Hybrid classical-quantum unsupervised multiclass classification

Abstract

To provide hybrid classical-quantum unsupervised multiclass classification.SOLUTION: A method may include obtaining a multi-dimensional training dataset that includes a plurality of pieces of data. Each piece of data may correspond to a number of quantum bits (qubits) and may represent a quantum state. The method may also include generating, using a quantum computing device, a Gram matrix based on the plurality of pieces of data. In addition, the method may include determining, using a classical computing device, a plurality of operators according to a constraint defined by the Gram matrix. Each of the operators may be configured as a proxy for a corresponding piece of data. Further, the method may include assigning, using the classical computing device, each of the operators to a label.SELECTED DRAWING: Figure 2

Description

[Technical field]

[0001] The present disclosure generally relates to systems and methods for hybrid classical-quantum unsupervised multi-class classification. [Background technology]

[0002] Unless otherwise indicated in this disclosure, the materials described in this disclosure are not prior art to the claims of this application and are not admitted to be prior art by inclusion in this section.

[0003] A classical computing device may use bits that can represent a data set as a sequence of 1s and 0s. A classical computing device may perform calculations to manipulate this data set, but the calculations performed by the classical computing device may be limited due to the dimensionality of the data set and / or the calculations. A quantum computing device may use quantum bits (qubits) that can represent a data set as a sequence of 1s, 0s, or 1s and 0s simultaneously. Quantum computing devices may perform some calculations more efficiently and / or accurately than classical computing devices. However, quantum computing devices may be costly to perform calculations at a scale large enough to perform multi-class classification.

[0004] The subject matter claimed in this disclosure is not limited to embodiments that solve any shortcomings or that operate only in environments such as those described above. Rather, this background is provided only to illustrate one example technology area where some embodiments described in this disclosure may be practiced. Summary of the Invention [Means for solving the problem]

[0005] According to an aspect of an embodiment, a method may include obtaining a multi-dimensional training data set including a plurality of data. Each data may correspond to a number of qubits and may represent a quantum state of a quantum system. The method may also include generating a Gram matrix based on the plurality of data, using a quantum computing device. In addition, the method may include determining a plurality of operators according to constraints defined by the Gram matrix, using a classical computing device. Each of the operators may be configured as a proxy for the corresponding data. Furthermore, the method may include assigning each of the operators to a label, using a classical computing device.

[0006] The object and advantages of the embodiments will be realized and attained at least by the elements, features, and combinations particularly pointed out in the appended claims. It is to be understood that both the foregoing general description and the following detailed description are explanatory but are not restrictive of the invention as claimed. [Brief description of the drawings]

[0007] Exemplary embodiments will be described and explained with additional specificity and detail through the accompanying drawings.

[0008] [Figure 1] FIG. 1 is a block diagram of an exemplary operating environment in which a quantum state of a quantum system may be detected.

[0009] [Diagram 2] The present invention illustrates a process flow by which data from a training dataset stored in qubits can be identified and assigned to various labels using quantum and classical computing devices.

[0010] [Diagram 3]FIG. 2 shows a flowchart of an exemplary method for labeling data of a quantum dataset using a quantum computing device and a classical computing device.

[0011] [Figure 4] 1 shows a flowchart of an exemplary method for classifying a multi-dimensional training dataset.

[0012] [Diagram 5] 1 illustrates an exemplary computer system in accordance with one or more embodiments of the present disclosure.

[0013] All according to at least one embodiment described in this disclosure. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0014] Classical computing devices use bits (e.g., binary bits) configured to store values ​​of 0 or 1 to represent a data set. Classical computing devices may perform various calculations using the bits to manipulate or store the data set. Classical computing devices may not be able to perform calculations when the data set includes a quantum data set (e.g., a multidimensional data set) that includes dimensions that cause exponentially sized calculations to be performed. For example, a quantum data set may include a large number of variables that characterize the data in the quantum data set.

[0015] Quantum computing devices use qubits that are configured to store values ​​of 0, 1, or a superposition of both 0 and 1. Because qubits can store multiple values ​​simultaneously, quantum computing devices may be able to manipulate quantum data sets despite the exponential size of the calculations. As a result, quantum computing devices may perform calculations involving quantum data sets more efficiently than classical computing devices.

[0016] In some embodiments, a quantum dataset may include data items each representing a data point or information associated with one or more topics. For example, each data item may represent a news article related to one or more topics of news. As another example, each data item may represent medical images associated with one or more medical conditions and / or medical conditions. As yet another example, each data item may represent microstructural and / or molecular data associated with one or more synthetic materials. Each data item may be stored as a number of qubits, and multi-class classification may be performed to label the data using a clustering algorithm.

[0017] Multi-class classification of data may be performed using semidefinite programming, which may result in exponentially sized computations due to the dimensionality of the data (e.g., the number of qubits corresponding to the data). Thus, a classical computing device may not be able to perform the computations or may take a significant amount of time to perform the computations. The time it takes a classical computing device to perform the computations may be prohibitive. A quantum computing device may use a quantum data set to perform the computations more quickly than a classical computing device. However, performing the entire multi-class classification on a quantum computing device may be complex and cost prohibitive.

[0018] Some embodiments described in this disclosure may include multi-class classification of a quantum dataset by a quantum-classical hybrid computing system including a classical computing device and a quantum computing device. The multi-class classification may be modeled as a semi-definite program that the quantum computing device can simplify to allow the quantum computing device to determine operators that act as surrogates for the data in the quantum dataset. The classical computing device may perform the multi-class classification using those operators to assign labels to the corresponding data.

[0019] In some embodiments, the quantum computing device may generate a Gram matrix based on the quantum dataset. The classical computing device may determine operators according to constraints defined by the Gram matrix. Each operator may act as a proxy to allow the classical computing device to label corresponding data. Each operator may include information about the corresponding data (e.g., quantum state) to be identified. The classical computing device may assign a label to each of the operators. The label may indicate the quantum state of the corresponding data.

[0020] At least some embodiments described in this disclosure may enable classification of data in a quantum dataset without incurring the cost of implementing multi-class classification solely on a quantum device and without incurring the delays that classical computing devices experience when operating on a quantum dataset. Additionally, at least some embodiments described in this disclosure may enable classical data to be encoded as quantum kernels to represent the classical data in a larger feature space.

[0021] These and other embodiments of the present disclosure will now be described with reference to the accompanying drawings, which are to be understood as diagrammatic and schematic representations of such exemplary embodiments, and are not limiting, and are not necessarily drawn to scale, in which features having similar numbering indicate similar structure and function unless otherwise noted.

[0022] 1 illustrates a block diagram of an example operating environment 100 in which a quantum state of a quantum system 103 may be detected, in accordance with at least one embodiment of the present disclosure. Environment 100 may include a preparation device 101, a quantum system 103, and a measurement device 105. In some embodiments, the quantum state of quantum system 103 may be detected by performing multi-class classification of quantum data that includes data representing different quantum states.

[0023] In some embodiments, the preparation device 101 and / or the measurement device 105 may include code and routines configured to enable a computing system to perform one or more operations. Additionally or alternatively, the preparation device 101 and / or the measurement device 105 may be implemented using hardware including a processor, a microprocessor (e.g., for performing or controlling the execution of one or more operations), a field programmable gate array (FPGA), or an application specific integrated circuit (ASIC). In some other cases, the preparation device 101 and / or the measurement device 105 may be implemented using a combination of hardware and software. In this disclosure, operations described as being performed by the preparation device 101 and / or the measurement device 105 may include operations that the preparation device 101 and / or the measurement device 105 may instruct one or more corresponding systems to perform. The measurement device 105 may be configured to perform a series of operations to detect the quantum state of the quantum system 103, as described in more detail below.

[0024] The preparation device 101 detects a set of quantum states |Ψ of a quantum system 103. j Each quantum state may be generated byi The set of quantum states may be represented by data of a quantum data set. Each data may correspond to a number of qubits. The measurement device 105 uses a set of measurement operators M={M1, M2, ..., M L} may be determined. In some embodiments, the sum of the measurement operators may be equal to an identity operator and may include a semi-positive definite property such that all values ​​of the measurement operators are greater than or equal to zero. In these and other embodiments, the identity operator may include a diagonal matrix that includes one values ​​on the diagonal. Additionally, the sum of the measurement operators may be equal to an identity operator to obtain a probability distribution over the measurement operators.

[0025] In some embodiments, the probability that the measurement device 105 identifies the current data based on the different data may be determined as defined in Equation 1.

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[0026] In some embodiments, the measurement device 105 may operate according to at least one probability function. For example, the measurement device 105 may operate according to a function of probability of correct detection, a function of probability of error, or a function of probability of indeterminate outcome. The function of probability of correct detection may be determined as defined in Equation 2.

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[0027] In some embodiments, the measurement device 105 may operate according to a minimum error strategy, an unambiguous strategy, or any other suitable strategy to identify data of the quantum data set. For a minimum error strategy, the measurement device 105 may identify data of the quantum data set to minimize the average error without identifying uncertain data. For example, the measurement device 105 may identify data such that an uncertain outcome function and an error probability function are minimized and a function of the probability of correct detection is maximized. For an unambiguous strategy, the measurement device 105 may identify data to eliminate errors but may include uncertain outcomes. For example, the measurement device 105 may identify data such that an error probability function is equal to zero and an uncertain outcome function is minimized greater than zero.

[0028] In some embodiments, the measurement device 105 may identify the data using a semidefinite program that maximizes a model function defined in Equation 5.

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[0029] In some embodiments, the measurement device 105 can identify the data using a reduced dual of semidefinite programming as defined in Equation 6.

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[0030] When measurement device 105 is implemented as a classical computing device, the amount of time it may take to determine an operator for each of the data representing the quantum states of quantum system 103 may be based on the exponential size of Equation 5 and / or Equation 6 and may be so long that it may not be possible to easily perform multi-class classification of quantum states using measurement device 105 implemented as a classical computing device that performs calculations according to Equation 5 or Equation 6.

[0031] Modifications, additions, or omissions may be made to environment 100 without departing from the scope of the present disclosure. For example, the designation of various elements in the manner described is intended to help illustrate the concepts described herein and is not intended to be limiting. For example, in some embodiments, preparation device 101, quantum system 103, and measurement device 105 are depicted in the particular manner described to help illustrate the concepts described herein, but such depictions are not intended to be limiting. Additionally, environment 100 may include any number of other elements or may be implemented in other systems or contexts other than those described.

[0032] 2 illustrates a process flow 200 in which data from a training dataset 202 stored on qubits may be identified and assigned to various labels using a quantum computing device 204 and a classical computing device 208, in accordance with at least one embodiment of the present disclosure. In some embodiments, the quantum computing device 204 may include a noisy intermediate-scale quantum (NISQ) device.

[0033] The quantum computing device 204 and the classical computing device 208 may classify the data in the training dataset 202 by assigning each of the data in the training dataset 202 to one of a specified number of labels using various techniques and clustering algorithms as described in this disclosure. In some embodiments, the specified number of labels may be predefined before the multi-class classification is performed. For example, a user may select the specific labels to be used for the multi-class classification. As another example, the quantum computing device 204 and / or the classical computing device 208 may select the specific labels based on one or more topics associated with the training dataset 202.

[0034] In some embodiments, the quantum computing device 204 may include code, routines, and quantum circuitry configured to enable the quantum computing system to perform one or more operations. In some other instances, the quantum computing device 204 may be implemented using a combination of hardware and software. The quantum computing device 204 may be configured to perform a set of operations with respect to the training dataset 202 and in connection with the example methods 300 and / or 400, as described in more detail below.

[0035] The quantum computing device 204 may receive the training dataset 202. In some embodiments, the training dataset 202 may include a multidimensional dataset. For example, the multidimensional dataset may be quantum data stored in qubits, where each of the data may correspond to a number of qubits. As a result, each of the data may be multidimensional. In some embodiments, the multidimensional data may include data stored in qubits that can simultaneously store values ​​as 0, 1, or a superposition of both 0 and 1. Additionally, each of the data may represent a quantum state of a quantum system. For example, each of the data may represent a quantum state of a corresponding number of qubits. The data may represent news articles, medical images, microstructures, molecular data, or any other suitable data.

[0036] In some embodiments, the classical computing device 208 may receive a two-dimensional training dataset (e.g., a classical dataset stored in classical bits). In these and other embodiments, the two-dimensional training dataset may include data, each of which may correspond to a number of classical bits. Additionally or alternatively, the classical computing device 208 may encode the two-dimensional training dataset to generate the training dataset 202 as a multidimensional training dataset. In some embodiments, the classical computing device 208 may encode each of the classical data to a different data of the training dataset 202.

[0037] The training data set 202 may be defined as defined in Equation 7.

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[0038] The quantum computing device 204 may generate the Gram matrix 206 based on the training dataset 202. For example, the quantum computing device 204 may construct the Gram matrix 206 using quantum circuitry within the quantum computing device 204. In some embodiments, the quantum computing device 204 may generate the Gram matrix 206 using one or more copies of the training dataset 202. The Gram matrix 206 may include multiple quantities or values. Each quantity or value of the Gram matrix 206 may include an inner product of qubits of different data in the training dataset 202. Due to the dimensionality of the qubits storing the different data, the Gram matrix 206 may include high dimensions, which may result in exponential calculations that may be performed by the quantum computing device 204.

[0039] In some embodiments, quantum computing device 204 may perform a Hadamard test to determine quantities for Gram matrix 206 (e.g., determine an inner product of qubits of different data in training data set 202). The Hadamard test may include a unitary operation on qubits of different data in training data set 202 based on a control qubit. In some embodiments, if the control qubit is equal to 1, a first transformation may be applied during the Hadamard test, if the control qubit is equal to 0, a second transformation may be applied during the Hadamard test, and if the control qubit is equal to a superposition of 0 and 1, a superposition of the first transformation and the second transformation may be applied during the Hadamard test. Quantum computing device 204 may output Gram matrix 206 and provide it to classical computing device 208.

[0040] In some embodiments, classical computing device 208 may include code and routines configured to enable a computing system to perform one or more operations. Additionally or alternatively, classical computing device 208 may be implemented using hardware including a processor, a microprocessor (e.g., for performing or controlling the execution of one or more operations), an FPGA, or an ASIC. In some other cases, classical computing device 208 may be implemented using a combination of hardware and software. In this disclosure, operations described as being performed by classical computing device 208 may include operations that classical computing device 208 may direct one or more corresponding systems to perform. Classical computing device 208 may be configured to perform a series of operations with respect to Gram matrix 206 and in connection with example methods 300 and / or 400, as described in more detail below.

[0041] The classical computing device 208 may include an operator module 210 configured to receive the Gram matrix 206. The operator module 210 may solve or determine operators 212 using a semidefinite program, such as a reduced semidefinite program, based on the Gram matrix 206. The operators 212 may act as proxies for corresponding data in the training data set 202. The operator module 210 may determine the operators 212 according to constraints defined by the Gram matrix 206, as described in more detail below.

[0042] In some embodiments, the classical computing device 208 may identify data in the training dataset 202 according to a simplified dual of semidefinite programming as defined in Equation 8.

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[0043] The operator module 210 may determine the operator 212 using a polynomial function such as that defined in Equation 9.

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[0044] The classical computing device 208 may include a multi-class classifier 214 configured to receive the operators 212. The multi-class classifier 214 may assign each of the operators 212 to a label using a clustering algorithm. In some embodiments, the clustering algorithm may include a K-means algorithm. The multi-class classifier 214 may perform a classification of the operators 212 and assign a label to each of the operators 212. As a result, each of the operators 212 serves as a surrogate for different data in the training dataset 202, such that the data are labeled.

[0045] The multi-class classifier 214 may output labeled data 216. In some embodiments, the labeled data 216 may include a labeled operator. In these and other embodiments, the labeled data 216 may include data from the training dataset 202 and corresponding labels.

[0046] Modifications, additions, or omissions may be made to process flow 200 without departing from the scope of the present disclosure. For example, the designation of various elements in the manner described is intended to aid in explaining the concepts described herein and is not intended to be limiting. For example, in some embodiments, training dataset 202, quantum computing device 204, Gram matrix 206, classical computing device 208, and labeled data 216 are depicted in the particular manner described to aid in explaining the concepts described herein, but such depictions are not intended to be limiting. Additionally, process flow 200 may include any number of other elements or may be implemented in other systems or contexts other than those described.

[0047] FIG. 3 illustrates a flowchart of an example method 300 of labeling data of a quantum dataset using the quantum computing device 204 and the classical computing device 208 of FIG. 2 in accordance with at least one embodiment of the present disclosure. The method 300 may be performed by any suitable system, apparatus, or device in association with labeling quantum data. For example, the quantum computing device 204 or the classical computing device 208 of FIG. 2 may perform or direct the performance of one or more of the operations associated with the method 300 in association with labeling data of the quantum dataset. The method 300 may include one or more blocks 302, 304, 306, 308, or 310. Although illustrated with discrete blocks, steps and operations associated with one or more of the blocks of the method 300 may be divided into additional blocks, combined into fewer blocks, or eliminated, depending on the particular implementation. Examples of labeling quantum data can include classifying drug compounds into classes containing similar properties, and classifying synthetic materials based on similarities in their microstructures using microscopic images of the synthetic materials.

[0048] In block 302, a quantum computing device may be used to construct a Gram matrix. The Gram matrix may be constructed based on a multi-dimensional training dataset. For example, the quantum computing device 204 of FIG. 2 may receive a training dataset 202 that may include data. In an example of classifying drug compounds, the training dataset may include data obtained from the GBD-13 database, which may represent one billion drug compounds. In an example of classifying synthetic materials, the training dataset may include data obtained from one or more sources, which may represent different microstructures.

[0049] At block 304, the Gram matrix may be used to construct a reduced semidefinite program. For example, the classical computing device 208 may use the Gram matrix 206 to construct a reduced semidefinite program according to Equation 6. At block 306, the semidefinite program may be solved to obtain an optimal solution. For example, the classical computing device 208 may solve the semidefinite program to obtain an operator (e.g., an optimal solution) according to Equation 8.

[0050] In block 308, the optimal solution may be used to classify the data into a specified number of labels. For example, the classical computing device 208 may use the operator 212 to classify the data of the training dataset 202. In block 310, each data and the cluster to which the data belongs may be output. For example, the classical computing device 208 may output labeled data 216 indicating the operator and the corresponding label, and / or the data and the corresponding label. In an example of classifying drug compounds, the classical computing device 208 may output labeled data 216 including data representing drug compounds and a label identifying the corresponding drug compound. In an example of classifying synthetic materials, the classical computing device 208 may output labeled data 216 including data representing various microstructures and a label identifying the corresponding synthetic materials, which may enable various simulations to be performed.

[0051] FIG. 4 illustrates a flowchart of an example method 400 for classifying a multidimensional training data set in accordance with at least one embodiment of the present disclosure. Method 400 may be performed by any suitable system, apparatus, or device in association with classifying multidimensional data. For example, measurement device 105 of FIG. 1, quantum computing device 204 of FIG. 2, or classical computing device 208 of FIG. 2 may perform or direct the performance of one or more of the operations associated with method 400 in association with classifying multidimensional data. Method 400 may include one or more blocks 402, 404, 406, or 408. Although illustrated with discrete blocks, steps and operations associated with one or more of the blocks of method 400 may be divided into additional blocks, combined into fewer blocks, or eliminated, depending on the particular implementation.

[0052] At block 402, a multidimensional training data set may be obtained that includes a plurality of data. In some embodiments, each of the data may correspond to a number of qubits. In these and other embodiments, each of the data sets may represent a quantum state of a plurality of quantum states. At block 404, a Gram matrix may be generated using a quantum computing device based on the plurality of data.

[0053] At block 406, a plurality of operators may be determined using a classical computing device according to the constraints defined by the Gram matrix. Each of the operators may be configured as a surrogate for the corresponding data. At block 408, each of the plurality of operators may be assigned to a label of the plurality of labels using a classical computing device.

[0054] Modifications, additions, or omissions may be made to method 400 without departing from the scope of the present disclosure. For example, the operations of method 400 may be performed in a different order. Additionally or alternatively, two or more operations may be performed simultaneously. Furthermore, the outlined operations and actions are provided only as examples, and some of the operations and actions may be optional, combined into fewer operations and actions, or expanded into additional operations and actions without detracting from the essence of the described embodiments.

[0055] 5 illustrates an exemplary computing system 500 in accordance with at least one embodiment of the present disclosure. The computing system 500 may include a processor 510, a memory 520, a data storage 530, and / or a communication unit 540, all of which may be communicatively coupled. Any or all of the environment 100 of FIG. 1 and the process flow 200 of FIG. 2 may be implemented as a computing system consistent with the computing system 500.

[0056] In general, the processor 510 may include any suitable special purpose or general purpose computer, computing entity, or processing device, including various computer hardware or software modules, and may be configured to execute instructions stored on any applicable computer-readable storage medium. For example, the processor 510 may include a microprocessor, a microcontroller, a digital signal processor (DSP), an ASIC, an FPGA, or any other digital or analog circuitry configured to interpret and / or execute program instructions and / or process data.

[0057] 5 as a single processor, it is understood that processor 510 may include any number of processors distributed across any number of networks or physical locations configured to individually or collectively perform any number of operations described in this disclosure. In some embodiments, processor 510 may interpret and / or execute program instructions and / or process data stored in memory 520, data storage 530, or memory 520 and data storage 530. In some embodiments, processor 510 may fetch program instructions from data storage 530 and load the program instructions into memory 520.

[0058] After the program instructions are loaded into memory 520, processor 510 may execute the program instructions, such as instructions for causing computing system 500 to perform the operations of methods 300 or 400 of Figures 3 and 4. For example, computing system 500 may execute the program instructions to obtain a multi-dimensional training, generate a Gram matrix, determine operators based on the Gram matrix, and then classify the operators.

[0059] Memory 520 and data storage 530 may include a computer-readable storage medium or one or more computer-readable storage media for storing computer-executable instructions or data structures. Such computer-readable storage media may be any available media that can be accessed by a general-purpose or special-purpose computer, such as processor 510. For example, memory 520 and data storage 530 may include training dataset 202, graph matrix 206, and / or labeled data 216 of FIG. 2. In some embodiments, computing system 500 may or may not include either memory 520 and data storage 530.

[0060] By way of example, and not limitation, such computer-readable storage media may include non-transitory computer-readable storage media including random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), compact disk read-only memory (CD-ROM) or other optical disk storage, magnetic disk storage or other magnetic storage devices, flash memory devices (e.g., solid-state memory devices), or any other storage medium that can be used to store desired program code in the form of computer-executable instructions or data structures and that can be accessed by a general-purpose or special-purpose computer. Combinations of the above may also be included within the scope of computer-readable storage media. Computer-executable instructions may include, for example, instructions and data configured to cause processor 510 to perform a certain operation or group of operations.

[0061] The communication unit 540 may include any component, device, system, or combination thereof configured to transmit or receive information over a network. In some embodiments, the communication unit 540 may communicate with other devices at other locations, at the same location, or even other components in the same system. For example, the communication unit 540 may include a modem, a network card (wireless or wired), an optical communication device, an infrared communication device, a wireless communication device (such as an antenna), and / or a chipset (such as a Bluetooth device, an 802.6 device (e.g., a metropolitan area network (MAN)), a WiFi device, a WiMax device, a cellular communication facility, etc.). The communication unit 540 may enable data to be exchanged with a network and / or any other device or system described in this disclosure. For example, the communication unit 540 may allow the system 500 to communicate with other systems, such as computing devices and / or other networks.

[0062] Those skilled in the art, after reviewing this disclosure, may recognize that modifications, additions, or omissions may be made to system 500 without departing from the scope of the disclosure. For example, system 500 may include more or fewer components than explicitly illustrated and described.

[0063] The foregoing disclosure is not intended to limit the disclosure to the precise form or specific field of use disclosed. Thus, various alternative embodiments and / or modifications to the disclosure are contemplated in light of the disclosure, whether expressly described or implied herein. Although embodiments of the disclosure have been thus described, it will be recognized that changes can be made in form and details without departing from the scope of the disclosure. Thus, the disclosure is limited only by the scope of the claims.

[0064] In some embodiments, the different components, modules, engines, and services described herein may be implemented as objects or processes (e.g., as separate threads) executing on a computing system. Although some of the systems and processes described herein are generally described as being implemented in software (stored and / or executed on general-purpose hardware), specific hardware implementations, or combinations of software and specific hardware implementations, are also possible and contemplated.

[0065] The terms used in this disclosure, and particularly in the appended claims (e.g., the body of the appended claims), are generally intended as "open terms" (e.g., the term "including" should be interpreted as "including, but not limited to").

[0066] Furthermore, if a specific number of claim recitations to be introduced is intended, such intent is expressly stated in the claim, and in the absence of such a statement, such intent does not exist. For example, as an aid to understanding, the following appended claims may include the use of the introductory phrases "at least one" and "one or more" to introduce claim recitations. However, the use of such phrases should not be construed as implying that the introduction of a claim recitation with the indefinite article "a" or "an" limits any particular claim that includes such an introduced claim recitation to an embodiment that includes only one such recitation, even if the same claim includes the introductory phrase "one or more" or "at least one" and an indefinite article such as "a" or "an" (e.g., "a" and / or "an" should be construed to mean "at least one" or "one or more"). The same applies to the use of definite articles used to introduce claim recitations.

[0067] In addition, even when a particular number of introduced claim recitations is explicitly recited, one of ordinary skill in the art will recognize that such recitation should be interpreted to mean at least the recited number (e.g., the recitation "two recitations" without other modifiers means at least two recitations, or more than two recitations). Furthermore, when idiomatic expressions similar to "at least one of A, B, and C, etc." or "one or more of A, B, and C, etc." are used, such constructions are generally intended to include A only, B only, C only, both A and B, both A and C, both B and C, or all of A, B, and C, etc.

[0068] Moreover, any disjunctive phrase preceding two or more alternative terms, whether in the present text, claims, or drawings, should be understood to contemplate the possibility of including one of those terms, either of those terms, or both of those terms. For example, the phrase "A or B" should be understood to include the possibilities of "A" or "B" or "A and B."

[0069] All examples and conditional language described in this disclosure are intended for educational purposes to help the reader understand the concepts contributed by the inventor to further the art, and should be interpreted without limitation to such specifically described examples and conditions. Although the embodiments of the present disclosure have been described in detail, various changes, substitutions, and alterations can be made without departing from the spirit and scope of the present disclosure.

[0070] The following supplementary notes are further disclosed regarding the embodiments including the above examples. (Appendix 1) obtaining a multidimensional training dataset including a plurality of data, each datum of the plurality of data corresponding to a number of quantum bits (qubits) and representing a quantum state of a plurality of quantum states; generating a Gram matrix based on the plurality of data using a quantum computing device; determining, using a classical computing device, a plurality of operators according to constraints defined by the Gram matrix, each operator of the plurality of operators being configured as a substitute for a corresponding datum of the plurality of data; and using the classical computing device, assigning each operator of the plurality of operators to a label of a plurality of labels. method. (Appendix 2) assigning each datum of the plurality of data to the label of the corresponding operator; outputting the plurality of data and the corresponding assigned labels. The method described in Appendix 1. (Appendix 3) The step of obtaining a multidimensional training dataset comprising the plurality of data comprises: obtaining a two-dimensional training data set including a plurality of classical data, each classical data of the plurality of classical data corresponding to a number of classical bits; generating the multidimensional training data set based on the two-dimensional training data set, wherein each classical data of the plurality of classical data is encoded into a different data of the plurality of data; The method described in Appendix 1. (Appendix 4) 2. The method of claim 1, wherein the Gram matrix comprises a plurality of quantities, each quantity in the plurality of quantities comprising an inner product of the qubits of a corresponding data in the plurality of data. (Appendix 5) 2. The method of claim 1, wherein the plurality of operators are further determined using a polynomial function. (Appendix 6) The polynomial function is

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[0071] 101 Preparation equipment 105 Measuring Equipment 202 training datasets 204 Quantum Computing Devices 206 Gram Matrix 208 Classical Computing Devices 210 Operator Module 212 Operators 214 Multi-class Classifier 212 Labeled Data 302 Constructing a Gram Matrix Using a Quantum Computing Device 304 Constructing Reduced Semidefinite Programs Using Gram Matrices 306 Solving Semidefinite Programming to Obtain the Optimal Solution 308 Classify data into a specified number of labels using the optimal solution 310 Output each data and the cluster to which it belongs Obtain a multidimensional training dataset that contains multiple data 404 Using a quantum computing device, generate a Gram matrix based on the plurality of data. 406 Using a classical computing device, multiple operators are determined subject to constraints defined by a Gram matrix. 408 Using a classical computing device, assigning each of the plurality of operators to a label among a plurality of labels. 500 Computing Systems 510 Processor 520 Memory 530 Data Storage 540 Communication Unit

Claims

1. obtaining a multidimensional training dataset including a plurality of data, each datum of the plurality of data corresponding to a number of quantum bits (qubits) and representing a quantum state of a plurality of quantum states; generating a Gram matrix based on the plurality of data using a quantum computing device; determining, using a classical computing device, a plurality of operators according to constraints defined by the Gram matrix, each operator of the plurality of operators being configured as a surrogate for a corresponding datum of the plurality of data; and using the classical computing device, assigning each operator of the plurality of operators to a label of a plurality of labels. method.

2. assigning each datum of the plurality of data to the label of the corresponding operator; outputting the plurality of data and the corresponding assigned labels. The method of claim 1.

3. The step of obtaining a multidimensional training dataset including the plurality of data comprises: obtaining a two-dimensional training dataset including a plurality of classical data, each classical data of the plurality of classical data corresponding to a number of classical bits; generating the multidimensional training data set based on the two-dimensional training data set, wherein each classical data of the plurality of classical data is encoded into a different data of the plurality of data. The method of claim 1.

4. 2. The method of claim 1, wherein the Gram matrix comprises a plurality of quantities, each quantity in the plurality of quantities comprising an inner product of the qubits of a corresponding data in the plurality of data.

5. The method of claim 1 , wherein the plurality of operators is further determined using a polynomial function.

6. The polynomial function is [0010] where N represents the number of data in the plurality of data, i represents a first counter variable, j represents a second counter variable, and q j represents the probability that the distinct quantum states contain a pure quantum state, and M i denotes the current measurement operator, The method according to claim 5.

7. The quantum computing device and the classical computing device [0025] where N represents a number of data in the plurality of data, i represents a first counter variable, j represents a second counter variable, and q j represents the probability that the distinct quantum states contain a pure quantum state, and Ψ j represents data corresponding to the different quantum states, and M i denotes the current measurement operator, The method of claim 1.

8. One or more non-transitory computer-readable storage media configured to store instructions that, in response to being executed, cause a system to perform operations, the operations including: obtaining a multidimensional training dataset including a plurality of data, each datum of the plurality of data corresponding to a number of quantum bits (qubits) and representing a quantum state of a plurality of quantum states; generating a Gram matrix based on the plurality of data using a quantum computing device; determining, using a classical computing device, a plurality of operators according to constraints defined by the Gram matrix, each operator of the plurality of operators being configured as a surrogate for a corresponding datum of the plurality of data; and using the classical computing device, assigning each operator of the plurality of operators to a label of a plurality of labels. One or more non-transitory computer-readable storage media.

9. The operations further include: assigning each datum of the plurality of data to the label of the corresponding operator; outputting the plurality of data and the corresponding assigned labels. One or more non-transitory computer-readable storage media as recited in claim 8.

10. The operation of obtaining a multidimensional training dataset including the plurality of data includes: obtaining a two-dimensional training dataset including a plurality of classical data, each classical data of the plurality of classical data corresponding to a number of classical bits; generating the multidimensional training data set based on the two-dimensional training data set, wherein each classical data of the plurality of classical data is encoded into a different data of the plurality of data. One or more non-transitory computer-readable storage media as recited in claim 8.

11. 10. The one or more non-transitory computer-readable storage media of claim 8, wherein the Gram matrix comprises a plurality of quantities, each quantity in the plurality of quantities comprising an inner product of the qubits of a corresponding data in the plurality of data.

12. The one or more non-transitory computer-readable storage media of claim 8 , wherein the plurality of operators are further determined using a polynomial function.

13. The polynomial function is [0030] where N represents the number of data in the plurality of data, i represents a first counter variable, j represents a second counter variable, and q j represents the probability that the distinct quantum states contain a pure quantum state, and M i denotes the current measurement operator, One or more non-transitory computer-readable storage media as recited in claim 12.

14. The quantum computing device and the classical computing device [0045] where N represents a number of data in the plurality of data, i represents a first counter variable, j represents a second counter variable, and q j represents the probability that the distinct quantum states contain a pure quantum state, and Ψ j represents data corresponding to the different quantum states, and M i denotes the current measurement operator, One or more non-transitory computer-readable storage media as recited in claim 8.

15. one or more processors; one or more non-transitory computer-readable storage media configured to store instructions; the instructions, in response to being executed, cause the system to perform an operation, the operation being: obtaining a multidimensional training dataset including a plurality of data, each datum of the plurality of data corresponding to a number of quantum bits (qubits) and representing a quantum state of a plurality of quantum states; generating a Gram matrix based on the plurality of data using a quantum computing device; determining, using a classical computing device, a plurality of operators according to constraints defined by the Gram matrix, each operator of the plurality of operators being configured as a surrogate for a corresponding datum of the plurality of data; and using the classical computing device, assigning each operator of the plurality of operators to a label of a plurality of labels. system.

16. The operations further include: assigning each datum of the plurality of data to the label of the corresponding operator; outputting the plurality of data and the corresponding assigned labels. The system of claim 15.

17. The operation of obtaining a multidimensional training dataset including the plurality of data includes: obtaining a two-dimensional training dataset including a plurality of classical data, each classical data of the plurality of classical data corresponding to a number of classical bits; generating the multidimensional training data set based on the two-dimensional training data set, wherein each classical data of the plurality of classical data is encoded into a different data of the plurality of data. The system of claim 15.

18. 16. The system of claim 15, wherein the Gram matrix comprises a plurality of quantities, each quantity in the plurality of quantities comprising an inner product of the qubits of a corresponding data in the plurality of data.

19. The system of claim 15 , wherein the plurality of operators are further determined using a polynomial function.

20. The polynomial function is [0050] where N represents the number of data in the plurality of data, i represents a first counter variable, j represents a second counter variable, and q j represents the probability that the distinct quantum states contain a pure quantum state, and M i denotes the current measurement operator, 20. The system of claim 19.

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