Calculation program, calculation method, and information processing device

Abstract

Reduce the amount of training data used in machine learning. [Solution] The processing unit 12 obtains geometric structure information that shows the geometric structure of a space including data of the first and second classical shadows corresponding to the first and second quantum data, each represented by multiple qubits, and lattice points associated with each qubit. Based on the geometric structure information, the processing unit 12 sets up a plurality of local subspaces, each having a set of local lattice points in that space. The processing unit 12 calculates the value of the first kernel function for each local subspace based on the first and second data elements corresponding to lattice points in the local subspace, from among the plurality of first data elements corresponding to multiple qubits included in the first classical shadow and the plurality of second data elements corresponding to multiple qubits included in the second classical shadow. Based on the value of the first kernel function corresponding to each local subspace, the processing unit 12 calculates the value of the second kernel function corresponding to the first and second classical shadows.

Description

[Technical Field]

[0001] This invention relates to a calculation program, a calculation method, and an information processing device. [Background technology]

[0002] A quantum computer is a computer based on the principles of quantum mechanics. By utilizing quantum superposition states, quantum computers can perform calculations faster than classical computers for certain problems. Classical computers are also called von Neumann architecture computers.

[0003] Here, kernel methods are known as a technique for solving nonlinear problems in machine learning. For example, when solving a nonlinear problem, the data is mapped to a high-dimensional feature space, and the linear problem in that feature space is solved, thereby solving the nonlinear problem in the original space. According to kernel methods, calculations can be performed without actually transforming the data into a high-dimensional space by using a kernel function. A kernel function is a function that represents the similarity between data.

[0004] One known technique for quantum machine learning using quantum computers is the quantum kernel method. The quantum kernel method is a kernel method that uses quantum computers to map data to a quantum feature space, enabling the solution of complex problems. The quantum feature space is generally difficult to simulate efficiently on classical computers. For this reason, quantum kernel methods are expected to have a quantum advantage, and exponential acceleration of computation has been shown for artificial problems. However, problems such as the vanishing similarity problem, where the kernel function values ​​concentrate at a certain point, have been pointed out, and quantum kernel methods have not yet been put into practical use.

[0005] Projection kernel methods and shadow kernel methods have been proposed as types of quantum kernel methods. Projection kernel methods are techniques that obtain reduced density matrices (quantum states reduced to subsystems) of quantum data on a quantum computer and compute kernels between these reduced density matrices on a classical computer. Shadow kernel methods are techniques that obtain classical approximations of quantum states called classical shadows on a quantum computer and compute kernels between classical shadows on a classical computer.

[0006] For example, a method has been proposed using quantum computing devices to compute the corresponding value of a kernel function based on a reduced density matrix for each pair of quantum data points in a training dataset. [Prior art documents] [Patent Documents]

[0007] [Patent Document 1] U.S. Patent Application Publication No. 2024 / 0046137 [Patent Document 2] U.S. Patent Application Publication No. 2023 / 0385674 [Patent Document 3] International Publication No. 2023 / 278462 [Patent Document 4] International Publication No. 2022 / 086918 [Non-patent literature]

[0008] [Non-Patent Document 1] Maria Schuld, et al., “Quantum Machine Learning in Feature Hilbert Spaces”, Physical Review Letters. 122, 040504, February 1, 2019. [Non-Patent Document 2] Yunchao Liu, et al., “A rigorous and robust quantum speed-up in supervised machine learning”, Nature Physics 17, 1013-1017 (2021), July 12, 2021. [Non-Patent Document 3] Hsin-Yuan Huang, et al., “Power of data in quantum machine learning”, Nature Communications 12, Article number: 2631 (2021), May 11, 2021. [Non-Patent Document 4] Hsin-Yuan Huang, et al., “Provably efficient machine learning for quantum many-body problems”, Science vol.377, Issue 6613, September 23, 2022. [Overview of the project] [Problems that the invention aims to solve]

[0009] In existing kernel computation methods such as the shadow kernel method described above, the dimension of the feature space becomes extremely large as the number of qubits increases. Generally, generalizing a learning model requires a number of training data points roughly equal to the number of features. Therefore, as the number of qubits increases, the amount of training data required for generalization also increases. On the other hand, in the near future, quantum computers may not have sufficient computational resources to generate large amounts of quantum data, making it difficult to prepare a sufficient number of training data points.

[0010] In one aspect, the present invention aims to reduce the amount of training data used in machine learning. [Means for solving the problem]

[0011] In one embodiment, a computation program is provided. The kernel computation program causes the computer to perform the following operations: The computer obtains data of a first classical shadow corresponding to first quantum data represented by multiple qubits, data of a second classical shadow corresponding to second quantum data represented by multiple qubits, and geometric structure information indicating the geometric structure of a space containing multiple lattice points associated with multiple qubits. Based on the geometric structure information, the computer sets up multiple local subspaces, each having a set of local lattice points among multiple lattice points in space. The computer calculates the value of a first kernel function for each of the multiple local subspaces, based on the first and second data elements corresponding to lattice points belonging to the local subspace, from among the multiple first data elements corresponding to multiple qubits contained in the first classical shadow and the multiple second data elements corresponding to multiple qubits contained in the second classical shadow. Based on the values ​​of the multiple first kernel functions corresponding to the multiple local subspaces, the computer calculates the value of a second kernel function corresponding to the first and second classical shadows.

[0012] In one embodiment, a method of computation performed by a computer is provided. In one embodiment, an information processing device having a storage unit and a processing unit is provided. [Effects of the Invention]

[0013] One aspect of this is that it can reduce the amount of training data used for machine learning. [Brief explanation of the drawing]

[0014] [Figure 1] This is a diagram illustrating the information processing device of the first embodiment. [Figure 2] This is a flowchart showing an example of processing performed by an information processing device. [Figure 3] This figure shows an example of a quantum computing system according to the second embodiment. [Figure 4] This figure shows an example of the hardware of a quantum computing system. [Figure 5] This figure shows an example of mapping data to a high-dimensional feature space. [Figure 6] This diagram shows examples of the functions of a classical computer. [Figure 7] This figure shows an example of the geometric structure of space. [Figure 8] This figure shows an example of a quantum circuit used to obtain classical shadows. [Figure 9] This figure shows an example of a local subspace. [Figure 10] This is a diagram illustrating the geometric local shadow kernel. [Figure 11] This figure shows an example of adaptive selection of a local subspace. [Figure 12] This is a flowchart showing an example of the learning process. [Figure 13] This is a flowchart showing an example of inference processing. [Figure 14] This is a diagram illustrating the geometric structure used in the experiment. [Figure 15] This figure shows an example of a local subspace. [Figure 16] This is a diagram illustrating support vector machines. [Figure 17] This figure shows an example of the prediction accuracy results. [Figure 18] This figure shows an example of the empirical results of an adaptive algorithm. [Figure 19] This figure shows an example of the empirical results of an adaptive algorithm. [Modes for carrying out the invention]

[0015] This embodiment will be described below with reference to the drawings. [First Embodiment] A first embodiment will be described.

[0016] Figure 1 is a diagram illustrating an information processing device according to the first embodiment. The information processing device 10 is used for machine learning based on quantum data. Quantum data is data represented by multiple qubits. The information processing device 10 is connected to a quantum computer 20. The quantum computer 20 performs quantum operations based on quantum circuits. The quantum computer 20 performs quantum operations using qubits.

[0017] The information processing device 10 has a storage unit 11 and a processing unit 12. The storage unit 11 may be a volatile semiconductor memory such as RAM (Random Access Memory), or a non-volatile storage such as an HDD (Hard Disk Drive) or flash memory. The processing unit 12 is a processor such as a CPU (Central Processing Unit), GPU (Graphics Processing Unit), or DSP (Digital Signal Processor). However, the processing unit 12 may also include application-specific electronic circuits such as an ASIC (Application Specific Integrated Circuit) or FPGA (Field Programmable Gate Array). The processor executes programs stored in memory such as RAM (which may also be the storage unit 11). A collection of multiple processors is sometimes called a "multiprocessor" or simply a "processor".

[0018] Here, examples of quantum kernel methods include shadow kernel methods and projection kernel methods. For example, the shadow kernel method is a technique that obtains the classical shadow of quantum data on a quantum computer 20 and calculates the kernel function values ​​between the classical shadows on a classical computer. A classical shadow is data that combines the measurement results of a random basis for a quantum state ρ with the basis itself. By using classical shadows, it is possible to efficiently approximate certain properties of a quantum state.

[0019] The kernel function used in the shadow kernel method is called the shadow kernel. (shadow) This is expressed by equation (1).

[0020]

Number

[0021] S T (ρ), S T (ρ ~ ) are the classical shadows corresponding to the quantum data ρ, ρ ~ respectively. Here, for example, the notation "ρ ~ " represents the character with a tilde symbol "~" attached above "ρ". S T (ρ) = {σ i (t)}. S T (ρ ~ ) = {σ ~ i (t)}. n is the number of qubits included in the quantum data. {σ i (1) , σ i (2) , …, σ i (T)} records are the data elements of the classical shadow corresponding to the i-th qubit of the quantum data ρ. T is the maximum value of the number of qubit measurements. σ i (1) , σ i (2) , …, σ i (T) Each of them is a 2×2 matrix. τ and γ are real hyperparameters.

[0022] Here, the information processing device 10 can obtain the classical shadow for the quantum data ρ by performing a random Pauli measurement of the quantum data ρ using the quantum computer 20. Specifically, it is as follows.

[0023] First, the quantum computer 20 prepares the quantum data ρ on the quantum computer 20. Next, the quantum computer 20 measures each qubit of ρ in a random basis of X, Y, Z. More specifically, the quantum computer 20, for each qubit of ρ, {I, H, HS† A quantum gate randomly selected from} is applied. I is the identity gate. H is the Hadamard gate. S is the phase gate. At this time, the quantum gate applied to the i-th qubit is U i Let's assume that the quantum computer 20 measures all qubits in the Z basis and projects the state of the i-th qubit as |b i >Let's assume that, then σ i This is expressed by equation (2).

[0024]

number

[0025] In equation (2), I is a 2x2 identity matrix. The quantum computer 20 repeats the above operation on ρ T times. The σ obtained in the tth measurement is i is, σ i (t) It is written as follows. The feature space of the shadow kernel contains a reduced density matrix of any order in any subspace, making it possible to learn the nonlinear properties of quantum states. Furthermore, the shadow kernel method avoids the vanishing similarity problem in some cases because it calculates the shadow kernel after reducing the information to a classical approximation.

[0026] However, with existing methods such as shadow kernel methods and projection kernel methods, the dimension of the feature space restricted to a finite subsystem size and order becomes very large as the number of qubits increases. As a result, the amount of training data required for generalization increases with the number of qubits. Therefore, the information processing device 10 calculates the kernel function as follows.

[0027] The processing unit 12 obtains the data of the first classical shadow corresponding to the first quantum data represented by multiple qubits, and the data of the second classical shadow corresponding to the second quantum data represented by multiple qubits. As described above, the processing unit 12 can obtain the data of the first classical shadow and the data of the second classical shadow using the quantum computer 20. The first classical shadow contains multiple first data elements corresponding to multiple qubits. For example, the first data element corresponding to the i-th qubit is {σ i (1) ,σ i (2) ,…,σ i (T) The second classical shadow contains multiple second data elements corresponding to multiple qubits. For example, the second data element corresponding to the i-th qubit is {σ ~ i (1) ,σ ~ i (2) ,…,σ ~ i (T)}

[0028] Furthermore, the processing unit 12 acquires geometric structure information that shows the geometric structure of a space containing multiple lattice points associated with multiple qubits. The processing unit 12 stores the data of the first classical shadow, the data of the second classical shadow, and the geometric structure information in the storage unit 11. The storage unit 11 stores the data of the first classical shadow, the data of the second classical shadow, and the geometric structure information.

[0029] Here, the geometric structure of space refers to the spatial arrangement of constituent particles such as atoms and molecules in molecules and solids, such as one-dimensional chains, two-dimensional square lattices, and three-dimensional cubic lattices. A one-dimensional chain can also be said to be a one-dimensional lattice. The geometric structure of a material greatly influences its properties. Therefore, incorporating information about the geometric structure into a model can be expected to improve the accuracy of learning. In particular, for many materials, the correlation function decays exponentially as the distance between lattice points in space increases. Therefore, by focusing on local subsystems in space, it becomes possible to efficiently learn the properties of a material.

[0030] However, the functions of the information processing device 10 can be used for purposes other than those mentioned above, such as local pattern recognition and local error detection of images, videos, and quantum data output from the quantum computer 20.

[0031] In one example, geometric structure 30 is a two-dimensional square lattice. In this case, multiple lattice points in the two-dimensional square lattice are pre-associated one-to-one with multiple qubits. The lattice points correspond to the points associated with the qubits.

[0032] The processing unit 12 sets up multiple local subspaces, each having a set of local lattice points in space, based on geometric structure information. Set of local subspaces A GL It is defined by equation (3).

[0033]

number

[0034] The local subspaces 31, 32, 33, ... are examples of local subspaces identified by the identifiers A1, A2, A3, ... For example, each of the local subspaces 31, 32, 33, ... is a square region containing two lattice points on each side. In this case, one local subspace has four lattice points. For example, the processing unit 12 can define the local subspaces 31, 32, 33, ... by shifting the square region by one lattice point along the sides of the geometric structure 30. The union of the local subspaces 31, 32, 33, ... corresponds to the entire space of the geometric structure 30. Note that local subspaces may also be called "local subsystems".

[0035] The shape of a local subspace can be a region with only one lattice point, a linear region with only one edge, or a shape corresponding to the shape of a lattice represented by geometric structure information, such as a square, rectangle, cube, or cuboid, depending on the dimension of the space represented by the geometric structure information.

[0036] The processing unit 12 identifies the first and second data elements that correspond to lattice points belonging to the local subspace from among the multiple first data elements included in the first classical shadow and the multiple second data elements included in the second classical shadow. The "first and second data elements that correspond to lattice points belonging to the local subspace" can also be said to be the "first and second data elements that correspond to qubits associated with lattice points belonging to the local subspace". The processing unit 12 calculates the value of the first kernel function based on the first and second data elements identified for each of the multiple local subspaces.

[0037] For example, with respect to the local subspace 31, the processing unit 12 obtains four lattice points belonging to the local subspace 31. These four lattice points are identified by the qubit indices a, b, c, and d. In this case, the first data element corresponding to the lattice points belonging to the local subspace 31 is {σ a (t)},{σ b (t)},{σ c (t)},{σ d (t)}. The second data element corresponding to a lattice point belonging to the local subspace 31 is {σ ~ a (t)},{σ ~ b (t)},{σ ~ c (t)},{σ ~ d (t)}. Here, for example, {σ a (t) The description in} is {σ a (1) ,σ a (2) ,…,σ a (T) This should indicate}.

[0038] The processing unit 12, for example, with respect to the local subspace 31, {σ a (t)},{σ b(t) , {σ c (t) , {σ d (t)} and {σ ~ a (t) , {σ ~ b (t) , {σ ~ c (t) , {σ ~ d (t)}, and based on this, the value of the first kernel function is calculated. The first kernel function is, for example, a shadow kernel. The processing unit 12 calculates the value of the first kernel function for the local subspaces 32, 33,... in the same way as for the local subspace 31.

[0039] Then, the processing unit 12 calculates the value of the second kernel function corresponding to the first classical shadow and the second classical shadow based on the values of the plurality of first kernel functions corresponding to the plurality of local subspaces.

[0040] For example, when using a shadow kernel as the first kernel function, the second kernel function is called a geometrically local shadow kernel (GLSK), and k GLSK is expressed as. k GLSK is defined as the sum or product of the shadow kernels for each local subspace. The GLSK defined by the sum is expressed by Equation (4).

[0041] [Equation]

[0042] |A| is the number of qubits included in the local subspace A. τ A is a hyperparameter that adjusts the degree of contribution of the reduced density matrix in each local subspace. γ A is a hyperparameter that adjusts the degree of contribution of the spatial spread in each local subspace. c AThis is a hyperparameter that adjusts the contribution of each local subspace.

[0043] The GLSK defined by the product is expressed by equation (5).

[0044]

number

[0045] |A GL | represents the set A of local subspaces. GL This is the total number of local subspaces A1, A2, ... contained within. In GLSK, which is defined as a product, 1 / |A is used for normalization. GL A | symbol is added.

[0046] Note that the Σc in equation (4) A Σc in exp(...) A The exp(...) part after the exp(...) corresponds to the first kernel function for the local subspace A, and in the example of equation (4), it is the shadow kernel. The exp(...) part after the π in Πexp(...) in equation (5) corresponds to the first kernel function for the local subspace A, and in the example of equation (5), it is the shadow kernel.

[0047] The first kernel function defined for a local subspace A is not limited to the shadow kernel; it can also be a fidelity kernel or a projection kernel. Quantum states ρ and ρ ~ The fidelity kernel is their inner product tr(ρρ ~ It is defined as follows. For the projection kernel, refer to Non-Patent Document 3 and Patent Document 4 mentioned above.

[0048] The processing unit 12 acquires multiple classical shadow data for multiple quantum data included in the training dataset and calculates a GLSK value for each pair of classical shadows. Based on the GLSK values ​​calculated for each pair of classical shadows, the processing unit 12 can generate a model that outputs predicted values ​​such as classification results for classical shadows through machine learning. For machine learning, a kernel-based machine learning algorithm such as a support vector machine is used. The model may also be called a learning model, AI (Artificial Intelligence) model, machine learning model, or prediction model.

[0049] In "GLSK defined by summation," the feature vector determined by the kernel function is the "reduced density matrix ρ in each local subspace A," as shown in equation (6). A It includes the sum of polynomials.

[0050]

number

[0051] In "GLSK defined by product," the feature vector determined by the kernel function is the "reduced density matrix ρ of all local subspaces A1, A2, ..." as shown in equation (7). A_1 ρ A_2 It includes a polynomial of ..., where "A_1" represents A1.

[0052]

number

[0053] Therefore, the expressive power of feature vectors is "GLSK defined by sum < GLSK defined by product". For example, if it is known in advance that the features to be learned can be expressed by equation (6), then "GLSK defined by sum" can be used, and if it is not known, then "GLSK defined by product" can be used. If a problem can be solved with "GLSK defined by sum", then using "GLSK defined by sum" requires less training data than using "GLSK defined by product".

[0054] Furthermore, it may not be known in advance how the set of local subspaces should be defined. Therefore, the processing unit 12 may execute an adaptive algorithm that adaptively changes the size or shape of a local subspace to reduce the training error or validation error of the learned model obtained for a local subspace of a certain size or shape. The processing unit 12 can perform computations on any local subspace by using classical shadows. For this reason, the adaptive algorithm does not increase the quantum computation cost on the quantum computer 20.

[0055] An example of an adaptive algorithm is to "keep the size of the local subspace relatively small in the initial stages, and increase the size of the local subspace if the validation error exceeds a certain threshold." For example, the size of local subspaces 31, 32, 33, ... can be determined by the number of lattice points contained in one edge. In this case, the minimum number of lattice points contained in one edge is 1. When changing the size, the processing unit 12 may increase the number of lattice points contained in both the vertical and horizontal edges, or it may increase the number of lattice points contained in either the vertical or horizontal edge. For example, if it is known that there is a relatively strong correlation in a particular direction in the geometric structure, the local subspace may be expanded only in that direction. Furthermore, when changing the size of the local subspace, the processing unit 12 can also adaptively change the shape of the local subspace, such as expanding the local subspace in a different direction if expanding only in a specific direction does not sufficiently improve the prediction accuracy of the learning model.

[0056] Next, the processing procedure of the information processing device 10 will be explained. Figure 2 is a flowchart showing an example of processing performed by an information processing device. (S1) The processing unit 12 uses the quantum computer 20 to obtain classical shadows for two quantum data. Here, during machine learning, the processing unit 12 obtains classical shadows for each of the multiple quantum data given as the training dataset.

[0057] (S2) Processing unit 12 controls the set A of local subspaces A. GL The processing unit 12 sets the size of the local subspace relatively small in the initial stage. For example, the processing unit 12 may minimize the size of the local subspace in the initial stage. Then, the process proceeds to step S3a or step S3b. For example, whether to use step S3a or S3b is predetermined depending on the problem.

[0058] (S3a) The processing unit 12 computes a geometric local kernel defined as a sum. The geometric local kernel corresponds to the second kernel function. GLSK is an example of a geometric local kernel. For example, GLSK defined as a sum is expressed by equation (4). The processing unit 12 computes a geometric local kernel for each pair of classical shadows in the multiple classical shadows used as training data. Then the process proceeds to step S4.

[0059] (S3b) The processing unit 12 calculates a geometric local kernel defined by a product. For example, a GLSK defined by a product is expressed by equation (5). The processing unit 12 calculates a geometric local kernel for each pair of classical shadows in the multiple classical shadows used as training data. Then the process proceeds to step S4.

[0060] (S4) The processing unit 12 adaptively selects the size of the local subspace. Specifically, the processing unit 12 generates a learning model that outputs predicted values ​​for classical shadow inputs by machine learning using the calculated geometric local kernel values ​​and training data. The processing unit 12 changes the size of the local subspace based on the accuracy of the predicted values ​​by the learning model. For example, the training error using the training data or the validation error on multiple classical shadows prepared as validation data can be used to evaluate the accuracy of the predicted values. In one example, the processing unit 12 increases the size of the local subspace if the training error or validation error exceeds a certain threshold.

[0061] The processing unit 12 then executes steps S2, S3a, or S3b to generate a learning model, evaluate the accuracy of the predictions made by the learning model, and resize the local subspace. This cycle is repeated until the training error or validation error falls below a threshold. Once the training error or validation error falls below the threshold, the process terminates.

[0062] Furthermore, due to the possibility of overfitting, it is not always possible to accurately evaluate accuracy using the training data. For this reason, in the adaptive algorithm of step S4, it is preferable to use the validation error from the validation data rather than the training error.

[0063] Furthermore, as a method for executing steps S3a and S3b, the processing unit 12 may first execute step S3a, and if the prediction accuracy of the learning model does not reach the standard with the kernel defined as a sum, it may execute step S3b to improve the prediction accuracy with the kernel defined as a product.

[0064] The processing unit 12 stores the information of the finally obtained learning model in the storage unit 11. Then, the processing unit 12 can obtain predicted values ​​for new quantum data using the learning model. When performing inference using the learning model, the processing unit 12 can also set a local subspace of the size determined during training and compute a geometric local kernel such as GLSK.

[0065] According to the information processing device 10 of the first embodiment, data of a first classical shadow corresponding to first quantum data represented by multiple qubits and data of a second classical shadow corresponding to second quantum data represented by multiple qubits are obtained. Geometric structure information showing the geometric structure of a space including multiple lattice points associated with multiple qubits is obtained. Based on the geometric structure information, multiple local subspaces are set up, each having a set of local lattice points in the space. For each of the multiple local subspaces, the value of the first kernel function based on the first data elements and second data elements corresponding to lattice points belonging to the local subspace is calculated, from among the multiple first data elements corresponding to multiple qubits included in the first classical shadow and the multiple second data elements corresponding to multiple qubits included in the second classical shadow. Based on the values ​​of the multiple first kernel functions corresponding to the multiple local subspaces, the value of the second kernel function corresponding to the first classical shadow and the second classical shadow is calculated.

[0066] This allows the information processing device 10 to reduce the amount of training data used for machine learning. As mentioned above, for example, the geometric structure of space greatly influences the properties of a material. In many materials, the correlation function between lattice points decays exponentially as the distance between lattice points increases. Therefore, by focusing on local subsystems within space, i.e., local subspaces, it becomes possible to efficiently learn the properties of a material. For this reason, by incorporating information about the geometric structure into the learning model, the accuracy of learning can be improved with less training data compared to when the geometric structure is not considered.

[0067] Furthermore, the functions of the information processing device 10 can be applied not only to materials, but also to local pattern recognition and local error detection of images, videos, quantum data output from quantum computers, and other materials. In these applications as well, the information processing device 10 can reduce the amount of training data used for machine learning.

[0068] [Second Embodiment] Next, a second embodiment will be described. Figure 3 shows an example of a quantum computing system according to the second embodiment. The quantum computing system 300 is a computer system using quantum devices. The quantum computing system 300 has a classical computer 100 and a quantum computer 200. Terminal devices 401, 402, ... are connected to the classical computer 100 via a network 40. Terminal devices 401, 402, ... are computers operated by users of the quantum computing system 300. The classical computer 100 receives computation requests including quantum circuits from terminal devices 401, 402, ... A quantum circuit indicates the sequence of operations on qubits by the arrangement of elements such as quantum gates. A qubit is a bit that can represent a superposition state of a "0" state and a "1" state.

[0069] The classical computer 100 instructs the quantum computer 200 to perform quantum computation according to the quantum circuits received from terminal devices 401, 402, ... The classical computer 100 also obtains the measurement results of each qubit from the quantum computer 200. The classical computer 100 is an example of the information processing device 10.

[0070] The quantum computer 200 has multiple qubits and devices for manipulating each of the multiple qubits. The multiple qubits of the quantum computer 200 can be realized using methods such as superconductivity, ion traps, or diamond spins. The quantum computing system 300 may have multiple quantum computers, including the quantum computer 200.

[0071] Figure 4 shows an example of the hardware of a quantum computing system. The classical computer 100 is controlled as a whole by a processor 101. The processor 101 is connected to memory 102 and several peripheral devices via a bus 109. The processor 101 may be a multiprocessor. The processor 101 is, for example, a CPU, an MPU (Micro Processing Unit), or a DSP. At least some of the functions realized by the processor 101 executing a program may be realized by electronic circuits such as ASICs or PLDs (Programmable Logic Devices). The processor that executes one of the multiple processes performed by the classical computer 100 may be different from the processor that executes a different process from the multiple processes. In addition, at least some of the processes described below may be executed in parallel using multiple processors or processor cores. The processor 101 may also be called "processor circuitry". The processor 101 is an example of the processing unit 12 of the first embodiment.

[0072] Memory 102 is used as the main memory of the classical computer 100. Memory 102 temporarily stores at least a portion of the OS (Operating System) program and application programs that are to be executed by the processor 101. Memory 102 also stores various data used for processing by the processor 101. For memory 102, a volatile semiconductor memory device such as RAM is used.

[0073] Peripheral devices connected to bus 109 include storage device 103, GPU 104, input interface 105, optical drive device 106, device connection interface 107, and network interfaces 108a and 108b.

[0074] The storage device 103 electrically or magnetically writes and reads data from its built-in recording medium. The storage device 103 is used as an auxiliary storage device for the classical computer 100. The storage device 103 stores the OS program, application programs, and various data. For example, an HDD or SSD (Solid State Drive) can be used as the storage device 103. The memory 102 or storage device 103 is an example of the storage unit 11 in the first embodiment.

[0075] The GPU104 is a processing unit that performs image processing. The GPU104 is an example of a graphics controller. A monitor 41 is connected to the GPU104. The GPU104 displays images on the screen of the monitor 41 according to instructions from the processor 101. The monitor 41 can be a display device using organic EL (Electro Luminescence) or a liquid crystal display device.

[0076] The input interface 105 is connected to a keyboard 42 and a mouse 43. The input interface 105 transmits signals from the keyboard 42 and mouse 43 to the processor 101. Note that the mouse 43 is just one example of a pointing device; other pointing devices can also be used. Other pointing devices include touch panels, tablets, touchpads, and trackballs.

[0077] The optical drive device 106 uses laser light or the like to read data recorded on the optical disc 44 or write data to the optical disc 44. The optical disc 44 is a portable recording medium on which data is recorded in a way that makes it readable by the reflection of light. Examples of optical discs 44 include DVD (Digital Versatile Disc), DVD-RAM, CD-ROM (Compact Disc Read Only Memory), and CD-R (Recordable) / RW (ReWritable).

[0078] The device connection interface 107 is a communication interface for connecting peripheral devices to the classical computer 100. For example, a memory device 45 and a memory reader / writer 46 can be connected to the device connection interface 107. The memory device 45 is a recording medium equipped with a communication function with the device connection interface 107. The memory reader / writer 46 is a device that writes data to or reads data from the memory card 47. The memory card 47 is a card-type recording medium.

[0079] The network interface 108a is connected to the network 40. The network interface 108a transmits and receives data to and from other computers or communication devices via the network 40. The network interface 108a is a wired communication interface, for example, connected by cable to a wired communication device such as a switch or router. Alternatively, the network interface 108a may be a wireless communication interface, connected by radio waves to a wireless communication device such as a base station or access point.

[0080] The network interface 108b is an interface for connecting to the quantum computer 200. The processor 101 transmits quantum circuits to the quantum computer 200 via the network interface 108b, causing the quantum computer 200 to perform quantum computations. The processor 101 also obtains the results of the quantum computations via the network interface 108b.

[0081] The classical computer 100 can realize the processing functions of the second embodiment using the hardware described above. The apparatus shown in the first embodiment can also be realized using hardware similar to that of the classical computer 100 shown in Figure 4.

[0082] The classical computer 100 implements the processing functions of the second embodiment by executing a program recorded on a computer-readable recording medium, for example. The program describing the processing to be executed by the classical computer 100 can be recorded on various recording media. For example, the program to be executed by the classical computer 100 can be stored in the storage device 103. The processor 101 loads at least a portion of the program in the storage device 103 into the memory 102 and executes the program. Alternatively, the program to be executed by the classical computer 100 can be recorded on a portable recording medium such as an optical disc 44, a memory device 45, or a memory card 47. The program stored on the portable recording medium becomes executable after being installed in the storage device 103, for example, under control from the processor 101. The processor 101 can also directly read and execute the program from the portable recording medium.

[0083] The quantum computer 200 comprises a control device 210 and a quantum device 220. The control device 210 performs gate operations on qubits within the quantum device according to a quantum circuit. The quantum device 220 has multiple qubits. The quantum device 220 is, for example, one or more QPUs (Quantum Processing Units).

[0084] Figure 5 shows an example of mapping data to a high-dimensional feature space. When solving nonlinear problems using machine learning techniques, the nonlinear problem in the original space can be solved by mapping the data to a high-dimensional feature space and solving the linear problem in that feature space.

[0085] For example, Graph 50 plots each data point using two features x and y. Data points plotted as black circles belong to one class. Data points plotted as white circles belong to another class. However, the black and white circles in Graph 50 cannot be classified using a linear function. In other words, the black and white circles in Graph 50 are not linearly separable.

[0086] On the other hand, Graph 51 plots each data point using three features x, y, and z. z = x 2 +y 2 Therefore, the black and white points represented in Graph 51 can be classified on a plane. That is, the black and white points represented in Graph 51 are linearly separable. In this way, by solving a linear problem in a feature space of a higher dimension than the original feature space, we can solve the nonlinear problem in the original feature space.

[0087] However, explicitly handling high-dimensional features can lead to computationally intensive tasks. Therefore, classical computer 100 can perform machine learning calculations without actually transforming the data into a high-dimensional feature space by using kernel methods.

[0088] Figure 6 shows an example of the functions of a classical computer. The classical computer 100 has an input data storage unit 110, a trained data storage unit 120, a classical shadow acquisition unit 130, a local subspace setting unit 140, a kernel calculation unit 150, a training processing unit 160, and an inference processing unit 170.

[0089] The input data storage unit 110 stores various types of data that are input to the classical computer 100. The data stored in the input data storage unit 110 includes geometric structure information that shows the geometric structure of space, and information that associates each qubit in the quantum data with a point in that space. The geometric structure of space is, for example, a one-dimensional chain, a two-dimensional square lattice, and a three-dimensional cubic lattice that show the spatial arrangement of constituent particles such as atoms and molecules in molecules and solids. The points associated with qubits in that space are, for example, lattice points in lattices such as a one-dimensional chain, a two-dimensional square lattice, and a three-dimensional cubic lattice. The data stored in the input data storage unit 110 includes classical shadows corresponding to the quantum data, which are obtained using the quantum computer 200.

[0090] The trained data storage unit 120 stores the trained data from the training processing unit 160. The trained data includes information about the trained model learned by the training processing unit 160. The classical shadow acquisition unit 130 uses the quantum computer 200 to acquire classical shadow data for the quantum data and stores it in the input data storage unit 110. Each of the multiple qubits representing the quantum data is pre-associated with a point in space indicated by geometric structure information, i.e., a lattice point.

[0091] During training, the classical shadow acquisition unit 130 acquires classical shadow data (training classical shadows) to be used as training data for each quantum data in the training quantum dataset. The classical shadow acquisition unit 130 also acquires classical shadow data (validation classical shadows) to be used as validation data for each quantum data in the validation quantum dataset. For example, in a classification problem, the training data includes multiple training classical shadows and the labels of the classes to which each training classical shadow belongs. The validation data also includes multiple validation classical shadows and the labels of the classes to which each validation classical shadow belongs. The classical shadow acquisition unit 130 acquires the training data and validation data, including the classical shadows, and stores them in the input data storage unit 110.

[0092] During inference, the classical shadow acquisition unit 130 acquires classical shadow data for the quantum data to be predicted. The local subspace setting unit 140 sets up a local subspace within the space indicated by the geometric structure information. As will be described later, the local subspace setting unit 140 adaptively determines the size of the local subspace according to the accuracy of the prediction by the learning model.

[0093] The kernel computation unit 150 computes the GLSK for the set local subspace and classical shadow pairs. During training, the kernel computation unit 150 computes the GLSK for each pair of training classical shadows in multiple training classical shadows.

[0094] During inference, the kernel computation unit 150 computes the GLSK for each pair of classical shadows to be predicted and each training classical shadow. Depending on the problem, the GLSK used may be either the "GLSK defined by sum" in equation (4) or the "GLSK defined by product" in equation (5).

[0095] The learning processing unit 160 uses GLSK for each pair of training classical shadows to learn training data using a predetermined machine learning algorithm and generates a learning model to be used for inference. For example, a kernel method-based algorithm such as a support vector machine can be used as the machine learning algorithm. Through machine learning, the learning processing unit 160 determines the parameter values ​​in the learning model. The learning processing unit 160 stores the information of the learning model, including the determined parameter values, as learned data in the learned data storage unit 120.

[0096] Here, the local subspace setting unit 140 evaluates the prediction accuracy of the learning model using the validation data. The local subspace setting unit 140 updates the size of the local subspace according to the prediction accuracy and resets the local subspace. Then, the local subspace setting unit 140 causes the kernel calculation unit 150 to perform the GLSK calculation during learning and the learning processing unit 160 to perform the learning of the training data again. When validation error is used to evaluate the prediction accuracy, the kernel calculation unit 150 calculates the GLSK value for each pair of training classical shadows and validation classical shadows, and uses these GLSK values ​​to calculate predicted values ​​such as labels based on the learning model for the validation classical shadows.

[0097] Thus, the cycle of updating the local subspace size by the local subspace setting unit 140, performing GLSK calculations by the kernel calculation unit 150, and performing learning processing by the learning processing unit 160 is repeated until the prediction accuracy of the learning model reaches a certain standard. When the prediction accuracy of the learning model reaches the standard, the cycle ends and the generation of the learning model is completed.

[0098] The inference processing unit 170 performs inference processing based on the learned model using GLSK for each pair of classical shadows to be predicted and each training classical shadow, and outputs predicted values ​​such as labels obtained for the classical shadows to be predicted. The inference processing unit 170 can obtain the learned model to be used for inference processing by referring to the trained data storage unit 120.

[0099] Figure 7 shows examples of geometric structures in space. Figure 7(A) illustrates a one-dimensional chain 61. A one-dimensional chain 61 can also be called a one-dimensional lattice. A one-dimensional chain 61 is a geometric structure in which lattice points are periodically arranged on line segments. In the figure, the points represented by white circles are lattice points. Lattice point 61a is one of the lattice points of the one-dimensional chain 61. Figure 7(B) illustrates a two-dimensional square lattice 62. A two-dimensional square lattice 62 is a geometric structure in which lattice points are periodically arranged at the vertices of a square. Figure 7(C) illustrates a three-dimensional cubic lattice 63. A three-dimensional cubic lattice 63 is a geometric structure in which lattice points are periodically arranged at the vertices of a cube. The geometric structure of space may also be an L-dimensional lattice, where L is an integer greater than or equal to 1. Note that the geometric structure may also be other structures such as face-centered cubic lattices, body-centered cubic lattices, or hexagonal close-packed structures.

[0100] The geometric structure of such a space corresponds, for example, to the arrangement of constituent particles such as atoms and molecules in a material. This geometric structure in a material greatly influences its properties. Therefore, incorporating information about the geometric structure into machine learning models can be expected to improve the accuracy of learning. In particular, in many materials, the correlation function decays exponentially as the distance between lattice points in space increases. For this reason, by focusing on local subsystems within space, i.e., local subspaces, it is possible to efficiently learn the properties of a material.

[0101] However, the capabilities of classical computer 100 can be used for purposes other than those mentioned above, such as local pattern recognition and local error detection of images, videos, and quantum data output from quantum computers.

[0102] Geometric structure information, which shows the geometric structure of the space dealt with in the problem, and information showing the correspondence between lattice points and qubits, are pre-input into the classical computer 100 and stored in the input data storage unit 110.

[0103] Figure 8 shows an example of a quantum circuit used to acquire classical shadows. The classical shadow acquisition unit 130 uses the quantum computer 200 to acquire classical shadow data for quantum data ρ as follows. First, quantum data ρ is prepared on the quantum computer 200. Let n be the number of qubits in the quantum data.

[0104] Quantum computer 200 measures each qubit of the quantum data ρ with a random basis of X, Y, Z. Quantum circuit 230 illustrates a quantum circuit used in quantum computer 200 to obtain a classical shadow for the quantum data ρ. For each qubit of the quantum data ρ, quantum computer 200 measures {I, H, HS}. † A quantum gate U is randomly selected from}. i Apply the following: Quantum computer 200 measures all qubits in the Z basis. The state of the i-th qubit projected by the measurement in the Z basis is |b i >Then, for each i, the quantum computer 200 calculates the σ of equation (2) i Obtain it.

[0105] Quantum computer 200 repeats the above operations on quantum data ρ T times. T is called the number of classical shadow shots. The classical shadow obtained in the tth measurement is σ i (t) And write the classical shadow {σ} for the quantum data ρ. i (t) The data of {σ} (i=1~n, t=1~T) is sent to the classical computer 100. The classical shadow acquisition unit 130 receives the classical shadow {σ} from the classical computer 100. i (t) Retrieve the data for}.

[0106] Figure 9 shows an example of a local subspace. The local subspace setting unit 140 sets a set of local subspaces A based on geometric structure information. GL Set {A1, A2, A3, ...}. Figure 9 shows examples of local subspaces 71, 72, 73, ... defined for a two-dimensional square lattice 70. Local subspaces 71, 72, 73, ... are examples of local subspaces identified by the identifiers A1, A2, A3, ... respectively. Local subspaces may also be called local subsystems.

[0107] The local subspace setting unit 140 determines the size, width, and shape of the local subspace according to the characteristics of the data to be trained. For example, the shape of the local subspace may be rectangular for a two-dimensional square grid 70. The local subspace setting unit 140 may also adaptively change the width and shape of the local subspace to reduce training errors and validation errors.

[0108] Figure 10 is a diagram illustrating the geometric local shadow kernel. The kernel calculation unit 150 uses two quantum data ρ,ρ ~ Classic Shadow S T (ρ) = {σ i (t)} and S T (ρ ~ )={σ ~ i (t) For each local subspace, the kernel calculation unit 150 calculates shadow kernels for each of the local subspaces 71, 72, 73, ... in the two-dimensional square grid 70.

[0109] The kernel calculation unit 150 calculates the "GLSK defined by sum" by summing the shadow kernels for each local subspace, as shown in equation (4). Alternatively, the kernel calculation unit 150 calculates the "GLSK defined by product" by multiplying the shadow kernels for each local subspace, as shown in equation (5).

[0110] As mentioned above, equations (4) and (5) are hyperparameters τ that adjust the order of the reduced density matrix in the local subspace. A This includes. Equations (4) and (5) are hyperparameters γ that adjust the contribution of spatial extent in local subspaces. A This includes the following. Also, equation (4) is a hyperparameter c that adjusts the contribution of each local subspace. A Includes.

[0111] As explained in equations (6) and (7), the expressive power of feature vectors is "GLSK defined by sum < GLSK defined by product". If it is known in advance that the features to be learned can be written using equation (6), then "GLSK defined by sum" should be used; otherwise, "GLSK defined by product" should be used. If a problem can be solved using "GLSK defined by sum", then using "GLSK defined by sum" requires less training data than using "GLSK defined by product".

[0112] Figure 11 shows an example of adaptive selection of a local subspace. The local subspace setting unit 140 executes an adaptive algorithm that adaptively changes the size and shape of a local subspace to reduce the training error and validation error of the learned model obtained for a local subspace of a certain size or shape.

[0113] By using classical shadows, it is possible to perform GLSK computations on any local subspace. Therefore, this adaptive algorithm does not increase the quantum computation cost on the quantum computer 200.

[0114] An example of an adaptive algorithm is to "keep the size of the local subspace relatively small in the initial stages, and then increase the size of the local subspace if the validation error exceeds a certain threshold."

[0115] For example, the local subspace setting unit 140 determines the size of the local subspace for a two-dimensional square grid 70 as follows: First, the local subspace setting unit 140 sets the smallest size local subspace (step ST1). The number of grid points belonging to the smallest size local subspace is 1. Local subspace 71a is an example of the smallest size local subspace.

[0116] The local subspace setting unit 140 detects that the prediction accuracy of the learning model does not meet the standard, i.e., learning has failed, as a result of learning using a local subspace of the size set in step ST1. Then, the local subspace setting unit 140 expands the size of the local subspace (step ST2). For example, the local subspace setting unit 140 increases the number of grid points on the vertical and horizontal sides of the local subspace, which is represented as a square, by one each. As a result, the number of grid points on each side of the local subspace becomes 2. Also, the number of grid points belonging to the local subspace becomes 4. Local subspace 71b is an example of a local subspace having 4 grid points.

[0117] The local subspace setting unit 140 detects that the learning process has failed using a local subspace of the size set in step ST2. In response, the local subspace setting unit 140 expands the size of the local subspace (step ST3). For example, the local subspace setting unit 140 increases the number of grid points on both the vertical and horizontal edges of the local subspace by one. This results in the number of grid points on each edge of the local subspace becoming 3. Additionally, the total number of grid points belonging to the local subspace becomes 9. Local subspace 71c is an example of a local subspace having 9 grid points.

[0118] The local subspace setting unit 140 then detects that the prediction accuracy of the learning model meets the criteria as a result of learning using the local subspace of the size set in step ST3, i.e., that learning has been successful. The local subspace setting unit 140 then sets the size of the local subspace to its current size and terminates the above cycle.

[0119] Here, when changing the size of the local subspace, the local subspace setting unit 140 may increase the number of grid points included in all edges representing the shape of the local subspace, or it may increase the number of grid points included in some of those edges. For example, if it is known that there is a relatively strong correlation in a particular direction, the local subspace setting unit 140 may expand the local subspace only in that direction. Furthermore, when changing the size of the local subspace, the local subspace setting unit 140 can also adaptively change the shape of the local subspace by expanding it in a different direction if expanding it only in a particular direction does not sufficiently improve the prediction accuracy of the learning model.

[0120] Furthermore, when the local subspace coincides with the entire space represented by geometric structure information, the GLSK in equations (4) and (5) becomes the same as the shadow kernel. Next, we will explain the processing procedure of the quantum computing system 300. First, we will explain the procedure of the learning phase.

[0121] Figure 12 is a flowchart showing an example of the learning process. (S10) The quantum computer 200 acquires a training quantum dataset and a validation quantum dataset.

[0122] (S11) The quantum computer 200 acquires classical shadows of quantum data. Specifically, the quantum computer 200 acquires multiple classical shadows of training data to be used as training data for multiple training quantum data included in the training quantum dataset. The quantum computer 200 also acquires multiple classical shadows of verification data to be used as verification data for multiple verification quantum data included in the verification quantum dataset. The classical shadow acquisition unit 130 acquires the data of multiple classical shadows of training data and multiple classical shadows of verification data from the quantum computer 200.

[0123] (S12) The local subspace setting unit 140 sets a set of local subspaces A based on the geometric structure information stored in the input data storage unit 110. GLThe following is set. When step S12 is executed for the first time, the initial size of the local subspace is predetermined. The initial size may be, for example, the minimum size. When step S12 is executed for the second time or later, the size of the local subspace is expanded compared to the previous time. The geometric structure information corresponding to the problem is stored in advance in the input data storage unit 110. In addition, information on the correspondence between lattice points in the geometric structure and qubits in the quantum data is stored in advance in the input data storage unit 110.

[0124] (S13) The kernel calculation unit 150 sets the values ​​of the hyperparameters. For example, if equation (4) is used in the calculation of GLSK, the kernel calculation unit 150 sets the hyperparameter τ of equation (4). A γ A ,c A The value of is set. When equation (5) is used in the GLSK calculation, the kernel calculation unit 150 sets the hyperparameter τ of equation (5). A γ A Set the value.

[0125] (S14) The kernel calculation unit 150 calculates the GLSK between the training data, i.e., for each pair of training classical shadows. The learning processing unit 160 learns the training data using the GLSK. In this learning process, various combinations of values ​​may be tried as combinations of hyperparameter values ​​included in the learning model, for example by grid search.

[0126] (S15) The local subspace setting unit 140 calculates the prediction accuracy for the validation data using the trained model, i.e., the trained model. For example, the local subspace setting unit 140 uses the inference processing unit 170 to obtain the prediction result of the trained model for each validation classical shadow. The local subspace setting unit 140 verifies whether the prediction result matches the label of the validation classical shadow included in the validation data, i.e., whether the prediction result is correct. The local subspace setting unit 140 calculates the prediction accuracy as the percentage of validation classical shadows for which the prediction result was determined to be correct.

[0127] (S16) The local subspace setting unit 140 determines whether the prediction accuracy meets the criteria. If the prediction accuracy meets the criteria, the learning process ends. If the prediction accuracy does not meet the criteria, the process proceeds to step S17. For example, the local subspace setting unit 140 determines that the prediction accuracy meets the criteria if the prediction accuracy is greater than the threshold. The local subspace setting unit 140 determines that the prediction accuracy does not meet the criteria if the prediction accuracy is less than or equal to the threshold. A threshold representing the criteria that the prediction accuracy must meet is predetermined.

[0128] (S17) The local subspace setting unit 140 sets the set of local subspaces A GL The local subspace setting unit 140 updates the local subspace. For example, the local subspace setting unit 140 increases the size of the local subspace. Then the process proceeds to step S12.

[0129] In step S15, the local subspace setting unit 140 may, for example, use the proportion of verification classical shadows for which the prediction result was determined to be incorrect for all verification classical shadows as the prediction error to make the determination in step S16. In that case, the local subspace setting unit 140 determines that the criteria are met if the prediction error is less than the threshold, and that the criteria are not met if the prediction error is greater than or equal to the threshold.

[0130] In machine learning, data is sometimes divided into three types: training data, validation data, and test data. Training data is the data that the machine learning model directly learns from. Validation data is used to evaluate the accuracy of the model that has learned from the training data, and to perform hyperparameter tuning and model selection. Test data is used to evaluate the final accuracy of the model after the model has been trained and hyperparameters have been tuned using the training and validation data. In the accuracy evaluation in steps S16 and S17, there is a possibility of overfitting, so the training data may not be able to accurately evaluate the accuracy. For this reason, it is preferable to use validation data rather than training data for the accuracy evaluation in steps S16 and S17. However, training data may also be used for accuracy evaluation.

[0131] Next, we will explain the procedure for the inference phase. Figure 13 is a flowchart showing an example of the inference process. (S20) Quantum computer 200 acquires unknown quantum data to be predicted.

[0132] (S21) The quantum computer 200 acquires the classical shadow of the unknown quantum data. The classical shadow acquisition unit 130 acquires the classical shadow data corresponding to the unknown quantum data from the quantum computer 200.

[0133] (S22) The kernel calculation unit 150 calculates the GLSK between the training data and the unknown data. Specifically, the kernel calculation unit 150 calculates the GLSK for each pair of training classical shadows and each pair of unknown classical shadows. The formula used in step S13 is used to calculate the GLSK in step S22. The size of the local subspace is the size finally determined by the local subspace setting unit 140 during the learning phase.

[0134] (S23) The inference processing unit 170 performs a prediction on unknown data, i.e., an unknown classical shadow, based on the calculation results of GLSK and the trained parameters in the learning model. The inference processing unit 170 outputs the prediction result. Then the inference process ends.

[0135] Furthermore, in step S15 of Figure 12, the inference processing unit 170 can obtain prediction results for the validation data using the same procedure as in Figure 13. In that case, "unknown data" in Figure 13 should be read as "validation data".

[0136] Next, we will explain the processing of the classical computer 100 by illustrating a specific problem. Figure 14 illustrates the geometric structure used in the experiment. The geometric structure dealt with in the problem is a one-dimensional chain 80. The experiment was conducted to classify the quantum data corresponding to the one-dimensional chain 80 into two classes, A and B. Class A is: <z1z2z3>It is a random n-qubit state with value = +1. Class B is, <z1z2z3>This is a random n-qubit state where the value is -1. <z1z2z3>=+1 indicates that the expectation value of the state for three locally consecutive qubits is +1. <z1z2z3>=-1 indicates that the expectation value of the state for three locally consecutive qubits is -1.

[0137] Local subspace 81 is an example of a local subspace. In this problem, the local features of quantum data <z1z2z3>Since the class is determined by this, GLSK, which reflects the locality of the space, is expected to show superior results.

[0138] The number of training data points used in the experiment is N, and the number of test data points is 200. The training data is the training classical shadow data. The test data is the test classical shadow data, and as mentioned above, it is not used to generate the learning model. The validation data used for the adaptive algorithm is given separately. A portion of the test data may be used as validation data. Also, the number of shots for the classical shadow is T = 1000.

[0139] Figure 15 shows an example of a local subspace. Local subspaces 81, 82, 83, ... are examples of local subspaces in a one-dimensional chain 80. Local subspaces 81, 82, 83, ... are identified by the identifiers A1, A2, A3, ... respectively. w is the size of the local subspace. The i-th local subspace A i is a local subspace A i The set of indices of lattice points possessed by, or the local subspace A i It can be represented as a set of indices of qubits corresponding to lattice points in the local subspace A. i It is defined by equation (8) using size w.

[0140]

number

[0141] Equation (8) is, A i It can also be said that this is a set of indices of qubits corresponding to lattice points belonging to the . As shown in equation (8), the local subspace setting unit 140 may set the local subspace with boundary conditions that cycle from the terminal index to the first index.

[0142] Figure 16 illustrates a support vector machine. A support vector machine is an example of a machine learning algorithm used by classical computers. A support vector machine is a method for finding a hyperplane that classifies classes in a feature space. For example, the scikit-learn library in Python can be used as a support vector machine library.

[0143] The classical computer 100 classifies the aforementioned classes A and B using the hyperplane f(x)=0, as illustrated in graph 90. Graph 90 represents the feature space. Star-shaped points in graph 90 represent data classified as class A. Triangular points in graph 90 represent data classified as class B.

[0144] Here, the training data is {x i ,y i } i=1 N It is expressed as x i This is the input data. y i is x i This is the label of the class to which it belongs. i If it belongs to class A, then y i = 1 x i If it belongs to class B, then y i = -1

[0145] The distance between a hyperplane and the data point closest to that hyperplane is called the margin. The learning processing unit 160 determines the hyperplane to maximize the margin. In general, it is not always possible to accurately classify all data points using a hyperplane. Therefore, the learning processing unit 160 performs classification while tolerating misclassification. The parameter that controls the degree of misclassification tolerance is called the regularization parameter and is represented by C. The larger the regularization parameter C, the less misclassification is tolerated.

[0146] The margin maximization problem involves the variable α = (α1, ..., α N This can be rewritten as an optimization problem expressed by equation (9) relating k(x). Here, k(x i ,x j ) is x i and x j This is the kernel function between [the specified time period].

[0147]

number

[0148] In equation (9), "st" is an abbreviation for "subject to," indicating a constraint. Furthermore, the hyperplane f(x)=0 is expressed by the following equation (10).

[0149]

number

[0150] M is the set of indices defined by equation (11). |M| is the number of indices contained in M.

[0151]

number

[0152] The inference processing unit 170 processes the unknown data x * If you want to predict the class to which it will be classified, add x to equation (10). * Substitute this into the equation, and f(x * If ) > 0, we predict it to be class A, and f(x * If ) < 0, we predict it to be class B.

[0153] In the following, a method using the existing shadow kernel represented by equation (1) as the kernel function in equation (9) is used as a comparative example. In the experiment, the prediction accuracy of the learning model based on the comparative example method was compared with the prediction accuracy of the learning model based on the method using GLSK as the kernel function in equation (9). For GLSK, the "sum-defined GLSK" shown in equation (4) was used.

[0154] The values ​​of each parameter are as follows: The regularization parameter C=1. In the comparative example, the hyperparameters of the shadow kernel in equation (1) are τ=γ=1. The hyperparameters of GLSK in equation (4) are τ A =γ A =c A = 1

[0155] Figure 17 shows an example of the prediction accuracy results. Figure 17(A) illustrates graph 91, which corresponds to the comparative example (shadow kernel). Figure 17(B) illustrates graph 92, which corresponds to the case using GLSK. The horizontal axis of graphs 91 and 92 is the number of qubits n in the quantum data. The vertical axis of graphs 91 and 92 is the prediction accuracy of the learning model on the test data. Classical computer 100 calculated the prediction accuracy of the classification target (test accuracy) on the test data while changing the number of qubits in the problem. The size w of the local subspace is fixed at w=3.

[0156] According to Graphs 91 and 92, in the comparative example, the prediction accuracy of the classification target decreases as the number of qubits increases, but in the method using GLSK, the prediction accuracy increases. Graphs 91 and 92 show sequences corresponding to each case of the number of training data N = 40, 80, 120, 160, and 200. Graphs 91 and 92 show that by using GLSK, high classification performance can be achieved even with a small amount of training data.

[0157] Figure 18 shows an example of the results of demonstrating an adaptive algorithm. Graph 93 shows the prediction accuracy of a learning model based on the GLSK method when the size w of the local subspace is changed to w=1, 2, 3, and 4. The horizontal axis of Graph 93 is the number of qubits n. The vertical axis of Graph 93 is the prediction accuracy of the classification target for the test data (test accuracy). Graph 93 shows the sequences corresponding to each case of w=1, 2, 3, and 4.

[0158] According to Graph 93, when the size w of the local subspace is 2 or less, the prediction accuracy is only about 50%, indicating that learning is not going well. On the other hand, when w is 3 or greater, the prediction accuracy increases sharply. Thus, the classical computer 100 can determine an appropriate w by starting with w=1 and increasing w until the prediction accuracy exceeds a certain threshold.

[0159] Figure 19 shows an example of the results of demonstrating an adaptive algorithm. Graph 94 shows the prediction accuracy of a learning model based on the GLSK method when the size w of the local subspace is changed to w=1, 2, 3, and 4. The horizontal axis of Graph 94 is the number of qubits n. The vertical axis of Graph 94 is the prediction accuracy of the classification target for the training data (training accuracy). Graph 94 shows the sequences corresponding to each case of w=1, 2, 3, and 4.

[0160] As shown in Graph 94, the prediction accuracy for training data, like the prediction accuracy for test data, shows a sharp improvement when the size w of the local subspace is 3 or greater. On the other hand, for example, with n=4 qubits and w=1,2, the prediction accuracy for training data is approximately 70%, which is higher than the prediction accuracy for test data (approximately 50%) for n=4 and w=1,2 in Graph 93. This indicates that overfitting occurs when the number of qubits n is small. Thus, evaluating prediction accuracy using training data may lead to an inability to appropriately determine the size w of the local subspace due to overfitting. Therefore, in order to determine the size of the local subspace with high reliability, it is preferable for the classical computer 100 to use validation data different from the training data.

[0161] According to the classical computer 100 of the second embodiment, by using a kernel function that reflects the geometric structure of space and its locality, it becomes possible to efficiently learn the local properties of quantum data in problems such as materials and substances with a relatively small amount of training data.

[0162] Specifically, classical computer 100 computes shadow kernels on local subspaces of geometric structures and then adds or multiplies them while shifting the local subspaces across the entire space. For example, in geometric structures such as materials and substances, the relationships between distant lattice points are tenuous. Therefore, by limiting the calculation of shadow kernels to local regions, classical computer 100 avoids handling information unnecessary for learning, i.e., information about spatially non-local features. By focusing on features significant for learning, classical computer 100 can reduce the amount of training data required for generalization. Furthermore, classical computer 100 can improve the accuracy of learning.

[0163] Furthermore, it is sometimes impossible to know in advance what size the local subspace should be. Therefore, classical computer 100 can appropriately determine the size of the local subspace by adaptively setting the local subspace so that the training error and validation error are smaller than the standard. Note that a training error and validation error being smaller than the standard corresponds to a prediction accuracy higher than the standard for the training data or the validation data.

[0164] While the shadow kernel was used as an example of a kernel function for a local subspace, it is not limited to the shadow kernel. For example, the kernel function used for a local subspace could also be the fidelity kernel or projection kernel mentioned above.

[0165] As explained above, the classical computer 100 performs the following operations, for example. Processor 101 obtains data for the first classical shadow and data for the second classical shadow. The first classical shadow is a classical approximation corresponding to the first quantum data, which is represented by multiple qubits. The second classical shadow is a classical approximation corresponding to the second quantum data, which is represented by multiple qubits. Processor 101 obtains geometric structure information that shows the geometric structure of a space containing multiple lattice points associated with multiple qubits. Based on the geometric structure information, Processor 101 sets up multiple local subspaces, each having a set of local lattice points among the multiple lattice points in the space. For each of the multiple local subspaces, Processor 101 calculates the value of the first kernel function based on the first and second data elements corresponding to the lattice points belonging to the local subspace, from among the multiple first data elements corresponding to multiple qubits included in the first classical shadow and the multiple second data elements corresponding to multiple qubits included in the second classical shadow. Based on the values ​​of the multiple first kernel functions corresponding to the multiple local subspaces, Processor 101 calculates the value of the second kernel function corresponding to the first and second classical shadows. This allows classical computer 100 to reduce the amount of training data used for machine learning. The first kernel function can, for example, be a shadow kernel, a fidelity kernel, or a projection kernel.

[0166] The second kernel function is a function that takes the sum or product of the values ​​of multiple first kernel functions. GLSK in equations (4) and (5) is an example of a second kernel function. "GLSK defined by sum" in equation (4) is an example of a function that takes the sum of the values ​​of multiple first kernel functions. "GLSK defined by product" in equation (5) is an example of a function that takes the product of the values ​​of multiple first kernel functions. This allows classical computer 100 to appropriately calculate the value of the second kernel function.

[0167] Processor 101 calculates the value of the second kernel function for each pair of classical shadows in multiple classical shadows corresponding to multiple quantum data, including the first and second quantum data. Using the values ​​of the second kernel function calculated for each pair of classical shadows, Processor 101 performs machine learning on a model that outputs predicted values ​​in response to the classical shadow inputs. Processor 101 resizes the local subspace based on the accuracy of the model's predictions. This allows the classical computer 100 to appropriately determine the size of the local subspace.

[0168] More specifically, processor 101 increases the size of the local subspace when the accuracy of the model's predictions falls below a threshold. Processor 101 then performs machine learning on the model based on the modified size of the local subspace. This allows the classical computer 100 to perform machine learning with an appropriately sized local subspace, thereby improving the prediction accuracy of the model generated by the machine learning process.

[0169] Furthermore, the geometric structure of a space represented by geometric structure information is, for example, an L-dimensional (where L is an integer greater than or equal to 1) lattice. Multiple lattice points in that space are, for example, lattice points in an L-dimensional lattice. Classical computer 100 is particularly useful when dealing with geometric structures represented by an L-dimensional lattice, such as problems concerning matter and materials.

[0170] However, the functions of classical computer 100 can be used for purposes other than those mentioned above, such as local pattern recognition and local error detection of images, videos, and quantum data output from quantum computer 200. [Explanation of symbols]

[0171] 10 Information Processing Devices 11 Storage section 12 Processing Units 20 Quantum Computers 30 Geometric Structure 31,32,33 Local subspaces

Claims

1. On the computer, The first classical shadow data corresponding to the first quantum data represented by multiple qubits, the second classical shadow data corresponding to the second quantum data represented by the multiple qubits, and geometric structure information showing the geometric structure of a space including multiple lattice points associated with the multiple qubits are obtained. Based on the geometric structure information, a plurality of local subspaces are defined, each having a set of local lattice points among the plurality of lattice points in the space. For each of the multiple local subspaces, the value of the first kernel function based on the first data elements and second data elements corresponding to the lattice points belonging to the local subspace is calculated, among the multiple first data elements corresponding to the multiple qubits included in the first classical shadow and the multiple second data elements corresponding to the multiple qubits included in the second classical shadow. Based on the values ​​of the multiple first kernel functions corresponding to the multiple local subspaces, the values ​​of the second kernel functions corresponding to the first classical shadow and the second classical shadow are calculated. Calculation program.

2. The second kernel function is a function that takes the sum or product of the values ​​of a plurality of the first kernel functions. The calculation program according to claim 1.

3. For each pair of classical shadows in a plurality of classical shadows corresponding to a plurality of quantum data including the first quantum data and the second quantum data, the value of the second kernel function is calculated. Using the value of the second kernel function calculated for each pair of classical shadows, machine learning is performed on a model that outputs a predicted value in response to the input of the classical shadows. The size of the local subspace is changed based on the accuracy of the predicted values ​​obtained by the aforementioned model. A calculation program according to claim 1 that causes the computer to perform the processing.

4. In the aforementioned size change, if the accuracy of the predicted value by the model is lower than a threshold, the size is increased. The machine learning of the model is performed based on the local subspace of the modified size. A calculation program according to claim 3 that causes the computer to perform the processing.

5. The aforementioned geometric structure is an L-dimensional lattice (where L is an integer greater than or equal to 1), The plurality of grid points are grid points in the L-dimensional grid. The calculation program according to claim 1.

6. Computers The first classical shadow data corresponding to the first quantum data represented by multiple qubits, the second classical shadow data corresponding to the second quantum data represented by the multiple qubits, and geometric structure information showing the geometric structure of a space including multiple lattice points associated with the multiple qubits are obtained. Based on the geometric structure information, a plurality of local subspaces are defined, each having a set of local lattice points among the plurality of lattice points in the space. For each of the multiple local subspaces, the value of the first kernel function based on the first data elements and second data elements corresponding to the lattice points belonging to the local subspace is calculated, among the multiple first data elements corresponding to the multiple qubits included in the first classical shadow and the multiple second data elements corresponding to the multiple qubits included in the second classical shadow. Based on the values ​​of the multiple first kernel functions corresponding to the multiple local subspaces, the values ​​of the second kernel functions corresponding to the first classical shadow and the second classical shadow are calculated. Calculation method.

7. A storage unit that stores data of a first classical shadow corresponding to first quantum data represented by multiple qubits, data of a second classical shadow corresponding to second quantum data represented by multiple qubits, and geometric structure information indicating the geometric structure of a space including multiple lattice points associated with the multiple qubits. A processing unit that, based on the geometric structure information, sets up a plurality of local subspaces, each having a set of local lattice points among the plurality of lattice points in the space, calculates the value of a first kernel function for each of the plurality of local subspaces based on the first data elements and second data elements corresponding to the lattice points belonging to the local subspace, from a plurality of first data elements corresponding to the plurality of qubits included in the first classical shadow and a plurality of second data elements corresponding to the plurality of qubits included in the second classical shadow, and calculates the value of a second kernel function corresponding to the first and second classical shadows based on the plurality of first kernel function values ​​corresponding to the plurality of local subspaces, An information processing device having

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