Quantum state conversion device and quantum state conversion method

Abstract

A quantum state conversion device according to one aspect of the present disclosure is for converting independent identical distribution states of N initial states into independent identical distribution states of M target states, and comprises: a first determination unit that obtains the upper limit of a conversion rate representing the ratio of the number M of the target states to the number N of the initial states, and determines a value less than the upper limit as a conversion rate r; a second determination unit that determines the value of the number N of the initial states by using the conversion rate r; and a conversion unit that converts the independent identical distribution states of the N initial states into independent identical distribution states of M=r×N target states by using a unitary operation, which represents a symmetric operation corresponding to an element of a finite commutative group, and a covariate operation, which is a commutative operation.

Description

Quantum state conversion device and quantum state conversion method

[0001] This disclosure relates to a quantum state conversion device and a quantum state conversion method.

[0002] In recent years, there has been vigorous research into theoretical frameworks that determine what state transformations are possible using only operations called covariant operations (Non-Patent Documents 1-4). For example, Non-Patent Document 3 clarifies the necessary and sufficient conditions for a transformation from one pure state to another (i.e., a transformation between single states) to be possible using only covariant operations.

[0003] On the other hand, even if conversion between single states is not possible, conversion between states called independent and identically distributed states may be possible. In conversions between independent and identically distributed states, a value called the conversion rate is important, and the conversion rate in the simple case has been clarified (Non-Patent Literature 5).

[0004] Gilad Gour and Robert W. Spekkens, "The resource theory of quantum reference frames: Manipulations and monotones," New Journal of Physics 10, 033023 (2008).Iman Marvian, "Symmetry, Asymmetry and Quantum Information," PhD thesis, University of Waterloo, 2012.Iman Marvian and Robert W. Spekkens, "The theory of manipulations of pure state asymmetry: I. Basic tools, equivalence classes and single copy transformations," New Journal of Physics 15, 033001 (2013).Koji Yamaguchi and Hiroyasu Tajima, "Beyond iid in the Resource Theory of Asymmetry: An Information-Spectrum Approach for Quantum Fisher Information," Physical Review Letters 131, 200203 (2023).Iman Marvian, "Coherence distillation machines are impossible in quantum thermodynamics," Nature Communications, November 25, 2020.

[0005] However, since the conversion rate for more general cases is not clear, there are cases where the possibility of conversion between independent and identically distributed states cannot be determined, or where the conversion method is unknown.

[0006] This disclosure has been made in view of the above points and aims to realize transformations between independent and identically distributed states when the symmetry is described by a finite commutative group.

[0007] A quantum state transformation device according to one aspect of the present disclosure is a quantum state transformation device that transforms N independent identically distributed initial states into M independent identically distributed target states, comprising: a first determination unit that determines a value less than the upper limit of the transformation rate r by finding an upper limit of the transformation rate which represents the ratio of the number of target states M to the number of initial states N; a second determination unit that determines the value of the number of initial states N using the transformation rate r; and a transformation unit that transforms the N independent identically distributed initial states into M = r × N independent identically distributed target states using a covariant operation which is an operation that commutes with a unitary operation which represents a symmetric operation corresponding to an element of a finite commutative group.

[0008] This allows for transformations between independent and identically distributed states when the symmetry is described by a finite commutative group.

[0009] This figure shows an example of the configuration of the quantum state conversion device according to this embodiment. This figure shows an example of the hardware configuration of the control device according to this embodiment. This figure shows an example of the functional configuration of the control device according to this embodiment. This flowchart shows an example of the quantum state conversion process according to this embodiment.

[0010] Hereinafter, one embodiment of the present invention will be described in detail with reference to the drawings. In the following embodiment, a method for transforming independent and identically distributed states when the symmetry is described by a finite commutative group (hereinafter also referred to as the "proposed method") is proposed, and a quantum state transformation device 10 that transforms independent and identically distributed states using this proposed method will be described.

[0011] <Background of the Proposed Method> It is known that by effectively utilizing the principles of quantum mechanics, information processing that was previously difficult to achieve can be efficiently carried out. In particular, with the recent advancements in quantum information processing technology, development is progressing toward the realization of fault-tolerant quantum computers. By using a fault-tolerant quantum computer, in principle, any operation that is physically permissible becomes possible.

[0012] However, even if fault-tolerant quantum computers are realized in the future, depending on their configuration, symmetries may be imposed on the quantum system (hereinafter simply referred to as "the system"), restricting the operations that can be performed. In such cases, only operations that preserve symmetry among physically feasible operations are permitted. Operations that preserve symmetry are called "covariant operations."

[0013] In recent years, research into asymmetry resource theory has been actively pursued as a theoretical framework for determining what state transformations are possible using only covariant operations (Non-Patent Documents 1-4). For example, Non-Patent Document 3 clarifies the necessary and sufficient conditions for a transformation from one pure state to another (i.e., a transformation between single states) to be possible using only covariant operations.

[0014] On the other hand, even if conversion between single states is not possible, state conversion may be possible by providing copies of multiple states (these are called "independent and identically distributed states"). In such conversions between independent and identically distributed states, the conversion rate is important. The conversion rate is defined as M / N when N initial states can be converted to M target states.

[0015] For simple cases (for example, when the symmetry is U(1) symmetry (Non-Patent Literature 1, Non-Patent Literature 5), when the symmetry is Z2 symmetry (Non-Patent Literature 1)), the conversion rate values ​​for transforming between independent and identically distributed states have been determined. However, the conversion rates for more general symmetries have not been determined, and therefore, transforming between independent and identically distributed states may not be possible.

[0016] <Proposed Method> Below, assuming finite commutative symmetry, we propose a method for realizing transformations between independent and identically distributed states using only covariant operations that preserve finite commutative symmetry.

[0017] Let the initial state be |ψ〉 and the target state be |φ〉. The independent and identically distributed states of the initial state |ψ〉 and the target state |φ〉 are as follows:

[0018] Here, N is the number of copies of the initial state |ψ〉, and M is the number of copies of the target state |φ〉, and N and M are integers greater than or equal to 2. The conversion rate is expressed as M / N.

[0019] In this section, we will clarify the upper limit of the conversion rate achievable by covariant operations by applying the conditions for convertibility between single states, as revealed in Non-Patent Document 3, to the above-mentioned independent and identically distributed states. Furthermore, we will clarify the number of copies N and the covariant operations required to perform the conversion at a conversion rate r that is even slightly smaller than this upper limit.

[0020] The symmetry of the system is described by a finite commutative group G. The symmetry operations corresponding to elements g of the finite commutative group G are unitary operations U on the quantum state. ｇ Let it be represented as follows: In this case, the characteristic function χ of the pure state |ψ〉 in g∈G ψ (g) is defined by the following equation (1).

[0021] According to Non-Patent Document 3, a necessary and sufficient condition for a state transformation |ψ〉→|φ〉 to be possible by a covariant operation is that there exists a positive definite function f(g) such that the characteristic function χ ψ (g) can be expressed by the following equation (2).

[0022] By applying the conditions shown in equation (2) above to the independent and identically distributed states of state |ψ〉 and state |φ〉, the upper limit of the conversion rate can be determined by the following equation (3).

[0023] Note that e is the identity element of the finite commutative group G.

[0024] The conversion can be performed at any conversion rate r < R (ψ→φ) that is even slightly smaller than the value obtained by equation (3) above. In order to perform the conversion at this conversion rate r, the conversion rate must be at most N ｒ =-2log|G| / logr ｍａｘ It is sufficient to prepare copies of the initial state |ψ〉, where r ｍａｘ The following equation (4) is given.

[0025] Note that |G| is the number of elements in the finite commutative group G.

[0026] Also, at this time, the conversion between the independent and identically distributed states can be realized by introducing an auxiliary state |τ〉 with respect to the independent and identically distributed state of the initial state |ψ〉 and applying a unitary operation representing a covariant operation to the following composite system by using the method described in Non-Patent Document 2.

[0027] Specifically, according to the following formula (5), the independent and identically distributed state of the initial state |ψ〉 can be converted into the independent and identically distributed state of the target state |φ〉. However, N ≥ N ｒ , and M = r × N.

[0028] Here, tr ｅｎｖ is an operation of tracing out the auxiliary system. Also, V is a covariant unitary operation.

[0029] Hereinafter, a quantum state conversion device 10 for converting between independent and identically distributed states by the above proposed method will be described.

[0030] <Configuration example of quantum state conversion device 10> An example of the configuration of the quantum state conversion device 10 according to this embodiment will be described while referring to FIG. 1. FIG. 1 is a diagram showing an example of the configuration of the quantum state conversion device 10 according to this embodiment.

[0031] As shown in FIG. 1, the quantum state conversion device 10 according to this embodiment includes a control device 100 and a quantum processor 200.

[0032] The control device 100 transmits a control signal to the quantum processor 200 and acquires a calculation result from the quantum processor 200. Thereby, quantum calculation is performed. The control device 100 is realized by, for example, a classical computer or the like.

[0033] The quantum processor 200 constitutes a two-level quantum system called a quantum bit (physical qubit), and performs physical operations such as initialization, gate operations (unitary transformations), and measurements on the physical qubits in response to control signals from the control device 100. The quantum system for realizing the qubits is not particularly limited, and any quantum system may be used. For example, a quantum system realized by a superconducting circuit, an ion trap, photons, quantum dots, or the like can be used.

[0034] <Example of Hardware Configuration of Control Device 100> An example of the hardware configuration of the control device 100 according to the present embodiment will be described while referring to FIG. 2. FIG. 2 is a diagram showing an example of the hardware configuration of the control device 100 according to the present embodiment.

[0035] As shown in FIG. 2, the control device 100 according to the present embodiment includes an input device 101, a display device 102, an external I / F 103, a communication I / F 104, a RAM (Random Access Memory) 105, a ROM (Read Only Memory) 106, an auxiliary storage device 107, and a processor 108. These hardware components are communicably connected to each other via a bus 109.

[0036] The input device 101 is, for example, a keyboard, a mouse, a touch panel, a physical button, or the like. The display device 102 is, for example, a display, a display panel, or the like. Note that the control device 100 may not have at least one of the input device 101 and the display device 102, for example.

[0037] The external I / F 103 is an interface with an external device such as a recording medium 103a. Examples of the recording medium 103a include a CD (Compact Disc), a DVD (Digital Versatile Disk), an SD memory card (Secure Digital memory card), a USB (Universal Serial Bus) memory card, and the like.

[0038] The communication interface 104 is an interface for sending and receiving various signals with the quantum processor 200. The RAM 105 is a volatile semiconductor memory (storage device) that temporarily holds programs and data. The ROM 106 is a non-volatile semiconductor memory (storage device) that can retain programs and data even when the power is turned off. The auxiliary storage device 107 is a non-volatile storage device (storage device) such as an HDD (Hard Disk Drive), SSD (Solid State Drive), or flash memory. The processor 108 is an arithmetic unit such as a CPU (Central Processing Unit).

[0039] Note that the hardware configuration shown in Figure 2 is just one example, and the hardware configuration of the control device 100 is not limited to this. For example, the control device 100 may have multiple auxiliary storage devices 107 and multiple processors 108, it may not have some of the hardware shown, or it may have various hardware other than the hardware shown.

[0040] <Example of Functional Configuration of Control Device 100> An example of the functional configuration of the control device 100 according to this embodiment will be described with reference to Figure 3. Figure 3 is a diagram showing an example of the functional configuration of the control device 100 according to this embodiment.

[0041] As shown in Figure 3, the control device 100 according to this embodiment includes a conversion rate determination unit 110, an initial state number determination unit 111, an initial state preparation unit 112, and a state conversion unit 113. Each of these units is realized by a process in which one or more programs installed in the control device 100 cause a processor 108 or the like to execute.

[0042] The conversion rate determination unit 110 determines the upper limit R(ψ→φ) of the conversion rate using the above equation (3), and then determines a conversion rate r that satisfies r < R(ψ→φ). However, it should be noted that by definition, the conversion rate is r > 0. The conversion rate determination unit 110 may determine the conversion rate r as a value r < R(ψ→φ) specified by the user, or it may determine the conversion rate r as r = R(ψ→φ) - ε using a predetermined positive integer ε.

[0043] The initial state number determination unit 111 determines the number of copies N of the initial state |ψ〉 (where N ≥ N ｒ ). However, N ｒ is calculated using r shown in the above formula (4) as N ｍａｘ = -2log|G| / log r ｒ . Note that the initial state number determination unit 111 may determine the value N ≥ N ｍａｘ specified by the user as the copy number N, or may determine N = N ｒ as the copy number N, or may determine the copy number N by other methods. ｒ

[0044] The initial state preparation unit 112 prepares (prepares) an independent and identically distributed state represented by N copies of the initial state |ψ〉 (more precisely, the tensor product of N initial states |ψ〉) on the quantum processor 200.

[0045] The state conversion unit 113 prepares an auxiliary state |τ〉 for the independent and identically distributed state of N initial states |ψ〉 on the quantum processor 200, and then converts it into an independent and identically distributed state of M = r × N target states |φ〉 according to the above formula (5).

[0046] <Example of quantum state conversion process> An example of the quantum state conversion process according to this embodiment will be described while referring to FIG. 4. FIG. 4 is a flowchart showing an example of the quantum state conversion process according to this embodiment.

[0047] The conversion rate determination unit 110 obtains the upper limit R (ψ→φ) of the conversion rate according to the above formula (3), and then determines a conversion rate r that satisfies r < R (ψ→φ) (step S101).

[0048] The initial state number determination unit 111 determines the number of copies N of the initial state |ψ〉 (where N ≥ N ｒ ) (step S102).

[0049] The initial state preparation unit 112 prepares an independent and identically distributed state represented by N copies of the initial state |ψ〉 on the quantum processor 200 (step S103).

[0050] The state conversion unit 113 prepares auxiliary states |τ〉 for the independent and identically distributed states of N initial states |ψ〉 on the quantum processor 200, and then converts them into independent and identically distributed states of M = r × N target states |φ〉 using the above equation (5) (step S104). As a result, the independent and identically distributed states of N initial states |ψ〉 are converted into independent and identically distributed states of M target states |φ〉.

[0051] <Application Examples> Below, an example of an application of the quantum state conversion device 10 according to this embodiment will be described. However, each of the following application examples is merely an example, and the quantum state conversion device 10 according to this embodiment is not limited to the following application examples.

[0052] • Application Example 1: The quantum state conversion device 10 according to this embodiment can be applied to an error-tolerant quantum computer that uses a stabilizer-type quantum error correction code.

[0053] The code state of a quantum error correction code in stabilizer form is defined by a state s|ψ〉=|ψ〉 (∀s∈S) that is invariant under the stabilizer group S. The stabilizer group S is a finite commutative group Z. ２ ｎ It is isomorphic to the above. Furthermore, when the stabilizer operation is considered a symmetric operation, the covariant operations are logical operations and stabilizer operations. Therefore, by applying the quantum state conversion device 10 according to this embodiment, it becomes possible to determine the conversion rate r that can be converted using only logical operations and stabilizer operations, the number of copies N of the initial state required for that conversion, and the covariant operations.

[0054] • Application Example 2: The quantum state conversion device 10 according to this embodiment can be applied to determining the communication rate of communication using a state encoded by discrete rotation operations.

[0055] Consider a situation where sender A sends a message k ∈ {0, 1, ..., n-1} to receiver B.

[0056] At this time, sender A encodes message k into the quantum state of light |k〉 as follows:

[0057] However, the following is an operation that rotates the quantum state of light by 2kπ / n on the phase plane.

[0058] Note that a is the photon annihilation operator, a † This is the photon generation operator.

[0059] On the other hand, when receiver B receives a quantum state |k> from sender A, it decodes the message k from that quantum state |k>.

[0060] The symmetric operations performed by sender A for encoding are in the finite commutative group Z. ｎ This constitutes the following. Furthermore, the only operation permitted to receiver B in order not to destroy message k is a covariant operation. Therefore, by applying the quantum state conversion device 10 according to this embodiment, it is possible to determine how many times more communication is required when using a certain state compared to when using a certain reference state (hereinafter referred to as the "reference state"). Specifically, if one reference state |φ〉 is arbitrarily fixed, it can be determined that the number of communication required when transmitting message k using a certain state |ψ〉 is 1 / R(ψ→φ) times that of the reference state |φ〉. Thus, for example, by using a certain ideal state as the reference state |φ〉, it is possible to know how many times more communication is required when transmitting message k using a certain state |ψ〉 compared to the ideal state.

[0061] <Summary> As described above, the quantum state transformation device 10 according to this embodiment can determine the transformation rate r that can be transformed using only covariant operations, the number of copies N of the initial state and the covariant operations required for the transformation, when the symmetry of the system is described by a finite commutative group. Therefore, according to the quantum state transformation device 10 according to this embodiment, the transformation rate r makes it possible to transform the independent and identically distributed states of the initial state into independent and identically distributed states of the target state.

[0062] Furthermore, the quantum state conversion device 10 according to this embodiment may, for example, measure the target state to obtain the expected value of a desired physical quantity in that target state and output it to a predetermined output destination (e.g., other devices or equipment, programs, memory, etc.).

[0063] The present invention is not limited to the embodiments specifically disclosed above, and various modifications, changes, and combinations with known technologies are possible without departing from the scope of the claims.

[0064] 10 Quantum state conversion device 100 Control device 101 Input device 102 Display device 103 External I / F 103a Recording medium 104 Communication I / F 105 RAM 106 ROM 107 Auxiliary storage device 108 Processor 109 Bus 110 Conversion rate determination unit 111 Initial state number determination unit 112 Initial state preparation unit 113 State conversion unit

Claims

1. A quantum state transformation device that transforms N independent and identically distributed initial states into M independent and identically distributed target states, comprising: a first determination unit that determines a value less than the upper limit of the transformation rate r by finding an upper limit of the transformation rate which represents the ratio of the number of target states M to the number of initial states N; a second determination unit that determines the value of the number of initial states N using the transformation rate r; and a transformation unit that transforms the N independent and identically distributed initial states into M = r × N independent and identically distributed target states using a covariant operation which is an operation that commutes with a unitary operation which represents a symmetric operation corresponding to an element of a finite commutative group.

2. The quantum state transformation apparatus according to claim 1, wherein the first determination unit uses the first characteristic function in the finite commutative group of the initial state and the second characteristic function in the finite commutative group of the target state to determine the upper limit value which minimizes the ratio of the logarithm of the absolute value of the first characteristic function to the logarithm of the absolute value of the second characteristic function, and determines a value less than the upper limit as the transformation rate r.

3. The second determination unit, with G being the finite commutative group, determines N ｒ =-2log|G| / logr ｍａｘ (However, r ｍａｘ The quantum state transformation apparatus according to claim 2, wherein the value of the number of initial states N is determined to be a value greater than or equal to the maximum value of the ratio of the absolute value of the first characteristic function to r times the absolute value of the second characteristic function.

4. A quantum state transformation method for transforming N independent and identically distributed initial states into M independent and identically distributed target states, the method comprising: a first determination step of determining a value less than the upper limit of the transformation rate, which represents the ratio of the number of target states M to the number of initial states N, by finding an upper limit of the transformation rate; a second determination step of determining the value of the number of initial states N using the transformation rate r; and a transformation step of transforming the N independent and identically distributed initial states into M = r × N independent and identically distributed target states using a covariant operation that is a unitary operation commutative with a symmetric operation representing an element of a finite commutative group.

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