Quantum computing methods

Abstract

This invention provides a quantum computing method for efficiently estimating parameters embedded in the amplitude of a quantum state. [Solution] A quantum computation method for estimating parameters embedded in the amplitude of a quantum state, comprising: obtaining a first operator that performs the operation of embedding parameters in the amplitude of a quantum state; using the first operator, constructing a second operator that performs the operation of embedding parameters in the phase of a quantum state; applying the second operator in parallel to the entangled input state; measuring the quantum state after applying the second operator; estimating the phase based on the measurement result; and estimating the parameters using the phase estimation result.

Description

Technical Field

[0001] The present disclosure relates to a quantum computing method for estimating the amplitude of a quantum state.

Background Art

[0002] Quantum Amplitude Estimation (QAE) is a basic algorithm used in many quantum algorithms. This quantum amplitude estimation is essential for speeding up by a quantum computer in many applications including financial simulations. And techniques for providing an amplitude estimation method for efficiently executing amplitude estimation have been studied (for example, Patent Document 1).

[0003] In quantum amplitude estimation, a first operator U pr that embeds a desired value in the amplitude is given as a gate operation, and this is used to estimate the amplitude value a. This first operator U pr is represented by the following formula.

[0004]

Equation

[0005] The performance of quantum amplitude estimation is evaluated by the relationship (query complexity) between "the number of calls N of the first operator U pr " and "the amplitude estimation error ε". By using amplitude amplification using a Grover operator Q composed of the first operator U pr the query complexity can be made "N = O(1 / ε)" (Heisenberg scaling). Note that "O" is the Landau symbol indicating the growth rate of the computational complexity of an algorithm.

[0006] In conventional quantum amplitude estimation, for the estimation error ε, it is necessary to execute a quantum circuit in which the number of quantum gates arranged in the serial direction (Depth) is "O(1 / ε)". Furthermore, if the scaling of the depth of the quantum circuit executed in quantum amplitude estimation with respect to ε can be improved, it will be possible to achieve efficient use of computational resources, reduction in computation time, and reduction in the required coherence time. Non-patent document 1 proposes a method for improving the scaling of the depth of the quantum circuit with respect to ε in quantum amplitude estimation.

[0007] Methods for estimating parameters embedded in quantum systems, such as quantum amplitude estimation, can also be considered from the perspective of quantum metrology. There are attempts to optimize quantum amplitude estimation and expectation value estimation algorithms based on quantum metrology theory (e.g., Patent Document 1). In quantum metrology, a method called the parallel strategy is known, which improves estimation accuracy by entangled the input states and then applying quantum channels that embed parameters into the phase of the quantum states in parallel to the input states. As an algorithm for estimating parameters embedded in the phase of the quantum states (phase estimation) using the parallel strategy, methods such as the Robust phase estimation method can be used (e.g., Non-Patent Document 2).

[0008] As a quantum computing technique, quantum signal processing (QSP) has been proposed, in which operators are constructed by polynomially transforming the eigenvalues ​​of an operator given as a gate operation (e.g., Non-Patent Document 3). In quantum signal processing, transformation operations are controlled by certain parameters, and in order to achieve the desired transformation, it is necessary to find these parameters. Methods for calculating these parameters have also been investigated (e.g., Non-Patent Document 4). The algorithm in Non-Patent Document 4 uses two methods: "halving" and "capitalization". These methods allow for more stable parameter calculation than conventional methods. [Prior art documents] [Patent Documents]

[0009] [Patent Document 1] Japanese Patent Publication No. 2025-008164 [Non-patent literature]

[0010] [Non-Patent Document 1] Tudor Giurgica-Tiron et al., “Low depth algorithms for quantum amplitude estimation”, arxiv, [online], November 6, 2020, [Retrieved March 17, 2020], Internet<https: / / arxiv.org / abs / 2012.03348> [Non-Patent Document 2] Shelby Kimmel et al., “Robust Calibration of a Universal Single-Qubit Gate-Set via Robust Phase Estimation” arxiv, [online], February 9, 2015, [searched March 19, 2020], Internet<https: / / arxiv.org / abs / 1502.02677> [Non-Patent Document 3] GHLow et al., “Optimal Hamiltonian Simulation by Quantum Signal Processing” arxiv, [online], January 8, 2016, [searched March 17, 2020], Internet<https: / / arxiv.org / abs / 1606.02685> [Non-Patent Document 4] Rui Chao et al., “Finding Angles for Quantum Signal Processing with Machine Precision” arxiv, [online], March 17, 2020, [searched on March 8, 2020], Internet<https: / / arxiv.org / abs / 2003.02831> [Overview of the project] [Problems that the invention aims to solve]

[0011] In conventional quantum amplitude estimation, improving estimation accuracy while maintaining Heisenberg scaling requires increasing the depth of the quantum circuit with a scaling of O(1 / ε) relative to the estimation error ε. Furthermore, Non-Patent Document 1 presents a method to improve this depth scaling, but the technique described in this document worsens query complexity depending on the degree of improvement in depth scaling, resulting in N=O(1 / ε). 2 This approaches scaling. For this reason, an algorithm that significantly improves depth scaling while maintaining query complexity with minimal overhead in quantum amplitude estimation has not yet been realized. [Means for solving the problem]

[0012] The quantum computation method of this disclosure estimates parameters embedded in the amplitude of a quantum state. Here, a first operator is obtained that performs the operation of embedding the parameters in the amplitude of the quantum state, a second operator is constructed using the first operator that performs the operation of embedding the parameters in the phase of the quantum state, the entangled input state is applied to the second operator connected in parallel, the quantum state after the application of the second operator is measured, the phase is estimated based on the measurement result, and the parameters are estimated using the phase estimation result. [Effects of the Invention]

[0013] According to this disclosure, it is possible to efficiently estimate parameters embedded in the amplitude of a quantum state in quantum computing. [Brief explanation of the drawing]

[0014] [Figure 1] This is a schematic diagram of the system of an embodiment. [Figure 2] This is an explanatory diagram of the hardware configuration of the embodiment. [Figure 3] This is an explanatory diagram illustrating the outline of the processing procedure of the embodiment. [Figure 4] This is an explanatory diagram of the quantum circuit in the embodiment. [Figure 5] This is an explanatory diagram of the quantum circuit in the embodiment. [Modes for carrying out the invention]

[0015] An embodiment of the quantum computation method will be described below using Figures 1 to 5. In this embodiment, a second operator is constructed from a first operator that embeds parameters into the amplitude of the quantum state, and the second operator that embeds parameters into the phase of the quantum state is constructed from the first operator that embeds parameters into the amplitude of the quantum state. The second operator is then arranged in both parallel and series configurations, and the quantum state is measured, and the phase is estimated based on the measurement results. Then, the parameters are estimated using the phase estimation results. Here, the first operator is converted into a second operator to which the parallel strategy can be applied, and the phase estimation is performed using the parallel strategy.

[0016] (Description of hardware configuration) Figure 2 illustrates the hardware configuration of the information processing device H10, which comprises the user terminal 10 and the arithmetic unit 20 (management unit 211, classical calculation unit 212). The information processing device H10 includes a communication device H11, an input device H12, a display device H13, a storage device H14, and a processor H15. Note that this hardware configuration is just one example, and it can be implemented using other hardware.

[0017] Communication device H11 is an interface (such as a network interface or wireless interface) that performs data transmission and reception by establishing a communication path with other devices.

[0018] Input device H12 is a device that receives input from users, etc. (for example, a mouse or keyboard). Display device H13 is a device that displays various information (for example, a display).

[0019] The storage device H14 is a device that stores data and various programs for executing various functions of the user terminal 10 and the arithmetic unit 20. Examples of storage devices H14 include ROM, RAM, and hard disks.

[0020] The processor H15 controls various processes in the user terminal 10 and the arithmetic unit 20 by using programs and data stored in the memory device H14. Examples of processor H15 include CPUs and MPUs. This processor H15 executes various processes for each service by loading programs stored in ROM, etc., into RAM.

[0021] The processor H15 is not limited to performing all of its operations through software processing. For example, the processor H15 may have dedicated hardware circuits (e.g., application-specific integrated circuits) that perform hardware processing for at least some of the operations it performs. In other words, the processor H15 can be configured as a circuit including the following:

[0022] (1) One or more processors that operate according to a computer program (2) One or more dedicated hardware circuits that perform at least some of the various processes. (3) combinations of those A processor includes the CPU and memory such as RAM and ROM. Memory stores program code or instructions configured to cause the CPU to perform processing. Memory, or computer-readable media, includes any available media that can be accessed by a general-purpose or dedicated computer.

[0023] (System configuration of the arithmetic unit 20) Next, using Figure 1, we will explain the system configuration of the interconnected user terminals 10 and computing devices 20.

[0024] User terminal 10 is a computer terminal used by the user. The computing unit 20 is a computer for performing quantum amplitude estimation. This computing unit 20 performs the processes described later (such as the management stage, classical computation stage, and quantum computation stage). By executing the program for this purpose, the computing unit 20 functions as a management unit 211, a classical computation unit 212, and a quantum computation unit 22.

[0025] Management Unit 211 is responsible for managing quantum computing and classical computing. The classical computation unit 212 performs statistical processing in maximum likelihood estimation. The quantum computing unit 22 performs quantum computation. This quantum computing unit 22 includes an operation unit 221, a state holding unit 222, and a measurement unit 223.

[0026] The operation unit 221 performs quantum operations on the qubits of the state holding unit 222 according to the program. In this case, the operation unit 221 manipulates (creates) the state held by the state holding unit 222 by using a quantum circuit composed of quantum gates, etc.

[0027] The state-holding unit 222 comprises multiple qubits and holds any quantum state. Each qubit holds a superposition state of multiple values ​​in any physical state, such as an electron level, electron spin, ion level, individual spins, or a photon. The qubit is not limited to the above, as long as it can hold a superposition state.

[0028] The measurement unit 223 observes the eigenstates of the superposition state of the qubits in the state-holding unit 222. The measurement unit 223 records the number of hits according to the state of the qubits in the state-holding unit 222.

[0029] (Overview of quantum computing) Next, we will explain the overview of quantum computing using Figure 3. First, the arithmetic unit 20 processes the first operator U pr The Grover operator is constructed from (step S11). Here, in order to embed parameter a into the amplitude of the quantum state, the following first operator U pr This assumes that the data is obtained from user terminal 10.

[0030]

Number

[0031] Act on the initial state with the first operator U pr to obtain a superposition state of |ψ0>|0> and |ψ1>|1>. For this first operator U pr construct the following Grover operator Q.

[0032]

Number

[0033] Specifically, the operators U f , U0 are expressed as follows. Here, I is the identity operator.

[0034]

Number

[0035] Next, the arithmetic unit 20 uses a controlled Grover operator CQ in which the Grover operator Q is controlled by an auxiliary qubit to construct a second operator Q ph,T for which phase estimation capable of applying the Parallel strategy is possible (step S12). Here, the second operator Q ph,T is an operator that embeds a value dependent on Ta into the phase of the quantum state, and T is a parameter representing the signal amplification intensity included in the second operator Q ph,T .

[0036] First, the controlled Grover operator CQ is expressed as follows.

[0037]

Number

[0038] In this case, the second operator Qph,T This can be configured as follows using the quantum signal processing described in Non-Patent Document 3. Here, R x , R z H is the rotation operator for the X and Z axes of the Bloch sphere. ⊥ In a Hilbert space of n+2 qubits, |0> n+2 and |1> n+2 This refers to a subspace pasted by quantum states orthogonal to the matrix. L represents the number of iterations of the unit process in quantum signal processing. The matrix basis is |0> and |1>, and e iφu This is the eigenvalue corresponding to |u>. The second operator Q is obtained using quantum signal processing. ph,T When constructing this, the signal amplification strength T can be adjusted by changing the time evolution length in quantum signal processing.

[0039]

number

[0040] Second operator Q ph,T For a quantum state where "only the first bit is |1>, and all the other n+1 bits are |0>", Tφ a =2Tcos2θ is an operator that approximately gives the phase of "2T(1-2a)". Here, the rotation angles ξ1, ξ2, ..., ξ L This is a phase parameter obtained in quantum signal processing by conventional computer calculations (e.g., Non-Patent Document 4) in order to perform the desired transformation.

[0041] Next, the arithmetic unit 20 processes the second operator Q ph,T After constructing the first operator Q, a phase estimation is performed using a parallel strategy (step S13). This results in the second operator Q ph,T The value φ embedded in the phase by a We estimate this.

[0042] First, prepare the following quantum state as the input state.

[0043]

number

[0044] Here, P represents the number of qubits to be entangled, |GHZ P > represents the "Greenberger-Horne-Zeilinger" state in the following equation.

[0045]

number

[0046] For the input state, the second operator Q ph,T By arranging (520) in series with S and in parallel with P, the following equation is obtained: a The value Mφ is obtained by amplifying it by M (=P·S·T) times. a This can be approximately embedded in the phase of the quantum state.

[0047]

number

[0048] Next, the second operator Q is determined by phase estimation using the parallel strategy. ph,T φ is embedded in the phase of the quantum state by a We estimate an approximate value of Mφ. In phase estimation using the parallel strategy, operators are arranged in series and parallel directions and applied as described above. a The operation of approximately embedding into the phase of the quantum state and performing multiple measurements with an appropriate measurement basis is repeated while changing M. Then, using the measurement results for all M, φ a An approximate value of is estimated. Here, the maximum value of M is determined according to the required estimation error ε, and the number of measurements for each M is set to be sufficient so that the estimation can be performed successfully. As such a phase estimation method, for example, Robust phase estimation (Non-Patent Literature 2) can be used. Note that when changing M, the second operator Q may be used as needed. ph,TIt is necessary to change the signal amplification strength T included in the signal, in which case the second operator Q is used in the procedure described above. ph,T It needs to be remade.

[0049] Figure 4 shows the quantum circuit 500 used for this calculation. Quantum circuit 500 has an entanglement operator 510 and a second operator Q ph,T (520) The device includes a measurement process 530.

[0050] The entangle operator 510 is the aforementioned |GHZ P This is an operator that generates >. In quantum circuit 500, the second operator Q ph,T This is arranged as a matrix of serial number S and parallel number P. This is the second operator Q. ph,T The serial number S and the second operator Q ph,T The signal amplification intensity T included in the signal is upper bounded by the coherence time Tc, which is the limit of the time the quantum state can be maintained. Then, to satisfy the estimation accuracy (computation conditions), the serial number S, parallel number P, and second operator Q are used. ph,T The signal amplification intensity T included in the signal is determined.

[0051] Figure 5 shows the second operator Q. ph,T The structure of (520) is shown. Second operator Q ph,T (520) includes revolving gates 521, 522, and 523, as well as a Grover operator Q524 as a control gate. The revolving gates 521, 522, and 523 each rotate around the X-axis on the Bloch sphere, ξ i ", "π / 2" around the Z axis, and "-ξ" around the X axis. i Rotate only the "".

[0052] Second operator Q ph,T (520) contains L control Grover operators CQ. Here, increasing L gives a second operator Q ph,T The approximation error becomes smaller. Second operator Q ph,TA sufficiently large L is set so that the effect of the approximation error is sufficiently small compared to the estimation error ε required for amplitude estimation. Based on prior research on quantum signal processing (Non-Patent Literature 3), to satisfy this condition, it is sufficient to increase L with respect to ε, P, S, and T using the scaling L=O(T+log(PS / ε)). From this relationship, in order to reduce L in order to achieve an M(=P·S·T) amplification, it is preferable to increase the length of time evolution T rather than the number of series connections S in the series direction. However, when T is large, the calculation of the rotation angle ξ becomes difficult, so it is set to "1", for example.

[0053] Next, the arithmetic unit 20 calculates the φ estimated in the above procedure. a The amplitude a is estimated from the approximate value of =2(1-2a) (step S14). Then, the amplitude estimation result is output to the user terminal 10.

[0054] This embodiment provides the following benefits. In this embodiment, the arithmetic unit 20 controls the first operator U pr A Grover operator is constructed from (step S11), and a second operator Q to which the Parallel strategy can be applied is obtained from the controlled Grover operator constructed from the Grover operator by quantum signal processing. ph,T This constitutes (step S12). This second operator Q ph,T By using this, phase estimation using the parallel strategy can be performed (step S13). Furthermore, the amplitude a can be estimated from the estimated phase value (step S14). When the amplitude a is estimated using this procedure, the first operator U pr By arranging the first operator U not only in series but also in parallel, it becomes possible to improve estimation accuracy. pr By arranging them in a series direction, the depth can be reduced compared to arranging them in a series direction, thus improving the scaling of depth with respect to the estimation error ε. This enables efficient use of computing resources, reduction of computation time, and reduction of the required coherence time Tc.

[0055] In this case, the scaling of the depth of the quantum circuit with respect to ε and the first operator U pr The scaling (query complexity) of the number of calls N is as follows: However, S max and T max is the maximum value of S and T in the quantum circuit performed by estimation.

[0056]

number

[0057] This scaling of Depth is performed by the first operator U pr When arranging them only in parallel directions (S max ,T max When = 1, the query complexity becomes O(log(1 / ε)), which is exponentially smaller than the O(1 / ε) of the conventional method. Furthermore, this query complexity is O(ε) compared to the query complexity of the conventional method. * log( * It only increases by / ε) / ε). Since it is a logarithmic value with respect to the estimation error ε, it can be almost ignored. This is the second operator Q ph,T The first operator U required for the construction pr This is because the number L is equal to L = O(T + log(PS / ε)) with respect to the estimation error ε.

[0058] This embodiment can be implemented with the following modifications. This embodiment and the following modifications can be combined with each other to the extent that they do not contradict each other technically. In the above embodiment, the first operator U pr From this, a Grover operator is constructed, and from the controlled Grover operator constructed from the Grover operator, the second operator Q is obtained by quantum signal processing. ph,T This was constructed. Here, the first operator U pr The second operator Q can be used to apply the Parallel strategy. ph,T If it is possible to construct the operator conversion procedure, the procedure is not limited to this procedure. ph,TWhen constructing this, the entangle operators are |0>,|1> in the GHZ state, and the second operator Q in that case. ph,T This operator prepares a quantum state by replacing the two eigenstates that give the phase difference. In addition, the second operator Q is obtained in a procedure other than that described in the above embodiment. ph,Tを When configuring, the second operator Q ph,T The value φ embedded in the phase of the quantum state by this value a is, φ a =2(1-2a) is a different expression (for example, φ a = a) is possible.

[0059] Another procedure for constructing operators to which the Parallel strategy can be applied is to use Quantum Singular Value Transformation (QSVT). First operator U pr The following relationship is satisfied. Then, 1-2a is e by QSVT. iTa By converting to this, it is possible to approximately construct an operator to which the parallel strategy can be applied. Note that this procedure also applies to the second operator Q ph,T Even when configuring this, the computational cost is about the same as when using the QSP described in the above embodiment.

[0060]

number

[0061] In the above embodiment, the serial number is S, the parallel number is P, and the second operator is Q. ph,T The signal amplification intensity T included in the signal ensures estimation accuracy. If the number of parallel connections P is sufficient, the number of serial connections S and the second operator Q are used. ph,T There is no need to increase the signal amplification strength T included in the signal. Note that the number of series connections is S, the number of parallel connections is P, and the second operator is Q. ph,T If estimation accuracy cannot be guaranteed by the signal amplification intensity T included in the signal, then N = O(1 / ε 2 Computational resources are increased under the scaling of ). In this way, depending on the estimation accuracy, the serial number S, parallel number P, and second operator Q are flexibly used. ph,TThe signal amplification strength T included in the signal can be set.

[0062] In the above embodiment, estimation accuracy was used as a calculation condition, but it is not limited to estimation accuracy as long as it is a constraint condition related to the calculation. For example, the calculation time may be used as a calculation condition. [Explanation of symbols]

[0063] 10...User terminal, 20...Calculation unit, 211...Management unit, 212...Classical calculation unit, 22...Quantum calculation unit, 221...Operation unit, 222...State holding unit, 223...Measurement unit.

Claims

1. A quantum computing method performed by a computer to estimate parameters embedded in the amplitude of a quantum state, The aforementioned computer, A first operator is obtained that performs the operation of embedding the aforementioned parameter into the amplitude of the quantum state, Using the first operator, a second operator is constructed that performs the operation of embedding the parameter into the phase of the quantum state. The second operator is applied in parallel to the entangled input state, the quantum state after the application of the second operator is measured, and the phase is estimated based on the measurement result. A quantum computing method characterized by estimating the parameters using the phase estimation results.

2. The computer The Grover operator is constructed from the first operator described above. A controlled Grover operator is constructed from the aforementioned Grover operators. The quantum computation method according to claim 1, characterized in that the second operator is constructed from the control Grover operator using quantum signal processing.

3. The quantum computation method according to claim 1, characterized in that the number of parallel operations for applying the second operator in parallel is determined according to the given computation conditions.

4. The quantum computation method according to claim 3, characterized in that, depending on the number of parallel connections, the number of series connections in which the second operator is applied and the signal amplification intensity included in the second operator are determined so as to satisfy the calculation conditions.

5. The number of series connections of the second operator or the signal amplification intensity included in the second operator is determined. The quantum computation method according to claim 1, characterized in that, if the number of series connections or the signal amplification intensity does not satisfy the given calculation conditions, at least the number of parallel connections of the second operator connected in parallel is determined.

6. The quantum computing method according to any one of claims 3 to 5, characterized in that the calculation conditions include at least one of estimation accuracy and calculation time.

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