stored on disk storage 1124, is used to control and allocate the resources of computer 1112.

[0542] System application 1130 utilizes operating system 1128 to manage resources via program modules 1132 and program data 1134 stored, for example, on system memory 1116 or disk storage 1124. It should be understood that this disclosure can be implemented using various operating systems or combinations thereof. Users input commands or information into computer 1112 using input devices(s) 1136. Input devices 1136 include, but are not limited to, pointing devices such as mice, trackballs, styluses, touchpads, keyboards, microphones, joysticks, game controllers, satellite dishes, scanners, TV tuners, digital cameras, digital camcorders, webcams, etc. These and other input devices are connected to processing unit 1114 via system bus 1118 and through interface ports(s) 1138. Interface ports(s) 1138 include, for example, serial ports, parallel ports, game ports, and Universal Serial Bus (USB). Output devices(s) 1140 use ports of the same type as input devices(s) 1136. Therefore, for example, a USB port can be used to provide input to computer 1112 and to output information from computer 1112 to output device 1140. Output adapter 1142 is provided to illustrate the presence of some output devices 1140, such as monitors, speakers, and printers, as well as other output devices 1140 that may require special adapters. By way of example and not limitation, output adapter 1142 includes video and sound cards that provide connectivity between output device 1140 and system bus 1118. It should be noted that other devices and / or device systems provide both input and output capabilities, such as (multiple) remote computers 1144.

[0543] Computer 1112 can be logically connected to one or more remote computers, such as remote computers(1144), in a networked environment. Remote computers(1144) can be computers, servers, routers, network PCs, workstations, microprocessor-based devices, peer-to-peer devices, or other public network nodes, and typically may include many or all of the elements described relative to computer 1112. For simplicity, only remote computers(1144) are used to illustrate memory storage device 1146. Remote computers(1144) are logically connected to computer 1112 via network interface 1148 and then physically connected via communication connection 1150. Network interface 1148 encompasses wired and / or wireless communication networks, such as local area networks (LANs), wide area networks (WANs), cellular networks, and / or other wired and / or wireless communication networks. LAN technologies include Fiber Distributed Data Interface (FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ring, etc. WAN technologies include, but are not limited to, point-to-point links, circuit-switched networks (such as Integrated Services Digital Network (ISDN) and its variants), packet-switched networks, and Digital Subscriber Line (DSL). Multiple communication connections 1150 refer to the hardware / software used to connect network interface 1148 to system bus 1118. Although communication connection 1150 is shown internal to computer 1112 for clarity, it can also be external to computer 1112. The hardware / software used to connect to network interface 1148 may also include internal and external technologies, such as modems including conventional telephone-grade modems, cable modems and DSL modems, ISDN adapters, and Ethernet cards, for illustrative purposes only.

[0544] Now for reference Figure 12 The illustration depicts a cloud computing environment 1250. As shown, the cloud computing environment 1250 includes one or more cloud computing nodes 1210, and local computing devices used by cloud consumers, such as personal digital assistants (PDAs) or cellular phones 1254A, desktop computers 1254B, laptop computers 1254C, and / or automotive computer systems 1254N, which can communicate with one or more cloud computing nodes 1210. Although Figure 12While not illustrated, cloud computing node 1210 may also include a quantum platform (e.g., a quantum computer, quantum hardware, quantum software, and / or another quantum platform), with the local computing device used by the cloud consumer capable of communicating with the quantum platform. Nodes 1210 can communicate with each other. They can be physically or virtually grouped (not shown) in one or more networks (such as private clouds, community clouds, public clouds, or hybrid clouds or combinations thereof as described above). This allows cloud computing environment 1250 to provide cloud consumers with Infrastructure as a Service, Platform as a Service, and / or Software as a Service without requiring them to maintain resources on their local computing devices. It is understood that... Figure 12 The types 1254A-N of computing devices shown are intended to be illustrative only, and 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).

[0545] Now for reference Figure 13 This demonstrates the 1250 cloud computing environment ( Figure 12 This provides a collection of functional abstraction layers. It should be understood beforehand, such as... Figure 13 The components, layers, and functions shown are intended to be illustrative only, and embodiments of the invention are not limited thereto. As shown, the following layers and corresponding functions are provided:

[0546] The hardware and software layer 1360 includes hardware and software components. Examples of hardware components include: a mainframe 1361; a server 1362 based on a RISC (Reduced Instruction Set Computer) architecture; a server 1363; a blade server 1364; a storage device 1365; and network and networking components 1366. In some embodiments, software components include network application server software 1367, database software 1368, and quantum platform routing software (…). Figure 13 (not illustrated in the image) and / or quantum software ( Figure 13 (Not shown in the image).

[0547] The virtualization layer 1370 provides an abstraction layer from which examples of the following virtual entities can be provided: virtual servers 1371; virtual storage 1372; virtual networks 1373, including virtual private networks; virtual applications and operating systems 1374; and virtual clients 1375.

[0548] In one example, management layer 1380 may provide the following functionalities: Resource Configuration 1381 provides dynamic procurement of computing resources and other resources used to perform tasks within the cloud computing environment. Metering and Pricing 1382 provides cost tracking for the use of resources within the cloud computing environment, as well as billing or invoicing for the consumption of these resources. In one example, these resources may include application software licenses. Security provides authentication for cloud consumers and tasks, as well as protection for data and other resources. User Portal 1383 provides access to the cloud computing environment for consumers and system administrators. Service Level Management 1384 provides cloud resource allocation and management to meet the required service level. Service Level Agreement (SLA) Planning and Fulfillment 1385 provides pre-scheduling and procurement of cloud resources for anticipated future needs according to the SLA.

[0549] Workload layer 1390 provides examples of functionalities that can be used in a cloud computing environment. Non-limiting examples of workloads and functionalities that can be provided from this layer include: map creation and navigation 1391; software development and lifecycle management 1392; virtual classroom education delivery 1393; data analysis and processing 1394; transaction processing 1395; and quantum resource estimation software 1396.

[0550] This invention can integrate systems, methods, apparatuses, and / or computer program products at any possible level of technical detail. The computer program product may include a computer-readable storage medium (or medium) having computer-readable program instructions thereon for causing a processor to perform aspects of the invention. The computer-readable storage medium may be a tangible device that can retain and store instructions for use by an instruction execution device. The computer-readable storage medium may be, for example, but not limited to, electronic storage devices, magnetic storage devices, optical storage devices, electromagnetic storage devices, semiconductor storage devices, or any suitable combination thereof. A non-exhaustive list of more specific examples of computer-readable storage media may also include: portable computer floppy disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable optical disc read-only memory (CD-ROM), digital versatile disk (DVD), memory sticks, floppy disks, mechanical encoding devices (such as punched cards or raised structures in recesses) on which instructions are recorded, and any suitable combination thereof. The computer-readable storage medium used in this document should not be construed as a transient signal itself, such as a radio wave or other freely propagating electromagnetic wave, an electromagnetic wave propagating through a waveguide or other transmission medium (e.g., an optical pulse transmitted through an optical fiber cable), or an electrical signal transmitted through a wire.

[0551] The computer-readable program instructions described herein can be downloaded from a computer-readable storage medium to a corresponding computing / processing device, or downloaded to an external computer or external storage device via a network (e.g., the Internet, a local area network, a wide area network, and / or a wireless network). The network may include copper transmission cables, fiber optic transmission fibers, wireless transmissions, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer-readable program instructions from the network and forwards them to a computer-readable storage medium within the corresponding computing / processing device for storage. The computer-readable program instructions used to perform the operations of this invention may be assembly instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, status setting data, integrated circuit configuration data, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk, C++, etc., and procedural programming languages ​​such as the "C" programming language or similar programming languages. Computer-readable program instructions may execute entirely on the user's computer, partially on the user's computer, as a standalone software package, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In the latter case, 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, programmable logic circuitry, field-programmable gate arrays (FPGAs), or programmable logic arrays (PLAs) can execute computer-readable program instructions by utilizing state information of the computer-readable program instructions, thereby personalizing the electronic circuitry to perform aspects of the invention.

[0552] 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 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 to produce a machine, such that, when executed via the processor of the computer or other programmable data processing apparatus, the instructions create components for implementing the functions / behaviors specified in one or more blocks of the flowchart illustrations and / or block diagrams. These computer-readable program instructions can also be stored in a computer-readable storage medium that can instruct a computer, programmable data processing apparatus, and / or other device to operate in a particular manner, such that the computer-readable storage medium on which the instructions are stored includes an article of writing having instructions that implement the functional / behavioral aspects specified in one or more blocks of the flowchart illustrations and / or block diagrams. Computer-readable program instructions may also be loaded onto a computer, other programmable data processing apparatus or other equipment to cause a series of operational actions to be performed on the computer, other programmable apparatus or other equipment. These instructions refer to the computer-implemented process and cause the instructions, which execute on the computer, other programmable apparatus or other equipment, to implement the functions / behaviors specified in one or more boxes of a flowchart and / or block diagram.

[0553] The flowcharts and block diagrams in the accompanying drawings illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in a flowchart or block diagram may represent a module, segment, or instruction section comprising one or more executable instructions for implementing a specified logical function(s). In some alternative implementations, the functions recorded in the blocks may not appear in the order shown in the figures. For example, depending on the functions involved, two blocks shown consecutively may actually be executed substantially simultaneously, or these blocks may sometimes be executed in reverse order. It should also be noted that each block illustrated in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, may be implemented by a system based on dedicated hardware, which performs the specified function or behavior or executes a combination of dedicated hardware and computer instructions.

[0554] While the subject matter has been described above in the general context of computer executable instructions of a computer program product already running on one or more computers, those skilled in the art will recognize that this disclosure can also be implemented in combination with other program modules. Typically, program modules include routines, programs, components, data structures, and / or other program modules that perform a specific task and / or implement a specific abstract data type. Furthermore, those skilled in the art will understand that the computer implementation of the methods of the present invention can be practiced using other computer system configurations, including single-processor or multi-processor computer systems, microcomputing devices, mainframe computers, and computers, handheld computing devices (e.g., PDAs, telephones), microprocessor-based or programmable consumer or industrial electronic devices, etc. The illustrated aspects can also be practiced in a distributed computing environment, where tasks are performed by remote processing devices connected via a communication network. However, some, if not all, aspects of this disclosure can be practiced on a standalone computer. In a distributed computing environment, program modules can reside in both local and remote memory storage devices. For example, in one or more embodiments, a computer executable component can be executed from a memory that may include or consist of one or more distributed memory cells. As used herein, the terms “memory” and “memory cell” are interchangeable. Furthermore, one or more embodiments described herein can execute code from a computer executable component in a distributed manner, for example, by multiple processors working together 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 located at one location, or multiple memory or memory units located at one or more locations.

[0555] As used herein, the terms “component,” “system,” “platform,” “interface,” etc., may refer to and / or include entities relating to a computer or to an operating machine having one or more specific functions. Entities disclosed herein may be hardware, a combination of hardware and software, software, or software being executed. For example, a component may be, but is not limited to, a process, processor, object, executable file, execution thread, program, and / or computer running on a processor. As an illustration, an application running on a server and a server may be components. One or more components may reside in a process and / or execution thread, and components may be localized on a single computer and / or distributed across two or more computers. In another example, a corresponding component may be executable from various computer-readable media having various data structures stored thereon. Components may communicate via local and / or remote processes, such as based on signals having one or more data packets (e.g., data from a component interacting with another component in a local system, a distributed system, and / or via signals interacting with other systems across a network such as the Internet). As another example, a component may be a device having specific functions provided by mechanical parts operated by electrical or electronic circuitry, which is operated by software or firmware applications executed by a processor. In this scenario, the processor can be internal or external to the device and can execute at least a portion of the software or firmware application. As another example, the component can be a device that provides specific functionality without mechanical parts, using electronic components, where the electronic components can include a processor or other components to execute software or firmware that at least partially grants the electronic components their functionality. In one aspect, the component can be, for example, within a cloud computing system, simulated via a virtual machine.

[0556] Additionally, the term "or" is intended to mean an inclusive "or" rather than an exclusive "or". That is, unless otherwise stated or explicitly indicated by the context, "X adopts A or B" is intended to mean any natural inclusive permutation. That is, "X adopts A or B" is satisfied in any of the foregoing if X adopts A; X adopts B; or X adopts both A and B. Furthermore, unless otherwise specified or explicitly indicated from the context as a singular form, the articles "a" and "an" used in the subject matter specification and figures should generally be interpreted as meaning "one or more". As used herein, the terms "example" and / or "exemplary" are used to indicate as an example, instance, or illustration. For the avoidance of doubt, the subject matter disclosed herein is not limited to such examples. Additionally, any aspect or design described herein as "example" and / or "exemplary" is not necessarily to be construed as superior to other aspects or designs, nor does it imply the exclusion of equivalent exemplary structures and techniques known to those skilled in the art.

[0557] As used in this subject matter specification, the term "processor" can refer to substantially any computing processing unit or device, including but not limited to a single-core processor designed to perform the functions described herein; a single processor with software multithreading capabilities; a multi-core processor; a multi-core processor with software multithreading capabilities; a multi-core processor with hardware multithreading technology; a parallel platform; and a parallel platform with distributed shared memory. Additionally, a processor can refer to an integrated circuit, an application-specific integrated circuit (ASIC), a digital signal processor (DSP), a field-programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), discrete gate or transistor logic, discrete hardware components, or any combination thereof. Furthermore, processors can utilize nanoscale architectures, such as, but not limited to, molecular and quantum dot-based transistors, switches, and gates, to optimize space utilization or improve the performance of user equipment. Processors can also be implemented as a combination of computing processing units. In this disclosure, terms such as "storage," "memory," "data storage," "data memory," "database," and any other substantially information storage component related to the operation and function of a component are used to refer to a "memory component," an entity embodied in "memory," or a component that includes memory. It should be understood that the memory and / or memory components described herein may be volatile or non-volatile memory, or may include both volatile and non-volatile memory. By way of illustration, but 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, for example, RAM may act as external cache memory. By way of illustration, but not limitation, RAM comes in various forms, such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM (RDRAM). Additionally, the memory components of the systems or computer-implemented methods disclosed herein are intended to include, but are not limited to, these and any other suitable memory types.

[0558] The above description includes only examples of systems and computer-implemented methods. It is certainly impossible to describe every imaginable combination of components or computer-implemented methods for the purposes of describing this disclosure; however, those skilled in the art will recognize that many further combinations and arrangements of this disclosure are possible. Furthermore, if terms such as “comprising,” “having,” “possessing,” etc., are used in the detailed description, claims, appendices, and drawings, such terms are intended to be inclusive in a manner similar to the term “comprising,” as “comprising” is interpreted as a transitional word in the claims.

[0559] The descriptions of various embodiments are for illustrative purposes and are not intended to be exhaustive or limited to 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 terminology used herein has been chosen to best explain the principles of the embodiments, practical applications of techniques found in the market, or improvements to techniques, or to enable those skilled in the art to understand the embodiments disclosed herein.

Claims

1. A quantum resource estimation system, comprising: A processor that executes computer-executable components stored in memory, the computer-executable components including: A reparameterization component that applies quantum operations to a first quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution. The application steps include: dT Gaussian operator The ndT qubits used in quantum computers are used to construct the first quantum state given by the following formula: , in It is in this multivariate standard Gaussian distribution 2 ndT The path of a discrete-time multivariable stochastic process running on one of the implementation methods, the Gaussian operator The Gaussian distribution is loaded into the n-qubit register, and Let n represent the corresponding probability, n represent the length of the quantum register of the quantum computer, d represent the number of assets, and T represent the number of time steps, where n, d, and T are positive integers. An affine transformation is performed to adjust each Gaussian distribution of the first quantum state, wherein the resulting second quantum state has a return path corresponding to the target probability distribution. Sum of probabilities , The reparameterization component is also configured to: Will The quantum register of the quantum computer is computed. ,in It is a payoff function; The value of the register is rotated to the amplitude of the auxiliary qubit to generate a second quantum state given by the following formula: ;as well as Perform amplitude estimation on the second quantum state to determine if the auxiliary qubit is The probability of.

2. The system of claim 1, wherein the reparameterization component performs the affine transformation to generate a second quantum state corresponding to the target probability distribution using at least one of the following: the mean of a defined target probability distribution; the standard deviation of a defined target probability distribution; or specifying one or more explicit parameters of the target probability distribution.

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

4. The system of claim 1, wherein the first quantum state is a variationally prepared quantum state prepared by executing a trained variational quantum circuit.

5. The system of claim 4, wherein the computer-executable component further comprises: A variational component trains the variational quantum circuit to prepare the first quantum state and reduces the computational cost of quantum arithmetic operations performed by the quantum computer to calculate the expected value of the stochastic process associated with the target probability distribution.

6. The system of claim 4, wherein the computer-executable component further comprises: A variational component that trains the variational quantum circuit to prepare a variationally prepared quantum state, wherein the training includes using a Hamiltonian operator to generate a ground state corresponding to the target probability distribution.

7. The system of claim 4, wherein the computer-executable component further comprises: An estimation component that estimates at least one defined criterion for a quantum computer, said at least one defined criterion being used to calculate the expected value of a stochastic process associated with the target probability distribution.

8. The system of claim 7, wherein the computer-executable component further comprises: An error analysis component that calculates one or more errors associated with at least one of the following: applying the quantum operation to the variationally prepared quantum state to generate the quantum state; estimating the at least one defined criterion; or calculating the expected value of the stochastic process associated with the target probability distribution.

9. The system of claim 7, wherein the at least one defined criterion is selected from the group consisting of attributes, conditions, properties, parameters, or configurations of the quantum computer, the at least one defined criterion enabling the quantum computer to achieve a defined quantum advantage when computing the expected value of the stochastic process associated with the target probability distribution, and wherein the probability distribution includes a standard normal probability distribution and the target probability distribution includes a normal probability distribution.

10. The system according to claim 1, wherein the quantum operation is a fault-tolerant operation.

11. The system of claim 7, wherein the at least one defined criterion is selected from the group consisting of attributes, conditions, properties, parameters, or configurations of the quantum computer, the at least one defined criterion causing the quantum computer to achieve a defined quantum advantage when computing the expected value of the stochastic process associated with the target probability distribution, and wherein the probability distribution includes a standard normal probability distribution and the target probability distribution includes a normal probability distribution.

12. A computer-implemented method for estimating quantum resources, comprising: By coupling quantum operations to a system that is coupled with processor operations, quantum computations are applied to a first quantum state corresponding to a probability distribution to generate a quantum state corresponding to a target probability distribution. The application steps include: dT Gaussian operator The ndT qubits used in quantum computers are used to construct the first quantum state given by the following formula: , in It is in this multivariate standard Gaussian distribution 2 ndT The path of a discrete-time multivariable stochastic process running on one of the implementation methods, the Gaussian operator The Gaussian distribution is loaded into the n-qubit register, and Let n represent the corresponding probability, n represent the length of the quantum register of the quantum computer, d represent the number of assets, and T represent the number of time steps, where n, d, and T are positive integers. An affine transformation is performed to adjust each Gaussian distribution of the first quantum state, wherein the resulting second quantum state has a return path corresponding to the target probability distribution. Sum of probabilities , The method further includes: Will The quantum register of the quantum computer is computed. ,in It is a payoff function; The value of the register is rotated to the amplitude of the auxiliary qubit to generate a second quantum state given by the following formula: ;as well as Perform amplitude estimation on the second quantum state to determine if the auxiliary qubit is The probability of.

13. The computer-implemented quantum resource estimation method of claim 12, wherein the affine transformation is performed to generate a second quantum state corresponding to the target probability distribution using at least one of the following: the mean of a defined target probability distribution; the standard deviation of a defined target probability distribution; or specifying one or more explicit parameters of the target probability distribution.

14. The computer-implemented quantum resource estimation method according to claim 12, wherein the first quantum state is a superposition of possible paths of the discrete-time multivariate stochastic process.

15. The computer-implemented quantum resource estimation method of claim 12, wherein the first quantum state is a variationally prepared quantum state prepared by executing a trained variational quantum circuit.

16. The computer-implemented quantum resource estimation method according to claim 15, further comprising: The system trains the variable quantum circuit to prepare the first quantum state and reduces the computational cost of quantum arithmetic operations performed by the quantum computer to calculate the expected value of the stochastic process associated with the target probability distribution.

17. The computer-implemented quantum resource estimation method according to claim 15, further comprising: The system trains the variational quantum circuit to prepare the variationally prepared quantum state, wherein the training includes using a Hamiltonian operator to generate a ground state corresponding to the target probability distribution.

18. The computer-implemented quantum resource estimation method according to claim 15, further comprising: The system estimates at least one defined criterion for a quantum computer, which is used to calculate the expected value of a stochastic process associated with the target probability distribution.

19. The computer-implemented quantum resource estimation method according to claim 18, further comprising: The system calculates one or more errors associated with at least one of the following: applying the quantum operation to the variationally prepared quantum state to generate the quantum state; Estimate the at least one defined criterion; or calculate the expected value of the stochastic process associated with the target probability distribution.

20. The computer-implemented quantum resource estimation method of claim 18, wherein the at least one defined criterion is selected from the group consisting of attributes, conditions, properties, parameters, or configurations of the quantum computer, the at least one defined criterion enabling the quantum computer to achieve a defined quantum advantage when calculating the expected value of the stochastic process associated with the target probability distribution, and wherein the probability distribution includes a standard normal probability distribution and the target probability distribution includes a normal probability distribution.

21. The computer-implemented quantum resource estimation method according to claim 12, wherein the quantum operation is a fault-tolerant operation.

22. The computer-implemented quantum resource estimation method of claim 18, wherein the at least one defined criterion is selected from the group consisting of attributes, conditions, properties, parameters, or configurations of the quantum computer, the at least one defined criterion enabling the quantum computer to achieve a defined quantum advantage when calculating the expected value of the stochastic process associated with the target probability distribution, and wherein the probability distribution includes a standard normal probability distribution and the target probability distribution includes a normal probability distribution.

23. A computer program product comprising a computer-readable storage medium having program instructions embodied therein, the program instructions being executed by a processor to cause the processor to: Applying quantum operations to the first quantum state corresponding to the probability distribution generates a quantum state corresponding to the target probability distribution. The application steps include: dT Gaussian operator The ndT qubits used in quantum computers are used to construct the first quantum state given by the following formula: , in It is in this multivariate standard Gaussian distribution 2 ndT The path of a discrete-time multivariable stochastic process running on one of the implementation methods, the Gaussian operator The Gaussian distribution is loaded into the n-qubit register, and Let n represent the corresponding probability, n represent the length of the quantum register of the quantum computer, d represent the number of assets, and T represent the number of time steps, where n, d, and T are positive integers. An affine transformation is performed to adjust each Gaussian distribution of the first quantum state, wherein the resulting second quantum state has a return path corresponding to the target probability distribution. Sum of probabilities , The program instructions are also executed by the processor to cause the processor to: Will The quantum register of the quantum computer is computed. ,in It is a payoff function; The value of the register is rotated to the amplitude of the auxiliary qubit to generate a second quantum state given by the following formula: ;as well as Perform amplitude estimation on the second quantum state to determine if the auxiliary qubit is The probability of.

24. The computer program product of claim 23, wherein the program instructions are further executed by the processor to cause the processor to: The affine transformation is performed to generate a second quantum state corresponding to the target probability distribution using at least one of the following: the mean of the target probability distribution as defined; the standard deviation of the target probability distribution as defined; or one or more explicit parameters of the target probability distribution.

25. The computer program product of claim 23, wherein the first quantum state is a superposition of possible paths of the discrete-time multivariable stochastic process.

26. The computer program product of claim 23, wherein the first quantum state is a variationally prepared quantum state prepared by executing a trained variational quantum circuit.

27. The computer program product of claim 26, wherein the program instructions are further executed by the processor to cause the processor to: The variable quantum circuit is trained to prepare the first quantum state, and the computational cost of quantum arithmetic operations performed by the quantum computer to calculate the expected value of the stochastic process associated with the target probability distribution is reduced.

28. The computer program product of claim 26, wherein the program instructions are further executed by the processor to cause the processor to: The variational quantum circuit is trained to prepare a variationally prepared quantum state, wherein the training includes using a Hamiltonian operator to generate a ground state corresponding to the target probability distribution.

29. The computer program product of claim 26, wherein the program instructions are further executed by the processor to cause the processor to: An at least one defined criterion for estimating a quantum computer, said at least one defined criterion being used to calculate the expected value of a stochastic process associated with the target probability distribution.

30. The computer program product of claim 29, wherein the program instructions are further executed by the processor to cause the processor to: Calculate one or more errors associated with at least one of the following: applying the quantum operation to the variationally prepared quantum state to generate the quantum state; estimating the at least one defined criterion; or calculating the expected value of the stochastic process associated with the target probability distribution.

31. The computer program product according to claim 23, wherein the quantum operation is a fault-tolerant operation.

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