Operation generation device, operation generation method and program

Abstract

To perform quantum phase estimation operations more efficiently than conventional techniques.SOLUTION: An operation generation device for generating configuration information for operations used for performing quantum phase estimation comprises: an input unit for accepting input of a Hermitian matrix and creating a linear combination of first Pauli matrices from the Hermitian matrix; a decomposition unit for, based on a set of the first Pauli matrices obtained by the input unit, generating a set of second Pauli matrices configured to cover the set of the first Pauli matrices, and generating configuration information for a select operation from configuration information for the set of second Pauli matrices; and an output unit for outputting the configuration information for the select operation.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to the technical field of quantum phase estimation. [Background technology]

[0002] Quantum computers are a technology that performs calculations by utilizing the principle of superposition in quantum mechanics, and are expected to be able to quickly solve problems such as prime factorization and quantum chemical calculations, so their development is being actively pursued around the world. The most famous quantum algorithm that can only be realized using quantum computers is quantum phase estimation. [Prior art documents] [Non-patent literature]

[0003] [Non-Patent Document 1] Low, GH, & Chuang, IL (2019), "Hamiltonian simulation by qubitization.", Quantum, 3, 163. Summary of the Invention [Problem to be solved by the invention]

[0004] As a method for constructing an efficient operation for quantum phase estimation, a technique called Qubitization is known, which is disclosed in Non-Patent Document 1. However, there is a demand for making the operation of quantum phase estimation more efficient than the conventional technique.

[0005] The present invention has been made in view of the above-mentioned points, and has an object to provide a technique that enables quantum phase estimation operations to be performed more efficiently than conventional techniques. [Means for solving the problem]

[0006] According to the disclosed technology, there is provided an operation generation device that generates configuration information of an operation used for performing quantum phase estimation, comprising: an input unit for inputting a Hermitian matrix and creating a linear combination of a first Pauli matrix from the Hermitian matrix; a decomposition unit that generates a set of second Pauli matrices configured to cover the set of first Pauli matrices based on the set of first Pauli matrices obtained by the input unit, and generates configuration information of a select operation from configuration information of the set of second Pauli matrices; an output unit that outputs configuration information of the selection operation; An operation generating device is provided, comprising: Effect of the Invention

[0007] The disclosed technology provides a technology that allows the operation of quantum phase estimation to be performed more efficiently than conventional techniques. [Brief description of the drawings]

[0008] [Figure 1] FIG. 2 is a configuration diagram of an operation generating device. [Diagram 2] 13 is a flowchart illustrating an operation of the operation generation device. [Diagram 3] FIG. 1 is a configuration diagram of a quantum computing device. [Figure 4] FIG. 2 illustrates an example of a hardware configuration of an operation generation device. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0009] Hereinafter, an embodiment of the present invention (the present embodiment) will be described with reference to the drawings. The embodiment described below is merely an example, and the embodiment to which the present invention is applied is not limited to the following embodiment.

[0010] In the following description, references are indicated by numbers such as [1], and the names of the references corresponding to the numbers are listed at the end of the specification. Note that the aforementioned Non-Patent Document 1 corresponds to Reference [1].

[0011] In the following, first, the conventional technology and its problems will be described in detail, and then the technology according to the present invention will be described. Note that the contents described in the documents referenced in the following description of the conventional technology and the quantum phase estimation itself are publicly known technology, but the following description of the conventional technology includes the considerations and analysis that led to the present invention, so the contents of the following description of the conventional technology are not publicly known technology. Furthermore, the description of the problems is not publicly known technology either.

[0012] (Regarding the Prior Art) Quantum computers are a technology that performs calculations by utilizing the principle of superposition in quantum mechanics. They are expected to be able to quickly solve problems such as prime factorization and quantum chemical calculations, and their development is being actively pursued around the world.

[0013] A quantum computer is a computer that is based on quantum bits. In addition to the 0 and 1 that normal bits can represent, quantum bits can also represent superpositions of these two states. A system consisting of n quantum bits can have a 2 n The bit sequence of the pattern is superimposed, and the state is 2 n dimensional complex vector

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[0015] Operations on quantum bits can be roughly divided into two categories: measurements and unitary operations. Measurement is a procedure for extracting classical information from a quantum state under a certain reaction. Unitary operations are the following procedures.

[0016] The operation of transposing a matrix and taking the complex conjugate is called a Hermitian adjoint. For a complex square matrix U, the Hermitian adjoint U † A matrix such that is its own inverse is called a unitary matrix. Unitary operations are n ×2 n A quantum state transition is characterized by a unitary matrix U in which n is the quantum state transition vector |ψ〉 to the quantum state corresponding to the state vector U|ψ〉. A typical quantum computer can realize unitary operations that act non-trivially on only a constant number of n in a constant time.

[0017] One of the most famous quantum algorithms that can be realized for the first time using a quantum computer is quantum phase estimation. Quantum phase estimation is the following algorithm. In the following, we will consider 2 for a positive integer n. n ×2 n Consider a complex matrix H of the Hermitian adjoint H † A matrix that is equal to itself is called a Hermitian matrix. n ×2 n The Hermitian matrix of is 2 n The normalized eigenvectors of |φ i 〉 and the corresponding real eigenvalues ​​λ i That is, for i∈[0,2 n -1], H|φ i 〉=λ i |φ i 〉. At this time, 2 n A set of eigenvectors of types {|φ i 〉} is 2 n It forms an orthonormal basis for the dimensional complex vector space.

[0018] Now, 2 n Consider a normalized complex vector |ψ〉 of dimension {|φ i 〉} is a basis, so |ψ〉=Σ i α i |φ i 〉 and the complex number α i Then, from the normalization condition, Σ i |αi | 2 = 1. Quantum phase estimation is a method to estimate the Hermitian matrix H that satisfies several conditions described below, i | 2 With probability λ i is a procedure for obtaining m-digit accuracy in binary representation. Quantum phase estimation can be realized by combining basic operations on quantum bits that depend polynomially on the value n that characterizes the dimension. Analysis of eigenvalues ​​using such a procedure appears in a wide range of scientific fields, so it has a wide range of applications.

[0019] To perform the same procedure naively on a normal computer, for example, 2 n We need to diagonalize a matrix of size 2 3n Basic operations proportional to n are required. Therefore, the calculation time increases exponentially with n, and when n exceeds a certain value, a normal computer cannot complete the calculation in a realistic time. On the other hand, the calculation cost of a quantum computer increases polynomially with n, so for a sufficiently large n, a quantum computer can achieve significantly faster speeds than a normal computer.

[0020] Below, we explain the necessary conditions for quantum phase estimation to be possible. The first condition is that a quantum state with |ψ〉 as the state vector can be prepared. The second condition is that the unitary operation control-e defined by the Hermitian matrix H can be used. iH This can be realized by basic operations that grow polynomially with respect to n. Here, control-e iH is a unitary matrix defined by the following formula:

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[0022] Although quantum phase estimation has a wide range of applications if it can be realized, it is not self-evident that the above two conditions can be satisfied in practical problems. Therefore, attempts have been made for a long time to expand the range in which the above two conditions can be satisfied. One method that can efficiently satisfy the second condition, in particular, is called Qubitization [1]. Qubitization is as follows.

[0023] A 2 × 2 complex matrix defined in the following form is called a Pauli matrix.

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[0027] Using the Qubitization framework, the procedure for satisfying the second condition can be reduced to two operations called PREPARE and SELECT. Here, the PREPARE operation U P is |0〉 = (1, 0, 0) T This is a unitary operation that, when applied to an L-dimensional complex vector, results in the following:

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[0030] In some fields such as condensed matter physics, PREPARE is easy to create, so the challenge is how to create an operation called SELECT efficiently. In the following, we will explain the SELECT operation.

[0031] A naive implementation of the SELECT operation requires a number of unitary operations proportional to L·logL, but with some ingenuity, a method is known to realize it with operations proportional to L [2]. Therefore, in applications in the field of condensed matter physics, SELECT, an element of Qubitization, can be realized with a cost proportional to L. If L grows polynomially with respect to n, the second condition necessary for implementing quantum phase estimation can be satisfied.

[0032] (About the assignment) While the construction of the quantum phase estimation based on the above-mentioned Qubitization is considered to be efficient, a method for realizing the SELECT operation more efficiently has been desired in order to handle quantum computers at higher speeds. The technology according to the present embodiment constructs a SELECT operation faster than previously thought possible for Hermitian matrices that satisfy certain conditions. The technology according to the present embodiment makes it possible to realize quantum phase estimation faster than previously thought possible, and therefore quantum computers can be used in a wider range of applications.

[0033] Hereinafter, the principle of the technology according to the present embodiment will be described first, and then a specific device configuration and device operation will be described as an example.

[0034] (Technical principle according to the present embodiment) The set of L Pauli matrices of interest is {P i} i In the technology according to the present embodiment, when the set of Pauli matrices can be rewritten in a specific format described below, a SELECT operation can be realized with a cost smaller than L.

[0035] A set of m indices X=(x0,...x m-1 ) These subscripts are x i ∈[0..y i -1], 0 or more y i Furthermore, let us take the integers less than . k ⊆X is (S0...S K-1) are created. Here, |S k | to S k It is defined as the product of the range of indices belonging to S. k of

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[0038] Here, the set {P i} i Consider a set that covers , satisfying the following conditions:

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[0046] Therefore, the technology according to the present embodiment is useful only in cases where the cost of PREPARE is not a problem under such modifications. However, in the Hamiltonian in condensed matter physics, the coefficients are homogeneous, so even if such modifications are made, the SELECT operation typically has a higher calculation time cost, so this does not pose a problem. In addition, particularly when the redundantly added term is a unit matrix I, the eigenvalues ​​are merely shifted, so it is also possible to deal with this by subtracting an appropriate value from the eigenvalues ​​obtained as a result of the calculation.

[0047] Any {P i} i There exists at least one configuration that satisfies the above constraints. The trivial configuration is given by: m=K=1, y0=L, S0=X={x0},

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[0049] Another trivial construction is m=K=n, and y i =4, S k ={x k},

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[0052] In the general case, say for a positive integer c, L=c m , K=m, and y i = c, and S i ={x i}, if the above holds true, the order of the required cost is L=c mto mc, which corresponds to the cost being reduced to approximately the m-th root. At least cases where this condition holds can be artificially constructed, and the Schwinger model, which corresponds to m=2, is the subject of analysis in quantum physics.

[0053] The most efficient implementation of a SELECT operation is to use the given set of Pauli matrices {P i} i In contrast,

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[0056] (Example) Next, an embodiment of the above-described technology will be described. FIG. 1 is a configuration diagram of an operation generation device 100 in the present embodiment. As shown in FIG. 1, the operation generation device 100 includes an input unit 110, a decomposition unit 120, a correction unit 130, and an output unit 140. FIG. 1 shows the information output from each unit. The operations of each unit of the operation generation device 100 will be described according to the procedure shown in the flowchart of FIG. 2.

[0057] <S101: Input> In S101, the Hamiltonian H (Hermitian matrix) is input to the input unit 110. The input unit 110 represents the input Hamiltonian H as a sum (linear combination) of Pauli matrices, i.e., H = Σ i a i P i in the form of. The input unit 110 sends the list of coefficients {a i} to the correction unit 130 and the set of Pauli matrices {P i} to the decomposition unit 120.

[0058] <S102: Decomposition Unit> In S102, the decomposition unit 120 decomposes the set of Pauli matrices {P i} i to construct a configuration that satisfies i} i from

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[0062] <S103: Correction> In S103, when the correction unit 130 determines that the covering performed by the decomposition unit 120 is redundant based on the configuration information received from the decomposition unit 120, the correction unit 130 corrects the coefficient sequence {b i} of the PREPARE operation in such a way as to construct it from {a i}. After that, the correction unit 130 constructs the PREPARE operation. The construction of the PREPARE operation itself is an existing technology. For example, the PREPARE operation is constructed using the same method as the method disclosed in reference [2]. The correction unit 130 sends the configuration information of the constructed PREPARE operation to the output unit 140.

[0063] <S104: Output> In S104, the output unit 140 outputs the configuration information of the SELECT operation received from the decomposition unit 120 and the configuration information of the PREPARE operation received from the correction unit 130 as a pair. From these information, the quantum phase estimation algorithm can be implemented according to the information in references [1, 2].

[0064] (Configuration example of a quantum computing device) A configuration example of a quantum computing device 400 that actually implements the quantum phase estimation algorithm using quantum bits using the operation information obtained by the operation generation device 100 is shown in FIG. 3. Note that the "quantum computing device" may also be referred to as a "quantum computer", a "quantum computing system", a "quantum computer", etc.

[0065] 3, the quantum computing device 400 includes a control device 200 and a quantum processor 300. The control device 200 transmits control signals and the like to the quantum processor 200 in accordance with a quantum phase estimation algorithm, and calculates quantum phase estimation results (i.e., eigenvalues) by acquiring measurement results from the quantum processor 300. The control device 200 can be realized by, for example, a classical computer.

[0066] The operation generation device 100 and the control device 200 may be separate devices, or the function of the operation generation device 100 may be included in the control device 200.

[0067] The quantum processor 300 has a physical system for realizing a quantum system (qubit). The physical system for realizing the quantum system is not limited to a specific one, but for example, microwave photons, superconducting circuits, ion traps, quantum dots, etc. can be used.

[0068] Furthermore, the quantum bits in the quantum processor 300 are not limited to those using an actual physical system. For example, the quantum bits may be on a simulator realized by software. In this case, the quantum processor 300 functions as a simulator of the quantum bits. This simulator may be provided inside the control device 200 or on the cloud.

[0069] (Hardware configuration example) The operation generation device 100 described in this embodiment can be realized by, for example, causing a computer (classical computer) to execute a program. This computer may be a physical computer or a virtual machine on the cloud.

[0070] That is, the operation generation device 100 can be realized by executing a program corresponding to the processing performed by the operation generation device 100 using hardware resources such as a CPU and memory built into a computer. The program can be recorded on a computer-readable recording medium (such as a portable memory) and stored or distributed. The program can also be provided via a network such as the Internet or email.

[0071] Fig. 4 is a diagram showing an example of the hardware configuration of the computer. The computer in Fig. 4 has a drive device 1000, an auxiliary storage device 1002, a memory device 1003, a CPU 1004, an interface device 1005, a display device 1006, an input device 1007, an output device 1008, etc., which are mutually connected by a bus BS. The computer may further include a GPU.

[0072] The program for realizing the processing on the computer is provided by a recording medium 1001 such as a CD-ROM or a memory card. When the recording medium 1001 storing the program is set in the drive device 1000, the program is installed from the recording medium 1001 to the auxiliary storage device 1002 via the drive device 1000. However, the program does not necessarily have to be installed from the recording medium 1001, but may be downloaded from another computer via a network. The auxiliary storage device 1002 stores the installed program as well as necessary files, data, etc.

[0073] The memory device 1003 reads out and stores the program from the auxiliary storage device 1002 when an instruction to start the program is received. The CPU 1004 realizes functions related to the operation generation device 100 in accordance with the program stored in the memory device 1003. The interface device 1005 is used as an interface for connecting to a network or the like. The display device 1006 displays a GUI (Graphical User Interface) or the like according to a program. The input device 1007 is composed of a keyboard and mouse, buttons, a touch panel, or the like, and is used to input various operation instructions. The output device 1008 outputs the results of calculations.

[0074] (Summary of the embodiment, effects) As described above, in this embodiment, the procedure required for the SELECT operation is decomposed to achieve further efficiency. That is, the technique according to this embodiment makes it possible to implement a SELECT circuit faster than the technique previously considered. This makes it possible to implement quantum phase estimation faster than the technique previously considered, and quantum computers can be used for a wider variety of applications.

[0075] The following supplementary notes are further disclosed regarding the above embodiment.

[0076] <Additional Notes> (Additional note 1) An operation generation device for generating configuration information of an operation used for performing quantum phase estimation, comprising: an input unit for inputting a Hermitian matrix and creating a linear combination of a first Pauli matrix from the Hermitian matrix; a decomposition unit that generates a set of second Pauli matrices configured to cover the set of first Pauli matrices based on the set of first Pauli matrices obtained by the input unit, and generates configuration information of a select operation from configuration information of the set of second Pauli matrices; an output unit that outputs configuration information of the selection operation; An operation generating device comprising: (Additional note 2) When the covering in the decomposition unit is redundant, configuration information for a prepare operation is generated by using a coefficient sequence configured based on a coefficient sequence of a linear combination of the Hermitian matrices so as to correct the redundancy. Item 1. An operation generating device according to item 1. (Additional note 3) The output unit outputs configuration information for the select operation and configuration information for the prepare operation. Item 3. An operation generating device according to item 2. (Additional note 4) The configuration information of the set of second Pauli matrices includes the number of subscripts, a list of the range of subscripts, the number of sets of subscripts, a list of the sets of subscripts, and the set of second Pauli matrices, and the second Pauli matrix is ​​a Pauli matrix determined by the set of subscripts. 4. An operation generating device according to any one of claims 1 to 3. (Additional note 5) An operation generation method executed by an operation generation device for generating configuration information of an operation used for quantum phase estimation, comprising: an input step of inputting a Hermitian matrix and generating a linear combination of a first Pauli matrix from the Hermitian matrix; a decomposition step of generating a set of second Pauli matrices configured to cover the set of first Pauli matrices based on the set of first Pauli matrices obtained by the input step, and generating configuration information of a select operation from configuration information of the set of second Pauli matrices; an output step of outputting configuration information of the selection operation; An operation generation method comprising: (Additional note 6) A non-transitory storage medium storing a program for causing a computer to function as each unit in the operation generation device described in any one of appended claims 1 to 4.

[0077] Although the present embodiment has been described above, the present invention is not limited to such a specific embodiment, and various modifications and variations are possible within the scope of the gist of the present invention described in the claims.

[0078] [References] [1] Low, GH, & Chuang, IL (2019). Hamiltonian simulation by qubitization. Quantum, 3, 163. (url: https: / / arxiv.org / abs / 1610.06546v3) [2] Babbush, R., Gidney, C., Berry, DW, Wiebe, N., McClean, J., Paler, A., ... & Neven, H. (2018). Encoding electronic spectra in quantum circuits with linear T complexity. Physical Review X, 8(4), 041015. (url: https: / / arxiv.org / abs / 1805.03662) [Explanation of symbols]

[0079] 100 Operation generator 110 Input section 120 Disassembly section 130 Correction section 140 Output section 200 Control device 300 Quantum Processors 400 Quantum computing device 1000 Drive Device 1001 Recording media 1002 Auxiliary storage device 1003 Memory device 1004 CPU 1005 Interface device 1006 Display device 1007 Input Device 1008 Output device

Claims

1. An operation generation device for generating configuration information of an operation used for performing quantum phase estimation, comprising: an input unit for inputting a Hermitian matrix and generating a linear combination of a first Pauli matrix from the Hermitian matrix; a decomposition unit that generates a set of second Pauli matrices configured to cover the set of first Pauli matrices based on the set of first Pauli matrices obtained by the input unit, and generates configuration information of a select operation from configuration information of the set of second Pauli matrices; an output unit that outputs configuration information of the selection operation; An operation generating device comprising:

2. When the covering in the decomposition unit is redundant, configuration information for a prepare operation is generated by using a coefficient sequence configured based on a coefficient sequence of a linear combination of the Hermitian matrices so as to correct the redundancy. The operation generating device according to claim 1 .

3. The output unit outputs configuration information for the select operation and configuration information for the prepare operation. The operation generating device according to claim 2 .

4. The configuration information of the set of second Pauli matrices includes the number of subscripts, a list of subscript ranges, the number of subscript sets, a list of subscript sets, and the set of second Pauli matrices, and the second Pauli matrix is ​​a Pauli matrix determined by the set of subscripts. The operation generating device according to claim 1 .

5. An operation generation method executed by an operation generation device for generating configuration information of an operation used for quantum phase estimation, comprising: an input step of inputting a Hermitian matrix and generating a linear combination of a first Pauli matrix from the Hermitian matrix; a decomposition step of generating a set of second Pauli matrices configured to cover the set of first Pauli matrices based on the set of first Pauli matrices obtained by the input step, and generating configuration information of a select operation from configuration information of the set of second Pauli matrices; an output step of outputting configuration information of the selection operation; An operation generation method comprising:

6. A program for causing a computer to function as each unit in the operation generating device according to any one of claims 1 to 4.

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