Quantum computing device, quantum computing method, and program

Abstract

This quantum computing device is provided with: an expectation value calculation unit that uses a hybrid tensor network method to calculate an expectation value of a physical quantity with respect to a quantum state; and a noise removal unit that uses a coefficient based on the probability of noise being generated in the quantum state to calculate an error suppressed expectation value obtained by removing the influence of noise from the expectation value.

Description

Quantum computing device, quantum computing method, and program

[0001] The present invention relates to quantum error suppression techniques in hybrid tensor network methods.

[0002] A hybrid tensor network method exists as a conventional technique for simulating a quantum system by combining a conventional tensor network, which enables high-speed simulation of physical property calculations, with a quantum tensor (i.e., a wave function) output from a quantum computer. Non-Patent Document 1 discloses such a hybrid tensor network method.

[0003] The hybrid tensor network method makes it possible to calculate physical quantities corresponding to quantum systems larger than the size of actual quantum hardware by constructing tree-type hybrid tensors.

[0004] Xiao Yuan, Jinzhao Sun, Junyu Liu, Qi Zhao, and You Zhou. "Quantum simulation with hybrid tensor networks." Physical Review Letters 127, no. 4 (2021): 040501.

[0005] In current quantum devices, the influence of noise from the environment cannot be ignored, and the influence of calculation errors must be considered. In addition, the quantum state simulated by the hybrid tensor network method is naturally influenced by noise because it uses the output from the quantum device. Therefore, removing the influence of noise is important in utilizing the hybrid tensor network method.

[0006] The present invention has been made in consideration of the above points, and aims to provide a technique for suppressing the influence of noise when calculating an expected value of a physical quantity using a hybrid tensor network method.

[0007] According to the disclosed technology, a quantum computing device is provided that includes an expected value calculation unit that calculates an expected value of a physical quantity for a quantum state using a hybrid tensor network method, and a noise removal unit that calculates an error suppression expected value by removing the influence of noise from the expected value using a coefficient based on the probability of noise occurring in the quantum state.

[0008] The disclosed technology provides a technology for suppressing the influence of noise when calculating an expected value of a physical quantity using a hybrid tensor network method.

[0009] 1 is a diagram showing an example of the configuration of a quantum computing device 300. FIG. 2 is a diagram showing an example of the functional configuration of the quantum computing device 300. FIG. 3 is a flowchart showing the overall flow of processing of the quantum computing device 300. FIG. 4 is a diagram showing an example of a two-layer tree hybrid tensor state. FIG. 5 is a diagram showing a graphical diagram showing the procedure of contraction. FIG. 6 is a diagram showing an example of a quantum circuit. FIG. 7 is a diagram showing an example of a noise model. FIG. 8 is a diagram showing an example of a noise model. FIG. 9 is a diagram showing an example of an Ansatz. FIG. 10 is a diagram showing the ratio between noiseless values ​​and noisy values. FIG. 11 is a diagram showing an example of the hardware configuration of the device.

[0010] Hereinafter, an embodiment of the present invention 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.

[0011] As mentioned above, the hybrid tensor network method disclosed in Non-Patent Document 1 makes it possible to calculate physical quantities corresponding to quantum systems larger than the size of actual quantum hardware by constructing tree hybrid tensors.

[0012] The quantum state simulated by the hybrid tensor network method is constructed using the output from a quantum device, and is therefore subject to noise (computational errors). Therefore, eliminating the effects of noise is important for utilizing the hybrid tensor network method. However, in the prior art, there was no appropriate method for eliminating the effects of noise in the hybrid tensor network method. Below, we will explain the technology according to this embodiment for solving this problem.

[0013] (Outline of the embodiment) In this embodiment, a quantum computing device 300 (described later) uses a hybrid tensor network method to measure (calculate) the expected value of a physical quantity for a certain quantum state. This expected value is rescaled due to the influence of noise (i.e., the influence of calculation errors). Therefore, the quantum computing device 300 suppresses the influence of noise by amplifying the calculation result by the reciprocal of the rescaling factor. Note that the physical quantity may also be called an observable.

[0014] (Example of Overall Configuration of Device) Fig. 1 shows an example of the configuration of a quantum computing device 300 according to this embodiment. The "quantum computing device" may also be called a "quantum computer" or a "quantum computing system."

[0015] As shown in Fig. 1, the quantum computing device 300 includes a control device 100 and a quantum processor 200. The control device 100 performs quantum computing by transmitting control signals and the like to the quantum processor 200 and obtaining calculation results (measurement results) from the quantum processor 200. The control device 100 can be realized by, for example, a classical computer. Hereinafter, "computer" means "classical computer."

[0016] The quantum processor 200 has a physical quantum system. In this embodiment, for example, a two-level quantum bit, a bosonic quantum bit capable of performing continuous quantum computation, or the like can be used as the quantum system. For example, a superconducting circuit, an ion trap, a quantum dot, a microwave photon, or the like can be used as a physical system for realizing the quantum system.

[0017] The quantum processor 200 may be referred to as a quantum device, a quantum processing unit (QPU), etc. The quantum processor 200 may include multiple quantum devices (multiple QPUs).

[0018] Furthermore, the quantum bits in this embodiment are not limited to those using actual physical systems. For example, the quantum bits may be on a simulator implemented by software on a classical computer. In this case, the quantum processor 200 functions as a quantum bit simulator. This simulator may be provided inside the control device 100.

[0019] (Example of Functional Configuration of Quantum Computing Device 300) The "control device 100 and quantum processor 200" included in the quantum computing device 300 cooperate to realize the functions for quantum computing by the quantum computing device 300.

[0020] An example of the functional configuration of the quantum computing device 300 of this embodiment is shown in Fig. 2. As shown in Fig. 2, the quantum computing device 300 has a quantum state preparation unit 310, an expected value calculation unit 320, and a noise removal unit 330. The operation of each unit will be described later. Since the control device 100 is responsible for the control of quantum computing, the functional configuration shown in Fig. 2 may be considered to be the functional configuration of the control device 100.

[0021] It should be noted that the "quantum state preparation unit 310, expected value calculation unit 320, and noise removal unit 330" in the quantum computing device 300 do not need to be provided within a single device. For example, the "quantum state preparation unit 310 and expected value calculation unit 320" may exist within a single device, and the noise removal unit 330 may exist within one or more devices located elsewhere.

[0022] (Example: Operation of quantum computing device 300) The operation of the quantum computing device 300 will be described below as an example. FIG. 3 shows the overall processing flow of the quantum computing device 300. In S1, the quantum state preparation unit 310 prepares a quantum state. In S2, the expectation calculation unit 320 calculates the expectation of the quantum state using the hybrid tensor network method. In S3, the noise removal unit 330 performs processing to remove the influence of noise (error) from the expectation calculated by the expectation calculation unit 320, calculates an expectation from which the influence of noise has been removed, and outputs the expected value. A more specific description will be given below.

[0023] Here, a two-layer tree-type hybrid tensor network will be considered as an example. An overview of the tree-type tensor in the hybrid tensor network method will be explained. Note that the hybrid tensor network method itself is a well-known technology disclosed in Non-Patent Document 1, etc., so only an overview will be explained here.

[0024] In the following description, the superscript or subscript in the superscript or subscript is not written as a superscript or subscript in the text of the specification, but it is clear from the context that it indicates a superscript or subscript. For example, "i ｋ " using "φ ｉｋ " is an example.

[0025] In addition, in the text of the specification, symbols that precede characters (e.g., →, ~) are placed at the top left of the character or before the character. → i ｋ ", "ψ →ｉｋ " is an example.

[0026] In the absence of noise, this two-level tree hybrid tensor state is as follows: Quantum state preparation 310 prepares a quantum state shown in the following formula (1). Figure 4 shows the configuration of this quantum state.

[0027] i ｋ (k=1, 2, . . . N) takes a value of 0 or 1, and the quantum state |ψ> shown below is the quantum state of an N-qubit quantum device.

[0028] Here, |φ ｉｋ >(k=1, 2, ......N) is a quantum state output from a quantum device with a K qubit system, the expected value calculation unit 320 calculates |ψ corresponding to the N K qubit system by the hybrid tensor network method. ｈｙｂ >, we can calculate the expected value corresponding to

[0029] An example of the calculation procedure will be described below. First, consider measuring the physical quantity O of the N K qubit system expressed by the following equation (2).

[0030] When measuring the physical quantity O, the expectation value calculation unit 320 calculates the first-level quantum tensor |φ ｉｋ For k = 1, 2, ..., N, first calculate the following matrix using a quantum device.

[0031] The expected value calculation unit 320 further utilizes the fact that the quantity shown in the following equation (3) is a Hermitian matrix, i.e., a physical quantity, to measure the expected value of the quantity shown in equation (3) with respect to |ψ>, thereby obtaining |ψ ｈｙｂ Expected value <O> of the physical quantity O shown in equation (2) for ｈｙｂ Calculate.

[0032] Here, |φ ｉｋ > (k=1, 2, . . . N) and |ψ> shown below are assumed to be affected by global depolarizing noise with probability p.

[0033] In this case, the expected value calculated using the noisy hybrid tensor is denoted by <O> ｈｙｂ (p), the approximation shown in the following equation (4) holds for tree tensors of two or more layers.

[0034] <O> in formula (4) ｈｙｂ (0) is the expected value in the absence of noise, and N ｑｔｅｎｓｏｒis the number of quantum tensors, i.e., the number of (noisy) wave functions used in the hybrid tensor state (quantum state in the hybrid tensor).

[0035] Here, r(p) = (1 - p) Ｎｑｔｅｎｓｏｒ where r(p) is the rescaling factor.

[0036] The noise removal unit 330 calculates the following equation (5): ｈｙｂ (p) is multiplied by the inverse of the rescale factor.

[0037] <O> ｈｙｂ (p) / r(p) (5) By using equation (5), the noiseless expected value <O> ｈｙｂ It is possible to obtain a value that approximates (0), that is, an expected value with error (noise) suppressed.

[0038] [Detailed Example] The hybrid tensor network method and the calculation method of the rescale factor and the like described above will be described in more detail below. Note that in the detailed example below, the numbers of the equations are renumbered starting from (1). For convenience, the numbers of the equations may not be consecutive. For example, the equation following equation (11) is equation (34).

[0039] (About Hybrid Tensor Networks) Classical tensor network (TN) methods efficiently describe quantum states of interest in a much smaller subset of Hilbert space based on physical observations. For example, in matrix product state (MPS) Ansatz, the state is expressed as follows:

[0040] In the case of quantum systems, j ｋ takes 0 or 1, and A １ and A Ｍ is a rank-2 tensor, and A ｋ is a rank-3 tensor. With this Ansatz structure, the number of parameters representing the state is O(Mκ ２ ) to O(2 Ｍ) The MPS representation can capture weakly and locally entangled quantum systems, but cannot simulate strongly entangled quantum dynamics.

[0041] While classical simulation of quantum systems requires efficient Ansatz representations, quantum computers allow the efficient generation of rank-n tensors, which motivates the introduction of quantum tensors.

[0042] Quantum tensors are generally represented as ψj1,j2,...,jMi1,i2,...,iN, where the lower index (i １ , i ２ , ..., i Ｎ ) is the quantum index, and the upper index (j １ , j ２ , ..., j Ｍ ) are classical indices. A quantum tensor is generated from a quantum device and the corresponding quantum state is:

[0043] The quantum index necessarily relates to a "physical" quantum device. ｋ For ∈{0, 1}∀k, |i １ , i ２ , ..., i Ｎ > is the computational basis state of a qubit-based quantum computer. Tensor contraction of quantum indices is performed through measurements of quantum states.

[0044] As a simple example of the HTN formalism, the rank-1 classical tensor α ｊ tensor for ～ ψi1,i2,...,iN=Σ ｊ α ｊ ψ ｊ Consider i1, i2, ..., iN. This hybrid tensor is |ψ ｊ >=Σ ｉ１，ｉ２，...，ｉＮ ψ ｊ i1,i2,...,iN|i １ , i ２ , ..., i Ｎ Quantum state with >| ～ ψ>=Σｊ α ｊ |ψ ｊ >corresponds to

[0045] Linear combinations of quantum states with classical coefficients are a generalization of quantum subspace expansion, which can enhance the representability of quantum states and thus serve as a quantum error suppression method.

[0046] Condition | ～ To calculate the expectation value of an observable O with respect to ψ, ～ψ =Σ ｉｊ α ｉ＊ α ｊ M ｉ，ｊ where M ｉ，ｊ =<ψ ｉ |O|ψ ｊ > is evaluated on a quantum computer. For example, P ｊ is the Pauli operator, and the quantum tensor is |ψ ｊ >=P ｊ |ψ ０ >, M ｉ，ｊ =<ψ ０ |P ｉ OP ｊ |ψ ０ > is obtained. Observable (physical quantity) is O=Σ ｋ f ｋ P ｋ When linearly decomposed as M ｉ，ｊ can be evaluated only by measuring the Pauli operators. （ｊ） Using |ψ （ｊ） >=U ｊ |ψ0〉, M ｉ，ｊ =<ψ ０ |U （ｉ）† OU （ｊ） |ψ ０ >=Σ ｋ f ｋ <ψ ０ |U （ｉ）† P ｋ U （ｊ） |ψ ０ > is obtained. ０ |U （ｉ）† P ｋ U （ｊ） |ψ０ > can be calculated with a Hadamard test circuit.

[0047] Here, we describe hybrid tree tensor networks (HTTNs) as an important class of hybrid tensor networks. A two-layer tree tensor network state consisting of quantum tensors can be expressed as follows:

[0048] |ψ ｉｋ ｋ >(k=1,...,N) is a quantum tensor with K qubits, ψ ｉ１，...，ｉＮ = <i １ , ..., i ２ Let |ψ〉 be the probability amplitude of the N-qubit system and C be a normalization constant, then |ψ ＨＴ > is an NK qubit system. ＨＴ It has been shown that the expectation value of the observable for > can only be calculated for a max{N, K} qubit system. ｋ For , the observables of the N K qubits are expressed as follows:

[0049] Next, the expected value including the normalization constant is calculated as follows:

[0050] and

[0051]

[0052] M ｉｋ，ｉ´ｋ ｋ =<ψ ｉｋ ｋ |O ｋ |ψ ｉ´ｋ ｋ > and S ｉｋ，ｉ´ｋ ｋ =<ψ ｉｋ ｋ |ψ ｉ′ｋ ｋ Therefore, <O> ψＨＴ can be evaluated as follows:

[0053] First, the local quantum tensor {|ψ ｉｋ> ｋ} ｋ By contraction of {M ｋ} Ｎ ｋ＝１ ({S ｋ} Ｎ ｋ＝１ ) is calculated. ｋ} ｋ ({S ｋ} ｋ ) are Hermitian operators, so the unitary operator U ｋ (V ｋ ) and the diagonal operator Λ ｋ (Λ´ ｋ ) for the spectral decomposition M ｋ =U † ｋ Λ ｋ U ｋ (S ｋ =V † ｋ Λ´ ｋ V ｋ ) is obtained.

[0054] Applying the above, measuring in the Z basis, and assigning Λ to the measurement result ｋ (Λ´ ｋ ) for the state |ψ〉

[0055] The expected value of can be measured. A graphical diagram showing the reduction procedure is shown in Figure 5.

[0056] Figure 5(a) shows a graphical representation of the two-layer hybrid tree tensor network expressed in Equation (3). The upper quantum tensors ψi1,i2,...,iN = <i １ , i ２ , ..., i Ｎ |U|0 Ｎ > represents the nonlocal correlation between subsystems, and the lower quantum tensor ψ ｉｋ ｊ１，...，ｊＫ =<j １ , ..., j Ｋ |ψ ｉｋ ｋ >(k=1,...,N) represents the quantum state of each subsystem.ＨＴ When calculating the expectation value of the observable for >, (b1) first reduce the local quantum system, and then (b2) use the Hermitian operator {M ｋ} Ｎ ｋ＝１ Using the quantum circuit in (c) corresponding to (b2), the reduction of the entire system is completed.

[0057] Although we have considered HTN states consisting only of quantum tensors above, it is also possible to use HTN models that alternate between quantum tensors and classical tensors.

[0058] Each M ｋ (S ｋ ) is calculated efficiently by computing the set of quantum states {|ψ ｉｋ ｋ > ｉｋ It depends heavily on how you set up M. ｋ (S ｋ To see the detailed procedure for computing the K-qubit local tensor ψ →ｉｋ ｋ Consider the more general scenario of contracting ψ →ｉｋ ｋ is the index of the parent tensor → i ｋ ∈{0, 1} τ and local observable O ｋ is connected to M →ｉｋ，→ｉ´ｋ ｋ =<ψ →ｉｋ ｋ |O ｋ |ψ →ｉ´ｋ ｋ > is obtained.

[0059] ψ →ｉｋ is a classical tensor (i.e., the set of states {|ψ →ｉｋ ｋ > →ｉｋ is prepared by a classical computer), and M ｋ (S ｋ ) are calculated. For example, we use the MPS representation to express the local tensor as |ψ →ｉｋ ｋ >=Σ ｊ１，ｊ２，...，ｊＫ Tr[A →ｉｋ，ｊ１ １ Aｊ２ ２ ...A ｊＫ Ｋ |j １ , j ２ , ..., j Ｋ >, the classical contraction of the MPS state gives the matrix element M →ｉｋ，→ｉ´ｋ ｋ =<ψ →ｉｋ ｋ |O ｋ |ψ →ｉ´ｋ ｋ On the other hand, ψ →ｉｋ is a quantum tensor (i.e., the set of states {|ψ →ｉｋ ｋ > →ｉｋ is prepared by a quantum computer (quantum device), M can be obtained by performing measurements on the quantum computer. ｋ (S ｋ ) is calculated. The specific calculation is →ｉｋ ｋ > →ｉｋ Since it depends on the definition of → For the following four cases where ik is associated, M ｋ (S ｋ ) will be explained.

[0060] (i) Initial state:

[0061] (ii) Projection:

[0062] iii) Pauli operator:

[0063] (iv) Unitary gate:

[0064] Although a type can be expressed mathematically in the framework of another type (e.g., types (i) and (iii) can be considered as special instances of type (iv)), here: → i ｋ State due to |ψ →ｉｋ ｋ >Different evaluations of ｋ (S ｋThe separation of each type is due to practical reasons, such as the need for procedures to calculate

[0065] <A. M in type (i) ｋ and S ｋ Calculation of > |ψ →ｉｋ ｋ >=U ｋ | → i ｋ >| → 0>, the Hermitian matrix M ｋ Each element of is expressed by the following formula.

[0066] Here, U ｋ (●) = U ｋ (●) U † ｋ is a K-qubit unitary channel. For convenience of description, U ｋ (●) = U ｋ (●) U † ｋ The letter U on the left side of is the same as that on the right side, but it is intended to be the letter U in equation (6). The same applies hereafter. Here, the following equations are defined.

[0067] | → i' ｋ >< → i ｋ Since | can be expressed as a linear combination of the eigenstates of the Pauli operator, various input |a ｋ >Measurement result E ｋ (|a ｋ By appropriately combining ｋ In the following, we will explain the case where τ = 1. First, we consider the input state |a ｋ 6(a) with the observable O for the state prepared by the quantum circuit of FIG. ｋ By measuring the expected value of M ｋ Find the diagonal elements of M ０，０ ｋ = E ｋ (|0>) and M １，１ ｋ = E ｋ(|1>). Next, M is calculated using the quantum circuit in FIG. 6(a) by the following formula: ｋ Calculate the off-diagonal elements of M １，０ ｋ = M ０，１＊ ｋ is.

[0068] Here, |+>, |->, |+i>, |-i> are defined as eigenvectors with eigenvalues ​​+1 and -1 of the Pauli operators X and Y, respectively. ｋ (|0>) + E ｋ (|1>) = E ｋ (|+>)+E ｋ (|->) = E ｋ (|+i>)+E ｋ (|-i>), the calculation term E in equation (7) ｋ (|->) and E ｋ (|-i>) can be abbreviated as follows:

[0069] For any k, S ｋ = I, and as a result, C ２ = 1, so the matrix M Ａｋ There is no need to calculate.

[0070] <B. M in type (ii) ｋ and S ｋ Calculation of > |ψ ｋ > is the (K+τ) qubit state, and |ψ →ｉｋ ｋ >=< → i ｋ |ψ ｋ >, the Hermitian matrix M ｋ Each element of is expressed as follows:

[0071] Here, U ｋ (| → 0>< → 0|) is U ｋ (| → 0>< → 0|) = |ψ ｋ ><ψ ｋ | is a (K+τ) qubit quantum channel that operates as | → i ｋ ><→ i' ｋ can be decomposed into a linear combination of Pauli operators, so various Pauli observables

[0072] Measurement results for

[0073] By combining ｋ The following explanation will be given taking the case of τ=1 as an example. The state |ψ prepared by the quantum circuit in FIG. 6(b) ｋ ><ψ ｋ |

[0074] By measuring the expected value of ０，０ ｋ and M １，１ ｋ That is, M ０，０ ｋ = {E ｋ (I) + E ｋ (Z)} / 2 and M １，１ ｋ = {E ｋ (I)-E ｋ (Z)} / 2). E ｋ (ω ｋ ) from the linearity of M ０，１ ｋ = {E ｋ (X) + iE ｋ (Y)} / 2 and M １，０ ｋ = M ０，１＊ ｋ Similarly, M ｋ The calculation is O ｋ By replacing with I, S ｋ can be calculated.

[0075] <C. M in type (iii) ｋ and S ｋ Calculation of >

[0076] is the tensor product of the Pauli operators, and |ψ →ｉｋ ｋ >=P →ｉｋ ｋ |ψ ｋ>, the Hermitian matrix M ｋ Each element of

[0077] It is expressed as:

[0078] Here, U ｋ (| → 0>< → 0|) is U ｋ (| → 0>< → 0|) = |ψ ｋ ><ψ ｋ is a K-qubit quantum channel acting as | ＾ O →ｉｋ，→ｉ´ｋ =P →ｉｋ ｋ O ｋ P →ｉ´ｋ ｋ is.

[0079] ＾ O →ｉｋ，→ｉ´ｋ =Σ ｌｋ f ｌｋ P ｌｋ (Fig. 6(c)) as an observable to a linear combination of Pauli operators ＾ O →ｉｋ，→ｉ´ｋ By decomposing the Pauli operator P ｌｋ Direct measurement of M →ｉｋ，→ｉ´ｋ ｋ can be evaluated, and M →ｉ´ｋ，→ｉｋ ｋ = M →ｉｋ，→ｉ´ｋ＊ ｋ By M →ｉ´ｋ，→ｉｋ ｋ can be obtained.

[0080] S ｋ In the calculation of each diagonal element S →ｉｋ，→ｉ´ｋ ｋ is 1. Each off-diagonal element S →ｉｋ，→ｉ´ｋ ｋ is |ψ ｋ ><ψ ｋ |, the Pauli observable P →ｉｋ P →ｉ´ｋ can be calculated by performing a direct measurement of

[0081] D. M in type (iv) ｋ and S ｋ Calculation of > |ψ →ｉｋ ​ｋ >=U →ｉｋ ｋ | → 0>, the Hermitian matrix M ｋ Each element is expressed by the following formula.

[0082] Diagonal element M →ｉｋ，→ｉｋ ｋ As shown in FIG. 6(d1), the state |ψ →ｉｋ ｋ ><ψ →ｉｋ ｋ |=U →ｉｋ ｋ (| → 0>< → 0|) for observable O ｋ All we need to do is measure the expected value of

[0083] Next, M ｋ Consider the procedure for estimating the off-diagonal elements of M →ｉｋ，→ｉ´ｋ ｋ The quantum circuit for estimating is shown in Figure 7(d2). Here, α is the phase, and the initial state is set as follows:

[0084] If we set α=0 for the parameterized initial state, we can measure the expected value of Re(M →ｉｋ，→ｉ´ｋ ｋ ) is obtained.

[0085] Similarly, by setting α=π / 2 and calculating the expected value shown below, Im(M →ｉｋ，→ｉ´ｋ ｋ ) is obtained.

[0086] Combining the real and imaginary parts, M →ｉｋ，→ｉ´ｋ ｋ and M →ｉ´ｋ，→ｉｋ ｋ = M →ｉｋ，→ｉ´ｋ＊ ｋ We obtain the matrix S ｋ In the estimation of ｋ is replaced with I to estimate the off-diagonal elements.

[0087] (Noise Analysis in Hybrid Tensor Networks) Next, we will explain noise analysis in hybrid tensor networks. One of the notable features of HTN is the step-by-step construction of new observables, which requires not only classical post-processing but also quantum computation. When applying HTN to practical applications, one of the most important issues to consider is that the presence of physical noise in quantum computation affects the overall output. More specifically, during the tensor contraction process, hardware noise may change the intended ideal state into a non-physical state (e.g., a state with negative eigenvalues). This situation ultimately causes serious problems in variational calculus. For example, when calculating the expectation value of a given Hamiltonian H using a trace-one Hamiltonian operator that is not necessarily positive, the calculated energy expectation value may be different from the Hamiltonian E. ｍｉｎ is not guaranteed to always be larger than the smallest eigenvalue of the Hamiltonian operator ～ There is a possibility that ρ exists.

[0088] Therefore, when simulating large-scale systems with the support of classical post-processing, it becomes important to understand which type of state preparation of hybrid states exhibits resilience to physical noise.

[0089] Here, we focus on simulations that use HT-TNs Ansatz to estimate the expectation value of the Hamiltonian, and analytically investigate whether multiple types of HTTN states tend to be induced into unphysical states under the influence of hardware noise.

[0090] (Operator-Based Representation of HTN) In the prior art disclosed in Non-Patent Document 1 and elsewhere, HTN states are represented by classical and quantum tensors with classical and quantum indices. For example, the graph representation of a two-layer HTTN state in Figure 5 is consistent with the conventional TN format. This representation rule explicitly shows the index connection relationship between quantum tensors and classical tensors, but it may not be suitable for analyzing noise states represented by density operators. Below, we introduce a representation format for HTN states based on expansion operators to capture the effects of physical noise.

[0091] First, consider the two-layer HTTN state with the language of HTN described below.

[0092] where: → ik∈{0,1} τｋ (k=1,...,N) is a classical bit string, and Φ →ｉ１，...，→ｉＮ =< → i １ , ..., → i Ｎ |Φ> is (Σ Ｎ ｋ＝１ τ ｋ ) is the probability amplitude of the qubit system, and |ψ →ｉｋ ｋ >(k=1,...,N) is the K ｋ is the quantum or classical tensor of the qubit, C is a normalization constant, and |ψ ＨＴ > denotes the κ qubit state, where κ = Σ ｋ K ｋ Both the parent tensor and the local tensor can be classical or quantum tensors. For example, we can use the MPS representation to convert the parent tensor into a classical tensor:

[0093] If the parent tensor is a quantum tensor, the contraction can be realized by measurement on a quantum computer, as shown in equations (4) and (5).

[0094] Here, we introduce the expansion operator in the following formula:

[0095] Superscript ξ ｋ is the state |ψ →ｉｋ ｋ represents the different preparation methods of ρ > and takes five values: 1, 2, 3, 4, and c, corresponding to the types (i) to (iv) mentioned above and the classically prepared state. Using the expansion operator, we can obtain the HTN state ρ corresponding to equation (35). ＨＴ =|ψ ＨＴ ><ψ ＨＴ | can be expressed as follows:

[0096] Equation (35) and equation (36) are mathematically equivalent. In this form, the operator A （ξｋ） ｋ extends the system size of the parent operator |Φ><Φ|.

[0097] System|ψ ＨＴ ><ψ ＨＴ When evaluating the expectation of the following observables against |,

[0098] As follows, {M （ξｋ） ｋ} ｋ ({S （ξｋ） ｋ} ｋ ) to obtain the

[0099]

[0100] A in formula (37) ｋ and O ｋ From {M （ξｋ） ｋ} ｋ ({S （ξｋ） ｋ} ｋ The process of generating the tensor vectors corresponds to the contraction of indices in a hybrid tensor network.

[0101] Using the above formalism, we consider the situation where physical noise affects the quantum computation process. To concretely formulate the problem setting, we consider that the hardware noise is a Hermitian observable {M （ξｋ） ｋ} ｋ ({S（ξｋ） ｋ} ｋ ) with noise { ～ M （ξｋ） ｋ} ｋ ({ ～ S （ξｋ） ｋ} ｋ ) and change the pure state |Φ〉<Φ| to a mixed state ρ. Then, we assume that the noisy expectation value < ～ O> ρＨＴ can be expressed as

[0102] Noise Condition ～ ρ ＨＴ The observable O in is expressed by the following equation:

[0103] where ～ ρ ＨＴ denotes a noisy valid HTTN state caused by physical noise.

[0104] How physical noise accumulates in quantum computing is explained by ξ ｋ In the following, we consider the noisy valid HTTN state ～ ρ ＨＴ Regarding the properties of ρ, ＨＴ is the density operator (i.e., Tr( ～ ρ ＨＴ ) = 1 and ～ ρ ＨＴ ≥ 0). ～ ρ （ｘ） ＨＴ Here, for each subsystem k, the ideal unitary channel of Figures 6(a) to 7(d2) is modified into an undesired channel.

[0105] Furthermore, the ideal channel ０ (●) is noise channel W ０ Here, we assume that the unitary channel U ０ (●)|Φ><Φ|:=U０ (| → 0>< → 0|), and the noise channel W ０ (●) to ρ:=W ０ (| → 0>< → 0|).

[0106] <B. Type (i)> K in FIG. ｋ Quantum bit ideal unitary channel U ｋ (●) = U ｋ (●) U † ｋ But, K ｋ Undesired channel W of the qubit ｋ (●) and the noisy Hermitian operator ～ M （１） ｋ can be written as follows:

[0107] where: ～ M （１） ｋ The superscripts in indicate operators corresponding to type (i) state preparation, in this case for all k ～ S （１） ｋ Set as follows:

[0108] This noisy operator ～ M （１） ｋ and operator S （１） ｋ For , the following equation holds:

[0109] where: ～ A （１） ｋ (●) is an operator defined as follows:

[0110] where: ～ A （１） ｋ (●) is the operator A （１） ｋ In fact, the ideal channel U ｋ (●) = U ｋ (●) U† ｋ W in equation (43) ｋ When replaced with (●), the noise-free operator A （１） ｋ (●) = A （１） ｋ (●) A （１）† ｋ get.

[0111] By applying equation (42) to equation (39), the following equation can be derived.

[0112] Therefore, from the correspondence between equations (40) and (44), the following equation is obtained:

[0113] Given the following channels are fully positive and trace preserving (CPTP) maps: ～ ρ （１） ＨＴ is the density operator.

[0114] <C. Type (ii)> As shown in FIG. 6(b), ｋ +τ ｋ ) qubit ideal unitary channel U ｋ (●) = U ｋ (●) U † ｋ But (K ｋ +τ ｋ ) the undesired channel W of the qubit ｋ It is assumed that it will be changed to (●).

[0115] Noisy Hermitian operators ～ M （２） ｋ is written as follows:

[0116] where σ ｋ =W ｋ | → 0>< → 0|). ～ S （２） ｋ can be expressed by substituting the following equation for O in equation (46):

[0117] Similar to type (i), the following transformation can be performed:

[0118] ～ A （２） ｋ (●) is an operator defined as follows:

[0119] Here, |> ● is the projection vector acting on the input state ●, |> σｋ is σ ｋ represents the projection vector acting on . From equation (47), the following equation is obtained:

[0120] where: → i = ( → i １ , ..., → i Ｎ ), | → i> σ = | → i １ > σ１ ... | → i Ｎ > σＮ is the projection vector acting on all subsystems from k = 1 to k = N, and | → i> ρ is the projection vector acting on the state ρ. From the correspondence between equations (40) and (49), we obtain the following equation:

[0121] ～ A （２） ｋ (●) is a completely positive (CP) map for all k. ～ ρ （２） ＨＴ is the density operator.

[0122] <D. Type (iii)> K in FIG. 6(c) ｋ Quantum bit ideal unitary channel U ｋ (●) = U ｋ (●) U † ｋ But, K ｋ Undesired channel W of the qubit ｋ Assume the noise model is changed to (●). ​

[0123] and the noisy Hermitian operator ～ M （３） ｋ can be expressed as follows:

[0124] where σ ｋ :=W ｋ (| → 0>< → 0|). ～ S （３） ｋ is the observable O in equation (51). ｋ is obtained by setting

[0125] As explained above, the following equation holds:

[0126] ～ A （３） ｋ (●) is as follows.

[0127] Here, | → ik> ● is the projection vector acting on the input state, and P →ｉｋ ｋ is σ ｋ By applying equation (52) to equation (39), we obtain the noisy expectation value < ～ O> ρＨＴ is obtained as the following formula:

[0128] where: → i = ( → i １ , ..., → i Ｎ ) and | → i> ρ is the projection vector acting on the state ρ, and P →ｉ is defined by the following formula:

[0129] From the calculation results using equation (40), the effective density state ～ ρ （３） ＨＴ can be expressed as follows:

[0130] ～ A （３） ｋ Since (●) is the CP map for all k, ～ ρ （３） ＨＴ The state is the density operator.

[0131] <E. Type (iv)> In this case, simply reducing the local subsystem results in a noisy hybrid tensor network state ～ ρ （４） ＨＴ is not always a positive operator in general. To observe this, under a simple noise model and the assumption of observability, ～ ρ （４） ＨＴ Show that can have negative eigenvalues.

[0132] As mentioned above, two types of quantum circuits are required to contract local observables. （４） ｋ (S （４） ｋ ) and a circuit for calculating the diagonal elements of M shown in FIG. （４） ｋ (S （４） ｋ ) We need a circuit to calculate the off-diagonal elements of the equation. Here, we set up a noise model for each type of quantum circuit.

[0133] M （４） ｋ (S （４） ｋ ) for the circuit that computes the diagonal elements of →ｉｋ ｋ (●) = U →ｉｋ ｋ (●) U →ｉｋ† ｋ Consider a noise model acting immediately after the corresponding depolarizing noise channel N →ｉｋ ｋ is as follows:

[0134] Here, p →ｉｋ ​is 0≦p →ｉｋ σ is the noise rate that takes ≦1. ｉｎ is the input state. The noise model in this case is shown in Figure 8.

[0135] M （４） ｋ (S （４） ｋ ) for the circuit that estimates the off-diagonal elements of each control U →ｉｋ ｋ Consider a noise model that operates after the operation. The corresponding noise channel ＾ N →ｉｋ ｋ is assumed to work as follows:

[0136] Here, q →ｉｋ is 0≦q →ｉｋ ≦1, and q ｉｎ is the input state. The noise model in this case is shown in Figure 9.

[0137] Regarding observable O,

[0138] is the subsystem's observable O ｋ is a Pauli sequence consisting of the tensor product of

[0139] is a Pauli sequence excluding the following:

[0140] A simple calculation reveals that the noisy Hermitian operator ～ M ｋ and ～ S ｋ is obtained by using equations (58) and (59): ～ M →ｉｋ，→ｉ´ｋ k = r →ｉｋ，→ｉ´ｋ ｋ M →ｉｋ，→ｉ´ｋ ｋ and ～ S →ｉｋ，→ｉ´ｋ ｋ =s →ｉｋ，→ｉ´ｋ ｋ S →ｉｋ，→ｉ´ｋ ｋ It can be expressed as:

[0141]

[0142] Here, M （４） ｋ and S （４） ｋ The following function Γ works for any matrix of the same size as ｋ (●) and Λ ｋ Define (●).

[0143]

[0144] Here, e →ｉｋ，→ｉ´ｋ is all → i ｋ and → i' ｋ is any complex number for the function Γ ｋ and Λ ｋ is the operator M （４） ｋ and S （４） ｋ are the noisy operators ～ M （４） ｋ =Γ ｋ (M （４） ｋ ) and ～ S （４） ｋ =Λ ｋ (S （４） ｋ ) into equation (39). ｋ and Λ ｋ Introducing (k=1, . . . , N), we obtain the following equation:

[0145] effective density operator ～ ρ （４） ＨＴ The component separated from the observable O is expressed by the following equation (62).

[0146]

[0147] is not always a positive semidefinite operator in general, so ～ ρ ＨＴ We can conclude that can have negative eigenvalues.

[0148] (Analysis of noise propagation for specific noise models) Below, we delve further into the expansion operator by specifying a noise model in the global depolarizing model for each quantum tensor, and show that the expectation value of the observable vanishes exponentially with the number of quantum tensors.

[0149] A. Explicit Expression of the Expansion Operator In preparation for the following analysis, the expansion operator A for the k-th subsystem is （ξｋ） ｋ (●) (ξ ｋ = 1, 2, 3). This operator is ｋ Let K be the quantum bit input system. ｋ In type (i), the expansion operator A （１） ｋ is expressed as follows:

[0150] Here, | → 0>< → 0 | is (K ｋ -τ ｋ ) qubit state, and U ｋ (●) = U ｋ (●) U † ｋ Is K ｋ In type (ii), the expansion operator A （２） ｋ is expressed by the following equation:

[0151] Here, P （２） ｋ =Σ →ｉｋ∈｛０，１｝τｋ | → i ｋ > ● | → i ｋ > σ２ and σ （２） ｋ (K ｋ +τ ｋ ) qubit system, and | → i ｋ > ● is the projection vector acting on the input state ●,

[0152] is σ （２）ｋ In type (iii), the expansion operator A （３） ｋ is written as follows:

[0153]

[0154] σ （３） ｋ Is K ｋ It is a qubit system, P →ｉｋ ｋ is σ （３） ｋ As already explained, using the quantum circuits of Figs. 6 and 7, ｋ Each expansion operator A （ξｋ） ｋ is the Hermitian operator M （ξｋ） ｋ can be reduced to

[0155] B. Exponential Decay of Observables We consider global depolarizing noise for hybrid tree-tensor networks. Here, we assume that the global depolarizing channel acts immediately after the unitary channel in Figure 6. In this case, the expansion operator for the noisy quantum tensor is ～ A （ξｋ） ｋ (ξ ｋ = 1, 2, 3) can be expressed as follows:

[0156] where ε ｋ is the noise rate (probability of noise occurrence), and the following formula holds:

[0157] We assume that the HTN state consists only of noisy quantum tensors, and for simplicity, ε ｋ By setting ε, the following equation is obtained:

[0158] where

[0159] is a binomial distribution, and B ｍ is error K （ξｋ）ｋ is a process that occurs m times. Here, the quantum process E ｋ For (k=1, 2, ..., N) the following holds:

[0160] Process B ｍ makes the HTN state highly mixed, allowing us to neglect the expectation values ​​of nonlocal Pauli observables.

[0161] Considering an L-layer TTN consisting of noisy N-rank quantum tensors, the total number of reduced quantum tensors is:

[0162] Therefore, the following formula is obtained:

[0163] With type (i) state preparation,

[0164] a noisy rank-N tensor ρ for →ｉ，→ｉ´ We numerically verify equation (91) by simulating an L-layer TTN consisting of: Each tensor is generated by a two-layer hardware-efficient Ansatz with global depolarizing noise, as shown in Figure 10. To efficiently simulate a noisy TTN, the matrix condensed at a layer ～ Once M is obtained, the following quantities are used for reduction in the next layer:

[0165] In Figure 10, to simulate a type (i) contraction, the first register is set to i = 0, 1. To avoid extremely small calculation results that can cause numerical errors, the variational rotation angles are set to [π / 10 ３ , π / 10 ３ ]. Also, d=2 is set.

[0166] FIG. 11 shows the ratio r between noiseless and noisy values ​​for N=10 and L=4, 5, 6. Ｌ (ε)=<O> ｎｏｉｓｙ / <O> ＥＦIn all cases,

[0167] agrees exactly with the numerical result, which shows that equation (91) is a very good approximation.

[0168] If the value of ε can be characterized, then ｎｏｉｓｙ of ～ r Ｌ Dividing by (ε) gives us the following equation:

[0169] This effectively acts as quantum error suppression, where the error suppression result <O> ＱＥＭ The variance of is as follows:

[0170] To achieve the same accuracy as in the error-free case, ～ r Ｌ (ε) ２ You'll need twice as many samples.

[0171] In other words, the scale at which quantum systems can be simulated can be greatly expanded, but the resources required are large in the number of contracted quantum tensors. To avoid this problem, one of the following strategies must be adopted:

[0172] (1) Adjust the number of classical and quantum tensors depending on the magnitude of the error.

[0173] (2) The error rate itself is suppressed by error suppression techniques such as quantum error correction and dynamic decoupling. Regarding (1), if quantum tensors are replaced with classical tensors in the L-layer tree tensor, for example, the following equation is obtained, which alleviates the problem of exponential decay of observables.

[0174]

[0175] (Example of hardware configuration)

[0176] (Example of Hardware Configuration of Device) Any of the devices described in this embodiment (control device 100, quantum computing device 300) can be realized by causing a computer to execute a program. This computer may be a physical computer or a virtual machine on the cloud.

[0177] That is, the device can be realized by executing a program corresponding to the processing performed by the device 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.

[0178] Fig. 12 is a diagram showing an example of the hardware configuration of the computer. The computer in Fig. 12 includes 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, and the like, all of which are interconnected by a bus BS. The computer may further include a GPU.

[0179] The program that realizes 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.

[0180] The memory device 1003 reads 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 device 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, the quantum processor 200, etc. The display device 1006 displays a GUI (Graphical User Interface) or the like according to the program. The input device 1007 is composed of a keyboard, mouse, buttons, a touch panel, etc., and is used to input various operation instructions. The output device 1008 outputs the calculation results.

[0181] (Effects of the Embodiment) As described above, the technique described in the present embodiment makes it possible to suppress the influence of noise when calculating the expected value of a physical quantity using the hybrid tensor network method.

[0182] The following additional notes are provided regarding the above-described embodiments.

[0183] <Additional Notes> (Additional Item 1) A quantum computing device comprising: an expected value calculation unit that calculates an expected value of a physical quantity for a quantum state using a hybrid tensor network method; and a noise removal unit that calculates an expected error reduction value by removing the influence of noise from the expected value using a coefficient based on the probability of noise occurrence in the quantum state. (Additional Item 2) The quantum computing device according to Additional Item 1, wherein the noise removal unit uses, as the coefficient, an inverse of a rescale factor that represents the influence of noise on the expected value, and calculates the expected error reduction value by multiplying the expected value by the coefficient. (Additional Item 3) A quantum computing method executed by a quantum computing device, comprising: an expected value calculation step that calculates an expected value of a physical quantity for a quantum state using a hybrid tensor network method; and a noise removal step that calculates the expected error reduction value by removing the influence of noise from the expected value using a coefficient based on the probability of noise occurrence in the quantum state. (Additional Item 4) A non-transitory storage medium storing a program for causing a computer to function as each unit in the quantum computing device according to Additional Item 1 or 2.

[0184] Although the present embodiment has been described above, the present invention is not limited to such a specific embodiment, and various modifications and changes are possible within the scope of the gist of the present invention described in the claims.

[0185] REFERENCE SIGNS LIST 100 Control device 200 Quantum processor 300 Quantum computing device 310 Quantum state preparation unit 320 Expected value calculation unit 330 Noise removal unit 1000 Drive device 1001 Recording medium 1002 Auxiliary storage device 1003 Memory device 1004 CPU 1005 Interface device 1006 Display device 1007 Input device 1008 Output device

Claims

1. A quantum computing device comprising: an expected value calculation unit that calculates an expected value of a physical quantity with respect to a quantum state using a hybrid tensor network method; and a noise removal unit that calculates an error suppression expected value obtained by removing the influence of noise from the expected value using a coefficient based on the probability of noise generation in the quantum state.

2. The quantum computing device according to claim 1, wherein the noise removal unit calculates the error suppression expected value by multiplying the expected value by a coefficient that is the reciprocal of a rescale factor representing the influence of noise on the expected value.

3. A quantum computing method executed by a quantum computing device, the method comprising: an expected value calculation step of calculating an expected value of a physical quantity with respect to a quantum state using a hybrid tensor network method; and a noise removal step of calculating an error suppression expected value obtained by removing the influence of noise from the expected value using a coefficient based on the probability of noise generation in the quantum state.

4. A program for causing a computer to function as each unit in the quantum computing device according to claim 1 or 2.

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