A hybrid classical-quantum computer system for quantum-assisted data evaluation

Abstract

In a general aspect, quantum-assisted two-sample test is presented. In some implementations, a hybrid computer system configured to evaluate data points in a dataset includes a quantum computing system and a classical computing system. The classical computing system is configured to cause the quantum computing system to execute a quantum logic circuit to encode the data points from the dataset in the quantum logic circuit which includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtain measurements of expectation values of quantum states generated by executing the quantum logic circuit on the quantum computing system; determining a quantum density operator associated with the dataset, the quantum density operator being determined based on the measurements; and determining a data characteristic of the dataset based on the quantum density operator.

Description

A Hybrid Classical-Quantum Computer System for Quantum-Assisted DataEvaluationCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 620,512, filed January 12, 2024, entitled "Classification of Probability Distributions;" and U.S. Provisional Patent Application No. 63 / 569,428, filed March 25, 2024, entitled "Quantum-Assisted Data Evaluation." The above-referenced priority documents are incorporated herein by reference.TECHNICAL FIELD

[0002] The following description relates generally to evaluating data points from datasets using a hybrid classical-quantum computer system.BACKGROUND

[0003] Quantum computers can perform computational tasks by storing and processing information within quantum states of quantum systems. For example, qubits (i.e., quantum bits) can be stored in, and represented by, an effective two-level sub-manifold of a quantum coherent physical system. A variety of physical systems have been proposed for quantum computing applications. Examples include superconducting circuits, trapped ions, spin systems, and others.BRIEF DESCRIPTION OF THE DRAWINGS

[0004] FIG. 1 is a block diagram of an example computing environment.

[0005] FIG. 2 is a flow chart showing aspects of an example process for data points evaluation.

[0006] FIG. 3 is a schematic diagram showing an example quantum logic circuit.

[0007] FIG. 4 is a flow chart showing aspects of an example process for data points evaluation.

[0008] FIG. 5A is a flow chart showing aspects of an example process for data preprocessing.

[0009] FIG. 5B includes schematic diagrams showing an example data pre-processing applied on a single dataset.

[0010] FIG. 5C is a schematic diagram showing an example data pre-processing applied on multiple datasets.

[0011] FIG. 6 is a flow chart showing aspects of an example process for classification of distributions of data points in two datasets.

[0012] FIG. 7 is a plot of sample generated datasets showing asset price as a function of time.

[0013] FIG. 8A includes a plot showing mean values of a MMD distance obtained using a classical MMD approach as a function of dataset index pairs and a plot showing mean values of a Frobenius distance obtained using the example process shown in FIG. 6 as a function of the index pairs.

[0014] FIG. 8B is a table showing ratios of mean distance values for different index pairs obtained using the classical MMD approach and the example process shown in FIG. 6.DETAILED DESCRIPTION

[0015] In some aspects of what is described here, data points in a dataset can be evaluated by performing a quantum-assisted data evaluation method using a hybrid classical-quantum computer system. A quantum density operator associated with the dataset can be determined by the quantum-assisted data evaluation method. The quantum density operator can be used as input to a machine learning model for a variety of applications. For example, the quantum-assisted data evaluation method can be used to compare multiple datasets to determine a classification of distributions of datapoints in the respective datasets. Data points in the multiple datasets may either be ordered or not; and the multiple datasets may have unequal size. In some instances, a quantum logic circuit can be used to encode data points from a dataset into quantum states, translating lowdimensional features of classical data points to high-dimensional features in the Hilbertspace. In some instances, the quantum logic circuit is determined or obtained according to the number of features and a range of feature values of the data points in the dataset. In some instances, the quantum logic circuit which is applied to a number of qubits defined by qubit devices in a quantum computing system includes parametric quantum logic gates which are defined by gate parameters. In some instances, the quantum logic circuit includes multiple alternating layers of single-qubit rotation gates for encoding data in rotation angles, and layers of two-qubit quantum logic gates for creating entanglement. In some implementations, the quantum logic circuit may be executed multiple times and measure an observable on the quantum states to obtain multiple measured quantum states. The multiple measured quantum states can be used to calculate an expectation value of the same observable for the quantum states. In some instances, the quantum logic circuit can be modified and further executed when the process is repeated on measurement of other observables, e.g., first order or higher order (e.g., as shown in operations 628, 638 in the example process 600 shown in FIG. 6). In some instances, a quantum density operator can be constructed based on the measured expectation values of the observables on the quantum states. When multiple datasets are evaluated, multiple quantum density operators can be obtained and compared by evaluating a distance metric between the respective quantum density operators. In certain examples, a classification of the respective datasets can be returned.

[0016] In some instances, the quantum-assisted two-sample test may be based on estimating a Frobenius distance on a quantum computer. The quantum-assisted data evaluation method can be used for solving probability distribution classification problems. For example, the methods and techniques presented here can be used, in finance, to identify two time series whether they are from two different distributions, to identify a structural break between in-sample and out-of-sample investment strategy performance (graduation testing), as well as being used for monitoring of alpha decay (investment strategies in live trading). Further use cases of classification with unbalanced data sets include fraud detection and anti-money laundering and counter-terrorist financing. The methods and techniques presented here may be used in machine learning applications to allow a comparison of samples drawn from distributions generated by machine learningalgorithms in order to implement their learning mechanisms and / or their classification, clustering, prediction or data generation capabilities, and allows a comparison of the closeness of a training distribution with the distribution which a machine learning algorithm has learnt. This test can be used within machine learning algorithms as an objective function or distance metric.

[0017] Other examples of applications where the methods and techniques presented here can be used include bioinformatics, in situations where identical samples are processed by different laboratories, to determine whether systematic differences have been introduced or whether results may be analyzed jointly; detecting whether tissue samples from cancer subtypes may be treated as indistinguishable from a diagnosis perspective, or to detect differences between healthy and diseased tissue; when merging information from different databases, database attribute matching requires determining whether fields from the different databases are the same or not, which can be achieved by examining their distributions; comparison of medical imaging data (e.g. from MRI, ECG, CT, X-ray, etc.) to determine whether data sets have the same statistical properties. In some instances, the methods and techniques presented here can be used in other applications.

[0018] In some implementations, the systems and techniques described here can provide technical advantages and improvements. The systems and techniques presented here can achieve high discriminatory power. The systems and techniques can scale linearly with the number of data points versus a quadratic scaling for the classical benchmark. In some cases, a combination of these and potentially other advantages and improvements may be obtained.

[0019] FIG. 1 is a block diagram of an example computing environment 100, according to an example embodiment. The example computing environment 100 shown in FIG. 1 includes a computing system 101 and user devices 110A, HOB, HOC. A computing environment may include additional or different features, and the components of a computing environment may operate as described with respect to FIG. 1 or in another manner.

[0020] The example computing system 101 includes classical and quantum computing resources and exposes their functionality to the user devices 110A, HOB, HOC (referred to collectively as "user devices 110"). The computing system 101 shown in FIG. 1 includes one or more servers 108, quantum computing systems 103A, 103B, a local network 109, and other resources 107. The computing system 101 may also include one or more user devices (e.g., the user device 110A) as well as other features and components. A computing system may include additional or different features, and the components of a computing system may operate as described with respect to FIG. 1 or in another manner.

[0021] The example computing system 101 can provide services to the user devices 110, for example, as a cloud-based or remote-accessed computer system, as a distributed computing resource, as a supercomputer or another type of high-performance computing resource, or in another manner. The computing system 101 or the user devices 110 may also have access to one or more other quantum computing systems (e.g., quantum computing resources that are accessible through the wide area network 115, the local network 109, or otherwise).

[0022] The user devices 110 shown in FIG. 1 may include one or more classical processors, memory, user interfaces, communication interfaces, and other components. For instance, the user devices 110 may be implemented as laptop computers, desktop computers, smartphones, tablets, or other types of computer devices. In the example shown in FIG. 1, to access computing resources of the computing system 101, the user devices 110 send information (e.g., programs, instructions, commands, requests, input data, etc.) to the servers 108; and in response, the user devices 110 receive information (e.g., application data, output data, prompts, alerts, notifications, results, etc.) from the servers 108. The user devices 110 may access services of the computing system 101 in another manner, and the computing system 101 may expose computing resources in another manner.

[0023] In the example shown in FIG. 1, the local user device 110A operates in a local environment with the servers 108 and other elements of the computing system 101. For instance, the user device 110A may be co-located with (e.g., located within 0.5 to 1 km of) the servers 108 and possibly other elements of the computing system 101. As shown in FIG.1, the user device 110A communicates with the servers 108 through a local data connection.

[0024] The local data connection in FIG. 1 is provided by the local network 109. For example, some or all of the servers 108, the user device 110A, the quantum computing systems 103A, 103B, and the other resources 107 may communicate with each other through the local network 109. In some implementations, the local network 109 operates as a communication channel that provides one or more low-latency communication pathways from the server 108 to the quantum computing systems 103A, 103B (or to one or more of the elements of the quantum computing systems 103A, 103B). The local network 109 can be implemented, for instance, as a wired or wireless Local Area Network, an Ethernet connection, or another type of wired or wireless connection. The local network 109 may include one or more wired or wireless routers, wireless access points (WAPs), wireless mesh nodes, switches, high-speed cables, or a combination of these and other types of local network hardware elements. In some cases, the local network 109 includes a software-defined network that provides communication among virtual resources, for example, among an array of virtual machines operating on the server 108 and possibly elsewhere.

[0025] In the example shown in FIG. 1, the remote user devices HOB, HOC operate remotely from the servers 108 and other elements of the computing system 101. For instance, the user devices 110B, HOC may be located at a remote distance (e.g., more than 1 km, 10 km, 100 km, 1,000 km, 10,000 km, or farther) from the servers 108 and possibly other elements of the computing system 101. As shown in FIG. 1, each of the user devices 110B, 110C communicates with the servers 108 through a remote data connection.

[0026] The remote data connection in FIG. 1 is provided by a wide area network 115, which may include, for example, the Internet or another type of wide area communication network. In some cases, remote user devices use another type of remote data connection (e.g., satellite-based connections, a cellular network, a virtual private network, etc.) to access the servers 108. The wide area network 115 may include one or more internet servers, firewalls, service hubs, base stations, or a combination of these and other types ofremote networking elements. Generally, the computing environment 100 can be accessible to any number of remote user devices.

[0027] The example servers 108 shown in FIG. 1 can manage interaction with the user devices 110 and utilization of the quantum and classical computing resources in the computing system 101. For example, based on information from the user devices 110, the servers 108 may delegate computational tasks to the quantum computing systems 103A, 103B and the other resources 107; the servers 108 can then send information to the user devices 110 based on output data from the computational tasks performed by the quantum computing systems 103A, 103B, and the other resources 107.

[0028] As shown in FIG. 1, the servers 108 are classical computing resources that include classical processors 111 and memory 112. The servers 108 may also include one or more communication interfaces that allow the servers to communicate via the local network 109, the wide area network 115, and possibly other channels. In some implementations, the servers 108 may include a host server, an application server, a virtual server, or a combination of these and other types of servers. The servers 108 may include additional or different features and may operate as described with respect to FIG. 1 or in another manner.

[0029] The classical processors 111 can include various kinds of apparatus, devices, and machines for processing data, including, by way of example, a microprocessor, a central processing unit (CPU), a graphics processing unit (GPU), an FPGA (field programmable gate array), an ASIC (application specific integrated circuit), or combinations of these. The memory 112 can include, for example, a random-access memory (RAM), a storage device (e.g., a writable read-only memory (ROM) or others), a hard disk, or another type of storage medium. The memory 112 can include various forms of volatile or non-volatile memory, media, and memory devices, etc.

[0030] Each of the example quantum computing systems 103A, 103B operates as a quantum computing resource in the computing system 101. The other resources 107 may include additional quantum computing resources (e.g., quantum computing systems, quantum simulators, or both) as well as classical (non-quantum) computing resources suchas, for example, digital microprocessors, specialized co-processor units (e.g., graphics processing units (GPUs), cryptographic co-processors, etc.), special purpose logic circuitry (e.g., field programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), etc.), systems-on-chips (SoCs), etc., or combinations of these and other types of computing modules.

[0031] In some implementations, the servers 108 generate programs, identify appropriate computing resources (e.g., a QPU or QVM) in the computing system 101 to execute the programs, and send the programs to the identified resources for execution. For example, the servers 108 may send programs to the quantum computing system 103A, the quantum computing system 103B, or any of the other resources 107. The programs may include classical programs, quantum programs, hybrid classical / quantum programs, and may include any type of function, code, data, instruction set, etc.

[0032] In some instances, programs can be formatted as source code that can be rendered in human-readable form (e.g., as text) and can be compiled, for example, by a compiler running on the servers 108, on the quantum computing systems 103, or elsewhere. In some instances, programs can be formatted as compiled code, such as, for example, binary code (e.g., machine-level instructions) that can be executed directly by a computing resource. Each program may include instructions corresponding to computational tasks that, when performed by an appropriate computing resource, generate output data based on input data. For example, a program can include instructions formatted for a quantum computer system, a simulator, a digital microprocessor, coprocessor or other classical data processing apparatus, or another type of computing resource.

[0033] In some cases, a program may be expressed in a hardware-independent format. For example, quantum machine instructions may be provided in a quantum instruction language such as Quil, described in the publication “A Practical Quantum Instruction Set Architecture,” arXiv:1608.03355v2, dated Feb. 17, 2017, or another quantum instruction language. For instance, the quantum machine instructions may be written in a format that can be executed by a broad range of quantum processing units or simulators. In some cases, a program may be expressed in high-level terms of quantum logic gates or quantumalgorithms, in lower-level terms of fundamental qubit rotations and controlled rotations, or in another form. In some cases, a program may be expressed in terms of control signals (e.g., pulse sequences, delays, etc.) and parameters for the control signals (e.g., frequencies, phases, durations, channels, etc.). In some cases, a program may be expressed in another form or format. In some cases, a program may utilize Quil-T, described in the publication "Gain deeper control of Rigetti quantum processing units with Quil-T," available at https: / / medium.com / rigetti / gain-deeper-control-of-rigetti-quantum-processors-with- quil-t-ea8945061e5b dated Dec. 10, 2020, which is hereby incorporated by reference in the present disclosure.

[0034] In some implementations, the servers 108 include one or more compilers that convert programs between formats. For example, the servers 108 may include a compiler that converts hardware-independent instructions to binary programs for execution by the quantum computing systems 103A, 103B. In some cases, a compiler can compile a program to a format that targets a specific quantum resource in the computer system 101. For example, a compiler may generate a different binary program (e.g., from the same source code) depending on whether the program is to be executed by the quantum computing system 103A or the quantum computing system 103B.

[0035] In some cases, a compiler generates a partial binary program that can be updated, for example, based on specific parameters. For instance, if a quantum program is to be executed iteratively on a quantum computing system with varying parameters on each iteration, the compiler may generate the binary program in a format that can be updated with specific parameter values at runtime (e.g., based on feedback from a prior iteration, or otherwise); the parametric update can be performed without further compilation. In some cases, a compiler generates a full binary program that does not need to be updated or otherwise modified for execution.

[0036] In some implementations, the servers 108 generate a schedule for executing programs, allocate computing resources in the computing system 101 according to the schedule, and delegate the programs to the allocated computing resources. The servers 108 can receive, from each computing resource, output data from the execution of each program. Based on the output data, the servers 108 may generate additional programs thatare then added to the schedule, output data that is provided back to a user device 110, or perform another type of action.

[0037] In some implementations, all or part of the computing system 101 operates as a hybrid computing environment. For example, quantum programs can be formatted as hybrid classical / quantum programs that include instructions for execution by one or more quantum computing resources (e.g., the quantum-based algorithms) and instructions for execution by one or more classical resources. The servers 108 can allocate quantum and classical computing resources in the hybrid computing environment, and delegate programs to the allocated computing resources for execution. The quantum computing resources in the hybrid environment may include, for example, one or more quantum processing units (QPUs), one or more quantum virtual machines (QVMs), one or more quantum simulators, or possibly other types of quantum resources. The classical computing resources in the hybrid environment may include, for example, one or more digital microprocessors, one or more specialized co-processor units (e.g., graphics processing units (GPUs), cryptographic co-processors, etc.), special purpose logic circuitry (e.g., field programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), etc.), systems-on-chips (SoCs), or other types of computing modules.

[0038] In some cases, the servers 108 can select the type of computing resource (e.g., quantum or classical) to execute an individual program, or part of a program, in the computing system 101. For example, the servers 108 may select a particular quantum processing unit (QPU) or other computing resource based on availability of the resource, speed of the resource, information or state capacity of the resource, a performance metric (e.g., process fidelity) of the resource, or based on a combination of these and other factors. In some cases, the servers 108 can perform load balancing, resource testing and calibration, and other types of operations to improve or optimize computing performance.

[0039] Each of the example quantum computing systems 103A, 103B shown in FIG. 1 can perform quantum computational tasks by executing quantum machine instructions (e.g., a binary program compiled for the quantum computing system). In some implementations, a quantum computing system can perform quantum computation by storing and manipulating information within quantum states of a composite quantumsystem. For example, qubits (i.e., quantum bits) can be stored in, and represented by, an effective two-level sub-manifold of a quantum coherent physical system. In some instances, quantum logic can be executed in a manner that allows large-scale entanglement within the quantum system. Control signals can manipulate the quantum states of individual qubits and the joint states of multiple qubits. In some instances, information can be read out from the composite quantum system by measuring the quantum states of the qubits. In some implementations, the quantum states of the qubits are read out by measuring the transmitted or reflected signal from auxiliary quantum devices that are coupled to individual qubits.

[0040] In some implementations, a quantum computing system can operate using gatebased models for quantum computing. For example, the qubits can be initialized in an initial state, and a quantum logic circuit comprised of a series of quantum logic gates can be applied to transform the qubits and extract measurements representing the output of the quantum computation. Individual qubits may be controlled by single-qubit quantum logic gates, and pairs of qubits may be controlled by two-qubit quantum logic gates (e.g., entangling gates that are capable of generating entanglement between the pair of qubits). In some implementations, a quantum computing system can operate using adiabatic or annealing models for quantum computing. For instance, the qubits can be initialized in an initial state, and the controlling Hamiltonian can be transformed adiabatically by adjusting control parameters to another state that can be measured to obtain an output of the quantum computation.

[0041] In some models, fault-tolerance can be achieved by applying a set of high-fidelity control and measurement operations to the qubits. For example, quantum error correcting codes can be deployed to achieve fault-tolerant quantum computation. Other computational regimes may be used; for example, quantum computing systems may operate in non-fault-tolerant regimes. In some implementations, a quantum computing system is constructed and operated according to a scalable quantum computing architecture. For example, in some cases, the architecture can be scaled to a large number of qubits to achieve large-scale general purpose coherent quantum computing. Otherarchitectures may be used; for example, quantum computing systems may operate in small- scale or non-scalable architectures.

[0042] The example quantum computing system 103A shown in FIG. 1 includes a quantum processing unit 102A and a control system 105A, which controls the operation of the quantum processing unit 102A. Similarly, the example quantum computing system 103B includes a quantum processing unit 102B and a control system 105B, which controls the operation of a quantum processing unit 102B. A quantum computing system may include additional or different features, and the components of a quantum computing system may operate as described with respect to FIG. 1 or in another manner.

[0043] In some instances, all or part of the quantum processing unit 102A functions as a quantum processing unit, a quantum memory, or another type of subsystem. In some examples, the quantum processing unit 102A includes a superconducting quantum circuit system. The superconducting quantum circuit may include data qubit devices, stabilizer qubit devices, coupler devices, readout devices, and possibly other devices that are used to store and process quantum information. In some cases, multiple data qubit devices are operatively coupled to a single stabilizer check qubit device through respective coupler devices. In some implementations, the quantum processing unit 102A is implemented utilizing aspects designed or generated from the components and processes shown in FIGS. 2-4, or in another manner. In certain examples, the qubit devices and the coupler devices are implemented as superconducting quantum circuit devices that include Josephson junctions, for example, in Superconducting QUantum Interference Device (SQUID) loops or other arrangements, and are controlled by radio-frequency signals, microwave signals, and bias signals delivered to the quantum processing unit 102A.

[0044] In some instances, the quantum processing modules can include a superconducting quantum circuit that includes one or more quantum circuit devices. For instance, a superconducting quantum circuit may include qubit devices, readout resonator devices, Josephson junctions, or other quantum circuit devices. In some implementations, quantum circuit devices in a quantum processing unit can be collectively operated to define a single logical qubit. A logical qubit includes a quantum register, for instance multiple physical qubits or qudits, and associated circuitry, that supports physical operations whichcan be used to detect or correct errors associated with logical states in a quantum algorithm. Physical operations supported by the quantum register associated with a logical qubit may include single-qubit or multi-qubit quantum logic gates and readout mechanisms. Error detection or correction mechanisms associated with a logical qubit may be based on quantum error correction schemes such as the surface code, color code, Bacon- Shor codes, low-density parity check codes (LDPC), some combination of these, or others.

[0045] The quantum processing unit 102A may include, or may be deployed within, a controlled environment. The controlled environment can be provided, for example, by shielding equipment, cryogenic equipment, and other types of environmental control systems. In some examples, the components in the quantum processing unit 102A operate in a cryogenic temperature regime and are subject to very low electromagnetic and thermal noise. For example, magnetic shielding can be used to shield the system components from stray magnetic fields, optical shielding can be used to shield the system components from optical noise, thermal shielding and cryogenic equipment can be used to maintain the system components at controlled temperature, etc.

[0046] In some implementations, the example quantum processing unit 102A can process quantum information by applying control signals to the qubits in the quantum processing unit 102A. The control signals can be configured to encode information in the qubits, to process the information by performing quantum logic gates or other types of operations, or to extract information from the qubits. In some examples, the operations can be expressed as single-qubit quantum logic gates, two-qubit quantum logic gates, or other types of quantum logic gates that operate on one or more qubits. A quantum logic circuit, which includes a sequence of quantum logic operations, can be applied to the qubits to perform a quantum algorithm. The quantum algorithm may correspond to a computational task, a hardware test, a quantum error correction procedure, a quantum state distillation procedure, or a combination of these and other types of operations.

[0047] The example control system 105A includes controllers 106A and signal hardware 104A. Similarly, control system 105B includes controllers 106B and signal hardware 104B. All or part of the control systems 105A, 105B can operate in a roomtemperature environment or another type of environment, which may be located near therespective quantum processing units 102A, 102B. In some cases, the control systems 105A, 105B include classical computers, signaling equipment (microwave, radio, optical, bias, etc.), electronic systems, vacuum control systems, refrigerant control systems, or other types of control systems that support operation of the quantum processing units 102A, 102B.

[0048] The control systems 105A, 105B may be implemented as distinct systems that operate independent of each other. In some cases, the control systems 105A, 105B may include one or more shared elements; for example, the control systems 105A, 105B may operate as a single control system that operates both quantum processing units 102A, 102B. Moreover, a single quantum computing system may include multiple quantum processing units, which may operate in the same controlled (e.g., cryogenic) environment or in separate environments.

[0049] The example signal hardware 104A includes components that communicate with the quantum processing unit 102A. The signal hardware 104A may include, for example, waveform generators, amplifiers, digitizers, high-frequency sources, DC sources, AC sources, etc. The signal hardware may include additional or different features and components. In the example shown, components of the signal hardware 104A are adapted to interact with the quantum processing unit 102 A. For example, the signal hardware 104A can be configured to operate in a particular frequency range, configured to generate and process signals in a particular format, or the hardware may be adapted in another manner.

[0050] In some instances, one or more components of the signal hardware 104A generate control signals, for example, based on control information from the controllers 106A. The control signals can be delivered to the quantum processing unit 102A during operation of the quantum computing system 103A. For instance, the signal hardware 104A may generate signals to implement quantum logic operations, readout operations, or other types of operations. As an example, the signal hardware 104A may include arbitrary waveform generators (AWGs) that generate electromagnetic waveforms (e.g., microwave or radiofrequency) or laser systems that generate optical waveforms. The waveforms or other types of signals generated by the signal hardware 104A can be delivered to devices inthe quantum processing unit 102A to operate qubit devices, readout devices, bias devices, coupler devices, or other types of components in the quantum processing unit 102A.

[0051] In some instances, the signal hardware 104A receives and processes signals from the quantum processing unit 102A. The received signals can be generated by the execution of a quantum program on the quantum computing system 103A. For instance, the signal hardware 104A may receive signals from the devices in the quantum processing unit 102A in response to readout or other operations performed by the quantum processing unit 102A. Signals received from the quantum processing unit 102A can be mixed, digitized, filtered, or otherwise processed by the signal hardware 104A to extract information, and the information extracted can be provided to the controllers 106A or handled in another manner. In some examples, the signal hardware 104A may include a digitizer that digitizes electromagnetic waveforms (e.g., microwave or radiofrequency) or optical signals, and a digitized waveform can be delivered to the controllers 106A or to other signal hardware components. In some instances, the controllers 106A process the information from the signal hardware 104A and provide feedback to the signal hardware 104A; based on the feedback, the signal hardware 104A can in turn generate new control signals that are delivered to the quantum processing unit 102A.

[0052] In some implementations, the signal hardware 104A includes signal delivery hardware that interfaces with the quantum processing unit 102A. For example, the signal hardware 104A may include filters, attenuators, directional couplers, multiplexers, diplexers, bias components, signal channels, isolators, amplifiers, power dividers, and other types of components. In some instances, the signal delivery hardware performs preprocessing, signal conditioning, or other operations to the control signals to be delivered to the quantum processing unit 102A. In some instances, signal delivery hardware performs preprocessing, signal conditioning, or other operations on readout signals received from the quantum processing unit 102A.

[0053] The example controllers 106A communicate with the signal hardware 104A to control the operation of the quantum computing system 103A. The controllers 106A may include classical computing hardware that directly interfaces with components of the signal hardware 104A. The example controllers 106A may include classical processors,memory, clocks, digital circuitry, analog circuitry, and other types of systems or subsystems. The classical processors may include one or more single- or multi-core microprocessors, digital electronic controllers, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit), or other types of data processing apparatus. The memory may include any type of volatile or non-volatile memory or another type of computer storage medium. The controllers 106A may also include one or more communication interfaces that allow the controllers 106A to communicate via the local network 109 and possibly other channels. The controllers 106A may include additional or different features and components.

[0054] In some implementations, the controllers 106A include memory or other components that store quantum state information, for example, based on qubit readout operations performed by the quantum computing system 103A. For instance, the states of one or more qubits in the quantum processing unit 102A can be measured by qubit readout operations, and the measured state information can be stored in a cache or other type of memory system in one or more of the controllers 106A. In some cases, the measured state information is subsequently used in the execution of a quantum program, a quantum error correction procedure, a quantum processing unit (QPU) calibration or testing procedure, or another type of quantum process.

[0055] In some implementations, the controllers 106A include memory or other components that store a quantum program containing quantum machine instructions for execution by the quantum computing system 103A. In some instances, the controllers 106A can interpret the quantum machine instructions and perform hardware-specific control operations according to the quantum machine instructions. For example, the controllers 106A may cause the signal hardware 104A to generate control signals that are delivered to the quantum processing unit 102A to execute the quantum machine instructions.

[0056] In some instances, the controllers 106A extract qubit state information from qubit readout signals, for example, to identify the quantum states of qubits in the quantum processing unit 102A or for other purposes. For example, the controllers may receive the qubit readout signals (e.g., in the form of analog waveforms) from the signal hardware 104A, digitize the qubit readout signals, and extract qubit state information from thedigitized signals. In some cases, the controllers 106A compute measurement statistics based on qubit state information from multiple shots of a quantum program. For example, each shot may produce a bitstring representing qubit state measurements for a single execution of the quantum program, and a collection of bitstrings from multiple shots may be analyzed to compute quantum state probabilities.

[0057] In some implementations, the controllers 106A include one or more clocks that control the timing of operations. For example, operations performed by the controllers 106A may be scheduled for execution over a series of clock cycles, and clock signals from one or more clocks can be used to control the relative timing of each operation or groups of operations. In some implementations, the controllers 106A may include classical computer resources that perform some or all of the operations of the servers 108 described above. For example, the controllers 106A may operate a compiler to generate binary programs (e.g., full or partial binary programs) from source code; the controllers 106A may include an optimizer that performs classical computational tasks of a hybrid classical / quantum program; the controllers 106A may update binary programs (e.g., at runtime) to include new parameters based on an output of the optimizer, etc.

[0058] The other quantum computing system 103B and its components (e.g., the quantum processing unit 102B, the signal hardware 104B, and controllers 106B) can be implemented as described above with respect to the quantum computing system 103A; in some cases, the quantum computing system 103B and its components may be implemented or may operate in another manner.

[0059] In some implementations, the quantum computing systems 103A, 103B are disparate systems that provide distinct modalities of quantum computation. For example, the computer system 101 may include both an adiabatic quantum computing system and a gate-based quantum computer system. As another example, the computer system 101 may include a superconducting circuit-based quantum computing system and an ion trap-based quantum computer system. In such cases, the computer system 101 may utilize each quantum computing system according to the type of quantum program that is being executed, according to availability or capacity, or based on other considerations.

[0060] In some instances, one or more components of the computing system 101 shown in FIG. 1 are configured to perform the operations of the example processes shown in FIGS.2, 4, 6, or another process. For example, a classical computing system [e.g., the classical processors 111 in the servers 108) can be configured to pre-process received datasets to determine a number of features and a range of feature values; design and construct a quantum logic circuit according to the number of features and the range of feature values; translate the quantum logic circuit into a sequence of native gates that can be executed on the quantum processing unit 102 in the quantum computing system 103; communicate control signals to the quantum processing unit 102 for executing the sequence of native gates to encoding datapoints from datasets into quantum states; obtain and process measured quantum states; determine respective quantum density operators for respective datasets; and return a classification. For example, a quantum processing unit may be configured to execute the sequence of native gates.

[0061] FIG. 2 is a flow chart showing aspects of an example process 200. The example process 200 can be used to evaluate datapoints in a dataset by constructing a quantum feature map that encodes the data points into quantum states by executing a quantum logic circuit on a quantum computing system [e.g., the quantum computing system 103 in FIG. 1); determining a quantum density operator [e.g., a complete form, a reduced form, or a partial view of a density matrix or another type of a quantum density operator) derived from the measurements of observables of the quantum states associated with the dataset; and determining a data characteristic of the dataset based on the quantum density operator. The quantum density operator may be used in supervised learning, unsupervised learning, or other machine learning applications. The example process 200 may include additional or different operations, and the operations may be performed in the order shown or in another order. In some cases, operations in the example process 200 can be combined, iterated or otherwise repeated, or performed in another manner.

[0062] In some instances, the example process 200 can be used for evaluating data points in multiple distinct datasets. In some implementations, the example process 200 is used for solving a multivariate distribution classification problem by constructing a quantum feature map that encodes data points from classical datasets into quantum statesby executing a quantum logic circuit on a quantum computing system (e.g., the quantum computing system 103 in FIG. 1), and determining a quantum density operator (e.g., a complete form, a reduced form, or a partial view of a density matrix) derived from the measurements of observables of the quantum states associated with the respective classical datasets. For example, the quantum density operator can be used to determine a classification by performing a quantum two-sample test to determine a distance value between the density matrices by operation of a computing system (e.g., the computing system 100 in FIG. 1); and return the classification. In some instances, the quantum density operator can be used in regression, clustering, anomaly detection, generative modelling e.g. as the comparator in a generative adversarial network (GAN), and other machine learning applications. In some instances, a classification may include a quantitative evaluation of probability distributions of the classical datasets; whether the two classical datasets are drawn from the same or different dataset, or other type of output. In some implementations, the example process 200 is used to perform a quantum two-sample test by constructing an injective quantum feature map that encodes two classical datasets into two respective density matrices; and calculating a Frobenius distance between the two respective density matrices.

[0063] At 202, a dataset is obtained. The dataset includes a series of data points. Each data point in the dataset is specified by different features or characteristics. In some instances, more than one dataset may be received. For example, a first dataset and a second dataset are received. In some instances, the first and second datasets are received by the computing system 101 from the user device 110 via the wide area network 115 as shown in FIG. 1. Let A and B be two datasets, sampled, respectively, from the probability distributions P(x) and Q(x), x G D c R”1. The cardinality of A and B can be indicated as [A] and [B]. Data points of the respective datasets a G A and b G B are of the form at; =the number of features.

[0064] At 204, the data points in the dataset are encoded in quantum logic circuit. In some implementations, a quantum feature map can be performed to map a vector of features of a datapoint into quantum states. In some implementations, a quantum logic circuit is constructed and executed. In some instances, a quantum logic circuit for quantumfeature mapping may include at least one layer of single-qubit quantum logic gates, and at least one layer of multi-qubit quantum logic gates. In some instances, the quantum logic circuit includes parametric quantum logic gates with gate parameters configured to encode each data point. For example, a quantum logic circuit for quantum feature mapping may include multiple layers of single-qubit rotation gates with rotation angles as gate parameters for encoding the vectors of the features of data points; and multiple layers of multi-qubit quantum logic gates configured to create entanglement. In some instances, the quantum logic circuit may be implemented as the quantum logic circuit 300 shown in FIG. 3 or in another manner.

[0065] In some implementations, a quantum logic circuit is used to efficiently encode relatively low-dimensional classical data points into extremely high-dimensional quantum states, thus creating a quantum feature map. In this case all quantum states are initialized as 0, and the initial state isThe vector of classical features, x, is mapped into a set of gate parameters of the quantum logic circuit. In some instances, an angle encoding method is used, where the classical features are mapped to rotation angles of single-qubit quantum logic gates of the quantum logic circuit. In some instances, different schemes to map the classical features of the data to parameters of the quantum logic circuit may be used. For example, amplitude encoding, basis encoding, reuploading encoding, or other types of encoding schemes can be used to map classical features onto quantum states. In certain instances, classical features can be encoded on parameters of the multi-qubit quantum logic gates, parameters on both the single-qubit quantum logic gates and the multi-qubit quantum logic gates, or in another manner. The final state, \xpf), thus can encode the classical data points in a dataset and the subsequent sample classification can be performed in the high-dimensional Hilbert space. In some instances, the quantum logic circuit can provide more expressive power than equivalent classical models when only a polynomial number of parameters is allowed.

[0066] When two datasets are received, the data points in the first and second datasets are encoded in quantum states. In some instances, a first quantum logic circuit for encoding the data points in the first dataset may be different from a second quantum logic circuit used for encoding the data points in the second dataset. For example, the firstand secondquantum logic circuits for encoding the data points in the first and second datasets may have different gate parameter values, different numbers of layers of quantum logic gates, different number of qubits, etc. In some instances, the data points in the first and second datasets may be encoded using the same quantum logic circuit, for example, when the first and second datasets include the same number of features and the features are within the same range of feature values. In the case of two datasets are received, executing the quantum logic circuit can provide a transformation cp that maps the data points and by in the two datasets into vectors <p(az) and <p(by) of a Hilbert space J-C . On the quantum computer, the transformation <p is realized via a unitary transformation U applied to the state |0) = |0)®nand |<p(x)) = t / (x)|0) = |x). Note that if the transformation (p is specified on n qubit devices, the dimensionality of the Hilbert space H * is 2".

[0067] As shown in FIG. 2, operation 202 includes suboperations 212 during which the quantum logic circuit is obtained; suboperation 214 during which the quantum logic circuit is executed by operation of a quantum computing system; and suboperation 216 during which quantum states are measured.

[0068] At 212, the quantum logic circuit is obtained. In some instances, the quantum logic circuit may be constructed based on the dataset and the quantum computing hardware. In some instances, a quantum logic circuit may be determined by the depth and shape of the quantum logic circuit, the number of layers of the single-qubit quantum logic gates and multi-qubit quantum logic gates in the quantum logic circuit, the number of qubits on which the quantum logic circuit is applied, types of quantum logic gates, the ansatz (e.g., manner the quantum logic gates are connected together), measurement basis, features in the high-dimensional Hilbert space, and other parameters.

[0069] For example, the depth and number of layers and the gates used within the quantum logic circuit may be varied. For example, a multi-qubit quantum logic gate to obtain entanglement may include a controlled-Z (CZ) gate, an imaginary SWAP (iSWAP) gate, a controlled-NOT (CNOT) gate, a controlled-phase (CPHASE) gate, or another multiqubit quantum logic gate. In some instances, circuit shape can be determined in order to best match the qubit topology of the quantum processing unit, e.g., connectivity of qubit devices. Features in the high-dimensional Hilbert space used may be Pauli observables offirst order, or of first and second order. In some examples, other Pauli observables may be selected as features; or completely different features may also be utilized. In some instances, utilizing expectation values of Pauli observables as features results in an algorithm that is impacted less by readout errors. This makes it more robust than algorithms which require measurement of specific bitstring values (e.g. the |0) state). In some instances, this results in improved speed and improved scaling with number of qubits.

[0070] In some implementations, a number of features of the data points are determined, e.g., by using feature extraction techniques or Random Fourier transforms. A range of feature values for each feature of the data points in the dataset is calculated. The quantum logic circuit is determined according to the number of features and the range of feature values. For example, a number of layers of the single-qubit quantum logic gates and a number of qubits to which the quantum logic circuit is applied are determined. In some instances, when the quantum logic circuit includes parametric single-qubit rotation gates, rotation angle values of the single-qubit rotation gates are determined according to the number of layers of the single-qubit quantum logic gates and the number of qubits to which the quantum logic circuit is applied.

[0071] At 214, the quantum logic circuit is executed. In some instances, the quantum logic circuit is executed on the qubits by operation of a hybrid quantum-classical computing system, e.g., the quantum computing system (e.g., the quantum processing unit 102 in FIG. 1) and the classical computing system (e.g., the control system 104 and the classical processors 111 or other classical computing resources in FIG. 1). In some instances, the quantum logic circuit may be converted, translated, or otherwise compiled into a native gate sequence which can be directly executed on quantum circuit devices of a quantum processing unit. The native gate sequence may include parametric single-qubit quantum logic gate. In some instances, the native gate sequence may include nonparametric quantum logic gates.

[0072] In some instances, prior to executing the quantum logic circuit, the data points are pre-processed by standardizing the feature values according to the range of feature values. In some instances, the data points may be pre-processed in another manner. Forexample, the data points from multiple datasets may be split into multiple subsets and data points from different subsets may be resampled and interleaved together. In some instances, the pre-processing operation may be implemented as the operations in the example process 500 shown in FIG. 5 or in another manner.

[0073] In some instances, signatures can be used to encode data streams (timeserieslike data). This encoding can be employed before and / or after the quantum feature mapping. For example, deploying signature encoding before the quantum feature mapping occurs when windows and truncated signatures are generated based on the original input data stream, and these signature features then become input features to the quantum algorithm. In some instances, this could occur in example process 402. Alternatively, in some instances, quantum features can be calculated first (for example in process 404 and 406), thereafter (for example, as a final part of process 406) windows are then applied to these features, and the signature transformation performed, resulting in a new set of data features. In certain instances, the signatures are not calculated explicitly but use Salvi’s kernel signature approach, e.g., by applying the MMD formula for kernels to the signature kernel. In some instances, Salvi’s signature kernel is the solution of a hyperbolic Goursat partial differential equation (PDF). This may not require explicit computation of signatures and can be solved efficiently using numerical PDF solvers, providing a kernel trick for the untruncated signature kernel. In some instances, one or more repetitions of quantum feature generation followed by signature feature generation may be performed. In some instances, one or more repetitions of signature feature generation followed by quantum feature generation may be performed.

[0074] At 216, quantum states are measured. After the execution of the quantum logic circuit, the quantum states are measured by operation of the quantum processing unit. During the measurement, the quantum states of the qubits are collapsed to one of their respective basis states (|0) or |1)J.

[0075] In some instances, readout error mitigation techniques were applied. Readout error mitigation can be used to reduce the errors which occur when measuring the quantum state of a qubit. For example, randomized readout error mitigation can be implemented on the quantum computer hardware / software stack.

[0076] Operations 214 and 216 can be repeated to execute the quantum logic circuit and measure the observable on the quantum states along the same computational basis multiple times to obtain multiple measured quantum states. In some instances, operations 212, 214, 216 may be repeated to obtain a modified quantum logic circuit; execute the modified quantum logic circuit; and measure different observables on the quantum states (e.g., along a different computational basis). The modified quantum logic circuit can be executed multiple times and measured multiple times to obtain multiple measured quantum states. The multiple measured quantum states can be used to calculate expectation values of different observables on the quantum states. In some instances, observables correspond to measurement along specific axis of the Bloch sphere (e.g., different measurement bases), such as the Pauli Matrices. For example, the operations 212, 214, 216 in the example process 200 may be repeated on different computational basis to obtain multiple sets of expectation values as shown in the example process 600 in FIG. 6 or in another manner.

[0077] In some implementations, the methods and techniques presented here can produce a large degree of linear independence, which corresponds to comparing a larger number of components of the distributions in a functional space.

[0078] At 206, a quantum density operator associated with the dataset is determined. In some instances, the quantum density operator may include a density matrix, a reduced form of a density matrix, or another type of quantum density operator. In some instances, when two datasets are received, two density matrices encoding the probability distributions P(x) and Q(x) corresponding to the two datasets can be constructed as

[0079] The density matrices p and q and the projectors |x)(x| that form them are all vectors in the space H *0of linear operators on H (note that projectors - the "measurement" operators - are linear but not unitary operators). Multiprocessing and multithreading parallel processing techniques are used to speed up the classical calculations. In some instances, a density matrix includes the average expectation values ofall Pauli values across the dataset, e.g., as described in operation 610 of the example process 600 shown in FIG. 6 or in another manner.

[0080] At 208, a data characteristic of the dataset is determined. In some instances, the data characteristic can be determined by evaluating the quantum density operator as an input to a machine learning model. In some instances, the quantum density operator can be used in regression, clustering, anomaly detection, generative modelling e.g. as the comparator in a generative adversarial network (GAN), and other machine learning applications. For example, when the machine learning model may include a generative model; and the quantum density operator may be used to generate a new dataset. In some implementations, the quantum density operator is used by a machine learning model to perform a quantitative evaluation of a distribution of the data points in the dataset.

[0081] In some instances, when the two datasets are received and compared, the quantum density operators can be used to determine a classification by performing a quantum two-sample test to determine a distance value between the quantum density operators by operation of a computing system. In some instances, a distance d is introduced on H *0and can be used to measure the closeness of p and q. This is a proxy for whether A and B are sampled from the same probability distribution.

[0082] In the limit case of infinitely many samples, it is easy to see how d(p, q) = 0 iff P(x) = Q(x). In fact, in this case p q = dxPDwhich is equal to 0 iff P(x) — Q(x) — 0, given the assumed injectivity of U (the injective quantum feature map is one that does not map different data points to the same point in the target space) or, equivalently, iff P(x) = (?(x), Vx. Given d is a distance, then d(p, q) = 0 iff P(x) = Q(x),Vx, since d(p, q) = O iffp = q for any distance d.

[0083] In the case of finite A and B, a distance d > 0 would in general be measured even if A and B contained elements sampled from P(x) and Q(x) with P = Q. This discrepancy from d = 0 is proportional to the variance of the distribution and inversely proportional to the cardinality of the datasets.

[0084] In some instances, the distance may be a Frobenius distance which can be expressed as:

[0085] In some instances, other metrics can be used to evaluate the probablity distribution, including the Jensen-Shannon distance, the trace distance, the Bures metric, the Bellinger distance, the geometric distance, or other metrics.

[0086] In some instances, the output of the quantum calculation, e.g., the classical vector of estimated expectations of Pauli observables can be fed into additional classical or quantum calculations to develop new features. In some instances, classical MMD RBF (radial basis functions) may be applied to the vectors of Pauli observables as a postprocessing step. In some instances, the vector of Pauli observables may be used as an input to another layer of quantum feature generation. In certain examples, the final layer of the parameterized quantum circuit or the classical results obtained post readout, can be recombined by using weights which are optimized (in a linear combination, or by other methods) to maximize the performance of the quantum feature map.

[0087] In some instances, classification of unbalanced data sets can be performed by training using the large dataset class in order to determine the means of the feature vectors. The distance to feature vectors obtained from outlier samples (or samples in the small dataset) can then be used within classification algorithms and outlier detectors to determine their distances from the means, and thereby perform classification.

[0088] The multiple measured quantum states can be used to calculate an expectation value of the same observable for the quantum states. In some instances, an Approximate Quantum State technique can be used for estimating Pauli expectations. The Approximate Quantum State technique enables estimation of numerous Pauli expectation values using a fixed set of bitstring samples obtained from a quantum computer. In some instances, 1000 shots per data point are used. This enables the possibility of scaling linearly with the number of data points. In some instances, calculating expectation values of Pauli observables as features in the high-dimensional Hilbert space can make the process or algorithm less affected by readout errors. This makes it more robust than algorithms whichrequire measurement of specific bitstring values (e.g. the |0> state). This results in improved speed and improved scaling with number of qubits. In some instances, other techniques can be used to estimate Pauli expectations.

[0089] In some instances, a classification result may be returned. For example, the classification result may be returned to the user device from the computing system. In some instances, the classification result includes a quantitative or qualitative evaluation of the probability distributions of the two datasets, e.g., the difference between the probability distributions, whether or not the two datasets are drawn from the same dataset, etc.

[0090] FIG. 3 is a schematic diagram showing an example quantum logic circuit 300. In some implementations, the example quantum logic circuit 300 corresponds to part of a hybrid computing program for performing a quantum two-sample test or another computing program. The example quantum logic circuit 300 includes unitary operations applied to qubits defined qubit devices of a quantum processing unit (e.g., the quantum processing unit 102 in FIG. 1). In some implementations, the example quantum logic circuit 300 may be performed on a superconducting quantum processing unit (e.g., the example superconducting quantum processing unit 102A in FIG. 1) or other superconducting quantum processing unit or other quantum processor modalities. The quantum logic circuit 300 is configured to implement a quantum feature map. In other words, the quantum logic circuit 300 maps each data point from a classical dataset into corresponding quantum states.

[0091] In some implementations, the quantum logic circuit 300 may include singlequbit quantum logic gates, two-qubit quantum logic gates, or other multi-qubit quantum logic gates applied on a number of qubits. In some implementations, layers of the single qubit quantum logic gates in the quantum logic circuit 300 are data encoding layers; and layers of the multi-qubit quantum logic gates are entanglement-creating layers. In some implementations, the number of qubits where the quantum logic circuit 300 is applied is determined by the number of features that represent different aspects or attributes of the data points, or dimensionality of the data points in the dataset. In some instances, the number of features of data points in the dataset can be determined by implementingfeature selection and extraction techniques. In some instances, the number of features may be equal to the number (n) of qubit devices (e.g., quantum registers) where the quantum logic circuit for performing quantum feature mapping is applied. In other words, each feature of data points in a dataset is encoded to a quantum state of a qubit device. In certain instances, the number of features may be different from the number of qubit devices. In this case, two or more features of data points may be encoded in a quantum state of a single qubit device; or quantum states of two or more qubit devices may be used to encode a single feature.

[0092] In some implementations, the quantum logic circuit 300 is configured to efficiently encode a relatively low-dimensional classical data point into extremely highdimensional quantum states, thus realising a quantum feature map. For example, all quantum states may be intialised as |0), and the initial state is = |0) 0 ••• 0 |0) = 10)®71. The vector of features, x, can be mapped into a set of gate parameters of the quantum logic circuit. For example, when the single-qubit quantum logic gates in the quantum logic circuit are parametric single-qubit rotation gates, the vectors of features can be mapped onto rotation angles of the parametric single-qubit rotation gates, e.g., an angle encoding method. The final state,, thus encodes the classical data point and the subsequent sample classification is performed in the high-dimensional Hilbert space.

[0093] In some instances, the number of layers of the single-qubit quantum logic gates and multi-qubit logic gates can be defined by the amount of entanglement required, the fidelity of the single-qubit and multi-qubit gates used in the circuit, the topology of the quantum device, the native gates available on the quantum device, and whether data reupload techniques are required, or in another manner.

[0094] As shown in FIG. 3, the example quantum logic circuit 300 includes a sequence of alternating sets of single-qubit rotation gates 302 and two-qubit iSWAP gates 304 applied on 4 qubits 312. Specifically, the quantum logic circuit 300 includes a first set of singlequbit rotation gates 302A on the X basis, Rx(fT), applied on the 4 qubits at a first time step tx, each of which is configured to apply a specific phase rotation to a qubit around the X axis of the Bloch sphere. The quantum logic circuit 300 includes a first set of two-qubit iSWAP gates 304A at a second time step t2, creating entanglement between qubit 312-1 and qubit312-2 and between qubit 312-3 and qubit 312-4. The quantum logic circuit 300 includes a second set of single-qubit rotation gates 302B on the Z basis, Rz(0), applied on the 4 qubits at a third time step t3, each of which is configured to apply a specific phase rotation to a qubit around the Z axis of the Bloch sphere. The quantum logic circuit 300 includes a second set of two-qubit iSWAP gates 304B at a fourth time step t4, creating entanglement between qubit 312-3 and qubit 312-2 and between qubit 312-1 and qubit 312-4. The quantum logic circuit 300 includes a third set of single-qubit rotation gates 302C on the ¥ basis, Ry(0), applied on the 4 qubits at a fifth time step t5, each of which is configured to apply a specific phase rotation to a qubit around the Y axis of the Bloch sphere. The quantum logic circuit 300 includes a third set of two-qubit iSWAP gates 304C at a sixth time step t6, creating entanglement between qubit 312-1 and qubit 312-2 and between qubit 312-3 and qubit 312-4. The quantum logic circuit 300 includes a fourth set of singlequbit rotation gates 302D on the Z basis, / ?z(0), applied on the 4 qubits at a seventh time step t7, each of which is configured to apply a specific phase rotation to a qubit around the Z axis of the Bloch sphere. The quantum logic circuit 300 includes a fourth set of two-qubit iSWAP gates 304D at an eighth time step t8, creating entanglement between qubit 312-3 and qubit 312-2 and between qubit 312-1 and qubit 312-4. The quantum logic circuit 300 includes a fifth set of single-qubit rotation gates 302E on the X basis, Rx(0), applied on the 4 qubits at a ninth time step t9, each of which is configured to apply a specific phase rotation to a qubit around the X axis of the Bloch sphere. The quantum logic circuit 300 includes a fifth set of two-qubit iSWAP gates 304E at a tenth time step t10, creating entanglement between qubit 312-1 and qubit 312-2 and between qubit 312-3 and qubit 312-4. The quantum logic circuit 300 includes a sixth set of single-qubit rotation gates 302F on the Z basis, Rz(fT), applied on the 4 qubits at an eleventh time step tT1, each of which is configured to apply a specific phase rotation to a qubit around the Z axis of the Bloch sphere. The quantum logic circuit 300 includes a sixth set of two-qubit iSWAP gates 304F at a twelfth time step t12, creating entanglement between qubit 312-3 and qubit 312- 2 and between qubit 312-1 and qubit 312-4. The quantum logic circuit 300 includes a seventh set of single-qubit rotation gates 302G on the Y basis, Ry(0), applied on the 4qubits at a thirteenth time step t13, each of which is configured to apply a specific phase rotation to a qubit around the Y axis of the Bloch sphere.

[0095] As shown in FIG. 3, a data point in a dataset can be encoded through one-qubit rotation operations which are rotations of the individual quantum states around the X, Y, or Z axes. Each data points can be represented by a vector of four features, and each feature is mapped into the corresponding rotation angle. The circuit consists of four quantum registers and seven data encoding layers. This specifies the feature mapping scheme where each qubit device encodes one feature (e.g., one-to-one mapping between feature and qubit) and all rotation angles are defined on the interval [0, TT / 7]. TO be more specific, the proposed angle encoding scheme is given by the following expression:1 = 1, ; = 1, ...,4. where min(F( / )) and max(F( / )) are the minimum and maximum values of feature F(y) across all data points, N is the total number of data points in a dataset.

[0096] The final quantum state, IV7 / }, is obtained after application of the n-qubit quantum logic circuit - a sequence of quantum logic gates (linear unitary operators ( / ) controlled by parameters 01(... , 0m- to the initial quantum state, |V>0):

[0097] In some implementations, final quantum states of the qubit devices 312 are measured. The example quantum logic circuit 300 may include a set of measurement, each of which is configured to extract classical information from the qubits 312 causing collapsing of the superposition of a qubit 312 into a respective definite output represented by a classical bit (0 or 1). The measurement produces a classical bitstring, which is a data point from the probability distribution encoded in the final quantum state (each quantum state encodes a probability distribution).

[0098] As shown in FIG. 3, the quantum logic circuit 300 includes six layers of two-qubit iSWAP gates 304 creating entanglement; alternating with seven layers of single-qubit Rx,Ryand Rzgates 302 performing repeated data encoding. The quantum logic circuit 300 is configured to ensure that data points in datasets are not concentrated in any one region of the system’s Hilbert space. The quantum logic gates in the quantum logic circuit may be organized as the example quantum logic circuit 300 shown in FIG. 3 or in another manner. For example, a quantum logic circuit may include a different number of layers of singlequbit rotation gates and two-qubit iSWAP gates. For another example, the seven layers of the single-qubit rotation gates may be organized in another order. In some instances, the quantum logic circuit may include other types of parametric quantum logic gates and the features can be encoded into gate parameters of the quantum logic circuits in another manner. In some instances, other multi-qubit gates different from the iSWAP gate may be used.

[0099] FIG. 4 is a flow chart showing aspects of an example process 400 of classification of probability distributions. The example process 400 is used to evaluate datapoints in a dataset by constructing a quantum feature map that encodes the data points into quantum states by executing a quantum logic circuit on a quantum computing system (e.g., the quantum computing system 103 in FIG. 1); determining a quantum density operator (e.g., a complete form, a reduced form, or a partial view of a density matrix) derived from the measurements of observables of the quantum states associated with the dataset; and determining a data characteristic of the dataset based on the quantum density operator. The quantum density operator may be used in supervised learning, unsupervised learning, or other machine learning applications. The example process 400 may include additional or different operations, and the operations may be performed in the order shown or in another order. In some cases, operations in the example process 400 can be combined, iterated or otherwise repeated, or performed in another manner.

[0100] In some instances, the example process 400 is a process of evaluating distribution of data points in multiple different datasets. In some implementations, the example process 400 is used for solving a multivariate distribution classification problem by constructing a quantum feature map that encodes data points from the distinct datasets into quantum states by executing a quantum logic circuit on a quantum computing system (e.g., the quantum computing system 103 in FIG. 1), determining a quantum densityoperator based on measured expectation values of the quantum states associated with the respective datasets; determining a classification by performing a quantum-assisted data evaluation to determine a distance value between the quantum density operators by operation of a computing system (e.g., the computing system 100 in FIG. 1); and returning the classification. In some instances, a classification may include a quantitative evaluation of distributions of the respective datasets; whether the distinct datasets are drawn from the same or different dataset, or other type of output. In some implementations, the example process 400 is used to perform a quantum two-sample test by constructing an injective quantum feature map that encodes two classical datasets into two respective density matrices; and calculating a Frobenius distance between the two respective density matrices.

[0101] In some instances, operations 404, 408, 410 in the example process 400 may be implemented as the respective operations 204, 206, 208 in the example process 200. The example process 400 includes additional operation 402 for data pre-processing and operation 406 for results post-processing. In some implementations, the operation 402 may be implemented as the operations in the example process 500 shown in FIG. 5 or in another manner. These additional operations in the example process 400 may be included to compensate for time-related changes in the performance of the quantum computing systems. For example, quantum computers require a calibration (re-tune) process to be run regularly in order to determine the control pulses needed to control the device, perform the numerous quantum operations, and to perform readout / measurement. After a re-tune, the performance can slowly degrade as the system gets impacted by changes in the environment. This drift can impact results when the time taken to perform many quantum calculations and obtain results is lengthy. In some implementations, the methods and techniques presented here can be used to ensure that the drift impact is felt similarly by results from all datasets.

[0102] FIG. 5A is a flow chart showing aspects of an example process 500 for data preprocessing. In some instances, the example process 500 is used to ensure the drift impact of the quantum computing system is spread similarly across all datapoints in a dataset.

[0103] At 502, original data points in a dataset are split into multiple subsets. As shown in FIG. 5B, the dataset 512 includes 15 data points, e.g., 1, 2, 3, 4, ..., 15; the 15 data points are split into three subsets 514 each including 5 data points. Specifically, a first subset 514A includes data points 1, 2, 3, 4, 5; a second subset 514B includes data points 6, 7, 8, 9, 10; and a third subset 514C includes data points 11, 12, 13, 14, 15. As shown in FIG. 50, three datasets 520A, 520B, 520C each includes 15 data points. Each of the three datasets 520 is split into three subsets 522A, 522B, 522C.

[0104] At 504, the data points from the multiple subsets are resampled. Referring to the example dataset shown in FIG. 5B, after the dataset 512 is split into three subsets 514A, 514B, 5140, data points from the three subsets are resampled and interleaved to one another to form a new dataset 516. As shown in FIG. 5B, the new dataset 516 includes data points in a different order from the original dataset 512. In this specific example, the first data point 1 from the first subset 514A, the first data point 6 from the second subset 514B, and the first data point in the third subset 5140 are ordered as the first three data points in the new dataset 516. In some instances, data points in the new dataset may be resampled and interleaved in another manner. Referring to the example datasets shown in FIG. 50, data points from different subsets and different datasets are resampled and interleaved to one another to form a new dataset 526, which includes data points from the three original datasets 520A, 520B, 5200 organized in a different order. The data preprocessing process 500 enables resampling of data points in the same dataset or different datasets; and allows a distribution of drift effect onto data points in the same datasets or across different datasets.

[0105] In response to a dataset being resampled and reorganization of datapoints of the dataset during a data pre-processing operation, a post-processing operation of the measurement results can be performed. As shown in FIG. 4, at operation 406, once all the data points in the new dataset 516 are encoded into quantum states and the measured expectation values of the quantum states are obtained, the measured expectation values of the quantum states may be reorganized to match the data points in the original dataset 512. For example, when expectation values of the quantum states corresponding to the second data points in the new dataset 516 is obtained, the expectation values of thequantum states are reorganized in the measurement results to a position corresponding to the datapoint 11 in the original dataset 512. In other words, the measurement results may be de-resampled and reorganized according to the resampling operation 504 and the split operation 502. After the post-processing operation in 406, the process 400 continues with operation 408, during which the respective quantum density operators of the original datasets can be determined according to the de-resampled and reorganized measurement results.

[0106] FIG. 6 is a flow chart showing aspects of an example process 600 for classification of probability distributions. The example process 600 is a process of evaluating distribution of data points in two different datasets. In some implementations, the example process 600 is used for solving a multivariate distribution classification problem by constructing a quantum feature map that encodes data points from classical datasets into quantum states by executing a quantum logic circuit on a quantum computing system (e.g., the quantum computing system 103 in FIG. 1), determining a density matrix based on measured expectation values of the quantum states associated with the respective classical datasets; determining a classification by performing a quantum two- sample test to determine a distance value between the density matrices by operation of a computing system (e.g., the computing system 100 in FIG. 1); and returning the classification. In some instances, a classification may include a quantitative evaluation of probability distributions of the classical datasets; whether the two classical datasets are drawn from the same or different dataset, or other type of output. In some implementations, the example process 600 is used to perform a quantum two-sample test by constructing an injective quantum feature map that encodes two classical datasets into two respective density matrices; and calculating a Frobenius distance between the two respective density matrices. The example process 600 may include additional or different operations, and the operations may be performed in the order shown or in another order. In some cases, operations in the example process 600 can be combined, iterated or otherwise repeated, or performed in another manner.

[0107] At 602, a quantum logic circuit is obtained. The quantum logic circuit is a parametric quantum logic circuit which includes one or more quantum logic gates definedby gate parameters. In some instances, the quantum logic circuit includes layers of one- qubit quantum logic gates for encoding sample and layers of two-qubit quantum logic gates for creating entanglement) is obtained. In some instances, the quantum logic circuit is applied to a number (n) of qubits The layers of single-qubit quantum logic gates alternate with the layers of two-qubit quantum logic gates in the quantum logic circuit. By default, the range of rotation angles should be set at [0,7i / m\, where m is the number of data encoding layers. In some instances, the quantum logic circuit may be obtained according to operation 212, 422 in the example process 200, 400 or in another manner. In some instances, the quantum logic circuit is implemented as the quantum logic circuit 300 shown in FIG. 3.

[0108] At 604, two datasets (A and B) are preprocessed. For example, features in the dataset can be calculated; and features can be standardised by determining a range of feature values (e.g., a maximum value and a minimum value). In some instances, the datasets can be preprocessed by performing the operations in the example process 500 of FIG. 5A or in another manner.

[0109] At 606, data points in the first dataset (dataset A) are encoded in quantum states. As shown in FIG. 6, operation 606 includes suboperations 622, 624, 626, and 628.

[0110] At 622, for each sample i = 1, ... , [A], the quantum logic circuit is executed for N times; after each execution of the quantum logic circuit, Pauli Zkis measured on all quantum registers, where k = 1, ...,n. When the Pauli Zkis measured as state |0), a value of + 1 is recorded; and when the Pauli Zkis measured as state 11), a value of -1 is recorded. An expectation value Zk^ is calculated as an average value of the recorded Zkacross all quantum logic circuit executions.

[0111] At 624, for each sample t = 1, ... , [A], the quantum logic circuit is modified by placing a H gate on all quantum registers before measurement; the modified quantum logic circuit is executed N times; after each execution of the modified quantum logic circuit, Pauli Xkis measured on all quantum registers k = 1, ... , n. When the Pauli Xkis measured as state 10), a value of +1 is recorded; and when the Pauli Xkis measured as state |1), a valueof -1 is recorded. An expectation value Xk^ is calculated as an average value of recorded Xkacross all the modified quantum logici circuit executions.

[0112] At 626, for each sample t = 1, ... , [A], the quantum logic circuit is modifed by placing HS^ gates on all quantum registers before measurement; the modified quantum logic circuit is executed N times; after each execution of the modified quantum logic circuit, Pauli Ykis measured on all quantum registers k — 1, ...,n. When the Pauli Ykis measured as state 10), a value of +1 is recorded; and when the Pauli Ykis measured as state |1), a value of -1 is recorded. An expectation value Yk^ is calculated as an average value of recorded Ykacross all the modified quantum logic circuit executions.

[0113] At 628, for each sample t = 1, ... , [A], the quantum logic circuit is modifed by placing either H or HS^ gates on a subset of quantum registers before measurement; the modified quantum logic circuit is executed N times; after each execution of the modified quantum logic circuit, multiple subsets of expectation values associated with distinct measurement bases for qubits, for example, either Pauli X or Pauli Y or Pauli Z is measured on the corresponding quantum registers. (The value +1 is recorded if we measure state |0) and the value — 1 is recorded if we measure state |1)). If gate H was placed on quantum register k before measurement and gates HS^ were placed on quantum register I before measurement, Xkand Ytcan be measured; the expectation value XkYq is then calculated as average value of the product XkYtof recorded values ofXkand yzacross all the modified quantum logic circuit executions.

[0114] At 608, data points in the second dataset (dataset B) are encoded in quantum states. Operation 608 includes suboperations 632, 634, 636, and 638.

[0115] At 632, for each sample j = 1, ... , [B], the quantum logic circuit is executed for N times; after each execution of the quantum logic circuit, Pauli Zkis measured on all quantum registers, where k = 1, ...,n. When the Pauli Zkis measured as state |0), a value of + 1 is recorded; and when the Pauli Zkis measured as state 11), a value of -1 is recorded. An expectation value ZkB. is calculated as an average value of the recorded Zkacross all quantum logic circuit executions.

[0116] At 634, for each sample j = 1, ... , [B], the quantum logic circuit is modified by placing a H gate on all quantum registers before measurement; the modified quantum logic circuit is executed N times; after each execution of the modified quantum logic circuit, Pauli Xkis measured on all quantum registers k = 1, ... , n. When the Pauli Xkis measured as state 10), a value of +1 is recorded; and when the Pauli Xkis measured as state 11), a value of -1 is recorded. An expectation value Xk® is calculated as an average value of recorded Xkacross all the modified quantum logici circuit executions.

[0117] At 636, for each sample j = 1, ... , [B], the quantum logic circuit is modifed by placing HS^ gates on all quantum registers before measurement; the modified quantum logic circuit is executed N times; after each execution of the modified quantum logic circuit, Pauli Ykis measured on all quantum registers k = 1, ...,n. When the Pauli Ykis measured as state |0), a value of +1 is recorded; and when the Pauli Ykis measured as state |1), a value of -1 is recorded. An expectation value Yk® is calculated as an average value of recorded Ykacross all the modified quantum logic circuit executions.

[0118] At 638, for each sample j = 1, ... , [B], the quantum logic circuit is modifed by placing either H or HS^ gates on a subset of quantum registers before measurement; the modified quantum logic circuit is executed N times; after each execution of the modified quantum logic circuit, either Pauli X or Pauli Y or Pauli Z is measured on the corresponding quantum registers. (The value +1 is recorded if we measure state |0) and the value — 1 is recorded if we measure state 11)) . If gate H was placed on quantum register k before measurement and gates B51" were placed on quantum register I before measurement, Xkand Ytcan be measured; the expectation value XkYt® is then calculated as average value of the product XkYtof recorded values of Xkand Ytacross all the modified quantum logic circuit executions.

[0119] At 610, the average expectation values across all data points in the datasets are calculated. Without loss of generality, assume that 3n second-order Paulis (i.e., the same number as all first order Paulis) are measured. A first array with a dimension of [A] x 6n containing expectation values of first and second order Paulis for dataset A is obtained; and a second array with a dimension of [B] x 6n containing expectation values of first andsecond-order Paulis for the dataset B is also obtained. For every column of these arrays, the average expectation values across all data points in the datasets are calculated using:

[0120] In some implementations, two 671-dimensional column vectors for datasets A and B are obtained:

[0121] At 612, the column vector of differences c is calculated by :

[0122] At 614, the estimate of the Frobenius distance is calcualted by:And a classification can be determined based on the Frobenius distance value. In some instances, the classification can be returned.

[0123] In some instances, a threshold on the distance metric is used to determine whether two data sets come from the same probability distribution. In some instances, the data sets may be classified as coming from the same distributions if the distance metric is less than the threshold. In some instances, the data sets may be classified as coming from different distributions if the distance metric is greater than the threshold. In some instances, the threshold is determined by splitting a data set into subsets and calculating distances between the subsets. This provides statistics on the distances obtained for samples from the same distribution. In certain examples, the threshold can be set based on these statistics. In some instances, the threshold may be set as a specified number of standard deviations away from the mean. In some instances, the threshold may be set using a convergence analysis and the central limit theorem. For example, one approach is a / 2- like test, where for n samples, if the statistics n|c|2is higher than the quantile of order 1 —a the hypothesis that the two distributions are equal can be rejected. In some instances, the threshold may be determined in another manner.

[0124] FIG. 7 is a plot 700 of sample generated datasets showing asset price as a function of time. FIG. 7 shows three simulated time series (A, B and C). The time series represent the asset price process, St, with variable degree of serial dependence controlled by the autocorrelation parameter p\zt-i>zt 7V(O,1) where p is the annualised drift (set equal to 0.1), a is the annualised volatility (set equal to 0.15), At is the year fraction (set equal to 1 / 250 = 1 business day). The autocorrelation was set at p = 0 for asset A, p = 0.2 for asset B, and p = 0.4 for asset C. The simulation is run for 5 years (1,250 observations). The task is to quantify the differences between the time series A, B and C and decide whether the null hypothesis of the time series being drawn from the same probability distribution can be rejected at the given confidence level.

[0125] Performance of the example process 600 and a classical MMD approach are compared with results shown in FIGS. 8A-8B. The two-sample classical and quantum tests described above do not make any assumptions about the nature of the datasets which they compare. Therefore, one could be forgiven for trying to run these tests on a single feature such as daily log-returns (in the spirit of a univariate Kolmogorov-Smirnov two-sample test - and with the same result, the Kolmogorov-Smirnov test is unlikely to work effectively).

[0126] Since financial asset returns are rarely, if ever, independent and identicaly distributed (i.i.d.), several features that would be able to capture such known effects as a serial dependence of returns (an up move is more likely to be followed by another up move) and a serial dependence of the absolute values of returns (a large amplitude move is more likely to be followed by another large amplitude move) can be constructed. To address these points, the day t sample, Ft, as a vector of the following four features includes:

[0127] Since the day t sample contains day t and day t — 1 log-returns, 625 independent samples can be obtained from the time series of 1,250 daily observations.Multiple simulations can be executed in order to produce robust estimates of the statistical errors.

[0128] FIG. 8A includes a plot 802 showing a MMD distance value obtained using a classical MMD approach as a function of index pairs and a plot 804 showing a Frobenius distance value obtained using the process 600 shown in FIG. 6 as a function of the index pairs. The classical (MMD) and quantum (Frobenius) distances were calculated for 100 simulated scenarios for assets A, B and C in FIG. 7. The black dots show the mean distances between two different asset datasets for all asset pairs and the black error bars display ±1 standard deviation around the mean values across all simulated scenarios.

[0129] On their own, these quantities can say only how close (or distant) the two asset datasets are relative to the distances between other asset pairs. To deduce whether the two given asset time series were drawn from the same probability distribution, the natural variance of the samples drawn from this probability distribution can be estimated. This is exactly what is shown in the coloured dots and error bars.

[0130] The coloured dots show the mean distances between the various realisations of the same asset dataset across all simulated scenarios and the coloured error bars show the±1 standard deviation intervals around the mean values. The ratios of mean distances are presented in FIG. 8B. FIG. 8B is a table 810 showing ratios of mean distances for different index pairs obtained using the classical MMD approach and the process 600 shown in FIG.6.

[0131] As shown in FIG. 8A, the classical MMD method (with the Gaussian kernel scaling parameter f = 1; larger values of scaling parameter do not improve discriminatory power and « 1 leads to poor MMD performance) allows clear discrimination between the time series of assets A (p = 0) and C (p = 0.4). For the two other asset pairs, A-B and B-C, the 1- standard deviation confidence intervals overlap and it is impossible to say with certainty whether the time series of asset returns were drawn from two different distributions. The number of core function calls (Gaussian kernel function) scales quadratically with the number of samples. The quantum algorithm performs better in terms of discriminatory power. There are clear gaps between the error bars and noticeably larger ratios of mean distances.

[0132] In some implementations, the quantum-assisted data evaluation method allows the number of quantum logic circuit runs to scale linearly with the number of data points. The Frobenius distance estimation algorithm requires every sample to be processed once. The quantum logic circuit (e.g., the quantum logic circuit 300 in FIG. 3) is executed by operation of the quantum computing system multiple times for every data point, but with a gate time of single-qubit quantum logic gates < 10-2microseconds (p.s) and a gate time of two-qubit quantum logic gate < 10-1p.s, the advantage of quadratic speedup may start to appear for datasets with the number of data points larger than the number of execution of the quantum logic circuit.

[0133] Some of the subject matter and operations described in this specification can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Some of the subject matter described in this specification can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions, encoded on a computer storage medium for execution by, or to control the operation of, data-processing apparatus. A computer storage medium can be, or can be included in, a computer-readable storage device, a computer-readable storage substrate, a random or serial access memory array or device, or a combination of one or more of them. Moreover, while a computer storage medium is not a propagated signal, a computer storage medium can be a source ordestination of computer program instructions encoded in an artificially generated propagated signal. The computer storage medium can also be, or be included in, one or more separate physical components or media.

[0134] Some of the operations described in this specification can be implemented as operations performed by a data processing apparatus on data stored on one or more computer-readable storage devices or received from other sources.

[0135] In a general aspect, quantum-assisted data evaluation is presented.

[0136] In a first example, a method of evaluating data points in a dataset includes encoding the data points from the dataset in a quantum logic circuit which includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtaining measurements of expectation values of quantum states generated by executing the quantum logic circuit on a quantum computing system; determining a quantum density operator associated with the dataset, the quantum density operator being determined based on the measurements; and determining a data characteristic of the dataset based on the quantum density operator.

[0137] Implementations of the first example may include one or more of the following features. The quantum density operator includes one of a density matrix; or a reduced form of a density matrix. The dataset is a first dataset. The quantum logic circuit is a first quantum logic circuit. The measurements are first measurements. The quantum density operator is a first quantum density operator. The data characteristic is a first data characteristic. The method includes encoding the data points from a second dataset in a second quantum logic circuit which includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; receiving second measurements of expectation values of quantum states generated by executing the second quantum logic circuit on the quantum computing system; determining a second quantum density operator associated with the second dataset, the second quantum density operator being determined based on the second measurements; and determining a second data characteristic of the second dataset based on the second quantum density operator. The first and second data characteristics include respective distributions of the first and seconddatasets. The method includes determining a distance between the first and second datasets by comparing the respective distributions; and returning a classification of the first and second datasets. The distance is a Frobenius distance, and determining the distance includes determining the Frobenius distance between the first and second quantum density operators using an approximate quantum state technique.

[0138] Implementations of the first example may include one or more of the following features. The method includes receiving the quantum density operator as an input to a machine learning model. The machine learning model is a generative model, and the method includes generating a second dataset, by operation of the generative model, based on the quantum density operator.

[0139] Implementations of the first example may include one or more of the following features. Encoding the data points in the quantum logic circuit includes calculating a number of features of the data points and a range of feature values; and determining the quantum logic circuit according to the range of feature values and the number of features. Determining the quantum logic circuit includes at least one of determining a number of layers of the single-qubit quantum logic gates; determining a number of qubits to which the quantum logic circuit is applied; and determining an ansatz of the quantum logic circuit. The method further includes prior to executing the quantum logic circuit, pre-processing the data points by standardizing the feature values according to the range of feature values. The quantum logic circuit includes parametric single-qubit quantum logic gates, and determining the quantum logic circuit includes determining rotation angle values of the single-qubit quantum logic gates according to the number of layers of the single-qubit quantum logic gates and the number of qubits to which the quantum logic circuit is applied.

[0140] Implementations of the first example may include one or more of the following features. The expectation values include multiple subsets of expectation values associated with distinct measurement bases for qubits in the quantum logic circuit. Determining the quantum density operator includes constructing the quantum density operator based on the measurements of the multiple subsets of expectation values. The multi-qubit quantum logic gates include at least one of two-qubit iSWAP gates, two-qubit Controlled-NOT gates,two-qubit Controlled-Z gates, or two-qubit Controlled-phase (CPHASE) gates. The singlequbit quantum logic gates include single-qubit rotation gates.

[0141] In a second example, a method of evaluating distributions of data points in a first dataset and a second dataset includes encoding the data points from respective datasets in respective quantum logic circuits each of which includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtaining measurements of expectation values of quantum states generated by executing the respective quantum logic circuits on a quantum computing system; determining respective quantum density operators associated with the respective datasets, the quantum density operators being determined by the measurements; determining a classification of the first and second datasets based on an evaluation of a distance between the respective quantum density operators; and returning the classification of the firstand second datasets.

[0142] Implementations of the second example may include one or more of the following features. The quantum density operator includes one of: a density matrix; or a reduced form of a density matrix. The distance is a Frobenius distance, and evaluating the distance includes determining the Frobenius distance between the first and second quantum density operators using an approximate quantum state technique. Encoding the data points in the respective quantum logic circuits includes calculating numbers of features and ranges of feature values for the respective datasets; and determining the respective quantum logic circuits according to the ranges of feature values and the numbers of features of the respective datasets. Determining a quantum logic circuit includes at least one of: determining a number of layers of the single-qubit quantum logic gates; determining a number of qubits to which the quantum logic circuit is applied; and determining an ansatz of the quantum logic circuit. The method further includes prior to executing the respective quantum logic circuits, pre-processing the data points by standardizing the feature values according to the range of feature values. The quantum logic circuit includes parametric single-qubit quantum logic gates, and determining the quantum logic circuit includes determining rotation angle values of the single-qubit quantum logic gates according to the number of layers of the single-qubit quantum logic gates and the number of qubits to which the quantum logic circuit is applied.

[0143] Implementations of the second example may include one or more of the following features. The expectation values include multiple subsets of expectation values associated with distinct measurement bases for qubits in the respective quantum logic circuits. Determining the quantum density operator includes constructing the quantum density operator based on the measurements of the multiple subsets of expectation values. The multi-qubit quantum logic gates include at least one of: two-qubit iSWAP gates, two- qubit Controlled-NOT gates, two-qubit Controlled-Z gates, or two-qubit Controlled-phase (CPHASE) gates. The single-qubit quantum logic gates include single-qubit rotation gates.

[0144] In a third example, a method of evaluating data points in a dataset includes encoding the data points from the dataset in a quantum logic circuit which includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtaining measurements of expectation values of quantum states generated by executing the quantum logic circuit on a quantum computing system; determining a quantum density operator associated with the dataset, the quantum density operator being determined based on the measurements; receiving the quantum density operator as an input to a generative model; and generating a second dataset, by operation of the generative model, based on the quantum density operator.

[0145] In a fourth example, a hybrid computer system configured to evaluate data points in a dataset includes a quantum computing system and a classical computing system. The classical computing system is configured to cause the quantum computing system to execute a quantum logic circuit to encode the data points from the dataset in the quantum logic circuit which includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtain measurements of expectation values of quantum states generated by executing the quantum logic circuit on the quantum computing system; determine a quantum density operator associated with the dataset, the quantum density operator being determined based on the measurements; and determine a data characteristic of the dataset based on the quantum density operator.

[0146] Implementations of the fourth example may include one or more of the following features. The quantum density operator includes one of a density matrix; or a reduced form of a density matrix. The dataset is a first dataset. The quantum logic circuit is a firstquantum logic circuit. The measurements are first measurements. The quantum density operator is a first quantum density operator. The data characteristic is a first data characteristic. The classical computing system is configured to encode the data points from a second dataset in a second quantum logic circuit which includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; receiving second measurements of expectation values of quantum states generated by executing the second quantum logic circuit on the quantum computing system; determining a second quantum density operator associated with the second dataset, the second quantum density operator being determined based on the second measurements; and determining a second data characteristic of the second dataset based on the second quantum density operator. The first and second data characteristics include respective distributions of the firstand second datasets. The classical computing system is configured to determine a distance between the first and second datasets by comparing the respective distributions; and returning a classification of the first and second datasets. The distance is a Frobenius distance, and determining the distance includes determining the Frobenius distance between the first and second quantum density operators using an approximate quantum state technique.

[0147] Implementations of the fourth example may include one or more of the following features. The classical computing system is configured to receive the quantum density operator as an input to a machine learning model. The machine learning model is a generative model, and the classical computing system is configured to generate a second dataset, by operation of the generative model, based on the quantum density operator.

[0148] Implementations of the fourth example may include one or more of the following features. Encoding the data points in the quantum logic circuit includes calculating a number of features of the data points and a range of feature values; and determining the quantum logic circuit according to the range of feature values and the number of features. Determining the quantum logic circuit includes at least one of determining a number of layers of the single-qubit quantum logic gates; determining a number of qubits to which the quantum logic circuit is applied; and determining an ansatz of the quantum logic circuit. The classical computing system is further configured to, prior to executing the quantumlogic circuit, pre-process the data points by standardizing the feature values according to the range of feature values. The quantum logic circuit includes parametric single-qubit quantum logic gates, and determining the quantum logic circuit includes determining rotation angle values of the single-qubit quantum logic gates according to the number of layers of the single-qubit quantum logic gates and the number of qubits to which the quantum logic circuit is applied.

[0149] Implementations of the fourth example may include one or more of the following features. The expectation values include multiple subsets of expectation values associated with distinct measurement bases for qubits in the quantum logic circuit. Determining the quantum density operator includes constructing the quantum density operator based on the measurements of the multiple subsets of expectation values. The multi-qubit quantum logic gates include at least one of two-qubit iSWAP gates, two-qubit Controlled-NOT gates, two-qubit Controlled-Z gates, or two-qubit Controlled-phase (CPHASE) gates. The singlequbit quantum logic gates include single-qubit rotation gates.

[0150] In a fifth example, a hybrid computer system configured to evaluate distributions of data points in a first dataset and a second dataset includes a quantum computing system; and a classical computing system. The classical computing system is configured to cause the quantum computing system to execute respective quantum logic circuits to encode the data points from respective datasets in the respective quantum logic circuits, each of which includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtain measurements of expectation values of quantum states generated by executing the respective quantum logic circuits on the quantum computing system; determine respective quantum density operators associated with the respective datasets, the quantum density operators being determined by the measurements; determine a classification of the first and second datasets based on an evaluation of a distance between the respective quantum density operators; and return the classification of the first and second datasets.

[0151] Implementations of the fifth example may include one or more of the following features. The quantum density operator includes one of a density matrix; or a reduced form of a density matrix. The distance is a Frobenius distance, and evaluating the distanceincludes determining the Frobenius distance between the first and second quantum density operators using an approximate quantum state technique. Encoding the data points in the respective quantum logic circuits includes calculating numbers of features and ranges of feature values for the respective datasets; and determining the respective quantum logic circuits according to the ranges of feature values and the numbers of features of the respective datasets.

[0152] In a sixth example, a hybrid computer system configured to evaluate data points in a dataset includes a quantum computing system; and a classical computing system. The classical computing system is configured to cause the quantum computing system to execute a quantum logic circuit to encode the data points from the dataset in the quantum logic circuit which includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtain measurements of expectation values of quantum states generated by executing the quantum logic circuit on the quantum computing system; determine a quantum density operator associated with the dataset, the quantum density operator being determined based on the measurements; receive the quantum density operator as an input to a generative model; and generate a second dataset, by operation of the generative model, based on the quantum density operator.

[0153] While this specification contains many details, these should not be understood as limitations on the scope of what may be claimed, but rather as descriptions of features specific to particular examples. Certain features that are described in this specification or shown in the drawings in the context of separate implementations can also be combined. Conversely, various features that are described or shown in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination.

[0154] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system components in the implementations described above should not be understood as requiring suchseparation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single product or packaged into multiple products.

[0155] A number of embodiments have been described. Nevertheless, it will be understood that various modifications can be made. Accordingly, other embodiments are within the scope of the following claims.

Claims

CLAIMSWhat is claimed is:

1. A hybrid computer system configured to evaluate data points in a dataset, the system comprising: a quantum computing system; and a classical computing system configured to: cause the quantum computing system to execute a quantum logic circuit to encode the data points from the dataset in the quantum logic circuit, the quantum logic circuit comprising at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtain measurements of expectation values of quantum states generated by executing the quantum logic circuit on the quantum computing system; determine a quantum density operator associated with the dataset, the quantum density operator being determined based on the measurements; and determine a data characteristic of the dataset based on the quantum density operator.

2. The hybrid computer system of claim 1, wherein the quantum density operator comprises one of: a density matrix; or a reduced form of a density matrix.

3. The hybrid computer system of claiml, wherein the dataset is a first dataset, the quantum logic circuit is a first quantum logic circuit, the measurements are first measurements, the quantum density operator is a first quantum density operator, the data characteristic is a first data characteristic, and the classical computing system is configured to: cause the quantum computing system to execute a second quantum logic circuit to encode the data points from a second dataset in the second quantum logic circuit, the second quantum logic circuit comprising at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates;receive second measurements of expectation values of quantum states generated by executing the second quantum logic circuit on the quantum computing system; determine a second quantum density operator associated with the second dataset, the second quantum density operator being determined based on the second measurements; and determine a second data characteristic of the second dataset based on the second quantum density operator.

4. The hybrid computer system of claim 3, wherein the first and second data characteristics comprise respective distributions of the first and second datasets, and the classical computing system is configured to: determine a distance between the first and second datasets by comparing the respective distributions; and return a classification of the firstand second datasets.

5. The hybrid computer system of claim 4, wherein the distance is a Frobenius distance, and determining the distance comprises: determining the Frobenius distance between the firstand second quantum density operators using an approximate quantum state technique.

6. The hybrid computer system of claim 1, wherein the classical computing system is configured to: receive the quantum density operator as an input to a machine learning model.

7. The hybrid computer system of claim 6, wherein the machine learning model is a generative model, and the classical computing system is configured to: generate a second dataset, by operation of the generative model, based on the quantum density operator.

8. The hybrid computer system of claim 1, wherein encoding the data points in the quantum logic circuit comprises: calculating a number of features of the data points and a range of feature values; and determining the quantum logic circuit according to the range of feature values and the number of features.

9. The hybrid computer system of claim 8, wherein determining the quantum logic circuit comprises at least one of: determining a number of layers of the single-qubit quantum logic gates; determining a number of qubits to which the quantum logic circuit is applied; and determining an ansatz of the quantum logic circuit.

10. The hybrid computer system of claim 8, wherein the classical computing system is configured to: prior to executing the quantum logic circuit, pre-process the data points by standardizing the feature values according to the range of feature values.

11. The hybrid computer system of claim 8, wherein the quantum logic circuit comprises parametric single-qubit quantum logic gates, and determining the quantum logic circuit comprises: determining rotation angle values of the single-qubit quantum logic gates according to the number of layers of the single-qubit quantum logic gates and the number of qubits to which the quantum logic circuit is applied.

12. The hybrid computer system of claim 1, wherein the expectation values comprise multiple subsets of expectation values associated with distinct measurement bases for qubits in the quantum logic circuit.

13. The hybrid computer system of claim 12, wherein determining the quantum density operator comprises: constructing the quantum density operator based on the measurements of the multiple subsets of expectation values.

14. The hybrid computer system of any one of claims 1 through 13, wherein the multiqubit quantum logic gates comprise at least one of: two-qubit iSWAP gates, two-qubit Controlled-NOT gates, two-qubit Controlled-Z gates, or two-qubit Controlled-phase (CPHASE) gates.

15. The hybrid computer system of any one of claims 1 through 13, wherein the singlequbit quantum logic gates comprise single-qubit rotation gates.

16. A hybrid computer system configured to evaluate distributions of data points in a first dataset and a second dataset, the system comprising: a quantum computing system; and a classical computing system configured to: cause the quantum computing system to execute respective quantum logic circuits to encode the data points from respective datasets in the respective quantum logic circuits, each quantum logic circuit comprising at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtain measurements of expectation values of quantum states generated by executing the respective quantum logic circuits on the quantum computing system; determine respective quantum density operators associated with the respective datasets, the quantum density operators being determined by the measurements; determine a classification of the first and second datasets based on an evaluation of a distance between the respective quantum density operators; and return the classification of the first and second datasets.

17. The hybrid computer system of claim 16, wherein the quantum density operator comprises one of: a density matrix; or a reduced form of a density matrix.

18. The hybrid computer system of claim 16, wherein the distance is a Frobenius distance, and evaluating the distance comprises: determining the Frobenius distance between the firstand second quantum density operators using an approximate quantum state technique.

19. The hybrid computer system of claim 16, wherein encoding the data points in the respective quantum logic circuits comprises: calculating numbers of features and ranges of feature values for the respectivedatasets; and determining the respective quantum logic circuits according to the ranges of feature values and the numbers of features of the respective datasets.

20. The hybrid computer system of claim 19, wherein determining the respective quantum logic circuits comprises at least one of: determining a number of layers of the single-qubit quantum logic gates; determining a number of qubits to which the quantum logic circuit is applied; and determining an ansatz of the quantum logic circuit.

21. The hybrid computer system of claim 19, wherein the classical computing system configured to: prior to executing the respective quantum logic circuits, pre-process the data points by standardizing the feature values according to the range of feature values.

22. The hybrid computer system of claim 19, wherein the quantum logic circuit comprises parametric single-qubit quantum logic gates, and determining the quantum logic circuit comprises: determining rotation angle values of the single-qubit quantum logic gates according to the number of layers of the single-qubit quantum logic gates and the number of qubits to which the quantum logic circuit is applied.

23. The hybrid computer system of any one of claims 16 through 22, wherein the expectation values comprise multiple subsets of expectation values associated with distinct measurement bases for qubits in the respective quantum logic circuits.

24. The hybrid computer system of any one of claims 16 through 22, wherein determining the quantum density operator comprises: constructing the quantum density operator based on the measurements of the multiple subsets of expectation values.

25. The hybrid computer system of any one of claims 16 through 22, wherein the multiqubit quantum logic gates comprise at least one of: two-qubit iSWAP gates, two-qubit Controlled-NOT gates.two-qubit Controlled-Z gates, or two-qubit Controlled-phase (CPHASE) gates.

26. The hybrid computer system of any one of claims 16 through 22, wherein the singlequbit quantum logic gates comprise single-qubit rotation gates.

27. A hybrid computer system configured to evaluate data points in a dataset, the system comprising: a quantum computing system; and a classical computing system configured to: cause the quantum computing system to execute a quantum logic circuit to encode the data points from the dataset in the quantum logic circuit, the quantum logic circuit comprising at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates; obtain measurements of expectation values of quantum states generated by executing the quantum logic circuit on the quantum computing system; determine a quantum density operator associated with the dataset, the quantum density operator being determined based on the measurements; receive the quantum density operator as an input to a generative model; and generate a second dataset, by operation of the generative model, based on the quantum density operator.

28. The hybrid computer system of claim 27, wherein the quantum density operator comprises one of: a density matrix; or a reduced form of a density matrix.

29. The hybrid computer system of claim 27, wherein encoding the data points in the respective quantum logic circuits comprises: calculating numbers of features and ranges of feature values for the respective datasets; and determining the respective quantum logic circuits according to the ranges of feature values and the numbers of features of the respective datasets.

30. The hybrid computer system of claim 29, wherein determining the respective quantum logic circuits comprises at least one of: determining a number of layers of the single-qubit quantum logic gates; determining a number of qubits to which the quantum logic circuit is applied; and determining an ansatz of the quantum logic circuit.

31. The hybrid computer system of claim 29, wherein the classical computing system configured to: prior to executing the respective quantum logic circuits, pre-process the data points by standardizing the feature values according to the range of feature values.

32. The hybrid computer system of claim 29, wherein the quantum logic circuit comprises parametric single-qubit quantum logic gates, and determining the quantum logic circuit comprises: determining rotation angle values of the single-qubit quantum logic gates according to the number of layers of the single-qubit quantum logic gates and the number of qubits to which the quantum logic circuit is applied.

33. The hybrid computer system of any one of claims 27 through 32, wherein the expectation values comprise multiple subsets of expectation values associated with distinct measurement bases for qubits in the respective quantum logic circuits.

34. The hybrid computer system of any one of claims 27 through 32, wherein determining the quantum density operator comprises: constructing the quantum density operator based on the measurements of the multiple subsets of expectation values.

35. The hybrid computer system of any one of claims 27 through 32, wherein the multiqubit quantum logic gates comprise at least one of: two-qubit iSWAP gates, two-qubit Controlled-NOT gates, two-qubit Controlled-Z gates, or two-qubit Controlled-phase (CPHASE) gates.

36. The hybrid computer system of any one of claims 27 through 32, wherein the singlequbit quantum logic gates comprise single-qubit rotation gates.

37. The hybrid computer system of any one of claims 27 through 32, wherein generating the second dataset based on the quantum density operator comprises: perform, by operation of the generative model, a quantitative evaluation of a distribution of the data points in the dataset.

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