Control of a quantum computing system for determining the quantity of objects

Abstract

The system may receive a function f(x) that describes the value of an object, the value of x, and a probability p(x) about the value of x. The system may determine a quantum operator (I) that, when executed by a quantum computing system, encodes an approximation of the function f(x) into the amplitude of a quantum state without computing |f(x)) for any of the values ​​of x. The system may instruct the quantum computing system to execute a quantum operator (including (I)) and generate quantum states for the registers of a qubit, one of which has an amplitude of quantum states that includes the probability p(x) about the value of x and the output value of an approximation of the function f(x) about the value of x. The system may determine the value of an object based on the generated quantum states.

Description

Technical Field

[0001] (Cross - reference to Related Applications) This application claims the priority and benefit of U.S. Provisional Patent Application No. 63 / 522,277, “Derivative Pricing using Quantum Signal Processing,” filed on June 21, 2023, Greek Patent Application No. 20230100473, filed on June 14, 2023, and U.S. Application No. 18 / 742,960, filed on June 13, 2024, each of which is incorporated by reference in its entirety.

[0002] (1. Technical Field) The present disclosure generally relates to controlling a quantum computing system, and more specifically to controlling a quantum computing system to determine the quantity of an object.

Background Art

[0003] (Background) (2. Description of Related Art) A quantum computing system can be controlled to determine the quantity of an object with probabilistic behavior (e.g., pricing of financial derivatives) using amplitude estimation (AE). These control methods can provide a quadratic acceleration compared to conventional classical Monte Carlo methods. However, the estimated resources required for the quantum circuits of these control methods are significantly more than what is considered possible using current and near - future quantum computing devices. Thus, there is a desire to determine new control methods that use fewer quantum resources.

Summary of the Invention

Means for Solving the Problems

[0004] (Introduction) This disclosure describes the control of quantum computing systems for determining the quantity of an object (e.g., a probabilistically changing quantity). These novel techniques use fewer quantum computational resources than conventional techniques. Among other advantages, these novel techniques can reduce computational resources by determining quantum operators, which enables the use of quantum signal processing (QSP) techniques to determine the quantity of an object (e.g., the value of a product). For ease of explanation, the following description describes the control of a quantum computing system in the context of determining pricing for financial derivatives. However, embodiments are not limited to this exemplary use case. For example, the techniques described herein can also be used to control a quantum computing system to determine initial states for differential equations, quantum simulations of field theory, and finite element analysis.

[0005] Embodiments of the present disclosure have other advantages and features that will become more readily apparent from the following detailed description and appended claims when considered in conjunction with the embodiments in the accompanying drawings. [Brief explanation of the drawing]

[0006] [Figure 1] Figure 1 shows the quantum operator of equation (9).

number

[0007] [Figure 2A] Figure 2A is an illustrative plot of the function f(x) of equation (30).

[0008] [Figure 2B] Figure 2B is an illustrative plot of the absolute error between f(x) (from Figure 2A) and the approximate G(x).

[0009] [Figure 3]Figure 3 is a flowchart illustrating an exemplary method for controlling a quantum computing system to determine the value of an object.

[0010] [Figure 4A] Figures 4A-4B are block diagrams of exemplary computing systems, including classical computing systems and quantum computing systems. [Figure 4B] Figures 4A-4B are block diagrams of exemplary computing systems, including classical computing systems and quantum computing systems.

[0011] [Figure 4C] Figures 4C-4D are block diagrams of exemplary components of a quantum computing system. [Figure 4D] Figures 4C-4D are block diagrams of exemplary components of a quantum computing system.

[0012] [Figure 4E] Figure 4E is a flowchart illustrating an exemplary execution of a quantum routine on a computing system.

[0013] [Figure 5] Figure 5 shows an exemplary architecture of a classical computing system. [Modes for carrying out the invention]

[0014] (Detailed explanation) The figures and the following description relate to preferred embodiments for illustrative purposes only. It should be noted from the following discussion that alternative embodiments of the structures and methods disclosed herein will be readily recognizable as viable alternatives that can be adopted without departing from the principles of the claimed ones.

[0015] In a quantum setting, the quantum circuit that influences the pricing process of financial derivatives may include two components: 1) loading a probability distribution across the paths of the random variables involved in the computation, and 2) encoding the payoff of each path into the amplitude of an ancila qubit. The first component is the random path, where each path arises with probability p(ω).

number

number

number

number

number

number

number

number

[0016] However, in equation (2)

number

number

number

number

number

[0017] (Overview) n-bit register |x> n And assuming a unitary quantum operator U that generates the state, the following holds (note that ω from the previous explanation is replaced with x),

number

number

number

number

[0018] Definition of projector

number

number

number

number

number

number

number

number

[0019] Next,

number

number

number

number

number

number

number

[0020] FIG. 1 is a circuit diagram of the operator of Equation (9) based on the unitary U of Equation (4). The selection of the angle [Number] determines the exact d-degree polynomial G applied to the state, and thus the probability of measuring |00> in the bottom two qubits when d is odd and |0> in the bottom n + 2 qubits when d is even [Number] will be determined by n+2 . [Number] .

[0021] (Implementation) This section shows a method for implementing the selection of the unitary U of Equation (4) and the polynomial G used in Equation (9) to construct the target state of Equation (5). This section assumes that the register |x> n is the n-bit two's complement fixed-point representation of the scalar x with p digits to the left of the binary point, i.e., [Number] .

[0022] First, we consider cases where x is a positive number such that x < 1 when p = 0.

number

number

number

number

[0023] Next, apply the following unitary operators:

number

number

number

number

number

number

number

[0024] When p > 0 such that x ≥ 1 (see equation (10)), the above unitary U is used, and the location of the binary decimal point p can be ignored, which in this case results in the following state:

number

[0025] 2 p Due to the excess factors, the polynomial G approximates the following function in this case to generate the target state of equation (5):

number

[0026] To handle negative values ​​of x, first we select values ​​s≧|x|≧0, and then |x> n |0> m →|x> n |x+s> m To implement this, apply a binary adder and |x+s> mThis is again a fixed-point register with the total number of digits of m and p to the left of the binary decimal point (thus increasing the x value so that all values ​​are either zero or positive). The unitary U in equation (17) is then applied using a second register as input to construct the following state:

number

number

[0027] For both positive and negative values ​​of x, the transformation in equation (19) is that the input to polynomial G is the domain

number

number

number

[0028] (applicable) This section explains how the explanation in the preceding section for preparing the conditions in equation (5) can be used in derivative pricing to calculate an exemplary payoff of form f(S)=max(0,SK) by considering a European call option. Assuming an exercise price K, the holder of a European call option contract will receive the following amount on a future date T (expiration date): f(S) = max(0, S) T -K) (21) S T is the market price of the underlying asset (which is an embodiment of the object) on date T. To determine the present value of this contract, the future price of the underlying asset is modeled as a random variable with a suitable probability distribution, and the price of the option is calculated by calculating the expected value of its payoff in equation (21).

[0029] Exemplary reparameterization method for derivative product pricing 3 (1) Load a discretized normal distribution, then perform an affine transformation to a register using quantum arithmetic, and consider the log-rate of profit r as a possibility in time T, each weighted by the probability of occurrence. i By generating superposition,

number

number

number

number

number

number

number

[0030] However, the above exemplary reparameterization method is resource-intensive, using quantum arithmetic (for example, using qubits like classical bits and performing the same arithmetic logic as in classical computing systems to digitally compute the values)

number

[0031] |r| in equation (22) i > n Let the registers be represented by their respective logarithmic profit margins in fixed-point representation, where s = ln(S0 / K). Comparison | r i >n |0>→|r i > n |r i Applying the circuit that enforces ≤ s > to equation (22), we obtain the following:

number

[0032] An m qubit register is added, and the second register is also a fixed-point representation, where m is r i Sufficiently large enough to hold the maximum value of +|s|, |r i > n |0> m →|r i > n |r i +|s|> m Calculate the following,

number

number

number

number

number

number

number

number

number

[0033] Figure 2A is a plot of the function f(x) of equation (30), used to calculate the payoff of the European call option, with d=16th order approximation Chebyshev polynomial G(x), which can be applied to quantum systems using the quantum signal processing unitary of equation (9). Figure 2B is a plot of the absolute error between f(x) and the approximated G(x). For both plots, the following values, i.e., S0=K=100 and p=3, are used.

number

[0034] (Example method) While the above description provides exemplary methods for controlling a quantum computing system to determine the value of an object (for example, the price of a financial derivative), the following paragraphs describe additional embodiments. The following description may omit the steps described above and / or include steps that are added to or alternative to the steps described above.

[0035] Figure 3 is a flowchart of an exemplary method 300 for controlling a quantum computing system to determine the value of an object (e.g., the price of a financial derivative). Note that the following description will refer several times to Figure 4, which is further described below. In the embodiment of Figure 3, method 300 is carried out from the perspective of a (e.g., classical) computing system (e.g., 410) that can instruct and / or control a quantum computing system (e.g., 420). Method 300 may include more or fewer steps than those described herein. In addition, the steps may be carried out in a different order or by different components than those described herein. In some embodiments, method 300 is carried out by a computing system that executes code stored on a (e.g., non-transient) computer-readable storage medium that causes the computing system to carry out the steps of method 300.

[0036] In step 310, the computing system receives a function f(x) that describes the value of an object (which is an embodiment of the quantity of the object), the value of x, and the probability p(x) about the value of x.

[0037] In step 320, the computing system generates a first quantum state characterized by the superposition of states, which encodes the x value, when executed by the first quantum computing system (e.g., 420), with the quantum operator P (also known as "

number

number

[0038] In step 330, the computing system encodes an approximation of the function f(x) into the amplitude of a second quantum state without computing |f(x)> for any of the values ​​of x (where |f(x)> is the binary representation of the output value of f(x) stored on the qubits of the quantum register), when executed by a quantum computing system (e.g., 420 or another quantum computing system), using a quantum operator.

number

number

[0039] Steps 310, 320, and 330 may be part of the exemplary step 460 in Figure 4E.

[0040] In step 340, the computing system generates a third quantum state with respect to the qubit's register using the quantum operator P and

number

number

number

[0041] In some respects, one of the amplitudes of the third quantum state is the square root of the weighted average of f(x) with respect to x values, where the weights are the probabilities p(x) with respect to the corresponding values ​​of x. In other words, one of the amplitudes is

number

number

[0042] Step 340 may be part of the exemplary step 465 in Figure 4E.

[0043] In step 350, the computing system determines the value of the object based on the third quantum state that has been generated.

[0044] Step 350 may be part of the exemplary step 475 in Figure 4E.

[0045] In some respects, method 300 does not involve calculating |f(x)>.

[0046] In some respects, method 300 further determines s (where s is based on the absolute value of the x-value) and performs the operation on a third quantum computing system, i.e., |x> n |0> m →|x> n |x+s> m In the formula, n and m are integers greater than zero, m > n, and |x> n |x+s> is a quantum state relating to an n-qubit register that stores the n-bit binary representation of the value of x, where |x+s> m This involves instructing a quantum binary adder circuit to implement a quantum state relating to a register of m qubits that stores a representation of the value of x+s, with the total number of digits of m and the number of digits of p to the left of the binary decimal point. In some aspects, s is based on an upper bound on the absolute value of the x value (e.g., s≧|x|≧0). For example, s is the absolute value of the smallest x value (however, s may be greater than this). For example, if the x value is in the range of 4 to -5, then s=5.

[0047] In some respects, the quantum binary adder circuit performs the operation of the quantum operator P, however the quantum operator

number

[0048] For additional information regarding s and the quantum binary adder circuit, see, for example, equations (18)-(19) and (27)-(29) and their associated explanations.

[0049] In some aspects, quantum operators

number

number

number

[0050] In some aspects, quantum operators

number

number

[0051] In some aspects, quantum operators

number

number

number

number

[0052] In some respects, the function f(x) is a normalized function with output values ​​ranging from 0 to 1.

[0053] In some aspects, the value of an object (a certain embodiment of the quantity of an object) varies probabilistically.

[0054] In some respects, determining the value of an object based on the generated quantum state involves performing an amplitude estimation algorithm to determine the object's value.

[0055] Some aspects of quantum computing systems (e.g., 420) are similar to those of classical computing systems (e.g., 410), namely,

number

number

number

number

[0056] Other aspects include components, devices, systems, improvements, methods, processes, applications, computer-readable media, and other technologies related to any of the above.

[0057] (Quantum signal processing) This section provides a general explanation of quantum signal processing (QSP). QSP is a technique that performs polynomial transformations on the singular values ​​of a matrix A that is block-encoded with unitary operators. Specifically, projectors

number

number

number

number

number

number

number

number

[0058] More formally, the operators are defined as follows:

number

number

number

number

[0059] In practice, d calls to U perform a d-degree polynomial transformation on A, and the phase factor

number

Number

Number

Number

Number

Number

[0060] (Description of the computing system) The embodiments described above can be implemented using one or more computing systems. An exemplary computing system is described below.

[0061] FIG. 4A is a block diagram illustrating an embodiment of a computing system 400. In the example of FIG. 4A, the computing system 400 includes a classical computing system 410 (also referred to as a non-quantum computing system) and a quantum computing system 420. However, the computing system may include only a classical computing system or only a quantum computing system. Certain embodiments of the classical computing system 410 are further described with respect to FIG. 5. Although the classical computing system 410 and the quantum computing system 420 are both illustrated, they may be physically separate systems. For example, FIG. 4B illustrates an exemplary cloud computing architecture in which the classical computing system 410 and the quantum computing system 420 communicate via a network 457. The computing system 400 may include elements different from or additional to those illustrated (e.g., multiple quantum computing systems 420).

[0062] The classical computing system 410 may control the quantum computing system 420. For example, the classical computing system 410 may generate and transmit instructions for the quantum computing system 420 to execute a quantum algorithm or a quantum circuit. Although only one classical computing system 410 is illustrated in FIG. 4A, any number of classical computing systems 410 or other external systems may be connected to the quantum computing system 420.

[0063] Figure 4C is a block diagram illustrating one embodiment of a quantum computing system 420. The quantum computing system 420 includes any number of qubits 450 and an associated qubit controller 440. As shown in Figure 4D, the qubits 450 may reside in a qubit register (or more registers) of the quantum computing system 420. Qubits are described further below. The qubit controller 440 is a module that controls one or more qubits 450. The qubit controller 440 may include one or more classical processors, such as one or more CPUs, one or more GPUs, one or more FPGAs, or any combination thereof. The qubit controller 440 may perform physical operations on one or more qubits 450 (for example, it may perform quantum gate operations on the qubits 440). In the embodiment shown in Figure 4C, a separate qubit controller 440 is illustrated for each qubit 450; however, a qubit controller 450 may control multiple (e.g., all) qubits 450 of the quantum computing system 420, or multiple controllers 450 may control a single qubit. For example, a qubit controller 450 can be a separate processor, parallel threads on the same processor, or some combination of both.

[0064] Figure 4E is a flowchart illustrating an exemplary execution of a quantum routine on computing system 400. Classical computing system 410 generates a quantum program (460) which will be executed or processed by quantum computing system 420. The quantum program may include instructions or subroutines which will be implemented by quantum computing system 420. In one embodiment, the quantum program is a quantum circuit. Quantum computing system 420 executes the program (465) and computes the result (referred to as a shot or invocation) (470). Computing the result may include performing a measurement of the quantum state generated by quantum computing system 420 as a result of executing the program. In practice, this may be done by measuring one or more values ​​of qubits 450. Quantum computing system 420 typically performs multiple shots and accumulates statistics from the probabilistic execution. The number of shots and any changes (e.g., parameter changes) that occur between shots may be referred to as a schedule. The schedule may be defined by a program. The results (e.g., quantum state data) (or accumulated results) are recorded by the classical computing system 410 (475). The results may be returned after a termination condition is met (e.g., a threshold number of shots have occurred). The classical computing system 410 may determine a quantity based on the received results.

[0065] The quantum computing system 420 makes effective use of the laws of quantum mechanics to perform computations. Quantum processing devices (QPUs), quantum computers, quantum processor systems, and quantum processing units are each embodiments of the quantum computing system. The quantum computing system 420 can be a universal or non-universal quantum computing system (a universal quantum computing system can execute any possible quantum circuit (which is constrained that the circuit does not use more qubits than the quantum computing system)). In some embodiments, the quantum computing system 420 is a gate-model quantum computer. As described above, the quantum computing system uses so-called qubits, i.e., quantum bits (e.g., 450A). While classical bits always have either a value of 0 or 1, qubits are quantum mechanical systems that can have a superposition of values ​​of 0, 1, or both. Exemplary physical implementations of qubits include superconducting qubits, spin qubits, trapping ions, arrays of neutral atoms, and photon systems (e.g., photons in waveguides). For the purposes of this disclosure, a qubit may be realized by a single physical qubit, or as an error-protected logical qubit comprising multiple physical qubits. In addition, this disclosure is not specific to qubits. This disclosure can be generalized to also apply to quantum computing systems 420 whose creation blocks are quantum continuous variables, not qudits (d-level quantum systems (when d>2)) or qubits.

[0066] Quantum operators (for example,

number

[0067] The parameters of a parameterized quantum circuit can refer to the parameters of a gate. For example, a gate that performs a rotation around the Y-axis can be parameterized by a real number that describes the angle of rotation.

[0068] Descriptions of quantum circuits to be executed on one or more quantum computing systems may be stored in a non-transient computer-readable storage medium. The term “computer-readable storage medium” should be interpreted as including one or more media capable of storing instructions (e.g., a centralized or distributed database, or associated caches and servers). The term “computer-readable medium” should also be interpreted as including any medium capable of storing instructions for execution by a quantum computing system, causing the quantum computing system to implement one or more of the methodologies disclosed herein. The term “computer-readable medium” includes, but is not limited to, data repositories in the form of solid-state memory, optical media, and magnetic media.

[0069] Figure 5 is an exemplary architecture of an exemplary classical computing system 500 (e.g., 410). A quantum computing system 420 may also have one or more components described with respect to Figure 5. Figure 5 depicts a high-level block diagram illustrating the physical components of a classical computing system used, in some embodiment, as some or all of the entities described herein. A classical computing system may have additions, fewer, or variations thereof of the components provided in Figure 5. Although Figure 5 depicts a classical computing system 500, the figure is intended not as a structural schematic of the implementation described herein, but as a functional description of the various features that may be present within the computing system. In practice, as will be recognized by those skilled in the art, articles shown separately may be combined, and some articles may be separated.

[0070] Figure 5 illustrates a processor 502 coupled to a chipset 504. Also coupled to the chipset 504 are memory 506, a storage device 508, a keyboard 510, a graphics adapter 512, a pointing device 514, and a network adapter 516. Although only a single processor 502 is illustrated in Figure 5, the classic computing system 500 may include multiple processors that operate individually or collectively and are capable of performing a process (e.g., one or more steps of method 300). A display 518 is coupled to the graphics adapter 512. In one embodiment, the functionality of the chipset 504 is provided by a memory controller hub 520 and an I / O hub 522. In another embodiment, the memory 506 is coupled directly to the processor 502 instead of the chipset 504. In some embodiments, the computer 500 includes one or more communication buses to interconnect these components. One or more communication buses optionally include a network of circuits (sometimes called a chipset) that interconnects and controls communication between system components.

[0071] The storage device 508 is any non-transient computer-readable storage medium such as a hard drive, compact disc read-only memory (CD-ROM), DVD, or solid-state memory device or other optical storage device, magnetic cassette, magnetic tape, magnetic disk storage device or other magnetic storage device, magnetic disk storage device, optical disk storage device, flash memory device, or other non-volatile solid-state storage device. Such a storage device 508 may also be referred to as persistent memory. The pointing device 514 may be a mouse, trackball, or other type of pointing device, used in combination with the keyboard 510 to input data to the computer 500. The graphics adapter 512 displays images and other information on the display 518. The network adapter 516 connects the computer 500 to a local area network or a wide area network.

[0072] Memory 506 holds instructions and data used by the processor 502. Embodiments of memory 506 may include non-persistent memory, such as high-speed random-access memory like DRAM, SRAM, DDR RAM, ROM, EEPROM, or flash memory.

[0073] As is known in the art, the computer 500 may have components different from or other than those shown in Figure 5. In addition, the computer 500 may lack certain illustrated components. In one embodiment, the computer 500 acting as a server may lack a keyboard 510, a pointing device 514, a graphics adapter 512, or a display 518. Also, the storage device 508 may be local or remote from the computer 500 (such as one embodied in a storage area network (SAN)).

[0074] As is known in the art, the computer 500 is adapted to execute computer program modules to provide the functionality described herein. As used herein, the term “module” refers to computer program logic used to provide the specified functionality. Thus, modules can be implemented in hardware, firmware, or software. In one embodiment, a program module is loaded into memory 506, stored on a storage device 508, individually or together, by one or more processors (e.g., 502), and executed.

[0075] (Additional considerations) The foregoing disclosures describe exemplary embodiments for illustrative purposes only. Any features described as essential, important, or otherwise implied to be required should be interpreted as being required only for that embodiment and not necessarily included in other embodiments.

[0076] Some portions of the above disclosure describe embodiments in terms of algorithmic processes or operations. These algorithmic descriptions and expressions are commonly used by those skilled in the field of computing to effectively communicate the nature of the work to others skilled in the field. These operations are described functionally, computationally, or logically, but are understood to be implemented by a computer program containing instructions for execution, individually or together, by one or more processors, equivalent electrical circuits, microcode, or equivalents. Furthermore, this also proves that it is sometimes convenient, without loss of generality, to refer to a sequence of functional operations as a module. In some cases, a module may be implemented in hardware, firmware, or software.

[0077] As used herein, any reference to “one embodiment” or “an embodiment” means that a particular element, feature, structure, or characteristic described in relation to an embodiment is included in at least one embodiment. Expressions of the phrase “in one embodiment” in various places within this specification do not necessarily all refer to the same embodiment. Similarly, the use of “a” or “an” preceding an element or component is for convenience only. This description should be understood to mean that one or more of those elements or components exist unless it is evident that this means otherwise. As used herein, the terms “comprises,” “comprising,” “includes,” “including,” “has,” “having,” or any other variations thereof are intended to encompass non-exclusive inclusion. For example, a process, method, article, or apparatus comprising a list of elements is not necessarily limited to those elements alone and may include other elements not explicitly enumerated or that are specific to such process, method, article, or apparatus. Furthermore, unless explicitly stated in contrast, “or” refers to a containing “or” and not an exclusive “or.” For example, condition A or B is satisfied by any one of the following: A is true (or exists) and B is false (or does not exist); A is false (or does not exist) and B is true (or exists); and both A and B are true (or exist).

[0078] In addition, the use of “a” or “an” is employed to describe elements and components of embodiments. This is done solely for convenience and to give the general meaning of this disclosure. This description should be read as including one or at least one singular noun, and plural nouns as well as singular nouns, unless it is obvious that they are meant otherwise. Where a value is described as being “approximate” or “substantially” (or a derivative thereof), such a value should be interpreted as being exactly + / - 10%, unless another meaning is obvious from the context. From the examples, “approximately ten” should be understood to mean “in a range from nine to eleven.”

[0079] Alternative embodiments are implemented within computer hardware, firmware, software, and / or combinations thereof. The implementation may be implemented within a computer program product, tangibly embodied in a machine-readable storage device for execution by a programmable processor system comprising one or more processors, individually or collectively, and the method steps may be carried out by a programmable processor system that performs functions by executing a program of instructions, acting with respect to input data, and generating outputs. Advantageously, embodiments may be implemented within one or more computer programs executable on a programmable system comprising one or more programmable processors coupled to receive data and instructions from a data storage system, at least one input device, and at least one output device, and to transmit data and instructions thereto. Each computer program may, as desired, be implemented within a higher-order procedural or object-oriented programming language, or in assembly or machine language, in either case the language may be a compiled language or an interpreted language. Preferred processors, as embodiments, include both general-purpose microprocessors and special-purpose microprocessors. Generally, a processor will receive instructions and data from read-only memory and / or random-access memory. Generally, a computer will include one or more mass storage devices for storing data files, such devices including magnetic disks, magneto-optical disks, and optical disks, such as internal hard disks and removable disks. Suitable storage devices for tangibly embodying computer program instructions and data include, in examples, semiconductor memory devices such as EPROMs, EEPROMs, and flash memory devices, magnetic disks, magneto-optical disks, and CD-ROM disks, as well as any form of non-volatile memory. Any of the foregoing can be complemented by or incorporated into ASICs (Application-Specific Integrated Circuits) and other forms of hardware.

[0080] While the above description contains many details, these should not be construed as limiting the scope of this disclosure, but merely as illustrating different embodiments. It should be understood that the scope of this disclosure includes other embodiments not discussed in detail above. Various other modifications, changes, and variations that would be obvious to those skilled in the art may also be made in the arrangement, operation, and details of the methods and apparatus disclosed herein without departing from the spirit and scope of this disclosure.

Claims

1. It is a method, Receiving an object value, a value of x, and a function f(x) that describes the probability p(x) related to the value of x, The first quantum computing system receives a quantum operator P that generates a first quantum state characterized by the superposition of states encoding the x value, wherein the amplitude with respect to the corresponding x value is the square root of the probability p(x) with respect to that x value. A quantum operator, when executed by a second quantum computing system, encodes an approximation of the function f(x) into the amplitude of the second quantum state without calculating |f(x)> for any of the values ​​of x. [Number 201] The determination is that the approximation is within the error threshold of the function f(x), The third quantum computing system includes the quantum operator P and [Number 202] The command to perform both of the above and generate a third quantum state relating to the register of the qubit, wherein one of the amplitudes of the third quantum state includes a probability p(x) relating to the value of x and the output value of the approximation of the function f(x) relating to the value of x, Based on the generated third quantum state, the value of the object is determined. Methods that include...

2. The method according to claim 1, wherein the method does not include calculating |f(x)>.

3. The quantum operator P and [Number 203] To instruct the third quantum computing system to perform both of the above is to instruct the quantum operator [Number 204] The method according to any of the preceding claims, comprising instructing the quantum operator P to be executed prior to executing the function.

4. The method according to any of the preceding claims, wherein one of the amplitudes of the third quantum state is the square root of the weighted average of the approximations of the function f(x) with respect to the x value, and the weights are the probabilities p(x) with respect to the corresponding values ​​of x.

5. The method according to any of the preceding claims, wherein the first quantum state is a probability-weighted superposition of all values ​​of x.

6. The first quantum state described above is, [Number 205] It is determined by |x> ｎ The method according to claim 5, wherein is a quantum state relating to an n-qubit register that stores an n-bit binary representation of the corresponding value of x.

7. The determination of s is based on the absolute value of the x value, The operation is performed on the third quantum computing system, i.e., |x> ｎ | 0 > ｍ →|x> ｎ |x+s> ｍ This involves instructing a quantum binary adder to apply, where n and m are integers greater than zero, m > n, and |x > ｎ |x+s> is a quantum state relating to an n-qubit register that stores the n-bit binary representation of the value of x, where |x+s> ｍ This is a quantum state relating to an m-qubit register that stores a representation of the value of x + s, including the total number of digits of m and the digit of p to the left of the binary decimal point. The method according to any of the prior claims, further comprising:

8. The method according to claim 7, wherein s is based on the upper bound of the absolute value of the x value.

9. The method according to claim 8, wherein s is the absolute value of the minimum x value of the x values.

10. The quantum binary adder circuit executes the quantum operator P, however the quantum operator [Number 206] The method according to claim 7 or 8, which is performed prior to the execution of the method.

11. The aforementioned quantum operator [Number 207] To determine this is [Number 208] This involves generating the initial quantum operator obtained by C, and C is a comparator quantum circuit defined by C: |a>|b>|0>→|a>|b>|a, [Number 209] The method according to any of the preceding claims, wherein is the identity matrix and H is a Hadamard gate.

12. The aforementioned quantum operator [Number 210] Determining further applies the initial quantum operator U to the state |x + s> ｍ |0> ｍ＋１ to obtain the following quantum state, namely, [Number 211] This includes generating |Ψ ０ > ｍ and |Ψ １ > ｍ The method according to claim 11, wherein is a normalized quantum state.

13. The aforementioned quantum operator [Number 212] The method according to claim 12, wherein determining further comprises performing a quantum signal processing (QSP) algorithm.

14. Implementing the aforementioned QSP algorithm means [Number 213] This includes determining the phase parameter, which represents a polynomial approximation that approximates, and A, B, and C are (1) based on the function f(x) and (2) [Number 214] The method according to claim 13, wherein the constant satisfies the conditions.

15. The method according to any one of the preceding claims, wherein the function f(x) is a normalized function with output values ​​in the range of 0 to 1.

16. The method according to any of the preceding claims, wherein the value of the object varies probabilistically.

17. The method according to any of the prior claims, wherein determining the value of the object based on the generated quantum state includes performing an amplitude estimation algorithm and determining the value of the object.

18. It is a method, In quantum computing systems, [Number 215] This involves preparing an approximation for the target quantum state obtained by, instructing a quantum operator to perform, where x is a scalar number and |x> ｎ is a quantum state relating to an n-qubit register of the quantum computing system that stores an n-bit binary representation of x, where n and a are integers greater than zero. [Number 216] This is a normalized quantum state, |0 ⊥ > ａ is, |0> ａ A normalized quantum state orthogonal to , where A, B, and C are [Number 217] It is a constant such that, and the above approximation is, [Number 218] A method that is within the error threshold.

19. The quantum computing system according to claim 18, wherein the quantum computing system performs the quantum operator.

20. A computing system configured to implement any of the limitations described in claims 1 to 19.

21. A non-transient computer-readable storage medium for storing instructions, wherein, when the instructions are executed by a computing system, the computing system causes the computing system to perform any of the operations described in claims 1 to 19.

Images