Quantum computing control program and quantum computing control method

Abstract

To improve the generation efficiency in a resource state.SOLUTION: A computer determines a physical rotation angle around a prescribed axis to execute for d first physical quantum bits 4 among multiple physical quantum bits 3 consisting of a logic quantum bit 2 based on a logic rotation angle for rotating the state of the logic quantum bit 2 around the prescribed axis. The computer instructs a quantum computer 1 with the multiple physical quantum bits 3 to execute the rotary gate operation which rotates each state of the d first physical quantum bits 4 at only the physical rotation angle around the prescribed axis. On this occasion, the computer specifies the application of m quantum bit rotary gate to rotate the state of m first physical quantum bits 4 at one rotary gate operation for physical quantum bit groups 4a, 4b coordinated by m bits among the d first physical quantum bits 4.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to a quantum computing control program and a quantum computing control method. [Background technology]

[0002] In calculations using a quantum computer, quantum calculations are performed according to quantum circuits by performing gate operations on quantum bits. A quantum bit is the smallest unit of information used in calculations, and is equivalent to a bit (classical bit) in a classical computer. However, unlike classical bits, quantum bits can also be in a superposition state of "0" and "1."

[0003] The information in quantum bits can be corrupted (errors can occur) due to interactions with the environment, errors in gate operation, etc. There are two ways to deal with errors: quantum error correction and quantum error mitigation.

[0004] Quantum error correction is a process of detecting and correcting errors by combining and encoding (redundant) multiple quantum bits. Hereafter, unencoded quantum bits are referred to as physical quantum bits, and a set of encoded quantum bits is referred to as logical quantum bits. Quantum error mitigation is a process of proceeding with calculations while including errors, and mitigating the effects of errors by modifying quantum circuits or extrapolating measurement results.

[0005] A quantum computer that performs quantum computations while performing quantum error correction on logical qubits is called a fault-tolerant quantum computer (FTQC). In an FTQC, any quantum computation can be performed by combining certain basic gates. The certain basic gates are the H gate, CNOT gate, S gate, and T gate. The H gate, CNOT gate, and S gate are quantum gates that perform Clifford operations, and the T gate is a quantum gate that performs non-Clifford operations. A set of these basic gates is called Clifford+T.

[0006] Among the basic gates of Clifford+T, the T gate uses a large number of physical qubits for error correction, so a FTQC that can perform useful computations requires a scale of around one million physical qubits.

[0007] As a technology to reduce the number of physical qubits used for error correction, a highly efficient phase rotation gate quantum computing architecture called the STAR (Space-Time Efficient Analog Rotation quantum computing) architecture has been proposed. In the STAR architecture, an arbitrary rotation gate is implemented using a predetermined resource state (also called an auxiliary state) and a gate teleportation circuit. The resource state is represented by redundant logical qubits. The process of preparing the resource state is called the "state injection protocol" or "state preparation protocol" (hereinafter referred to as the "state preparation protocol"). [Prior art documents] [Non-patent literature]

[0008] [Non-Patent Document 1] Yutaro Akahoshi, Kazunori Maruyama, Hirotaka Oshima, Shintaro Sato, and Keisuke Fujii, "Partially Fault-tolerant Quantum Computing Architecture with Error-corrected Clifford Gates and Space-time Efficient Analog Rotations", arXiv:2303.13181v1, 23 Mar 2023 [Non-patent document 2] Hyeongrak Choi, Frederic T. Chong, Dirk Englund, Yongshan Ding, "Fault Tolerant Non-Clifford State Preparation for Arbitrary Rotations", arXiv:2303.17380v1, 30 Mar 2023 Summary of the Invention [Problem to be solved by the invention]

[0009] When generating resource state using the state preparation protocol, syndrome measurements are performed several times during the resource state generation process. If an error is detected during syndrome measurement, the resource state generation process is restarted from the beginning. The more times the resource state generation process is restarted, the less efficient the resource state generation becomes.

[0010] In one aspect, the present invention aims to improve the efficiency of resource state generation. [Means for solving the problem]

[0011] In one proposal, a quantum computing control program is provided that causes a computer to perform the following processes. The computer determines a physical rotation angle about a predetermined axis to be performed on d first physical quantum bits among the plurality of physical quantum bits constituting the logical quantum bit, based on the logical rotation angle for rotating the state of the logical quantum bit encoded with a code distance d (d is an integer greater than or equal to 2) about the predetermined axis. The computer then specifies the application of an m-qubit rotation gate that rotates the states of the m first physical quantum bits with a single rotation gate operation for a physical quantum bit group consisting of m (m is an integer greater than or equal to 2 and less than or equal to d) of the d first physical quantum bits, and instructs a quantum computer having a plurality of physical quantum bits to perform a rotation gate operation that rotates the states of each of the d first physical quantum bits about the predetermined axis by the physical rotation angle. [Effects of the Invention]

[0012] According to one aspect, the efficiency of resource state generation is improved. [Brief explanation of the drawings]

[0013] [Figure 1] FIG. 2 is a diagram illustrating an example of a quantum computing control method according to the first embodiment. [Figure 2] FIG. 10 illustrates an example of a system configuration according to a second embodiment. [Figure 3] FIG. 1 is a diagram illustrating an example of hardware of a quantum computing system. [Figure 4] FIG. 1 illustrates the characteristics of a quantum bit. [Figure 5] FIG. 1 illustrates an example of quantum error correction. [Figure 6] FIG. 1 is a diagram illustrating an example of a basic gate used in error-tolerant quantum computing. [Figure 7] FIG. 1 is a diagram illustrating an example of a quantum circuit that performs gate operations of arbitrary rotation. [Figure 8] FIG. 10 is a diagram illustrating an example of a state preparation protocol. [Figure 9] FIG. 10 is a diagram showing an example of a combination pattern of physical quantum bits that perform gate operation of a transversal rotation gate. [Figure 10] FIG. 10 is a diagram illustrating an example of processing when an error occurs in the state preparation protocol. [Figure 11] FIG. 1 is a diagram showing an example of gate operation of a transversal revolving gate. [Figure 12] FIG. 10 is a diagram showing an example of syndrome measurement after transversal rotation. [Figure 13] FIG. 10 is a diagram illustrating an example of a post-selection situation when an error occurs. [Figure 14] FIG. 1 is a diagram showing an example of a transversal rotation gate that rotates two or more physical quantum bits together. [Figure 15]FIG. 1 illustrates an example of a quantum circuit for implementing a Multi-Z rotation gate. [Figure 16] FIG. 1 shows an example of a transversal rotary gate implemented with RZZ(θ). [Figure 17] FIG. 10 is a diagram showing an example of a transversal rotation gate in which multi-Z rotations of multiple weights are mixed. [Figure 18] FIG. 1 illustrates an example of a hybrid post-selection and error correction scheme for rotated surface codes. [Figure 19] FIG. 10 is a diagram showing an example of an error pattern that is suitable for handling by post-selection. [Figure 20] FIG. 10 is a diagram showing an example of an error pattern that can be avoided by performing post-selection. [Figure 21] FIG. 10 illustrates an example of error condition detection in a post-selection region. [Figure 22] FIG. 1 illustrates a first case in which an error condition is detected. [Figure 23] FIG. 10 illustrates a second case in which an error condition is detected. [Figure 24] FIG. 10 is a diagram illustrating an example of setting a post-selection region. [Figure 25] FIG. 1 is a block diagram showing an example of functions for quantum computing in a quantum computing system. [Figure 26] FIG. 10 is a block diagram showing an example of functions of an arbitrary rotation execution unit. [Figure 27] FIG. 10 is a sequence diagram showing an example of a calculation processing procedure of a quantum circuit. [Figure 28] 1 is a flowchart illustrating an example of a procedure for quantum computing processing in a classical computer. [Figure 29] 10 is a flowchart illustrating an example of a procedure for quantum circuit execution processing. [Figure 30] 10 is a flowchart illustrating an example of a procedure for a resource state generation process. [Figure 31]This figure shows an example of how to implement a transversal rotation gate using a Multi-Z rotation gate on hardware where two-qubit gates are restricted to only nearest-neighbor qubits. [Figure 32] FIG. 10 is a diagram illustrating an example of the success probability of resource state generation when there is no error. [Figure 33] 10A and 10B are diagrams illustrating an example of the success probability in the pass determination of resource state generation depending on the resource state generation method. [Figure 34] FIG. 10 is a diagram illustrating an example of a comparison result of the tracing distance. DETAILED DESCRIPTION OF THE INVENTION

[0014] The present embodiment will be described below with reference to the drawings. Note that each embodiment can be implemented in combination with a plurality of other embodiments within a range that does not contradict each other. [First embodiment] The first embodiment is a quantum computation control method for efficiently generating resource states to be used for gate teleportation.

[0015] Fig. 1 is a diagram illustrating an example of a quantum computing control method according to a first embodiment. Fig. 1 illustrates an information processing device 10 that implements the quantum computing control method. The information processing device 10 can implement the quantum computing control method by, for example, executing a quantum computing control program.

[0016] The information processing device 10 includes a storage unit 11 and a processing unit 12. The storage unit 11 is, for example, a memory or a storage device included in the information processing device 10. The processing unit 12 is, for example, a processor or an arithmetic circuit included in the information processing device 10.

[0017] The storage unit 11 stores, for example, a quantum computing control program, and also stores intermediate data generated in the process of executing the quantum computing control program. The processing unit 12 controls the quantum computer 1, for example, according to a quantum computation control program, to have the quantum computer 1 perform quantum computation and find a solution to the quantum computation. The processing unit 12 can perform gate operations for arbitrary rotation of logical quantum bits during the quantum computation process. The processing unit 12 causes the quantum computer 1 to perform gate operations for arbitrary rotation based on a gate teleportation circuit. To execute the gate teleportation circuit, the processing unit 12 prepares a resource state with logical quantum bits other than the logical quantum bits that are the target of the gate operations for arbitrary rotation. For example, the processing unit 12 generates the resource state using the following procedure.

[0018] For example, the processing unit 12 transitions the state of the logical quantum bit 2 coded with the code distance d (d is an integer equal to or greater than 2) to a resource state. To this end, the processing unit 12 determines a physical rotation angle around a predetermined axis to be executed on d first physical quantum bits 4 out of the plurality of physical quantum bits 3 constituting the logical quantum bit 2, based on the logical rotation angle for rotating the state of the logical quantum bit 2 around the predetermined axis. Here, the logical rotation angle is defined as "θ * " and the physical rotation angle is "θ".

[0019] The logical qubit 2 is encoded, for example, by a surface code. The first physical qubit 4 is a physical qubit that is arranged in a row from one side to the opposite side of a plurality of physical qubits 3 that are arranged, for example, in a lattice pattern. In the example of FIG. 1, the physical qubit in the top row is the first physical qubit 4.

[0020] Based on the determined physical rotation angle, the processing unit 12 instructs the quantum computer 1 to perform a gate operation of a transversal rotation gate. For example, the processing unit 12 instructs the quantum computer 1, which has multiple physical quantum bits 3, to perform a rotation gate operation that rotates the state of each first physical quantum bit 4 around a predetermined axis by the determined physical rotation angle.

[0021] At this time, the processing unit 12 specifies the application of an m-qubit rotation gate that rotates the states of m first physical quantum bits 4 with a single rotation gate operation to physical quantum bit groups 4a and 4b each including m (m is an integer of 2 or more and d or less) of the first physical quantum bits 4. In the example of FIG. 1, the processing unit 12 specifies the application of an m-qubit rotation gate that rotates the states of m first physical quantum bits 4 with a single rotation gate operation to each of two physical quantum bit groups 4a and 4b each including three first physical quantum bits. ZZZ The quantum computer 1 is instructed to perform a rotation gate operation using "(θ)".

[0022] For example, the processing unit 12 generates a quantum circuit 5 that realizes the gate operation of an m-qubit rotate gate. The quantum circuit 5 includes multiple CNOT gates and one 1-qubit rotate gate. The processing unit 12 then instructs the quantum computer 1 to execute the generated quantum circuit 5.

[0023] In quantum computer 1, a transversal rotation gate operation is performed in accordance with an instruction from processing unit 12. For example, quantum computer 1 executes a rotation gate operation on first physical quantum bit 4 in logical quantum bit 2 in the logical |+> state, rotating it around a predetermined axis by a physical rotation angle. In this case, quantum computer 1 rotates the states of each of physical quantum bit groups 4a and 4b with a single rotation gate operation.

[0024] For example, quantum computer 1 performs gate operations on each of physical quantum bit groups 4a and 4b according to quantum circuit 5. Quantum circuit 5 is a circuit equivalent to a three-qubit rotate gate. Quantum circuit 5 includes only one one-qubit rotate gate. The rotate gate operation performed according to quantum circuit 5 is only one one-qubit rotation. In other words, one rotate gate operation affects three physical quantum bits.

[0025] After performing the rotation gate operation, the quantum computer 1 performs error detection on the logical qubit 2. For example, the quantum computer 1 measures the eigenvalues ​​of the stabilizers of the measurement qubits (also called auxiliary qubits) connected to multiple physical qubits 3 using a syndrome measurement circuit.

[0026] The processing unit 12 acquires error detection results for multiple physical qubits 3 from the quantum computer 1. For example, if there is a measurement qubit in which the stabilizer eigenvalue is inverted, there is a possibility that an error has occurred in the physical qubit connected to that measurement qubit. The processing unit 12 can identify the position of the physical qubit in which the error has occurred based on the position of the measurement qubit in which the stabilizer eigenvalue is inverted.

[0027] The processing unit 12 divides the region where multiple physical qubits 3 exist into a post-selection region 6 and an error correction region 7. The post-selection region 6 is a region that includes at least the first physical qubit 4. The post-selection region 6 includes, for example, physical qubits within a range affected by an error that occurred in the first physical qubit 4.

[0028] When processing unit 12 detects an error that occurred in post-selection region 6, it instructs quantum computer 1 to perform a gate operation to return logical quantum bit 2 to the logical |+> state and to redo the rotation gate operation. In this case, quantum computer 1 follows the instructions to perform a gate operation to return logical quantum bit 2 to the logical |+> state, and then performs the rotation gate operation of first physical quantum bit 4 again.

[0029] If no error occurring in the post-selection region 6 is detected, the processing unit 12 sets the state of the logical quantum bit 2 to the resource state "|m θ* > L (The * following θ is a subscript of θ, and so on). θ* > L The subscript L in " indicates that the state is represented by a redundant logical qubit. Similarly, hereafter, the subscript L will be added to the state of a logical qubit.

[0030] Furthermore, when an error that has occurred in an error correction region 7 other than the post-selection region 6 among the regions in which the plurality of physical quantum bits 3 exist is detected, the processing unit 12 instructs the quantum computer 1 to correct the detected error. In accordance with the instruction, the quantum computer 1 performs a gate operation to correct the detected error.

[0031] The processing unit 12 instructs the quantum computer 1 to execute the gate teleportation circuit using the state of the logical quantum bit 2 after the error correction. Upon receiving the instruction to execute the gate teleportation circuit, the quantum computer 1 executes the gate teleportation circuit using the resource state.

[0032] In this way, if no error is detected in the post-selection region 6, the processing unit 12 accepts the state output from the quantum computer 1 after correcting the error in the error correction region 7 as the resource state. Note that if no error is detected in the error correction region 7 either, error correction processing is not necessary. Then, the processing unit 12 causes the quantum computer 1 to execute a gate teleportation circuit that uses the accepted resource state.

[0033] In this way, the resource state used to execute the gate teleportation circuit can be efficiently generated. That is, the states of the m first physical qubits 4 included in each of the physical qubit groups 4a and 4b are rotated around a predetermined axis by one rotation gate operation for each of the physical qubit groups 4a and 4b. This reduces the number of rotation gate operations, lowering the possibility of a failure in generating a resource state due to an error in the rotation gate operation. That is, the state of the logical qubit 2 after the rotation gate operation is more likely to be accepted as the resource state by post-selection. As a result, the number of times resource state generation needs to be retried is reduced, and resource states are generated efficiently.

[0034] Furthermore, the region of the multiple physical qubits 3 within the logical qubit 2 is divided into a post-selection region 6 and an error correction region 7, and if an error occurs in the error correction region 7, the error is corrected and the state of the logical qubit 2 is accepted as the resource state. This increases the likelihood that the state of the logical qubit 2 after the rotation gate operation will be accepted as the resource state by post-selection. As a result, the number of times resource state generation needs to be retried is reduced, and resource states are generated efficiently.

[0035] The post-selection region 6 is a region that includes a measurement qubit that is used to detect an error that occurs in the first physical qubit 4, among the multiple measurement qubits used to measure the syndromes of the multiple physical qubits. For example, the post-selection region 6 is a region that includes a first measurement qubit that can detect an error in the first physical qubit 4, and a second measurement qubit that can detect an error that occurs in a gate operation on the first measurement qubit. In this case, the error correction region 7 is a region that includes the third measurement qubit that is the remaining measurement qubit, excluding the first measurement qubit and the second measurement qubit, among the multiple measurement qubits.

[0036] By using such a post-selection region 6 and error correction region 7, it is possible to properly prevent the generation of a resource state containing an error by post-selection and to maximize the error correction region 7. The wide error correction region 7 increases the likelihood that, when an error occurs in the logical qubit 2, the error can be corrected without having to regenerate the resource state, improving the efficiency of resource state generation.

[0037] Second Embodiment The second embodiment is a quantum computing system that can efficiently generate resource states in a state preparation protocol.

[0038] FIG. 2 is a diagram showing an example of a system configuration according to the second embodiment. A quantum computing system 30 includes a classical computer 100 and a quantum computer 200. The classical computer 100 is a computer known as a von Neumann computer. The quantum computer 200 is a non-von Neumann computer that applies the principles of quantum mechanics. The classical computer 100 is connected to a terminal 29 via a network 20. The terminal 29 is a von Neumann computer used by a user.

[0039] A user uses terminal 29 to create a quantum circuit for solving a target problem using quantum computing. The created quantum circuit is sent from terminal 29 to quantum computing system 30. In quantum computing system 30, classical computer 100 and quantum computer 200 work together to execute quantum computation according to the acquired quantum circuit. Then, quantum computing system 30 sends the computation result to terminal 29.

[0040] FIG. 3 is a diagram illustrating an example of hardware for a quantum computing system. A classical computer 100 is entirely controlled by a processor 101. A memory 102 and multiple peripheral devices are connected to the processor 101 via a bus 109. The processor 101 may be a multiprocessor. The processor 101 is, for example, a CPU (Central Processing Unit), an MPU (Micro Processing Unit), or a DSP (Digital Signal Processor). At least some of the functions realized by the processor 101 executing a program may be realized by an electronic circuit such as an ASIC (Application Specific Integrated Circuit) or a PLD (Programmable Logic Device).

[0041] The memory 102 is used as a main storage device of the classical computer 100. The memory 102 temporarily stores at least a portion of the OS (Operating System) program and application programs to be executed by the processor 101. The memory 102 also stores various data used in processing by the processor 101. As the memory 102, for example, a volatile semiconductor storage device such as a RAM (Random Access Memory) is used.

[0042] The peripheral devices connected to the bus 109 include a storage device 103, a GPU (Graphics Processing Unit) 104, an input interface 105, an optical drive device 106, a device connection interface 107, and a network interface 108.

[0043] The storage device 103 writes and reads data electrically or magnetically to and from a built-in recording medium. The storage device 103 is used as an auxiliary storage device for the classical computer 100. The storage device 103 stores the OS program, application programs, and various data. Note that the storage device 103 may be, for example, an HDD (Hard Disk Drive) or an SSD (Solid State Drive).

[0044] The GPU 104 is an arithmetic unit that performs image processing. The GPU 104 is an example of a graphics controller. The GPU 104 is connected to a monitor 21. The GPU 104 displays an image on the screen of the monitor 21 in accordance with an instruction from the processor 101. The monitor 21 may be a display device using organic EL (Electro Luminescence) or a liquid crystal display device.

[0045] The input interface 105 is connected to a keyboard 22 and a mouse 23. The input interface 105 transmits signals sent from the keyboard 22 and the mouse 23 to the processor 101. The mouse 23 is an example of a pointing device, and other pointing devices can also be used. Examples of other pointing devices include a touch panel, a tablet, a touch pad, and a trackball.

[0046] The optical drive device 106 uses a laser beam or the like to read data recorded on an optical disc 24 or write data to the optical disc 24. The optical disc 24 is a portable recording medium on which data is recorded so that it can be read by reflected light. The optical disc 24 includes a DVD (Digital Versatile Disc), a DVD-RAM, a CD-ROM (Compact Disc Read Only Memory), a CD-R (Recordable) / RW (Rewritable), and the like.

[0047] The device connection interface 107 is a communication interface for connecting peripheral devices to the classical computer 100. For example, a memory device 25 or a memory reader / writer 26 can be connected to the device connection interface 107. The memory device 25 is a recording medium equipped with a function for communicating with the device connection interface 107. The memory reader / writer 26 is a device for writing data to the memory card 27 or reading data from the memory card 27. The memory card 27 is a card-type recording medium.

[0048] The network interface 108 is connected to the network 20. The network interface 108 transmits and receives data to and from other computers or communication devices via the network 20. The network interface 108 is a wired communication interface connected by a cable to a wired communication device such as a switch or a router. The network interface 108 may also be a wireless communication interface connected by radio waves to a wireless communication device such as a base station or an access point.

[0049] The quantum computer 200 shares a bus 109 with the classical computer 100. The quantum computer 200 can communicate information with each element in the classical computer 100 via the bus 109.

[0050] Quantum computer 200 has quantum processing unit 201 connected to bus 109. Quantum processing unit 201 performs gate operations on quantum bits according to quantum gates shown in the quantum circuit and measures the states of the quantum bits. Quantum processing unit 201 has quantum bit device 202 and quantum bit control signal generator 203. Quantum bit device 202 holds the states of multiple quantum bits and performs gate operations on those quantum bits. Quantum bit control signal generator 203 generates control signals that instruct gate operations or measurements on the quantum bits.

[0051] The quantum computing system 30 can realize the processing functions of the second embodiment by using the hardware described above. Note that the information processing device 10 shown in the first embodiment can also be realized by using the same hardware as the quantum computing system 30 shown in FIG.

[0052] The classical computer 100 realizes the processing functions of the second embodiment by executing a program recorded on, for example, a computer-readable recording medium. The program describing the processing to be executed by the classical computer 100 can be recorded on various recording media. For example, the program to be executed by the classical computer 100 can be stored in a storage device 103. The processor 101 loads at least a portion of the program in the storage device 103 into the memory 102 and executes the program. The program to be executed by the classical computer 100 can also be recorded on a portable recording medium such as an optical disk 24, a memory device 25, or a memory card 27. The program stored on the portable recording medium becomes executable after being installed on the storage device 103, for example, under the control of the processor 101. The processor 101 can also read and execute the program directly from the portable recording medium.

[0053] Next, we will provide an overview of error correction in quantum computing and explain the usefulness of the STAR architecture. FIG. 4 is a diagram showing the characteristics of a quantum bit. A quantum bit 41, which is the smallest unit of information in a quantum computer 200, can be in the state |0> or the state |1>, and can also be in a superposition state between these. In the superposition state, whether the value obtained by measuring the quantum bit 41 is |0> or |1> is determined probabilistically. For example, if the probability of |0> and the probability of |1> are the same, the superposition state of the quantum bit 41 is "2 -1 / 2 (|0>+|1>)".

[0054] The information held by such a quantum bit 41 can be destroyed (an error occurs) due to interactions with the environment or operational errors. For example, if an error destroys the superposition state, the state of the quantum bit 41 changes to a state like |0>. To improve calculation accuracy, it is necessary to detect quantum bits in which an error has occurred and correct the state of the quantum bit to the correct state.

[0055] To address this issue, a technology called quantum error correction has been proposed. In quantum error correction, multiple qubits are combined and encoded. When encoded, the state of one or more logical qubits is represented by the multiple physical qubits used in the encoding. Based on the overall state of the encoded multiple qubits, the qubit in which an error occurred is detected and corrected.

[0056] FIG. 5 is a diagram illustrating an example of quantum error correction. As shown in FIG. 5, a logical quantum bit 42 is defined by a plurality of physical quantum bits 42a, 42b, . . . , 42n. In the example of FIG. 5, when the states of the plurality of physical quantum bits 42a, 42b, . . . , 42n are all |0>, an error occurs, and the state of physical quantum bit 42b is inverted to |1>. In such a case, the error is detected based on information obtained from the states of each of the plurality of physical quantum bits 42a, 42b, . . . , 42n. Then, the physical quantum bit 42b in which the error occurred is identified, and the state of that physical quantum bit 42b is corrected.

[0057] By performing quantum error correction appropriately in this way, even if errors occur in the physical qubits, as long as the number of errors is within the allowable range, the logical qubit 42 will remain in a correct state. Quantum computation with quantum error correction for logical qubits can be realized by combining predetermined basic gates.

[0058] FIG. 6 shows an example of a basic gate used in error-tolerant quantum computing. The basic gates used in quantum computing with quantum error correction are an H gate 43a, a CNOT gate 43b, an S gate 43c, and a T gate 43d. The H gate 43a is called a Hadamard gate and is a quantum gate that rotates a state by 180 degrees around an axis tilted at 45 degrees between the Z axis and the X axis. The CNOT gate 43b leaves the state of the target quantum bit unchanged if the state of the control quantum bit is |0>, and inverts the state of the target quantum bit (from |0> to |1>, and from |1> to |0>) if the state of the control quantum bit is |1>. The S gate 43c is a quantum gate that rotates a state by π / 2 around the Z axis. The T gate 43d is a quantum gate that rotates a state by π / 4 around the Z axis.

[0059] Of these, the H gate 43a, CNOT gate 43b, and S gate 43c are called Clifford operators. In contrast, the T gate is called a non-Clifford operator. These basic gates are collectively called Clifford+T. Clifford+T in the calculations of the quantum computer 200 corresponds to AND, XOR, and NOT in the classical computer 100. In other words, by combining Clifford+T quantum gates, it is possible to perform any quantum calculation.

[0060] Here, in Clifford+T fault-tolerant quantum computing, more than one million physical qubits are used to perform useful calculations. A large proportion (e.g., more than 90%) of the large number of physical qubits is used to rotate the logical qubit by an arbitrary angle. Rotation by an arbitrary angle involves gate operations using a large number of T gates 43d. A large number of physical qubits are used for error correction of the T gates 43d (the error correction cost is high). This is a major factor in increasing the number of physical qubits required to realize FTQC.

[0061] Moreover, in order to realize a rotation operation of an arbitrary angle, in most cases, the gate operation of the T gate 43d must be repeated several tens of times. Therefore, realizing rotation of an arbitrary angle using the T gate 43d leads to a decrease in the execution efficiency of the quantum circuit.

[0062] Therefore, in the STAR architecture, a phase rotation gate 43e is used as a basic gate instead of the T gate 43d of Clifford+T. The phase rotation gate 43e in the STAR architecture rotates the phase of a given resource state "|m θ* > L This can be done using a gate teleportation circuit using θ * is an arbitrary rotation angle. θ* >" is "|m θ* >=R Z (θ * )|+>=2 -1 / 2 (e -iθ* / 2 |0>+e +iθ* / 2 |1>)". "R Z (θ * )" is "R Z (θ * )=e iθ*Z ” (Z is the Pauli Z operator).

[0063] 7 is a diagram showing an example of a quantum circuit that performs gate operations of arbitrary rotation. The gate teleportation circuit 50 performs gate operations of arbitrary rotation angle θ * Rotation (R Z (θ * )) is a quantum circuit that realizes the state of the object of operation, "|ψ>" L " is input, and the second qubit is given the resource state "|m θ* > L " is entered.

[0064] In the gate teleportation circuit 50, first, a CNOT gate 50a is operated with the second qubit as the control qubit and the first qubit as the target qubit. Then, a measurement 50b of the first qubit is performed, and if the measurement result is "+1", an X gate 50c is operated on the second qubit.

[0065] If the measurement result of the first qubit is "+1", the state of the second qubit is "R Z (θ * )|ψ> L If the measurement result of the first qubit is "0", the state of the second qubit is "R Z (-θ * )|ψ> L " In this way, after the gate operation of the gate teleportation circuit 50 for any rotation, "R Z (θ * )|ψ> L " or "R Z (-θ * )|ψ> L " is obtained. In other words, the output state is probabilistically reversed. If the output state is "R Z (θ * )|ψ> L " and "R Z (-θ * )|ψ> L The probability of each being "1 / 2".

[0066] In the quantum computing system 30, the output state of the gate teleportation circuit 50 will probabilistically result in the desired rotation (forward rotation) or reverse rotation, and so the quantum computing system 30 repeatedly executes the same gate operation until the desired rotation is successful.

[0067] For example, the target's logical rotation angle θ * The gate operation of the rotation of the * ), the quantum computing system 30 performs the next rotation gate operation at an angle 2θ * Perform a rotation through the angle 2θ * The rotation gate operation also failed, and the reverse rotation (-2θ* ), the sum of the two rotation operations is -3θ * In this case, the quantum computing system 30 performs the next gate operation of rotation, for example, at an angle of 4θ * Perform a rotation of

[0068] If the probability of a successful spin and the probability of an unsuccessful spin are 1 / 2, then the average number of times until a spin is successful is 1×(1 / 2)+2×(1 / 4)+=Σ n n2 -n =2". In other words, the quantum computing system 30 can realize any rotation by performing the gate operation of the gate teleportation circuit 50 an average of two times.

[0069] The gate teleportation circuit 50 receives the resource state "|m θ* > L " is used. Therefore, the resource state is generated before the gate teleportation is executed. The accuracy of the generated resource state affects the accuracy of the entire arbitrary rotation.

[0070] State preparation protocol methods for generating resource states include those disclosed in the aforementioned Non-Patent Documents 1 and 2. For example, the method disclosed in Non-Patent Document 2 is known to exhibit high performance for rotary gates with small rotation angles.

[0071] Fig. 8 is a diagram showing an example of a state preparation protocol. A logical qubit 51 is composed of a plurality of physical qubits 52. In the example of Fig. 8, the logical qubit 51 is made redundant with a surface code having a code distance of "3".

[0072] The quantum computing system 30 uses the logical qubit 51 to generate a logical |+> state "|+> L For example, the quantum computing system 30 prepares a logical Z operator "Z i =× i Z i” (× is an × inside a circle) is applied (i is the quantum bit number of the physical quantum bit that is the target of the gate operation).

[0073] The quantum computing system 30 performs a transversal rotation gate operation on the logical quantum bit 51. The transversal rotation gate is a quantum gate that simultaneously applies Z rotation by a physical rotation angle θ to a row of physical quantum bits that are arranged horizontally among the physical quantum bits that make up the logical Z operator. The transversal rotation gate to the logical |+> state is a gate that operates by "Π i R i (θ)|+> L " is expressed as:

[0074] The state represented by the logical qubit 51 is called "|m θ* > L " The logical rotation angle θ * depends on the physical rotation angle θ of the transversal rotating gate, * ~θ d " (d is the code distance). The quantum computing system 30 calculates θ * The value of is in the state |ψ> L The physical rotation angle θ is determined so that it becomes the target angle in the rotation gate operation to act on the logical quantum bit. Then, the quantum computing system 30 can generate an appropriate resource state by performing a gate operation to rotate the physical rotation angle using the transversal rotation gate.

[0075] The quantum computing system 30 performs syndrome measurement of the logical quantum bit 51 after the gate operation of the transversal rotary gate. Syndrome measurement is a special quantum measurement operation performed to detect errors that have occurred in the error-correcting code state. Syndrome measurement is performed using an auxiliary quantum bit (not shown). Syndrome measurement is performed based on a quantum circuit that extracts only error information so as not to destroy the quantum state encoded in the error-correcting code. The result of the syndrome measurement is given as a binary value, ±1. When an error occurs, the value of the syndrome measurement result associated with it is inverted.

[0076] The quantum computing system 30 performs post-selection on the state of the logical qubit 51 when no errors occur as a resource state. Post-selection is a process in which, after performing syndrome measurements, only trials that yield measurement results that satisfy specific conditions are extracted, and other trials are discarded. Post-selection allows for the acquisition of only quantum states with good properties that meet the objectives.

[0077] There are multiple combinations of physical qubits that perform gate operations when preparing the logical |+> state. Similarly, there are multiple combinations of physical qubits that are the targets of gate operations for the transversal rotation gate.

[0078] 9 is a diagram showing an example of a combination pattern of physical qubits that perform gate operation of a transversal rotation gate. In pattern 1, as shown in FIG. 8, three physical qubits arranged diagonally are the targets of gate operation of the transversal rotation gate. In pattern 2, three physical qubits arranged horizontally in the first row are the targets of gate operation of the transversal rotation gate. In pattern 3, three physical qubits arranged horizontally in the second row are the targets of gate operation of the transversal rotation gate.

[0079] In the state preparation protocol, errors can occur during the preparation of the logical |+> state or during the gate operation of the Transversal turnstile. Errors can also occur during syndrome measurement to detect the occurrence of errors. If an error occurs during the resource state generation process, the resource state generation process is restarted. When the resource state generation process is restarted, it starts over from the preparation of the logical |+> state.

[0080] 10 is a diagram showing an example of processing when an error occurs in the state preparation protocol. The quantum computing system 30 first performs preparation processing for the logical |+> state on the logical quantum bit 51 used to generate the resource state (step S11). Next, the quantum computing system 30 performs syndrome measurement on the logical quantum bit 51 (step S12). If the quantum computing system 30 detects an error in the syndrome measurement, it restarts the resource state preparation processing (step S13).

[0081] If no error is detected in the syndrome measurement after the preparation process for the logical |+> state, the quantum computing system 30 performs a gate operation of the transversal rotation gate on the logical quantum bit 51 (step S14). Next, the quantum computing system 30 performs a syndrome measurement for the logical quantum bit 51 (step S15). If the quantum computing system 30 detects an error in the syndrome measurement, it restarts the preparation process for the resource state (step S16).

[0082] If no error is detected in the first syndrome measurement after the transversal rotary gate is operated, quantum computing system 30 performs another syndrome measurement of logical quantum bit 51 (step S17). This assumes that an error occurs in the first syndrome measurement and the error cannot be correctly detected. If quantum computing system 30 detects an error in the second syndrome measurement, it restarts the resource state preparation process (step S18).

[0083] If no error is detected in the second syndrome measurement, quantum computing system 30 determines that the preparation of the resource state has been successful, and post-selects the state of logical quantum bit 51 at that time as the resource state (step S19).

[0084] Next, we will explain why an error is detected in the syndrome measurement after the gate operation of the transversal revolving gate. 11 is a diagram showing an example of a gate operation of a transversal rotation gate. When a transversal rotation gate is applied to a logical quantum bit 53, for example, a rotation gate operation of phase rotation is performed on a plurality of physical quantum bits in the first row. Focusing on the i-th physical quantum bit 53a to be operated on, the gate operation of phase rotation on that physical quantum bit 53a is expressed by the following equation (1).

[0085]

number

[0086] The same gate operation is performed on all physical qubits to be operated, and the transversal rotation gate is expressed by equation (2).

[0087]

number

[0088] Equation (2) shows that various Pauli sequence operations such as "II···I,ZI···I,ZZ···I,···,ZZ···Z" act as quantum superpositions by the gate operation of transversal rotation. The coefficient of each term, "u 00···0 ,u 10···0 ,···,u 11···1 " is a real number determined according to the value of θ and is defined as follows: u x =cos d-|x| (θ / 2)·(-i) |x| sin |x| (θ / 2) Here, x is an arbitrary d-digit bit string, and |x| represents the total number of 1s contained in the bit string x.

[0089] In this way, transversal rotation generates and superimposes multiple patterns of Pauli strings. By measuring the syndrome of the logical qubit 53 after this transversal rotation, the applied Pauli string operation is measured.

[0090] 12 is a diagram showing an example of syndrome measurement after transversal rotation. When syndrome measurement is performed after the transversal rotation gate is operated on the logical qubit 53 for generating resource states, the logical qubit 53 branches into one of multiple patterns of the Pauli series operation that is applied. At this time, the logical qubit 53 is in the ideal resource state |m θ* > L The only conditions that contribute to this are when all Pauli series operations are I (III···I) or all are Z (ZZZ···Z). The Pauli series operations "Pattern 1" and "Pattern n" (n is a natural number indicating the number of patterns) shown in Figure 12 are ideal resource states.

[0091] On the other hand, the Pauli sequence operation with mixed I and Z results in the wrong resource state |m θerror (The "error" following θ is a subscript of θ, and so on.) "Pattern 2" to "Pattern n-1" of the Pauli series operations shown in FIG. 12 are erroneous resource states.

[0092] Syndrome measurement is a process of measuring the stabilizer eigenvalues ​​using an auxiliary qubit. For example, the auxiliary qubit for syndrome measurement is connected to two adjacent physical qubits that are the target of transversal rotation manipulation. From the measurement results of the stabilizer eigenvalues ​​on each of the multiple auxiliary qubits, it is possible to detect whether I and Z are mixed in the Pauli series manipulation.

[0093] When syndrome measurement is performed, only one state where one type of Pauli series operation is applied is extracted. Which pattern of Pauli series operation is extracted is determined probabilistically. The ideal resource state |m θ* > L is detected, the state of the logical qubit 53 at that time is post-selected.

[0094] Incorrect resource state|m θerror > LIf detected, the resource state at that time is discarded and the resource state generation process is restarted. For the sake of efficiency, the ideal resource state |m θ* > L It is important to increase the probability that only the original image is correctly extracted.

[0095] When an error occurs in the gate operation of the Transversal turnstile, the incorrect resource state |m θerror > L However, the ideal resource state |m θ* > L may also be extracted.

[0096] Figure 13 shows an example of a post-selection situation when an error occurs. Here, for simplicity, the order of the probability branch due to syndrome measurement and the probability branch due to error occurrence is reversed (this reordering is justified because these probability events are independent of each other). For example, when the Pauli sequence operation is "Pattern 1," a Z error occurs in a physical qubit that is not the target of the transversal rotation gate operation. This Z error is detected by syndrome measurement. In this case, the state of the logical qubit 53 is rejected. If no error occurs in "Pattern 1," the state of the logical qubit 53 at that time is accepted as an ideal resource state.

[0097] Next, let us consider the case where a Z error occurs in a physical qubit when the Pauli series operation is in "Pattern 2." If the Z error occurs in a physical qubit that is not the target of the transversal rotation gate operation, the Z error is detected by syndrome measurement, and the state of logical qubit 53 at this time is rejected.

[0098] On the other hand, it is possible that the location where the Z error occurs is the target of the transversal rotary gate and is a physical qubit where an undesired Pauli series operation has occurred. In this case, the Z operation caused by the Pauli series operation and the Z operation caused by the error cancel each other out. This makes it impossible to distinguish between the "IIIII" and "IIIIII" states. As a result, despite the occurrence of an error, the state of logical qubit 53 at that time is accepted as the resource state. In other words, if a Z error occurs in a physical qubit that is the target of the transversal rotary gate and where an undesired Pauli series operation has occurred, the error cannot be detected correctly, which will cause an error in quantum computation using a gate teleportation circuit that uses the resource state.

[0099] Thus, when generating a resource state using a transversal rotation gate, various types of errors can cause errors in the output state of the target quantum computation. The error rate (probability of rejection) of the prepared resource state is proportional to the code distance d of the logical qubit 53. Furthermore, the failure probability of generating a resource state using the state preparation protocol (the probability that a resource state with an error is selected ex post) is proportional to the square of the code distance d. 2 is proportional to.

[0100] To use a phase rotation gate instead of the T gate of Clifford+T, it is important to reduce as many types of errors as possible in the resource state generation process, which are factors that cause errors in the output state, while also being able to demonstrate high performance even in areas with large code distances.

[0101] Therefore, the quantum computing system 30 shown in the second embodiment is improved mainly in the following two points. 1. The quantum computing system 30 realizes the transversal rotation gate not with a Z rotation gate (1-qubit gate) for each physical qubit, but with a multi-Z rotation gate, thereby achieving a reduction in the error rate and an improvement in the success probability.

[0102] 2. The quantum computing system 30 uses a hybrid method of post-selection and error correction to generate resource states, rather than simple post-selection, which improves the success rate. These improvements make it possible to fully utilize resource state preparation even in areas with large code distances.

[0103] 14 is a diagram showing an example of a transversal rotation gate that rotates two or more physical qubits together. In the logical qubit 54 shown in FIG. 14, the transversal rotation gate is operated on the nine physical qubits on the upper side of the multiple physical qubits that make up the surface code. In this case, when the rotation gate operation is performed individually, R is applied to each of the nine physical qubits related to the logical Z operator. Z (θ) rotation gate operation is performed.

[0104] On the other hand, in the quantum computing system 30, the transversal rotation gate is extended to perform m-weight Multi-Z rotation so as to fill up the quantum bits related to the logical Z operator. The 3-weight Multi-Z rotation is expressed by Equation (3).

[0105]

number

[0106] In general, the Multi-Z rotation is expressed by equation (4).

[0107]

number

[0108] For example, the quantum computing system 30 divides the nine physical qubits into groups of three, and the states of the three physical qubits are expressed as R ZZZBy realizing a transversal rotation gate using Multi-Z rotation, the tolerance to single qubit errors is increased and the error rate can be reduced even for logical qubits with large code distances.

[0109] 15 is a diagram showing an example of a quantum circuit for implementing a Multi-Z rotation gate. The quantum circuit 61 shown in FIG. 15 is an R ZZZ This is a quantum circuit for implementing (θ).

[0110] In quantum circuit 61, first, a CNOT gate 61a is arranged, with the first physical quantum bit as a control quantum bit and the second physical quantum bit as a target quantum bit. Next to CNOT gate 61a, a CNOT gate 61b is arranged, with the second physical quantum bit as a control quantum bit and the third physical quantum bit as a target quantum bit. Next, a phase rotation gate 61c is arranged at the third physical quantum bit.

[0111] Next to the phase rotation gate 61c is a CNOT gate 61d, which uses the second physical qubit as a control qubit and the third physical qubit as a target qubit. Finally, there is a CNOT gate 61e, which uses the first physical qubit as a control qubit and the second physical qubit as a target qubit.

[0112] By making the quantum computer 200 perform gate operations according to such a quantum circuit 61, R ZZZ In this way, the Multi-Z rotation gate is converted into a quantum circuit 61 that combines one phase rotation gate 61c and multiple CNOT gates 61a, 61b, 61d, and 61e during implementation. In the quantum circuit 61, the phase rotation gate operation is performed only once.

[0113] Figure 16 shows the R ZZThis is a diagram showing an example of a transversal rotation gate implemented with (θ). The specific implementation method of the transversal rotation gate can be expressed by the following three types of parameters. m: Z-weight number of the rotation gate (number of physical qubits to rotate simultaneously) k: total number of revolving gates d: Code distance The equation "d = m × k" holds. In the case of a surface code, d is equal to the length of the side of the square representing the surface code area (the number of data qubits on one side).

[0114] For example, when performing a transversal rotation gate operation with a one-qubit rotation gate on a logical qubit 55 encoded as "d=5", the transversal rotation gate is expressed as "(m, k, d)=(1,6,6)". In the operation of a one-qubit rotation gate, I or Z acts probabilistically on the one physical qubit being operated on.

[0115] Furthermore, when a two-qubit rotation gate is used to perform a transversal rotation gate operation on logical qubit 55, the transversal rotation gate is expressed as "(m, k, d) = (2, 3, 6)". In the gate operation of a two-qubit rotation gate, II or ZZ acts probabilistically on the two physical qubits being operated.

[0116] When a transversal rotation is performed using a 1-qubit rotation gate (1,6,6), the resource state is successfully generated if all six rotation gate operations on six physical qubits result in I or Z. That is, 2 6 Only two of the possible Pauli series operations are successful.

[0117] On the other hand, when a transversal rotation is performed using a two-qubit rotation gate (2, 3, 6), the resource state is successfully generated if all three rotation gate operations on two physical qubits result in I or Z. That is, 2 3 Only two of the possible Pauli sequence operations are successful. Therefore, the transversal rotation gate with "(m, k, d) = (2, 3, 6)" is more likely to succeed in generating a resource state than the transversal rotation gate with "(m, k, d) = (1, 6, 6)".

[0118] The probability of successfully creating a resource state can be calculated as follows: The action of the transversal rotation gate is as shown in equation (2). In the resource state generation process, the transversal rotation gate with the action shown in equation (2) is executed on the logical qubit in the logical |+> state. The target resource state at this time is as shown in equation (5).

[0119]

number

[0120] The probability of being projected onto the desired resource state (probability of success) is given by equation (6) in the limit where the effect of errors can be ignored.

[0121]

number

[0122] where |u 00···0 | 2 ,|u 11···1 | 2 are given by equations (7) and (8), respectively, depending on the number k of acted gates.

[0123]

number

[0124]

number

[0125] Considering that |cos(θ / 2)|≦1 and |sin(θ / 2)|≦1, the success probability p suc is a monotonically decreasing function of k. Therefore, the smaller the total number of rotation gates k (the larger the Z-weight number m), the higher the success probability.

[0126] Note that due to hardware limitations of the quantum computer 200, there is an upper limit to the Z-weight number m of the Multi-Z rotation gate. For example, if a Multi-Z rotation gate is decomposed into an equivalent circuit as shown in Figure 15, the number of quantum gates to be executed increases. As the number of quantum gates to be executed increases, errors accumulate and the calculation accuracy decreases. Furthermore, if the number of quantum gates to be executed increases too much, there is a possibility that the quantum calculation will not be completed within the coherence time (the time required to maintain quantum properties). Therefore, it is appropriate to set the Z-weight number m to a large value within the range allowed by the hardware of the quantum computer 200.

[0127] Also, transversal rotation gate operations may include a mixture of multi-Z rotations with different weights. FIG. 17 is a diagram showing an example of a transversal rotation gate in which a plurality of weighted Multi-Z rotations are mixed. In the example of FIG. 17, the quantum computing system 30 independently controls the rotation gate “R Z The quantum computing system 30 applies a Multi-Z rotation gate "R (θ)" with a weight of "2" to the remaining two physical quantum bits. ZZ (θ)" is applied.

[0128] Next, a hybrid method of post-selection and error correction will be described. 18 shows an example of a hybrid method of post-selection and error correction for a rotated surface code. For example, if all logical qubits 54 used for generating resource states are subject to post-selection, and an error is detected in any physical qubit within the logical qubits 54 by syndrome measurement, the state of the logical qubits 54 at that time is discarded. The resource state generation process using the state preparation protocol is then restarted.

[0129] In contrast, in the hybrid method, the region of the physical qubits that make up the logical qubit 54 is divided into a post-selection region 54a and an error correction region 54b. The post-selection region 54a is the region used for post-selection decisions. The error correction region 54b is the region that corrects errors detected by error correction if accepted in post-selection.

[0130] In the hybrid approach, if an error is detected in the post-selection region 54a by syndrome measurement, the current state of the logical qubit 54 is discarded. The resource state generation process is then restarted using the state preparation protocol. On the other hand, if no error is detected in the post-selection region 54a, but an error is detected only in the error correction region 54b, the current state of the logical qubit 54 is accepted. If the state of the logical qubit 54 is accepted, the error that occurred in the error correction region 54b is corrected by the error correction process.

[0131] The hybrid method increases the likelihood that the state of the transversal turnstile after gate operation will be accepted as the resource state. As a result, resource states can be generated efficiently. The effectiveness of the hybrid method will be specifically explained below with reference to Figures 19 and 20.

[0132] Figure 19 shows an example of an error pattern that is suitable for handling by post-selection. For example, "Pattern 1" in sampling after a Pauli series operation using a transversal rotation gate is the Pauli series operation "IZIII." This is an error state. Also, "Pattern 2" is the Pauli series operation "IIIII." This is an ideal state. If a syndrome measurement is then performed using a quantum circuit for syndrome measurement, an error may occur in the syndrome measurement.

[0133] For example, an error may occur in one of the physical qubits that are the target of the transversal rotation gate in the ideal state, "Pattern 2." In this case, it is not possible to distinguish whether the error state occurred during sampling of the Pauli series operation or during subsequent gate operation.

[0134] In this way, if an error occurs around the physical qubit that is the target of the transversal rotary gate operation, the possibility that an error has occurred in the sampling of the Pauli series operation cannot be ruled out. Therefore, in both "Pattern 1" and "Pattern 2," the state of logical qubit 55 is rejected in post-selection. In other words, the reason post-selection is performed is because the situation shown in Figure 19 exists.

[0135] FIG. 20 shows an example of an error pattern that can be avoided by post-selection. In the example of FIG. 20, the position of the error that occurred in "Pattern 2" is distant from the physical qubit that is the target of the transversal rotation gate operation. In such a case, it is possible to distinguish by syndrome measurement whether the error state occurred during sampling of the Pauli series operation or whether the error occurred during subsequent gate operation. In other words, in the case of "Pattern 2," it can be determined that the state of logical qubit 55 is an ideal resource state.

[0136] If it is known that the state of the logical qubit 55 is an ideal resource state, accepting that state as the resource state prevents the need to regenerate the resource state. However, if an error is left in the physical qubit that constitutes the logical qubit 55, it will cause an error in subsequent gate teleportation. For errors detected at a position away from the physical qubit that is the target of the transversal rotary gate, it is sufficient to perform error correction to cancel the error.

[0137] Here, it is not easy to uniformly determine how close the range of the post-selection region 54a should be from the physical qubit that is the target of the transversal rotary gate operation. For example, simulations can be performed using multiple patterns in which the range of the post-selection region 54a is changed, and the post-selection region 54a can be determined so that it is as narrow as possible under the constraint that "the original error rate is not changed."

[0138] Next, a problem that occurs when the post-selection region 54a is too narrow will be described. FIG. 21 is a diagram showing an example of error state detection in a post-selection region. For simplicity, a non-rotational surface code is assumed. In FIG. 21, among the physical qubits constituting the logical qubit 55 used to generate the resource state, data qubits representing the state of the logical qubit are indicated by white circles, and measurement qubits used for syndrome measurement are indicated by black circles. The measurement qubits in the hatched region correspond to the X stabilizer operator. The measurement qubits in the open region correspond to the Z stabilizer operator. The physical qubits in a row on the top side of the logical qubit 55 are the gate operation targets of the transversal rotation gate 56.

[0139] Here, we focus on the syndrome measurements for the two data qubits d1 and d3 that are the targets of transversal rotary gate 56's gate operation and their adjacent data qubit d2. Data qubits d1, d2, and d3 are connected to a measurement qubit m1. Data qubits d1 and d2 are also connected to a measurement qubit m2. Data qubit d2 is further connected to a measurement qubit d3.

[0140] Before executing syndrome measurement circuit 57 to measure the syndromes of data qubits d1 and d3, measurement qubits m1, m2, and m3 are initialized to |0>. In syndrome measurement circuit 57, Hadamard gates 57a and 57b are arranged for measurement qubits m1 and m3, respectively. However, to simplify the diagram, the circuit diagram is shown with CNOT gates connected to qubits other than those labeled.

[0141] Next, CNOT gate 57c is placed, with measurement qubit m1 as the control qubit and data qubit d1 as the target qubit. Next, CNOT gate 57d is placed, with measurement qubit m1 as the control qubit and data qubit d2 as the target qubit. Next, CNOT gate 57e is placed, with measurement qubit m3 as the control qubit and data qubit d2 as the target qubit. Next, CNOT gate 57f is placed, with data qubit d1 as the control qubit and measurement qubit m2 as the target qubit. Next, CNOT gate 57g is placed, with measurement qubit m1 as the control qubit and data qubit d3 as the target qubit. Next, CNOT gate 57h is placed, with data qubit d2 as the control qubit and measurement qubit m2 as the target qubit.

[0142] Finally, Hadamard gates 57i and 57j are arranged for measurement quantum bits m1 and m3, respectively, in syndrome measurement circuit 57. Then, measurement of measurement quantum bits m1, m2, and m3 is performed.

[0143] Now consider a situation where the Pauli operators Z and I are sampled for the data qubits d1 and d3, respectively, as a result of the transversal rotation gate. This state is an incorrect resource state, so it is appropriate to discard it. Below, we explain how to detect signs that the Pauli operators Z and I have been sampled.

[0144] FIG. 22 illustrates a first case in which an error state is detected. The first case is one in which no error occurs during execution of syndrome measurement circuit 57. In this case, the action of Pauli operator Z on data qubit d1 is transferred to measurement qubit m1 via CNOT gate 57c. The action of Pauli operator I on data qubit d3 does not affect measurement qubit m1, even via CNOT gate 57g. In this case, the eigenvalue of the Z stabilizer measured by measurement qubit m1 is inverted, and an error is detected. As a result, the state of logical qubit 55 is rejected as being inappropriate as a resource state.

[0145] FIG. 23 is a diagram showing a second case in which an error state is detected. The second case is when an error occurs during execution of syndrome measurement circuit 57. In the example of FIG. 23, it is assumed that a ZZ error occurs in CNOT gate 57d. In this case, the action of the Pauli operator Z on data qubit d1 is transferred to measurement qubit m1 via CNOT gate 57c, but the action of the Pauli operator Z is canceled out by an error that occurs in the gate operation of CNOT gate 57d. As a result, no error can be detected from the measurement result of measurement qubit m1.

[0146] On the other hand, due to an error occurring in CNOT gate 57d, data qubit d2 enters a state in which the Pauli operator Z has been applied after the gate operation of CNOT gate 57d. This state is transferred to measurement qubit m3 by CNOT gate 57e. Then, the eigenvalue of the Z stabilizer measured by measurement qubit m3 is inverted, and the error is detected.

[0147] In this way, when an error occurs in syndrome measurement circuit 57, it is possible to detect the occurrence of an error in syndrome measurement circuit 57 even if it is not possible to detect that the Pauli series operation by transversal rotary gate 56 has entered an error state. When an error occurs in syndrome measurement circuit 57, it cannot be guaranteed that the state of logical quantum bit 55 is an ideal resource state, and therefore the state of logical quantum bit 55 is discarded.

[0148] 23, when an error occurs in syndrome measurement circuit 57, signs of Pauli series operations other than II···I and ZZ···Z may not be detected by the stabilizer operator of the topmost measurement qubit alone. Therefore, to determine whether the state of logical qubit 55 has reached the ideal resource state, not only the topmost measurement qubit but also the measurement qubits up to the third row are used.

[0149] FIG. 24 is a diagram showing an example of post-selection region settings. Of the data qubits (white circles) that make up logical qubit 55, the topmost data qubit is the target of operation of the transversal rotary gate. Assuming the second case shown in FIG. 23, if there is an error in the resource state, it can be detected by the measurement qubits (black circles) in the first three rows from the top. Therefore, the region including the physical qubits in the first three rows from the top becomes post-selection region 55a, which rejects the state of logical qubit 55 when an error is detected. The region other than post-selection region 55a becomes error-correction region 55b, which does not use the detected error in determining post-selection but instead targets the error for error correction.

[0150] In this way, by defining the range in which the transversal rotary gate's operation affects the physical qubit to be operated as the post-selection region 55a, post-selection can be performed appropriately. Furthermore, by limiting the post-selection region 55a to such a range and defining the rest as the error correction region 55b, the size of the post-selection region 55a can be minimized, and the probability of successfully generating a resource state is improved.

[0151] Next, the function of the quantum computing system 30 that performs quantum computation using a phase rotation gate instead of the T gate of Clifford+T will be described in detail. 25 is a block diagram showing an example of functions for quantum computation in a quantum computing system. Classical computer 100 has a quantum computation request acceptance unit 110 and a quantum circuit execution control unit 120. Quantum computer 200 has a quantum bit initialization unit 210 and a quantum bit measurement unit 220. In addition, functions realized by the classical computer 100 and quantum computer 200 operating in conjunction with each other include a Clifford computation execution unit 31 and an arbitrary rotation execution unit 32.

[0152] The quantum computing request receiving unit 110 receives a quantum computing request from the terminal 29. The quantum computing request includes, for example, a quantum circuit corresponding to the problem to be solved. The quantum computing request receiving unit 110 transmits an execution command for the quantum circuit corresponding to the problem to be solved, which is indicated in the quantum computing request, to the quantum circuit execution control unit 120. Furthermore, when the quantum computing request receiving unit 110 obtains the result of the quantum computing by the quantum circuit from the quantum circuit execution control unit 120, it transmits the calculation result to the terminal 29.

[0153] The quantum circuit execution control unit 120 transmits quantum gate execution commands to the quantum computer 200 in the order indicated in the quantum circuit acquired as the execution target. When the quantum circuit execution control unit 120 acquires measurement results indicating the states of quantum bits after gate operations according to the quantum circuit from the quantum computer 200, it calculates a solution to the problem to be solved based on the measurement results. The quantum circuit execution control unit 120 then transmits the solution to the problem to be solved to the quantum computation request receiving unit 110 as the result of the quantum computation.

[0154] The quantum bit initialization unit 210 initializes the logical quantum bits in accordance with instructions from the quantum circuit execution control unit 120. For example, the quantum bit initialization unit 210 initializes the physical quantum bits that make up the logical quantum bits in the quantum bit device 202 to a predetermined state.

[0155] The quantum bit measurement unit 220 measures the state of the logical quantum bit. For example, the quantum bit measurement unit 220 measures the state of the physical quantum bits that make up the logical quantum bit, and determines the state of the logical quantum bit based on the measurement result. The quantum bit measurement unit 220 transmits the measured state of the logical quantum bit to the classical computer 100.

[0156] The Clifford operation execution unit 31 executes a Clifford operation on a logical quantum bit in cooperation with the classical computer 100 and the quantum computer 200. For example, in the quantum computer 200, the gate operation of a quantum gate corresponding to the Clifford operation is executed on the physical quantum bits that constitute the logical quantum bit. The Clifford operation execution unit 31 then performs syndrome measurement of the operated physical quantum bit. The Clifford operation execution unit 31 detects an error based on the result of the syndrome measurement. If the Clifford operation execution unit 31 detects an error, it determines the location of the error and performs a gate operation on the physical quantum bit at the error location to correct the error.

[0157] Of the functions of the Clifford operation execution unit 31, error detection and error location determination are executed by the classical computer 100. Gate operation of quantum gates corresponding to the Clifford operation, syndrome measurement, and error correction are executed by the quantum computer 200.

[0158] The arbitrary rotation execution unit 32 executes an arbitrary rotation gate operation on the logical quantum bit in cooperation with the classical computer 100 and the quantum computer 200. 26 is a block diagram showing an example of the functions of the arbitrary rotation execution unit 32. The functions realized by the classical computer 100 in the arbitrary rotation execution unit 32 include a physical rotation angle calculation unit 32a, a post-selection pass determination unit 32b, a syndrome measurement and recording unit 32c, an error location estimation unit 32d, and a gate teleportation success / failure determination unit 32e.

[0159] The physical rotation angle calculation unit 32a converts the logical rotation angle into a physical rotation angle. For example, the physical rotation angle calculation unit 32a calculates the physical rotation angle using the following formula:

[0160]

number

[0161] The physical rotation angle calculation unit 32a calculates the logical rotation angle θ based on the equation (9). * becomes the target angle. The physical rotation angle calculation unit 32a transmits the calculated value of the physical rotation angle θ to the transversal rotation gate execution unit 32h. The physical rotation angle calculation unit 32a also notifies the post-selection pass determination unit 32b that the setting of the physical rotation angle θ has been completed.

[0162] Furthermore, when the physical rotation angle calculation unit 32a receives an instruction to double the logical rotation angle from the gate teleportation success / failure determination unit 32e, it calculates the physical rotation angle θ based on the corrected logical rotation angle. Then, the physical rotation angle calculation unit 32a transmits the calculated value of the physical rotation angle θ to the transversal rotation gate execution unit 32h, and notifies the post-selection pass determination unit 32b that the setting of the physical rotation angle θ has been completed.

[0163] When the setting of the physical rotation angle θ is completed, the post-selection pass determination unit 32b transmits a preparation command for the logic |+> state to the logic |+> state preparation unit 32f. The post-selection pass determination unit 32b also acquires syndrome measurement information from the syndrome measurement recording unit 32c and determines whether or not an error exists based on the syndrome measurement information. If the post-selection pass determination unit 32b detects an error in the post-selection region, it instructs the logic |+> state preparation unit 32f to prepare the logic |+> state.

[0164] If the syndrome measurement information obtained from the first syndrome measurement unit 32g does not indicate the occurrence of an error in the post-selection region, the post-selection pass determination unit 32b notifies the transversal rotary gate execution unit 32h that the error detection process has been passed.If the syndrome measurement information obtained from the second syndrome measurement unit 32i does not indicate the occurrence of an error in the post-selection region, the post-selection pass determination unit 32b notifies the error correction unit 32j that the error detection process has been passed.

[0165] The syndrome measurement recording unit 32c acquires syndrome measurement information indicating the results of the syndrome measurement from the first syndrome measurement unit 32g or the second syndrome measurement unit 32i, and records the acquired syndrome measurement information in the memory 102 or the storage device 103.

[0166] The error location estimation unit 32d estimates the location of an error in the error correction region based on the syndrome measurement information and in accordance with the encoding method applied to encoding the logical quantum bit. When the error location estimation unit 32d identifies a physical quantum bit that is the location of an error, it transmits estimated error information indicating the position of the physical quantum bit and the error content to the error correction unit 32j.

[0167] The gate teleportation success / failure determination unit 32e determines whether the arbitrary rotation of the logical quantum bit by the gate teleportation unit 32k is a forward rotation “R Z (θ)|ψ> L " or reverse rotation "R Z (-θ)|ψ>L If the rotation is forward, the gate teleportation success / failure determination unit 32e outputs success / failure information indicating that the gate operation of the arbitrary rotation has been completed. If the rotation is reverse, the gate teleportation success / failure determination unit 32e instructs the physical rotation angle calculation unit 32a to double the physical rotation angle.

[0168] The functions realized by the quantum computer 200 in the arbitrary rotation execution unit 32 include a logical |+> state preparation unit 32f, a first syndrome measurement unit 32g, a transversal rotation gate execution unit 32h, a second syndrome measurement unit 32i, an error correction unit 32j, and a gate teleportation unit 32k.

[0169] The logical |+> state preparation unit 32f prepares the logical |+> state of the logical quantum bit in response to an instruction from the post-selection pass determination unit 32b. The first syndrome measurement unit 32g performs syndrome measurement on the logical quantum bit that has entered the logical |+> state. The first syndrome measurement unit 32g transmits syndrome measurement information indicating the results of the syndrome measurement to the syndrome measurement recording unit 32c.

[0170] When the first syndrome measurement unit 32g detects no error and determines that the data is passed, the transversal rotation gate execution unit 32h executes gate operation of the transversal rotation gate using a Multi-Z rotation gate. For example, the transversal rotation gate execution unit 32h decomposes the Multi-Z rotation gate into multiple CNOT gates and a phase rotation gate, and generates a quantum circuit (equivalent circuit) corresponding to the Multi-Z rotation gate. The transversal rotation gate execution unit 32h then executes gate operation using the equivalent circuit on the physical quantum bit gate that is the operation target of the transversal rotation gate.

[0171] The second syndrome measurement unit 32i performs syndrome measurement of the logical quantum bit after the transversal rotation gate is operated. The second syndrome measurement unit 32i transmits syndrome measurement information indicating the result of the syndrome measurement to the syndrome measurement recording unit 32c.

[0172] If an error is detected in the syndrome measurement by the second syndrome measurement unit 32i and the data passes through, the error correction unit 32j corrects the error location based on the estimated error information. For example, the error correction unit 32j performs an X-gate gate operation on the physical quantum bit in which an X-inversion error occurs. The error correction unit 32j also performs a Z-gate gate operation on the physical quantum bit in which a Z-inversion error occurs.

[0173] The gate teleportation unit 32k uses the resource state to perform an arbitrary rotation using the gate teleportation circuit on the quantum state of the logical quantum bit to be rotated. As a result, the logical quantum bit to be manipulated becomes the quantum state after the rotation. The gate teleportation unit 32k also notifies the gate teleportation success / failure determination unit 32e of information indicating the state of the logical quantum bit after the rotation.

[0174] Next, a processing procedure in the quantum computing system 30 when executing a quantum circuit configured by "Clifford+φ (arbitrary rotation)" will be described. 27 is a sequence diagram showing an example of a calculation processing procedure of a quantum circuit. In FIG. 27, the gate operations of the Clifford gate are omitted. First, the classical computer 100 and the quantum computer 200 work together to perform an initialization process of the logical quantum bits (step S10). Thereafter, the classical computer 100 and the quantum computer 200 work together to sequentially execute the operations of the quantum gates shown in the quantum circuit.

[0175] When the quantum gate to be executed is an arbitrary rotation gate, the classical computer 100 and the quantum computer 200 cooperate to perform an arbitrary rotation gate operation based on the STAR architecture (step S20). In the arbitrary rotation process, there is a 1 / 2 probability that the desired rotation will be reversed, so the rotation gate operation is repeated while updating the rotation angle until the desired rotation is achieved. The arbitrary rotation process is divided into a resource state preparation process (step S21) and a gate teleportation circuit execution process (step S26).

[0176] In the resource state preparation process (step S21), first, the classical computer 100 sends a preparation command for the logical |+> state to the quantum computer 200 (step S22). In accordance with the preparation command for the logical |+> state, the quantum computer 200 performs gate operations so that the logical qubit that performs arbitrary rotation is in the |+> state.

[0177] Next, the classical computer 100 transmits a transversal rotation gate execution command to the quantum computer 200 (step S23). The quantum computer 200 performs a gate operation of the transversal rotation gate on the logical quantum bit, and then performs a syndrome measurement of the logical quantum bit. The quantum computer 200 then transmits the syndrome measurement result to the classical computer 100 (step S24).

[0178] The classical computer 100 determines whether to pass the post-selection based on the syndrome measurement results. If the classical computer 100 detects an error, it sends an error correction command to the quantum computer 200 (step S25). In response to the error correction command, the quantum computer 200 performs a gate operation to correct the error that occurred.

[0179] The resource state preparation process is repeated until the post-selection is passed. When the resource state preparation process is completed, the classical computer 100 and the quantum computer 200 cooperate to execute the gate teleportation circuit (step S26).

[0180] The classical computer 100 and the quantum computer 200 execute an arbitrary rotation process (step S20) each time the timing for executing a quantum gate for arbitrary rotation in the quantum circuit arrives. Then, when all data operations in the quantum circuit are completed, the classical computer 100 and the quantum computer 200 work together to measure the quantum state (step S30). The classical computer 100 finds a solution to the problem to be solved from the measurement results of the quantum state and outputs the solution.

[0181] In this way, quantum computation is performed by cooperative processing between the classical computer 100 and the quantum computer 200. The processing performed by the classical computer 100 during quantum computation will be described below.

[0182] 28 is a flowchart showing an example of the procedure of quantum computing processing in a classical computer. The processing shown in FIG. 28 will be explained below in order of step number. [Step S101] When the quantum computing request receiving unit 110 receives a quantum computing request from the terminal 29, it decomposes the quantum gates (e.g., 3-qubit gates) in the quantum circuit to be computed and converts them into a quantum circuit that combines "Clifford + φ (arbitrary rotation)" quantum gates.

[0183] [Step S102] The quantum circuit execution control unit 120 performs initialization processing on the logical quantum bits. For example, the quantum circuit execution control unit 120 identifies the physical quantum bits to be used in executing the quantum circuit, and sends initialization instructions for those physical quantum bits to the quantum bit initialization unit 210 of the quantum computer 200. The quantum bit initialization unit 210 initializes the states of the physical quantum bits to predetermined states in accordance with the initialization instructions.

[0184] [Step S103] The quantum circuit execution control unit 120 performs quantum circuit execution processing upon receiving a response indicating the completion of initialization from the quantum bit initialization unit 210. The quantum circuit execution processing will be described in detail later (see FIG. 29).

[0185] [Step S104] When the quantum circuit execution control unit 120 completes the execution of the quantum circuit and acquires the measurement result of the final logical qubit state, it calculates a solution to the problem to be solved based on the measurement result. The quantum circuit execution control unit 120 then transmits the calculation result to the quantum computing request receiving unit 110. The quantum computing request receiving unit 110 transmits the calculation result to the terminal 29.

[0186] In this way, quantum computation using the quantum circuit is performed. Next, the quantum circuit execution process will be described in detail. 29 is a flowchart showing an example of the procedure of the quantum circuit execution process. The process shown in FIG. 29 will be explained below in order of step number.

[0187] [Step S201] The quantum circuit execution control unit 120 selects the next operation (gate operation or measurement) to be executed from the quantum circuit. [Step S202] The quantum circuit execution control unit 120 determines whether the selected operation is a gate operation of an arbitrary rotation quantum gate. If the next quantum gate is a gate operation or measurement of a Clifford gate, the quantum circuit execution control unit 120 proceeds to step S203. If the next quantum gate is an arbitrary rotation quantum gate, the quantum circuit execution control unit 120 proceeds to step S204.

[0188] [Step S203] The quantum circuit execution control unit 120 transmits an execution command for the next Clifford gate operation or measurement to the quantum computer 200. If the transmitted command is a Clifford gate operation command, the Clifford operation execution unit 31 in the quantum computer 200 executes the Clifford gate operation on the logical quantum bit. If the transmitted command is a measurement command, the quantum bit measurement unit 220 measures the state of the physical quantum bits that make up the logical quantum bit. The quantum bit measurement unit 220 transmits the measurement result to the quantum circuit execution control unit 120. The quantum circuit execution control unit 120 then proceeds to step S209.

[0189] [Step S204] The quantum circuit execution control unit 120 obtains the logical rotation angle from the arbitrary rotation execution unit 32. <Step S205> The arbitrary rotation execution unit 32 executes a resource state generation process, the details of which will be described later (see FIG. 30).

[0190] [Step S206] The arbitrary rotation execution unit 32 sends an execution command for the gate teleportation circuit to the quantum computer 200. In response, the gate teleportation unit 32k provided in the quantum computer 200 receives as input the logical quantum bit indicating the quantum state to be rotated and the logical quantum bit indicating the resource state, and executes gate operations in accordance with the gate teleportation circuit.

[0191] [Step S207] The gate teleportation success / failure determination unit 32e determines whether the arbitrary rotation using the gate teleportation circuit was successful. If forward rotation was performed, the gate teleportation success / failure determination unit 32e determines that it was successful, and proceeds to step S209. If reverse rotation was performed, the gate teleportation success / failure determination unit 32e determines that it was unsuccessful, and proceeds to step S208.

[0192] [Step S208] The gate teleportation success / failure determination unit 32e updates the logical rotation angle to twice the current value and transmits the updated logical rotation angle to the physical rotation angle calculation unit 32a. After that, the gate teleportation success / failure determination unit 32e proceeds to step S205.

[0193] [Step S209] The quantum circuit execution control unit 120 determines whether the final operation of the quantum circuit has been completed. If the final operation of the quantum circuit has been completed, the quantum circuit execution control unit 120 terminates the quantum circuit execution process. If there are unprocessed operations, the quantum circuit execution control unit 120 proceeds to step S201.

[0194] Next, the resource state generation process will be described in detail. 30 is a flowchart illustrating an example of a procedure for resource status generation processing. The processing shown in FIG. 30 will be described below in order of step number.

[0195] [Step S301] The physical rotation angle calculation unit 32a calculates a physical rotation angle based on the logical rotation angle. For example, the physical rotation angle calculation unit 32a calculates a physical rotation angle based on the logical rotation angle θ * The physical rotation angle θ is calculated so that

[0196] [Step S302] The post-selection pass determination unit 32b transmits a preparation command for the logical |+> state to the logical |+> state preparation unit 32f of the quantum computer 200. In response, the logical |+> state preparation unit 32f determines whether the state of the logical qubit that performs the arbitrary rotation is |+> L The gate operation is performed on at least some of the physical quantum bits that make up the logical quantum bit so that: Then, the first syndrome measurement unit 32g performs syndrome measurement on the logical quantum bit.

[0197] [Step S303] The post-selection pass determiner 32b acquires the syndrome measurement result from the first syndrome measurer 32g. [Step S304] The post-selection pass determination unit 32b determines whether or not there is an error in the post-selection region based on the syndrome measurement results. If there is an error, the post-selection pass determination unit 32b proceeds to step S302 and redoes preparation for the logical |+> state. If there is no error, the post-selection pass determination unit 32b proceeds to step S305.

[0198] [Step S305] The post-selection pass determination unit 32b sends an instruction to execute a transversal rotary gate using Multi-Z rotation to the transversal rotary gate execution unit 32h of the quantum computer 200. The transversal rotary gate execution unit 32h then executes the gate operation of the transversal rotary gate using the Multi-Z rotary gate. The second syndrome measurement unit 32i then repeatedly executes syndrome measurement of the logical quantum bit d times (d is the code distance). The syndrome measurement recording unit 32c records syndrome measurement information indicating the syndrome measurement result each time the second syndrome measurement unit 32i executes syndrome measurement.

[0199] [Step S306] The syndrome measurement recorder 32c acquires syndrome measurement information indicating the first syndrome measurement result from the second syndrome measurer 32i. Then, the post-selection pass determiner 32b acquires the first syndrome measurement result from the syndrome measurement recorder 32c.

[0200] [Step S307] The post-selection pass determination unit 32b determines whether or not there is an error in the post-selection region based on the result of the first syndrome measurement. If there is an error in the post-selection region, the post-selection pass determination unit 32b proceeds to step S302 and restarts the process from preparation of the logical |+> state. If there is no error in the post-selection region, the post-selection pass determination unit 32b proceeds to step S308.

[0201] [Step S308] The syndrome measurement recorder 32c acquires syndrome measurement information indicating the second syndrome measurement result from the second syndrome measurer 32i. The post-selection pass determiner 32b then acquires the second syndrome measurement result from the syndrome measurement recorder 32c.

[0202] [Step S309] The post-selection pass determination unit 32b determines whether or not there is an error in the post-selection region based on the second syndrome measurement result. If there is an error in the post-selection region, the post-selection pass determination unit 32b proceeds to step S302 and restarts the process from preparation of the logical |+> state. If there is no error in the post-selection region, the post-selection pass determination unit 32b proceeds to step S310.

[0203] The post-selection pass determination unit 32b performs error determination twice within the post-selection area based on the syndrome measurement results in order to prevent errors from being missed due to gate operation of the transversal rotating gate caused by errors that occur when the syndrome measurement circuit is executed.

[0204] [Step S310] The syndrome measurement and recording unit 32c acquires "d-2" syndrome measurement results (from the third to the dth measurement) from the second syndrome measurement unit 32i. Then, the error location estimation unit 32d acquires d syndrome measurement results in the error correction area from the syndrome measurement and recording unit 32c.

[0205] [Step S311] The error location estimation unit 32d determines whether or not there is an error in the error correction area based on the acquired syndrome measurement results. If there is an error, the error location estimation unit 32d proceeds to step S312. If there is no error, the error location estimation unit 32d ends the resource status generation process.

[0206] [Step S312] The error location estimation unit 32d sends estimated error information indicating the error location and error content to the error correction unit 32j of the quantum computer 200, and terminates the resource state generation process.

[0207] In this way, the gate operation of the transversal rotation gate is performed by the Multi-Z rotation gate, and a resource state is generated. By generating a resource state in this way, the probability of successfully generating a resource state is improved, and efficient arbitrary rotation gate operation is realized. Below, using a specific example, the effect of improving the probability of successfully generating a resource state is explained in detail.

[0208] FIG. 31 shows an example of how to implement a transversal rotation gate using a Multi-Z rotation gate on hardware where two-qubit gates are limited to only those between nearest neighbor qubits. The logical qubit 71 used for the resource state is coded with a code distance of "d=4". A Multi-Z rotation gate (R ZZ Assume that the gate operation of a transversal revolving gate is performed by (θ).

[0209] The transversal rotary gate operates on the topmost data qubit among the physical qubits that make up logical qubit 71. The identifiers of the seven physical qubits in the topmost row of logical qubit 71 are "q0, q1, . . . , q6." "q0, q2, q4, q6" are data qubits. "q1, q3, q5" are measurement qubits.

[0210] The transversal rotation gate can be implemented by quantum circuit 72. Quantum circuit 72 first includes a control Z gate 72a for data qubit "q0" and measurement qubit "q1," and a control Z gate 72b for data qubit "q4" and measurement qubit "q5."

[0211] Next to control Z gate 72a is CNOT gate 72c, which uses measurement qubit "q1" as the control qubit and data qubit "q2" as the target qubit. Next to control Z gate 72b is CNOT gate 72d, which uses measurement qubit "q5" as the control qubit and data qubit "q6" as the target qubit.

[0212] Next to CNOT gate 72c is a phase rotation gate 72e for data qubit "q2." Next to CNOT gate 72d is a phase rotation gate 72f for data qubit "q6."

[0213] Next to phase rotation gate 72e is CNOT gate 72g, which uses measurement qubit "q1" as the control qubit and data qubit "q2" as the target qubit. Next to phase rotation gate 72f is CNOT gate 72h, which uses measurement qubit "q5" as the control qubit and data qubit "q6" as the target qubit.

[0214] Next to CNOT gate 72g is a control Z gate 72i for data qubit "q0" and measurement qubit "q1". Next to CNOT gate 72h is a control Z gate 72j for data qubit "q4" and measurement qubit "q5".

[0215] The control Z gate 72a, the CNOT gate 72c, the phase rotation gate 72e, the CNOT gate 72g, and the control Z gate 72i form a two-qubit rotation gate “R Z0Z2 (θ)" (the number following Z is a subscript of Z). The control Z gate 72b, the CNOT gate 72d, the phase rotation gate 72f, the CNOT gate 72h, and the control Z gate 72j realize a two-qubit rotation gate "R Z4Z6 (θ)" (the number following Z is a subscript of Z) is realized.

[0216] FIG. 31 shows an example of an implementation of a two-qubit rotation gate, but by adding a controlled Z gate and a CNOT gate, a rotation gate of three or more qubits can be similarly implemented.

[0217] Here, we will explain how the success probability changes depending on the number of Z-weights (m) of the rotating gate under error-free conditions. FIG. 32 is a diagram showing an example of the success probability of generating a resource state when there is no error. In graph 81, the horizontal axis represents the code distance, and the vertical axis represents the probability of successfully generating a resource state. The probability of successfully generating a resource state is a value calculated based on an analytical formula. For example, when a transversal rotation gate specified by parameters (m, k, d) is executed, the success probability p (0) sample (θ * , k) is given by equation (10).

[0218]

number

[0219] As shown in equation (10), the success probability decreases monotonically with k. Moreover, since "k = d / m" holds, the success probability increases when Multi-Z rotation is performed with a larger Z-Weight (m).

[0220] The broken line 81a of the graph 81 is "(m,θ * )=1,10 -3 ) and the success probability. * )=2,10 -3 ) and the success probability. * )=3,10 -3 ) and the success probability. * )=1,10 -4) and the success probability. * )=2,10 -4 ) and the success probability. * )=3,10 -4 ) shows the relationship between the code distance and the success probability.

[0221] In graph 81, the logical rotation angle "θ=10 -3 In the case of "m=1", the broken line 81b of "m=2" is always above the broken line 81a of "m=1" (high probability of success), and the broken line 81c of "m=3" is always above the broken line 81b of "m=2" (high probability of success). -4 In this case, the broken line 81e for "m=2" is always higher than the broken line 81d for "m=1" (higher probability of success), and the broken line 81f for "m=3" is always higher than the broken line 81e for "m=2" (higher probability of success).

[0222] Next, the results of comparing the numerical values ​​of the pass probability of post-selection when an error exists will be described. 33 is a diagram showing an example of the success probability in the pass determination of resource state generation depending on the resource state generation method. In graph 82, the horizontal axis represents the error rate of the physical quantum bit, and the vertical axis represents the success probability in the pass determination of resource state generation. Graph 82 shows the success probability according to the error rate when simulating under the following conditions:

[0223] The logical qubits are encoded using an unrotated surface code. The code distance d is set to two values: "d=12" and "d=18". Errors are introduced based on a circuit-level noise model. The probability of success in pass determination is calculated through simulation for a method that does not use a Multi-Z rotation gate (m=1) and a method that uses a Multi-Z rotation gate (m=2). For post-selection, both simple post-selection (rejecting the state regardless of the position of the error) and a hybrid method (rejecting the state in the event of an error in some areas and correcting errors in other areas) are verified.

[0224] The results of the simulation under these conditions are shown in graph 82 by lines 82a to 82h (the dotted line represents simple post-selection, and the solid line represents the hybrid method). Line 82a represents the success probability when the hybrid method is applied with a code distance of "d=12" and a rotation gate Z-weight number of "m=1". Line 82b represents the success probability when simple post-selection is applied with a code distance of "d=12" and a rotation gate Z-weight number of "m=1". Line 82c represents the success probability when the hybrid method is applied with a code distance of "d=18" and a rotation gate Z-weight number of "m=1". Line 82d represents the success probability when simple post-selection is applied with a code distance of "d=18" and a rotation gate Z-weight number of "m=1".

[0225] Line 82e shows the success probability when the hybrid method is applied with a code distance of "d=12" and a rotation gate Z-weight number of "m=2". Line 82f shows the success probability when simple post-selection is applied with a code distance of "d=12" and a rotation gate Z-weight number of "m=2". Line 82g shows the success probability when the hybrid method is applied with a code distance of "d=18" and a rotation gate Z-weight number of "m=2". Line 82h shows the success probability when simple post-selection is applied with a code distance of "d=18" and a rotation gate Z-weight number of "m=2".

[0226] As shown in Graph 82, all other conditions being equal, the hybrid method (solid line) has a higher success rate than the simple post-selection method (dotted line). Even with the hybrid method, the success rate is further improved by implementing the transversal rotation gate with a multi-Z rotation gate (m=2).

[0227] Next, we explain the error reduction effect of implementing a transversal rotation gate with a Multi-Z rotation gate. There are two types of contributions to errors in a quantum state: diagonal and off-diagonal components. The type of error contribution that is incorporated varies depending on the index used to evaluate the error. Error evaluation indexes include infidelity and trace distance.

[0228] The non-fidelity is expressed by equation (11).

[0229]

number

[0230] F is the fidelity. ψ ideal is the ideal state. error is a state containing an error. Infidelity is an index that evaluates only diagonal errors. Therefore, the accuracy of error evaluation is not high. The trace distance D tr (ρ error , ψ) is expressed by equation (12).

[0231]

number

[0232] The trace distance is an index that evaluates the effect of off-diagonal errors. Therefore, we will explain the evaluation results of the error reduction effect using a more accurate trace distance. Note that the smaller the trace distance value, the less the influence of noise and the higher the accuracy of the resource status.

[0233] 34 is a diagram showing an example of a comparison result of the tracing distance. In graph 83, the horizontal axis represents the error rate of the physical quantum bit, and the vertical axis represents the tracing distance. Graph 83 shows the tracing distance as a function of the error rate when a simulation is performed under the following conditions:

[0234] The logical qubits are encoded with an unrotated surface code. The code distance d is set to "d=6". Errors based on a circuit-level noise model are introduced. The results of a numerical simulation of the error rates of the resource states output by a method that does not use a Multi-Z rotation gate (m=1) and a method that uses a Multi-Z rotation gate (m=2) are shown in graph 83 by lines 83a and 83b.

[0235] The broken line 83a shows the tracing distance when "m=2". The broken line 83b shows the tracing distance when "m=1". As can be seen from the graph 83, the tracing distance decreases when the Multi-Z rotation gate is used. In other words, an improvement in the error rate has been achieved.

[0236] Although the embodiments have been described above, the configuration of each part shown in the embodiments can be replaced with other parts having similar functions. Also, any other components or processes may be added. Furthermore, any two or more configurations (features) of the above-described embodiments may be combined. [Explanation of symbols]

[0237] 1. Quantum computers 2 logical qubits 3 Physical qubits 4. The First Physical Qubit 4a, 4b Physical qubit groups 5 Quantum circuit 6 Post-selection region 7 Error Correction Area 10. Information processing equipment 11 Storage section 12 Processing section

Claims

1. A quantum computing control program that causes a computer to execute a process of causing a quantum computer to execute a rotation operation on a target quantum bit using a resource state generated using a rotation gate and a gate teleportation circuit, comprising: The process of generating the resource state includes: determining, based on a logical rotation angle for rotating a state of a logical quantum bit encoded by a code distance d (d is an integer equal to or greater than 2) around a predetermined axis, a physical rotation angle to be performed on d first physical quantum bits among a plurality of physical quantum bits constituting the logical quantum bit around the predetermined axis; For a physical quantum bit group consisting of m (m is an integer of 2 or more and d or less) second physical quantum bits out of the d first physical quantum bits, specify the application of an m-qubit rotation gate that rotates the states of the m second physical quantum bits with a single rotation gate operation, and instruct the quantum computer having the plurality of physical quantum bits to perform a rotation gate operation that rotates the states of each of the d first physical quantum bits around the predetermined axis by the physical rotation angle. Quantum computing control program including processing.

2. In the process of instructing the execution of the rotation gate operation, a quantum circuit that realizes the gate operation of the m-qubit rotation gate using a plurality of CNOT gates and one 1-qubit rotation gate is generated, and the quantum computer is instructed to execute the generated quantum circuit. The quantum computing control program according to claim 1.

3. In the process of instructing the execution of the rotation gate operation, the quantum computer is caused to execute the rotation gate operation on the d first physical quantum bits of the logical quantum bit in a logical |+> state; moreover, obtaining an error detection result for the plurality of physical quantum bits from the quantum computer; when an error occurring within a first region including at least the d first physical quantum bits among regions in which the plurality of physical quantum bits exist is detected, instructing the quantum computer to perform a gate operation that returns the logical quantum bit to the logical |+> state and to redo the rotation gate operation; When an error that has occurred in a second region other than the first region among the regions in which the plurality of physical quantum bits exist is detected, instructing the quantum computer to correct the detected error.

2. The quantum computing control program according to claim 1, which causes the computer to execute a process.

4. if no error is detected for the plurality of physical quantum bits, causing the quantum computer to execute the gate teleportation circuit using the state of the logical quantum bit after the rotation gate operation; When an error occurring in the second region is detected, the quantum computer is caused to execute the gate teleportation circuit using the state of the logical quantum bit after correcting the error. The quantum computing control program according to claim 3.

5. the first region is a region including, among a plurality of measurement qubits used in syndrome measurement of the plurality of physical qubits, a first measurement qubit capable of detecting an error in the d first physical qubits and a second measurement qubit capable of detecting an error occurring in a gate operation on the first measurement qubit; the second region is a region including a third measurement qubit remaining from the plurality of measurement qubits excluding the first measurement qubit and the second measurement qubit; The quantum computing control program according to claim 3.

6. The d first physical quantum bits are d physical quantum bits that constitute a logical Z operator. The quantum computing control program according to claim 1.

7. A quantum computing control program that causes a computer to execute a process of causing a quantum computer to execute a rotation operation on a target quantum bit using a resource state generated using a rotation gate and a gate teleportation circuit, comprising: The process of generating the resource state includes: determining, based on a logical rotation angle for rotating the state of the encoded logical quantum bit around a predetermined axis, a physical rotation angle to be performed on at least a first physical quantum bit among a plurality of physical quantum bits constituting the logical quantum bit around the predetermined axis; instructing the quantum computer having the plurality of physical qubits to perform a rotation gate operation to rotate the state of each of the first physical qubits of the logical qubit by the physical rotation angle about the predetermined axis; obtaining an error detection result for the plurality of physical quantum bits from the quantum computer; When an error is detected in a first region including at least the first physical quantum bit among the regions in which the plurality of physical quantum bits exist, instructing the quantum computer to re-execute the rotation gate operation; When an error that has occurred in a second region other than the first region among the regions in which the plurality of physical quantum bits exist is detected, instructing the quantum computer to correct the detected error. Quantum computing control program including processing.

8. In the rotation gate operation, the state of each of the first physical qubits of the logical qubits in a logical |+> state is rotated; The generating the resource state further includes returning the state of the logical qubit to the logical |+> state before the re-execution. The quantum computing control program according to claim 7.

9. if no error is detected for the plurality of physical quantum bits, causing the quantum computer to execute the gate teleportation circuit using the state of the logical quantum bit after the rotation gate operation; When an error occurring in the second region is detected, the quantum computer is caused to execute the gate teleportation circuit using the state of the logical quantum bit after correcting the error. The quantum computing control program according to claim 7.

10. the first region is a region including, among a plurality of measurement qubits used in syndrome measurement of the plurality of physical qubits, a first measurement qubit capable of detecting an error in the first physical qubit and a second measurement qubit capable of detecting an error occurring in a gate operation on the first measurement qubit; the second region is a region including a third measurement qubit remaining from the plurality of measurement qubits excluding the first measurement qubit and the second measurement qubit; The quantum computing control program according to claim 7.

11. The logical quantum bit is encoded with a code distance d (d is an integer greater than or equal to 2), The plurality of first physical quantum bits are d physical quantum bits that constitute a logical Z operator. The quantum computing control program according to claim 7.

12. A quantum computing control method in which a computer executes a process of causing a quantum computer to perform a rotation operation on a target quantum bit using a resource state generated using a rotation gate and a gate teleportation circuit, comprising: The process of generating the resource state includes: determining, based on a logical rotation angle for rotating a state of a logical quantum bit encoded by a code distance d (d is an integer equal to or greater than 2) around a predetermined axis, a physical rotation angle to be performed on d first physical quantum bits among a plurality of physical quantum bits constituting the logical quantum bit around the predetermined axis; For a physical quantum bit group consisting of m (m is an integer of 2 or more and d or less) second physical quantum bits out of the d first physical quantum bits, specify the application of an m-qubit rotation gate that rotates the states of the m second physical quantum bits with a single rotation gate operation, and instruct the quantum computer having the plurality of physical quantum bits to perform a rotation gate operation that rotates the states of each of the d first physical quantum bits around the predetermined axis by the physical rotation angle. A quantum computing control method including processing.

13. A quantum computing control method in which a computer executes a process of causing a quantum computer to perform a rotation operation on a target quantum bit using a resource state generated using a rotation gate and a gate teleportation circuit, comprising: The process of generating the resource state includes: determining, based on a logical rotation angle for rotating the state of the encoded logical quantum bit around a predetermined axis, a physical rotation angle to be performed on at least a first physical quantum bit among a plurality of physical quantum bits constituting the logical quantum bit around the predetermined axis; instructing the quantum computer having the plurality of physical qubits to perform a rotation gate operation to rotate the state of each of the first physical qubits of the logical qubit by the physical rotation angle about the predetermined axis; obtaining an error detection result for the plurality of physical quantum bits from the quantum computer; When an error is detected in a first region including at least the first physical quantum bit among the regions in which the plurality of physical quantum bits exist, instructing the quantum computer to re-execute the rotation gate operation; When an error that has occurred in a second region other than the first region among the regions in which the plurality of physical quantum bits exist is detected, instructing the quantum computer to correct the detected error. A quantum computing control method in which the computer executes processing.

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