Magic-enhanced quantum feature generation

Abstract

In a general aspect, input datasets for an optimization model are processed using magic-enhanced quantum logic circuits. In some implementations, an input dataset includes a plurality of data points, and a quantum logic circuit includes multiple layers of subcircuit modules applied on a plurality of qubits. Each subcircuit module includes a first subset of fixed non-Clifford quantum logic gates and a second subset of parametric quantum logic gates that are parameterized by the plurality of data points. Bitstrings are obtained based on output quantum states generated by a quantum computing resource executing the quantum logic circuit. A quantum feature vector associated with the input dataset is determined based on the bitstrings, and the quantum feature vector is provided as an input to the optimization model.

Description

Attorney Docket No.: RIGET-134WO1 Magic-Enhanced Quantum Feature Generation CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 696,891, filed September 20, 2024, entitled “Quantum Feature Maps and Chebyshev Inequality.” The above-referenced priority document is incorporated herein by reference. TECHNICAL FIELD

[0002] The following description relates generally to generating magic-enhanced quantum features. 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 schematic diagram showing an example quantum logic circuit.

[0008] FIG.5 is a plot showing approximation and generalization tradeoff.

[0009] FIGS.6A-6B are plots with F1 scores using different classifiers for fraud detection under different procedures to generate datasets.Attorney Docket No.: RIGET-134WO1

[0010] FIG.7A is a table showing a performance comparison between quantum- enhanced signature kernel (QSK) Model against classical benchmarks across different feature dimensions for the prediction of the mid-price movements of the limit order book (LOB) of a stock.

[0011] FIG.7B is a table showing the performance of the QSK across different batches of data for the prediction of the mid-price movements of the LOB of a stock.

[0012] FIG.8 is a plot showing average AUC scores across various additive discrete logarithm problems. DETAILED DESCRIPTION

[0013] In some aspects of what is described here, data points in a dataset can be processed by performing a magic-enhanced quantum feature mapping method using a hybrid classical-quantum computer system. A classical feature vector associated with the input dataset can be transformed to a quantum feature vector that can be then provided to an optimization model, a machine learning model, or another learning pipeline for a variety of applications. For example, 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 learning algorithms 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.

[0014] In some implementations, the magic-enhanced quantum feature mapping includes a generation and an execution of a quantum logic circuit. In some instances, the quantum logic circuit may include a first subset of quantum logic gates that can be used to encode data points from a dataset into quantum states, translating low-dimensional features of classical data points to high-dimensional features in the Hilbert space. In some instances, the quantum logic circuit is determined or obtained according to the number of classical features and a range of classical feature values of the data points in the dataset. In some instances, the first subset of quantum logic gates which is applied to a number ofAttorney Docket No.: RIGET-134WO1 qubits defined by qubit devices in a quantum computing system includes parametric quantum logic gates which are defined by gate parameters.

[0015] In some instances, the quantum logic circuit can further include a second subset of quantum logic gates that are configured to create magic states. The quantum logic gates in the second subset can be implemented as fixed non-Clifford quantum logic gates. A fixed quantum logic gate is non-parametric, which means that the quantum logic gate is predetermined and is not parameterized by a variable. As such, each time the fixed quantum logic gate is applied during an execution of the quantum logic circuit, the fixed quantum logic gate remains the same regardless of input parameters or other variables. Clifford gates are quantum logic gates that map the Pauli group to itself under conjugation; Clifford gates can be efficiently simulated classically. Non-Clifford gates are the quantum logic gates that do not have the properties of Clifford gates and therefore are distinct from Clifford gates.. The non-Clifford quantum logic gate in the second subset can introduce the non-stabilizer nature of quantum states or operations that cannot be generated using Clifford gates alone. In some implementations, the quantum logic gates in the second subset create magic by transforming stabilizer states into non-stabilizer (“magic”) states. In certain examples, the quantum logic circuit may include a third subset of quantum logic gates that are configured to spread the magic across multiple different qubits. In this case, magic states are no longer localized to certain qubits alone. For example, applying a quantum logic gate in the third subset on two qubits that already have magic after applying the quantum logic gates in the second subset can combine the two magic states; can spread the magic states; and can create more complex and more deeply entangled magic states, e.g., two-qubit entangled magic states. In some instances, magic is a measure of non- Cliffordness of the transformation.

[0016] In some instances, the quantum logic circuit may be executed multiple times and quantum states can be measured to obtain multiple bitstrings. The multiple bitstrings can be fed to an estimator to calculate a quantum feature vector. In some instances, when the estimator is an estimator of Pauli expectation values, estimated Pauli-expectation values can be obtained based on the measured bitstring. The estimated Pauli-expectation valuesAttorney Docket No.: RIGET-134WO1 can be used to construct a quantum density operator; and the quantum feature vector in this case includes the quantum density operator.

[0017] In some implementations, the techniques and systems presented here can be used to detect credit card fraud with improved accuracy compared to classical methods. The output of the magic enhanced quantum logic circuit is then used by a classical machine learning detection algorithm to automatically classify transactions between allowed and fraudulent. In some instances, a classification algorithm based on a Chebyshev inequality can further improve the detection performance. In some implementations, the techniques and systems presented here can be used to learn problems that are difficult to classical approaches, e.g., the discrete logarithm function, or effectively handle complex input dataset, e.g., limit order book. In some implementations, the techniques and systems presented here can achieve improved or comparable performance to those based on classical approaches using a reduced subset of data points.

[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 presented here are general and independent of the chosen problem (input dataset), i.e. the model is not problem informed. The systems and techniques can scale linearly with the number of data points versus a quadratic scaling for the classical benchmark. In some instances, the systems and techniques presented here can result in higher magic quantum logic circuits, which exhibit higher (or more precisely slower decrease in) purity and therefore better generalization. They also exhibit higher expressivity. The systems and techniques presented here can enable feature maps with higher magic, and thus, enable higher approximation for the same generalization. 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, 110B, 110C. A computing environment may include additional or different features, and the components of aAttorney Docket No.: RIGET-134WO1 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, 110B, 110C (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.Attorney Docket No.: RIGET-134WO1

[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 110B, 110C operate remotely from the servers 108 and other elements of the computing system 101. For instance, the user devices 110B, 110C 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 connectionAttorney Docket No.: RIGET-134WO1 (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 of remote 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.Attorney Docket No.: RIGET-134WO1

[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 such as, 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, co- processor 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 SetAttorney Docket No.: RIGET-134WO1 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 quantum algorithms, 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.Attorney Docket No.: RIGET-134WO1

[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 that are 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.Attorney Docket No.: RIGET-134WO1

[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 quantum system. 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 gate- based 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 mayAttorney Docket No.: RIGET-134WO1 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. Other architectures 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 resonatorAttorney Docket No.: RIGET-134WO1 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 which can 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.Attorney Docket No.: RIGET-134WO1

[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 room- temperature environment or another type of environment, which may be located near the respective 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 102A. 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 arbitraryAttorney Docket No.: RIGET-134WO1 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 in the 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.Attorney Docket No.: RIGET-134WO1

[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.Attorney Docket No.: RIGET-134WO1

[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 the digitized 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 mayAttorney Docket No.: RIGET-134WO1 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 FIG. 2, 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 preprocess datapoints in an input dataset by performing a magic-enhanced quantum feature mapping. The example process 200 can transform classical features in the input dataset to quantum features, which may be used in various machine learning pipelines, including supervised learning, unsupervised learning, or other machine learning or optimization applications. The quantum feature map can transform an input vector of real numbers into quantum states; and then provide a mapping from quantum states to expectation values of observables on some or all of the ^ qubits. In some instances, the observables may include Pauli operators of first degree and second degree. The output of the quantum logic circuit (e.g., including the expectation values of observables, information derived from them, etc.) can provide quantum-enhanced features that can be used in an optimization model. The example process 200 may include additional or different operations, and the operations may be performed in the orderAttorney Docket No.: RIGET-134WO1 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] At 202, an input dataset is obtained. The input dataset includes a series of data points. Each data point in the input dataset is specified by different features or characteristics. In some instances, the input dataset may be received by the computing system 101 from the user device 110 via the wide area network 115 as shown in FIG.1.Each data point ^^ in the input dataset ^ ∈ ^ may be in the form of ^^: = (^^(1),… , ^^(^))^,where ^ is the number of classical features.

[0063] For example, the input dataset may be a numerical feature vector associated with each credit or debit card transaction, e.g., a transaction feature vector (FV). In some instances, each feature is a quantity crafted by domain experts that can be used to identify which transactions are more likely to have been authorized by the account owner and which are frauds performed by malicious actors.

[0064] For another example, the input dataset may include limit order book (LOB), which represents the raw data available in the market. The input dataset may be microstructure data consisting of a list of pairs: volume and price and a direction (buy or sell). The number of pairs retained is the depth of the LOB. In some instances, the input dataset includes the FI 2010 dataset publicly available in the publication (Ntakaris, et al., Benchmark Dataset for Mid-Price Forecasting of Limit Order Book Data with Machine Learning Methods, arXiv:1705.03233 [cs.CE], March 11, 2020). In some implementations, the FI 2010 dataset can be used to predict the mid-price movements ticks ahead. In some instances, the FI 2010 dataset may have a depth of 40 (which corresponds to 40 features) or another depth.

[0065] As another example, the input dataset for a discrete logarithm problem can be constructed by performing the following operations. Given a number of bits ^, a prime number can be sampled such that its binary representation requires at least ^ bits. A generator ^ of the additive group of integers modulo ^ can be computed, denoted by ^^∗. The numbers from the group ^^∗can be then sampled and c discrete logarithm in base ^ can be computed. A classification dataset can be built from the data sampled by using one of theAttorney Docket No.: RIGET-134WO1 bits (for example the second) of the discrete logarithm bit string as label. This naturally creates a balanced classification dataset. An unbalanced dataset can be artificially createdfrom the balanced one by sampling a proportion 1 − ^ of the data from one class and aproportion ^ of the data from the other class, for some small proportion ^. In some instances, the discrete logarithm may not be computed. Instead, since it is known that the discrete logarithm of a number in ^^∗is also in the same group, uniformly sampling in the group ^^∗and then multiply the sampled number by ^ can obtain the number in the group that would have given the sampled number as logarithm.

[0066] At 204, a quantum feature map is performed. In some implementations, a quantum feature map can be performed to map a vector of classical features (a classical feature vector) of a datapoint into quantum states, which can be further used to construct a vector of quantum features (a quantum feature vector). In other words, the classical features in the input dataset are transformed into quantum features which can be used in a machine learning problem, an optimization problem, or another learning pipeline. As shown in FIG.2, operation 204 includes suboperations 212 during which a quantum logic circuit is obtained; suboperation 214 during which the quantum logic circuit is executed by operation of a quantum computing resource; suboperation 216 during which quantum states are measured; and suboperation 218 during which a quantum feature vector is determined based on the measured quantum states. In some implementations, after operation 204, the classical feature vector of the input dataset is transformed to a quantum feature vector associated with the input data. In other words, the classical input dataset is encoded into the quantum domain.

[0067] At 212, a quantum logic circuit is obtained. In some instances, the quantum logic circuit may be constructed based on the input dataset and the quantum computing hardware. In some instances, a quantum logic circuit may be characterized by the depth and shape of the quantum logic circuit, the number of layers of subcircuit modules in the quantum logic circuit, the number of qubits to 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, and other parameters.Attorney Docket No.: RIGET-134WO1

[0068] In some implementations, a number of features of the data points are determined, e.g., by using feature extraction techniques. 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. 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.

[0069] In some instances, each subcircuit module (e.g., the subcircuit modules 310, 410 in the example quantum logic circuits 300, 400 shown in FIGS.3 and 4) includes multiple distinct subsets of quantum logic gates. In some implementations, each subcircuit module includes a first subset of quantum logic gates that, when being applied to the plurality of qubits, can create magic states. In some implementations, each quantum logic gate in the first subset includes a fixed non-Clifford quantum logic gate. For instance, the first subset of quantum logic gates can be implemented as the subset 302 in the example subcircuit module 310 shown in FIG.3; the first subset of quantum logic gates can be implemented as the subset 404A or the subset 404B in the example subcircuit module 410 shown in FIG.4; or the first subset may be implemented in another manner in some cases. In some implementations, each subcircuit module includes a second subset of quantum logic gates when applied to the plurality of qubits can encode datapoints in the input dataset. In some implementations, each quantum logic gate in the second subset is a parametric quantum logic gate. For instance, the second subset of quantum logic gates can be implemented as the subset 304 in the example subcircuit module 310 shown in FIG.3; the second subset of quantum logic gates can implemented as the subset 408 in the example subcircuit module 410 shown in FIG.4; or the second subset may be implemented in another manner in some cases.

[0070] Clifford gates are a well-known class of quantum logic gates that map the Pauli group to itself under conjugation; Clifford gates can be efficiently simulated classically. Well-known examples of Clifford gates include the Hadamard gate, the Pauli gates (X, Y, Z Paulis), and others. Non-Clifford gates do not have the properties of Clifford gates andAttorney Docket No.: RIGET-134WO1 therefore are distinct from Clifford gates. Examples of non-Clifford gates include the T gate and others.

[0071] In some implementations, when each fixed non-Clifford quantum logic gate in the first subset is a single-qubit T gate. The single-qubit T gate can apply a ^ / 4 phase rotation to the |1^ state and can be represented by the unitary operator matrix ^= ^1 00 ^^ / !".(1) The single-qubit T gate can create a property known as “magic” in a quantum state. Forexample, when a T gate is applied to a superposition state like |+^ = (|0^ + |1^) / √2, theresult is a qubit having a magic state: &|0^ + ^^ / !|1^' / √2. Inquantumlogic gate in the first subset may includenon-Clifford quantum logic gate configured to create magic states. For example, the multi-qubit fixed non-Clifford quantum logic gate may be a controlled-T gate. In some instances, each fixed non-Clifford quantum logic gate in the first subset may include other types of quantum logic gates. In some implementations, each subcircuit module includes a set of Hadamard gates (e.g., the subset 402A or 402B shown in FIG.4) prior to the application of the first subset of quantum logic gates. In some instances, a Hadamard gate can be expressed by the unitary operator matrix (Hadamard =1^1 1(2)

[0072] In someencode relatively low-dimensional classical features in the data points into extremely high- dimensional quantum states. In this case all quantum states are initialized as 0, and theinitial state is | / 0^ = |0^ ⊗ ⋯ ⊗ |0^ ≡ 0^⊗5. The vector of classical features, 6, can bemapped intoparametric quantum logic gates of the second subset. In some instances, an angle encoding method is used, where the classical features are mapped to rotation angles of parametric quantum logic gates of the second subset. In this case, each parametric quantum logic gate in the subcircuit module may be a single- qubit rotation gate with a rotation angle as a gate parameter for encoding the vectors of the features of data points. In some instances, the multiple layers of subcircuit modules may have the parametric single-qubit quantum logic gates that apply specific phase rotations toAttorney Docket No.: RIGET-134WO1 qubits around the same axis of the Bloch sphere, e.g., Z axis 78(9) (a single-qubit Z-rotation gate), X axis 7:(9) (a single-qubit X-rotation gate), or Y axis 7;(9) (a single-qubit Y- rotation gate).

[0073] In some instances, a single-qubit X-rotation gate can be expressed as the unitary operator matrix 7:(9) = < cos(9⁄ 2 ) −A sin(9⁄ 2 ) (3)−A sin(9⁄ 2 ) cos(9⁄ 2 ) D.In some instances, a single-matrix 7;(9) = <cos(9⁄ 2 ) − sin(9⁄ 2 ) D (4)sin(9⁄ 2 ) cos(9⁄ 2 ) .In some instances, a single-qubit Z-rotation gate can be expressed as the unitary operator matrix 7E(9) = <exp(−A 9⁄ 2 ) 00 ( )D. (5)exp A 9⁄ 2

[0074] In some instances, other 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 output quantum states, | / I〉, thus can encode the classical data points in the input dataset and the subsequentlearning problem 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.

[0075] In some implementations, each subcircuit module further includes a third subset of quantum logic gates. Each quantum logic gate in the third subset may include a multi- qubit non-parametric Clifford quantum logic gate configured to spread and entangle the created magic states across different qubits. Examples of multi-qubit non-parametricAttorney Docket No.: RIGET-134WO1 Clifford quantum logic gate that can be used to spread magic between qubits include the controlled-Z (CZ) gate, the imaginary SWAP (iSWAP) gate, the controlled-NOT (CNOT) gate, the controlled-phase (CPHASE) gate, and other multi-qubit Clifford quantum logic gates.

[0076] In some instances, the CZ gate can be represented by the unitary operator matrix 10 0 0(6) (CZ = M0 1 0 0 N

[0077] In some instances, thematrix 10 0 0(7) =M0 0 A 0

[0078] In some instances, thematrix 10 0 0(8) =1 0 0

[0079] In some instances, thematrix 10 0 01 0 0(9)

[0080] In some instances,qubit topology of the quantum processing unit, e.g., connectivity of qubit devices. In some implementations, the third subset of quantum logic gates is applied in the quantum logic circuit after the application of the first subset of quantum logic gates and prior to the application of the second subset of quantum logic gates. In other words, the magic states after being created by the first subset of quantum logic gates can be spread across different qubits by applying the third subset of quantum logic gates before encoding the data points by applying the second subset of quantum logic gates in each subcircuit module. ForAttorney Docket No.: RIGET-134WO1 instance, the third subset of quantum logic gates can be implemented as the subsets 406A and 406B in the example subcircuit module 410 shown in FIG.4; or the third subset may be implemented in another manner. In some instances, the different subsets of quantum logic gates in a subcircuit module may be organized or connected in another manner. In some instances, each subcircuit module in the quantum logic circuit may include other quantum logic gates. In some instances, the depth and number of layers of subcircuit modules and the type of quantum logic gates used within the quantum logic circuit may be varied.

[0081] In some implementations, the quantum logic circuit has a structure that can help the downstream learning, e.g., the detection method, to solve the classification problem more accurately, and that makes it inefficient for a classical computer to emulate to a good degree of approximation the same mathematical transformation: for an increasing number of qubits and longer gate sequences the computation cost, in memory and / or time for the classical computer increase at least super-polynomially – although in practice and for relevant problem instances, classical polynomial overheads may already make the emulation of the quantum feature mapping impractical, unfeasible or devoid of business relevance.

[0082] At 214, the quantum logic circuit is executed. In some instances, the quantum logic circuit is applied to and 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. In some instances, the quantum logic circuit may be executed on other quantum computing resources, e.g., a quantum simulator.

[0083] In some instances, prior to executing the quantum logic circuit, the data points may be 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. For example, the data points from multiple datasets may be split into multiple subsets and data points from different subsets may be resampled and interleaved together.Attorney Docket No.: RIGET-134WO1

[0084] At 216, quantum states are measured. After the execution of the quantum logic circuit, the quantum states are measured by operation of the quantum computing resource. During the measurement, the quantum states of the qubits are collapsed to one of their respective basis states (|0^or |1^). In some instances, readout error mitigation techniques can be 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.

[0085] During the measurement, the quantum states of the qubits are collapsed to one of their respective basis states (|0^ or |1^). In some implementations, the observables on the quantum states are measured. In some implementations, expectation values of different observables on the quantum states can be determined. For example, the multiple measured quantum states can be used to calculate the expectation values of different observables on the quantum states. In some instances, the expectation values of different observables on the quantum states may be calculated in another manner.

[0086] In some instances, a common choice for the observables {Y^} is tensor product of Pauli operators and identities. In some implementations, the expectation values of observables in the preprocessed dataset include all the Pauli operators of first degree on all (or a subset of) the qubits and Pauli operators of second degree on all (or a subset of) pairs of qubits. In some implementations, a Pauli operator of first degree represents measurements along the X, Y, or Z axis of the Bloch sphere, e.g., a single-body Pauli operator. When a Pauli operator of first degree is applied to a quantum state, it yields an eigenvalue corresponding to the possible outcomes of the measurement. In some implementations, a Pauli operator of second degree represents an observable related to a two-qubit system, e.g., a tensor product of Pauli operators of the first degree on two respective qubits in the two-qubit system or two-body Pauli operators. Generally, anobservable of degree ^ (where ^ > 1) means an operator acting non-trivially on a systemwith ^ qubits rather than on individual qubits.

[0087] In some implementations, quantum state measurement at 216 is a projective measurement that extracts classical information from a quantum system by collapsing its state into one of the possible outcomes defined by a measurement basis, e.g., theAttorney Docket No.: RIGET-134WO1 computational (Pauli-Z) basis. When a quantum state is measured in this manner, the result is probabilistic: each outcome occurs with a probability determined by the squared amplitude of its corresponding basis component in the state vector. For an ^-qubit system, a single measurement yields a bitstring of length ^, such as “0101,” representing the observed eigenstates of the measured qubits. 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. Repeating the measurement many times (shots) allows estimation of the underlying probability distribution over all possible bitstrings. 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. 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 degree. In some instances, a quantum density operator can be constructed based on the measured expectation values of the observables on the quantum states.

[0088] In some instances, the operations 214, 216 in the example process 200 may be repeated on different computational basis to obtain multiple sets of expectation values. Forexample, after each execution of the quantum logic circuit, either Pauli \ or Pauli ] or Pauli^ is measured on the corresponding qubits. (The value +1 is recorded if state |0^ ismeasured and the value −1 is recorded if state |1^is measured).

[0089] At 218, a quantum feature vector associated with the input dataset is determined. In some instances, the quantum feature vector is a quantum representation of the classical feature vectors of datapoints in the input dataset. In some implementations, the quantum feature vector is determined based on the obtained bitstrings from the execution of the quantum logic circuit. In some implementations, the quantum featureAttorney Docket No.: RIGET-134WO1 vector, obtained based on the execution of the quantum logic circuit, includes magic- enhanced quantum features associated with the input dataset.

[0090] In certain examples, magic-enhanced quantum features obtained from the measured quantum states in the high-dimensional Hilbert space may be Pauli observables of first degree, or of first and second degree. In some examples, other Pauli observables may be selected as features; or values of different observables on the quantum states 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 a number of qubits.

[0091] In some implementations, the bitstrings from the execution of the quantum logic circuit and measurement of quantum states are then fed to an estimator. In some implementations, the estimator is an estimator of Pauli-expectation values. In this case, the quantum feature vector produced by the estimator includes estimated Pauli-expectation values. For example, if only first-degree Pauli operators are used, the output of the estimator of Pauli expectation values is (<X_1>, <Y_1>, <Z_1>,...,<X_m>,<Y_m>,<Z_m>).

[0092] In some instances, a quantum density operator associated with the dataset may be further determined. In some instances, the quantum density operator may include a density matrix, a reduced form of a density matrix, a partial view of a density matrix, or another type of quantum density operator. In some instances, a density matrix encoding the input dataset can be constructed as [`]1 )

[0093] The density matrices ^are all vectors in the space ℋfof linear operators on ℋ (note that projectors – the "measurement" operatorslinear but notoperators). In some instances, a density matrix includes the average expectation values of all Pauli values across the dataset. In some instances, a reduced form of a density matrix may include multiple subsetsAttorney Docket No.: RIGET-134WO1 of expectation values associated with the same computational basis for the plurality of qubits in the quantum logic circuit. In certain examples, multiprocessing and multithreading parallel processing techniques can be used to speed up the classical calculations.

[0094] In some instances, signatures can be used to encode data streams (timeseries- like 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 part of the input features to the quantum algorithm. Alternatively, in some instances, quantum features can be determined first during operation 218, thereafter (for example, as a final part of process 204) windows are then applied to the quantum features, and the signature transformation performed, resulting in a new set of quantum features. In certain instances, the signatures are not calculated explicitly but use Salvi’s signature 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 (PDE). This may not require explicit computation of signatures and can be solved efficiently using numerical PDE 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.

[0095] From each quantum data point, the expectation values of certain observables can be determined. Using a quantum computer, these can be accurately estimated by collecting shots from the quantum data point. In some instances, weight-1 Pauli observables, commonly known as X, Y, Z, for each qubit and the weight-2 Pauli observables on adjacent qubits, obtained by the combinations of 2 weight-1 Pauli multiplied together can be determined. Each one of these expectation values becomes a component of the Quantum Feature Vector (QFV). The QFV can be used in the same way as the feature vector to train a classifier to detect transactions.Attorney Docket No.: RIGET-134WO1

[0096] At 206, the quantum feature vector is provided to an optimization model. In some instances, ab optimization model may be operated by a machine learning system, which may refer to a classical computing system that executes a class of algorithms that aims to generate a predictive output based on an input, referred to as features of the dataset. In supervised learning cases, machine learning systems operate by training on a dataset (a dataset of features and labels) to configure its internal operation to correctly apply a label to incoming data. Such training data may include a feature tuple and a label. During training, the machine learning system may execute on the training data one or more times until the internal operation can correctly label another set of training data (e.g., a test dataset) until some criteria have been met, e.g., the machine learning system may correctly predict the labels for a test dataset within a threshold accuracy. In certain examples, other machine learning systems may operate in unsupervised learning setting, which uses a different metric to train the machine learning model. In some instances, the preprocessed dataset may be used in a machine learning pipeline which may be a machine learning system that is applicable to an unsupervised learning, generative models, and reinforcement learning procedures. In some instances, a machine learning model includes linear and logistic regression, random forests, boosting algorithms (e.g., gradient boost, AdaBoost, XGBoost), neural networks and deep neural networks (e.g. feedforward networks, convolutional neural networks, recurrent neural networks), kernel methods (e.g., SVMs and SVRs with linear, polynomial, Gaussian, Laplacian, hyperbolic, sigmoid, ANOVA kernels), unsupervised algorithms (e.g., affinity propagation, agglomerative hierarchical clustering, BIRCH, DBSCAN, K-means, mini batch K-means, mean shift, OPTICS, spectral clustering, Gaussian Mixtures), generative models (e.g., GANs, VAEs, auto regressive models), reinforcement learning, signature kernels, etc. In some implementations, the machine learning pipeline may be a quantum machine learning pipeline and / or algorithm. In some instances, the machine learning pipeline may be operated, in whole or in part, by the computing systems shown in FIG.1, such as by the user devices 110, the servers 108, the control system 105, or any combination thereof.

[0097] In some instances, prior to computing the truncated signatures or the signature Kernel using the calculated quantum features, standard lead-lag pre-processing can beAttorney Docket No.: RIGET-134WO1 applied to the time series of quantum features. For example, the Goursat PDE can be solved to compute the Kernel directly.

[0098] In some instances, when the magic-enhanced quantum features associated with the input dataset are within the same domain as the original classical features, the quantum features are compatible with systems and operations designed to receive the original classical features, e.g., classical machine learning models or systems. In this case, the magic- enhanced quantum features can be used as an alternative to the original dataset in a classical machine learning pipeline. In other terms, with an opportune definition of the quantum logic circuit and of the observables, replacing the original dataset with one transformed with the quantum feature map can improve the performance of the machine learning algorithm that would have been applied to the original dataset. In some instances, the preprocessed dataset transformed by the quantum feature map may not need to be processed with a classical kernel, and can be sent as input to a different classical machine learning model, such as neural network, random forest, etc., enhancing the performance of the chosen classical machine learning model, similarly to the case of kernels.

[0099] In certain instances, when the magic-enhanced quantum features are not within the same domain as the original dataset, the machine learning systems may be modified or otherwise configured to accept the preprocessed dataset with the magic-enhanced quantum features. For example, the input dataset may be a categorical dataset where the quantum feature map transforms the input dataset to a real valued quantum feature space. In some instances, the machine learning system may be configured to operate on the preprocessed dataset from the new domain, e.g., the real valued quantum feature space. In other cases, the input dataset with original features and the preprocessed dataset with the quantum-enhanced features may be from the same domain but may differ in size.

[0100] In machine learning, the central problem is balancing approximation and generalization. Approximation refers to how well a machine learning model can represent the true relationship between inputs and outputs, which improves as models become more expressive. Generalization refers to how well the model’s learned patterns extend to unseen data, which typically worsens as models grow more complex and risk overfitting. The bias–variance tradeoff captures this tension: too simple a model yields high bias andAttorney Docket No.: RIGET-134WO1 underfitting, while too complex a model yields high variance and overfitting. The goal is to find the “sweet spot” where both approximation and generalization errors are low enough to achieve good predictive performance.

[0101] The challenge is that approximation and generalization are not independently controlled — improving one often worsens the other. However, advances in model architectures, regularization, optimization methods, and the use of more data can push this boundary outward, enabling models that both approximate complex patterns and generalize well. This shifting frontier explains why modern approaches, such as deep learning with regularization and inductive biases, can outperform older models: they achieve stronger function approximation without proportionally sacrificing generalization. The core problem, then, is designing learning systems that move along or beyond this frontier, capturing the richness of the data while remaining reliable on unseen examples.

[0102] FIG.5 is a plot 500 showing the approximation versus generalization. The graph illustrates the tradeoff between approximation (g) and generalization (h), showing that their product is bounded and you cannot minimize both simultaneously with finite data. The curve 502 (“Low Magic”) represents the classical limit of this tradeoff, while the curve 504 (“High Magic”) shows the presented techniques and systems can push the boundary outward, enabling improved approximation without sacrificing generalization, or vice versa. As shown in FIG.5, the boundary is defined by the magic gh ≤ 1 ℳ (11)where g is approximation, h isis magic. The arrows highlight this shift, illustrating how magic-enhanced quantum feature mapping based on the techniques and systems presented here can do. The boundary can be pushed out to greater approximation and greater generalization by increasing the magic. Greater approximation and generalization can effectively reduce errors introduced from approximation and generalization; therefore, a better performance can be achieved using the same optimization model or machine learning model. Example use cases including using Chebyshev inequality in fraud detection (FIGS.6A-6B), the prediction of the mid- price movements of the LOB of one stock (FIGS.7A-7B), and tackling a classificationAttorney Docket No.: RIGET-134WO1 problem based on the discrete logarithm function (FIG.8) show the advantages offered by the process 200 based on the quantum logic circuits 300, 400 shown in FIGS.3 and 4).

[0103] In some implementations, the optimization model is a Chebyshev inequality- based anomaly classifier, a probabilistic distance-based method, which can be used to process the received quantum feature vector. The optimization model can be used in detecting fraudulent transactions and other outlier detection problems.

[0104] n=1 Chebyshev inequality: let X be a random variable with mean U and standard deviation V, kl m(\ − n)o≥ rs ≤ 1 (12)r

[0105] n>1 Chebyshevvariance-covariance matrix V, kl((X − n)uvwc(X − n) ≥ r) ≤ ^r (13)

[0106] n>1 Chebyshevx(c), … , x(5), x(5yc) samples of \. In some instances, the mean and variance on the first ^be estimated. ooz{yc |&x({yc) − n{'} ^wc&x({yc) − n{' ≥ ~o ^ ≤ min ^1, ^^(^ − 1 + ^~ ) ^ (14)

[0107] thetraining set, which is a subset of x(c), … , x(5). In some instances, supervised (calculate themean and variance only on theunsupervised learning can be used. In some instances, the anomaly score can be defined as the upper bound of the Chebyshev inequality value, where ~ can be replaced with the value of the statistic. When the anomaly score is lower than a certain threshold value the data point can be labeled as anomaly. In some instances, the threshold value can be estimated by cross-validation as some quantile of the statistic or determined in another manner. In some instances, the Mahalanobis distance can be computed with the estimated mean and variance. The computed Mahalanobis distance (one per training point) can be substituted to ~oin the right-handAttorney Docket No.: RIGET-134WO1 side of the inequality. The quantile of these quantities for the outliers can be computed. For every new data point the same quantities are computed and compared with the estimated quantile to decide whether it is an inliner or outlier.

[0108] Because the statistic is a quadratic function of the random variable, the decision boundary (e.g., a contour shape that separates between normal data and anomalies) can be an ellipsoid in the space of the random variables. In some implementations, the systems and techniques presented here can generate more complex contour shapes that have higher capacity and can capture nonlinear relationships between normal and anomalous data.

[0109] Each sample / datapoint x, is a vector of real numbers representing either a feature vector or a quantum feature vector. A datapoint x can be assigned to one of the two classes, either fraudulent or legitimate. A new datapoint x(5yc)needs to be classified and atraining set of ^ datapoints, x(c), … , x(5), alreadybe used to compute thestatistics, that is mean U∑wc.

[0110] The quantum feature map can be defined by a unitary transformation parametrized by the data ^^ ((^)|0^^0|((^)^ (15)

[0111] The mean of themeaning of a density matrix of a mixed state. In a clustering problem, the mean of the feature map over the points of each cluster can be computed. { ^1 ^ ^(16)

[0112] In someusing the Frobenius norm. For example, (1,1)-norm can be used as a measure of magic of the unitary transformation associated with the quantum logic circuit. In some instances,Attorney Docket No.: RIGET-134WO1 the unitary transformation can be projected into Paulis observables of the first and second degree to reduce the dimension.

[0113] In some instances, the problem of detecting fraudulent transactions can be recast in a classification problem that can be solved by supervised or unsupervised machine learning methods and, in particular, binary classification algorithms. In some implementations, an optimization model includes any suitable classification algorithm.

[0114] During a training process, the feature vectors can be randomly split into train and test set according to stratified k-fold cross-validation (CV). In each training CV fold, a majority class containing the authorized transactions can be down-sampled as this is known, and in fact verified in this case to greatly help with classification problems with highly unbalanced data. In some instances, a down-sampling is configured to bring the ratio of fraudulent to legitimate transactions to a value in a range of between 1:1 to 1:5, from the original ratio of 1:578. Interestingly, as the total the train-test split used here is around 1:1 with stratification, and the number of CV folds is between 5 and 10. Each CV training fold is in a small sample regime that is around 100 datapoints. On the CV test folds, many metrics can be computed to evaluate the detection model quality. In some implementations, an Area Under the Curve (AUC) of the Receiving Operator Characteristic (ROC) is used.

[0115] The transformation of the feature vector to gate angle / parameter can be fitted to the train set, then the quantum computer processes all data to create the quantum feature vectors. For any classification algorithms chosen for detection, there are hyperparameters to be selected as random hyperparameter choices that will not lead to optimal results out of sample. In some instances, a Bayesian optimization method can be used. Operations can be repeated using different hyperparameter choices until model quality “has converged” as measured by the metric computed on the CV-test folds. In some implementations, each feature vector and corresponding quantum feature vector is labelled, and each label takes one of two values, such as 0, marking a legitimate transaction and 1, in case of a fraudulent transaction. Labels are assigned based on the ground truth.

[0116] During a deployment process, a transaction request can be received by the system. The feature vector can be calculated based on transaction data. The quantumAttorney Docket No.: RIGET-134WO1 computing resource receives the classical feature vector and generates the quantum feature vector by mapping the classical feature vector to the parameters of the quantum logic circuit, generating the quantum data point, computing the quantum feature vector from shots of the quantum data point. In some instances, the quantum feature vector is fed to the trained detector algorithm. The detector algorithm assigns a binary value to the quantum feature vector, that is, the transaction is flagged as either fraudulent or legitimate. Example results are shown in FIGS.5A-5B.

[0117] FIGS.6A-6B are plots showing the results. Kaggle “credit_card_fraud_detection” was used as the input dataset. Kaggle “credit_card_fraud_detection” contains 284,315 transactions and 492 frauds which is about 0.17% of the total number of transactions. Kaggle’s “credit_card_fraud_detection” dataset includes 24 features. Different detection algorithms (classifiers) including CBS+ (Chebyshed classifier described above 602, 612), SVC (a support vector classifier 606, 616), LGB (LightGBM implementation of Boosted Tree classifiers 604, 614) are used and compared. All the quantum logic circuits were simulated using classical hardware. Each quantum logic circuit includes 60 blocks (e.g., the repeat subcircuit modules 310, 410 in FIGS.3 and 4) repetitions.

[0118] As shown in FIGS.6A-6B, F-1 score of the detected fraud (y axis), grouped by the procedure utilized to generate the dataset (x axis) and the detection algorithm (classifier) used. Three data generation methods were used, including “Original” where the input classical feature vectors are utilized, “Q Inspired feature sets” where a benchmark provided by HSBC based on tensor network simulators, and “Quantum” where the input classical feature vectors are processed to obtain associated quantum feature vectors with a quantum feature mapping as described in the process 200.

[0119] The quantum logic circuit used the experiment (“Quantum”) shown in FIGS.6A- 6B, the data-encoding layer includes parameterized Rz rotations; and the parameter values FV components are rescaled, for example using a classic MinMax scaling on the training set over an interval centered around Pi / 2 and for a narrow width around that value, such as 0.001698981634. In certain instances, other scaling methods may be used. The quantum logic circuit used does not include a T-gate layer and the RZ rotations function of a MinMax rescaling of the FV that is centered around the T-gate value, that is Pi / 4 and an intervalAttorney Docket No.: RIGET-134WO1 around the center of (Pi / 4) / 5. It should be noted that although the obtained data (“Quantum”) are based on the quantum feature vectors generated using a quantum logic circuit without including T gates, the quantum logic circuit is equivalent to the quantum logic circuit with T gates up to a single qubit Clifford unitary; and they contain similar magic.

[0120] As shown in FIG.6A, when all data was used, the CBS+ 602 using QFV obtained by the “quantum” feature map classifiers shows the best performance on the test set. As shown in FIG.6B, when only the most valuable 24 thousand transactions were utilized and again the “quantum” QFM is associated with the best result, this time obtained using an SVC detector. As shown in FIGS.6A-6B, the method based on the quantum logic circuit described in the process 200 outperforms the classical and quantum inspired benchmarks in terms of the number of fraud detected (it contains higher proportion), the false positive rate (it has the lower rate), and the F1 score.

[0001] In some instances, the optimization model includes a kernel-based learning algorithm. For example, the optimization model may include quantum kernels, which are similar to classical kernels with the main difference being that the kernel (aka Gram) matrix is computed with a quantum computer, leveraging the large dimensional Hilbert space as the feature space. Quantum kernels can lead to an exponential speedup over classical machine learning algorithms on artificially generated tasks. In some instances, a quantum kernel may be a fidelity quantum kernel, a projected quantum kernel (PQK) or another quantum kernel. In some instances, the kernel-based learning algorithm can be applied to the LOB input dataset.

[0121] FIG.7A is a table 700 showing a performance comparison between QSK Model against classical benchmarks across different feature dimensions for the prediction of the mid-price movements of the LOB of one stock. The first column indicates whether the MLModel is classical or Quantum. The letters s.k. Stand for Signature Kernel. The notation ‘^ =^’ indicates that n features (which corresponds to choosing a LOB of depth ^ / 2) are usedand that the quantum logic circuit is applied on ^ qubits. The notation ‘^ = ^’ indicates thatthe depth of the quantum logic circuit (defined as the repetition of the fixed block 310, 410 as shown in FIGS.3 and 4) is equal to ^. The results correspond to the full original datasetAttorney Docket No.: RIGET-134WO1 is in the second column, as for example the ones published in Zhang, et al., DeepLOB: Deep Convolutional Neural Networks for Limit Order Books, arXiv:1808.03668 [q-fin.CP], 23 Jan 2020. The weighted version of the metrics, which are given in the last 3 columns, are obtained assuming that the 3 classes of the problem are balanced. Note that the original problem is unbalanced, the number of points corresponding to the stationary class is significantly smaller than the number of points in the other classes. For different (reduced) samples of the original LOB problem we have been able to find a bandwidth and depth such that quantum enhanced signature kernels outperform classical signatures and other benchmark models by a significant margin. Moreover, the performance of quantum enhanced signatures obtained when training on a small number of data points (1,000) is comparable with the performance of the benchmark model trained on the full dataset (200,000 Points).

[0122] FIG.7B is a table 710 showing the performance of a quantum-enhanced signature kernel (QSK) across different batches of data for the prediction of the mid-price movements of the LOB of one stock. All the results are obtained using 12 features and a quantum feature map generated by a quantum logic circuit applied to 12 qubits and depth 12 with the same bandwidth. The depth and bandwidth of the circuit have been chosen based on the performance of QSK on the first batch of data points. It is observed that t-hat ‘optimal’ circuit performance is robust across different batches of data (distant batches, we can be identified by the difference in the batches indices) perform less well suggesting different regimes). The ‘optimal’ circuit determined on a single batch of points also performs when training on a larger set of points.

[0123] In some instances, multi-dimensional Chebyshev inequality can be configured to provide an upper bound for the tail distribution of a random variable. The inequality only requires that the random variable is in ^o, and is expressed entirely using the mean and the variance-covariance matrix of the random variable. In practice the exact value of the mean and of the variance covariance matrix of a random variable are not known and need to be estimated. In some instances, a modified Chebyshev inequality using estimated variance covariance matrix can be determined.Attorney Docket No.: RIGET-134WO1

[0124] In some instances, the multi-dimensional Chebyshev inequality can be configured to use estimated mean and variance below. Let \c, ... , \5, \5yc, .. be a sequenceof i.i.d. random variables in 7^. Let k denote the probability distribution of the \^. \^5is the estimated mean using ^ samples, and v^5is the estimated variance covariance matrix using the ^ first samples: 5 \^(18 5= 1^ ^) \^)Then:k((\5yc − \^5)^v^5wc(\5yc − \^5) ≥ ~o)(20) ≤min ^ −

[0125] In some implementations, thealgorithm. The classification algorithm includes an hyperparameter, denoted ^^which is the empirical quantile threshold of the outliers. Given (^c, ^c), ... , (^5, ^5) training points.By construction we consider an anomaly detection problem. The label of the inliners is 1, whereas the label of the outliers is -1. In some instances, the conditional expectation and the conditional variance-covariance matrix of the inliners can be computed using: ^5̅c = 1(21) ^^ ^^^5̅c = 1^− 1 − −c^^^5:^^bc

[0126] The statistic in equation 10 can be computed for outliers only. In some instances, the empirical 1 − ^^ quantile ^^^ of the distribution of the random variable ((^^ −^5c̅)^(^5̅c)wc(^^ − ^5c̅), 1 ≤ A ≤ ^: ^^ = 1) can be computed. For any new sample of the^ can be labeled as an outlier if (^ − ^5c̅)^(^5̅c)wc(^ − ^5c̅) is higher than ^^^and as an inliner otherwise.Attorney Docket No.: RIGET-134WO1

[0127] FIG.8 is a plot 800 showing average AUC scores across 10 different additive discrete logarithm problems. Each discrete logarithm problem has a size of 10 as described above. The x axis represents the number of training points used for training. The total number of points in each of the 10 datasets depends on the prime number and is around. The problem is artificially converted into an outlier detection problem by undersampling one of the two classes. All three curves 802, 804, 806 use the outlier detection classifier based on Chebyshev inequality. The difference between the three curves resides in the feature map used prior to the classifier. A high magic feature map (curve 806) generated by process 200 and circuit 410, is compared against a quantum feature map (curve 804) generated by process 200 and a quantum logic circuit which is designed to have less magic than 410, and a purely classical feature map: Random Fourier Features (curve 802) of the same dimension of the quantum feature vector. As shown in FIG.8, all models learn the problem when a high number of training points is used. However, when trained on fewer points, a clear separation between the high-magic quantum feature map and other maps can be observed.

[0128] 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 quantum machine learning program. The example quantum logic circuit 300 includes unitary operations applied to qubits which may be defined by qubit devices of a quantum processing unit (e.g., the quantum processing unit 102 in FIG.1). In some instances, qubits may be represented by classical data structures and can be simulated on a quantum simulator. 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 magic-enhanced quantum feature map. In other words, the quantum logic circuit 300 maps classical features of each data point from a classical input dataset into corresponding quantum states, and further to magic-enhanced quantum features. The quantum logic circuit 300 can be executed by another type of quantumAttorney Docket No.: RIGET-134WO1 computing resource including a quantum simulator, a quantum computing system or a hybrid computing system.

[0129] As shown in FIG.3, the example quantum logic circuit 300 includes a subcircuit module 310 which can be repeated ^ times. Each subcircuit module 310 includes multiple subsets 302, 304 of quantum logic gates. Each subset 302, 304 may include one or more quantum logic gates applied on one or more qubits. In particular, a first subset of quantum logic gates 302 includes fixed non-Clifford quantum logic gates. Each quantum logic gate in the first subset 302 may be a single-qubit, a two-qubit, or a multi-qubit quantum logic gate. In some implementations, each quantum logic gate of the first subset 302 is configured to create magic. The quantum logic gate in the first subset 302 can introduce the non- stabilizer nature of quantum state or operations that cannot be generated using Clifford gates alone. In some implementations, the quantum logic gates in the first subset 302 create magic by transforming stabilizer states into non-stabilizer (“magic”) states. In some implementations, each quantum logic gate in the first subset 302 is a single-qubit T gate (e.g., the single-qubit T gates in the subsets 404A, 404B in FIG.4). In some instances, each quantum logic gate in the first subset 302 may include a multi-qubit fixed non-Clifford quantum logic gate configured to create magic states. For example, the multi-qubit fixed non-Clifford quantum logic gate may be a controlled-T gate. In some implementations, prior to the application of the first subset of quantum logic gates, each subcircuit module 310 includes a set of Hadamard gates prior to the application of the first subset 302 of quantum logic gates.

[0130] As shown in FIG.3, the example quantum logic circuit 300 includes a second subset 304 of quantum logic gates that can be used to encode data points from the input dataset into quantum states, translating low-dimensional classical features of data points to high-dimensional magic-enhanced quantum features in the Hilbert space. In some instances, the quantum logic circuit 300 is determined or obtained according to the number of classical features and a range of classical feature values of the data points in the dataset. In some instances, the second subset 304 of quantum logic gates 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.Attorney Docket No.: RIGET-134WO1

[0131] In some instances, an angle encoding method is used, where the classical features are mapped to rotation angles of parametric quantum logic gates of the second subset 304. In this case, each parametric quantum logic gate in second subset 304 may be a single-qubit rotation gate with a rotation angle as a gate parameter for encoding the vectors of the classical features of data points. In some instances, the multiple layers of subcircuit modules 310 may have the parametric single-qubit quantum logic gates that apply specific phase rotations to qubits around the same axis of the Bloch sphere, e.g., Z axis 78(9), X axis 7:(9), or Y axis 7;(9).

[0132] In some instances, other schemes to map the classical features of the data to parameters of the quantum logic circuit 300 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 this case, quantum logic gates in the second subset 304 may be in the form of other types of quantum logic gates. In certain instances, the quantum logic gates in the second subset 304 may include multi-qubit parametric quantum logic gates or a combination of single-qubit and multi-qubit quantum logic gates. In this case, classical features can be also 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.

[0133] In certain examples, the quantum logic circuit 300 may include a third subset of quantum logic gates that are configured to spread or entangle the magic across multiple different qubits. In this case, magic states are no longer localized to certain qubits where they are created. For example, applying a quantum logic gate in the third subset on two qubits that already have magic after applying the quantum logic gates in the second subset 304 can combine the two magic states; can spread the magic states, and create more complex and more deeply entangled magic states, e.g., two-qubit entangled magic states.

[0134] In some instances, the quantum logic gates in the first subset 302 may be divided into different layers. In some instances, the number of layers can be defined by the amount of entanglement required, the fidelity of the quantum logic gates used in the circuit, the topology of the quantum circuit devices, the native gates available on the quantum circuit device, and whether data re-upload techniques are required, or in another manner.Attorney Docket No.: RIGET-134WO1

[0135] FIG.4 is a schematic diagram showing an example quantum logic circuit 400. In some implementations, the example quantum logic circuit 400 corresponds to part of a quantum machine learning program. The example quantum logic circuit 400 includes unitary operations applied to qubits defined by 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 400 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 400 is configured to implement a quantum feature map. In other words, the quantum logic circuit 400 maps classical features of each data point from a classical input dataset into corresponding quantum states, and further to magic-enhanced quantum features. The quantum logic circuit 400 can be executed by another type of quantum computing resource including a quantum simulator, a quantum computing system or a hybrid computing system.

[0136] In some implementations, the number of qubits where the quantum logic circuit 400 is applied is determined based on 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 implementing feature selection and extraction techniques. In some instances, the number of features may be equal to the number (^) 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 can be 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.

[0137] As shown in FIG.4, the example quantum logic circuit 400 includes a subcircuit module 410 which can be iterated ^ times. The circuit depth (^) that corresponds to the optimal trade-off between expressivity and generalization for a given bandwidth can be determined experimentally. Each subcircuit module 410 includes multiple subsets 402A,Attorney Docket No.: RIGET-134WO1 404A, 406A, , 402B, 404B, 406B, 408 of quantum logic gates applied on four qubits 412-1, 412-2, 412-3, 412-4 (referred to collectively as “qubits 412”). Each subset shown in FIG.4 includes multiple quantum logic gates applied on one or more qubits 412. As shown in FIG. 4, subsets 402A, 402B, 402C in a subcircuit module 410 each includes multiple Hadamard (H) gates; subsets 404A, 404B in the subcircuit module 410 each includes multiple single- qubit T gates; subsets 406A, 406B in the subcircuit module 410 each includes multiple CZ gates. subset 408 in the subcircuit module 410 includes multiple single-qubit rotation gates.

[0138] As shown in FIG.4, the example quantum logic circuit 400 includes a sequence of alternating sets of T gates 404 and CZ gates 406 applied on four qubits 412. Specifically, the quantum logic circuit 400 includes a first layer of Hadamard gates (subset 402A) applied on the four qubits at a first time step ^c. The quantum logic circuit 400 includes a first layer of T gates (subset 404A) applied on the four qubits at a second time step ^o. The quantum logic circuit 400 includes a first layer of CZ gates (subset 406A) at a third time step ^^, creating entanglement and spreading the magic between qubit 412-1 and qubit 412-2 and between qubit 412-3 and qubit 412-4. The quantum logic circuit 400 includes a second layer of Hadamard gates (subset 402B) applied on the four qubits at a fourth time step ^!. The quantum logic circuit 400 includes a second set of T gates (subset 404B) applied on the four qubits at a fifth time step ^^. The quantum logic circuit 400 includes a second layer of two-qubit CZ gates (subset 406B) at a sixth time step ^^, creating entanglement and spreading magic between qubit 412-3 and qubit 412-2 and between qubit 412-1 and qubit 412-4. The quantum logic circuit 400 includes a third set of Hadamard gates (subset 402C) applied on the four qubits at a seventh time step ^^; and a set of single-qubit rotate gates (subset 408), each of which is configured to apply a specific phase rotation to a qubit around the Z axis of the Bloch sphere, 78(9).

[0139] In some implementations, the first and second layers of the single-qubit T gates (subsets 404A, 404B) in the quantum logic circuit 400 are magic creating layers; the first and second layers of the CZ gates (subsets 406A, 406B) are magic spreading layers; and the layer of single-qubit rotation gates (subset 408) is a data encoding (data uploading) layer.Attorney Docket No.: RIGET-134WO1 The quantum logic gates in the quantum logic circuit may be organized as the example quantum logic circuit 400 shown in FIG.4 or in another manner.

[0140] As shown in FIG.4, a data point in a dataset can be encoded through one-qubit rotation operations which are rotations of the individual quantum states around the Z axis. Each data point can be represented by a vector of four features, and each feature is mapped into the corresponding rotation angle. T

[0141] The final quantum state, | / I^, is obtained after application of the ^-qubit quantum logic circuit – a logic gates (linear unitary operators ()controlled by parameters Θc, … ,– the initial quantum state, | / 0^:| / I^ = ( (Θ ) … (o(Θo)(c(Θc)

[0142] In some implementations, final quantum states of themeasured. The example quantum logic circuit 400 may include a set of measurement, each of which is configured to extract classical information from the qubits 412 causing collapsing of the superposition of a qubit 412 into a respective definite output represented by a classical bit (0 or 1). The measurement produces a classical bitstring, which corresponds a data point from the probability distribution encoded in the final quantum state (each quantum state encodes a probability distribution).

[0143] 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 or destination of computer program instructions encoded in an artificially generatedAttorney Docket No.: RIGET-134WO1 propagated signal. The computer storage medium can also be, or be included in, one or more separate physical components or media.

[0144] 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.

[0145] In a general aspect, quantum-assisted data evaluation is presented.

[0146] In a first example, a method of processing an input dataset for an optimization model includes obtaining the input dataset comprising a plurality of data points; and causing a quantum computing resource to execute a quantum logic circuit. The quantum logic circuit includes multiple layers of subcircuit modules applied on a plurality of qubits. Each subcircuit module includes a first subset of fixed non-Clifford quantum logic gates and a second subset of parametric quantum logic gates that are parameterized by the plurality of data points. The method further includes obtaining bitstrings from the quantum computing resource based on output quantum states generated by executing the quantum logic circuit; determining a quantum feature vector associated with the input dataset based on the obtained bitstrings; and providing the quantum feature vector as an input to the optimization model.

[0147] Implementations of the first example may include one or more of the following features. Each fixed non-Clifford quantum logic gate in the first subset includes a single- qubit T gate. Each subcircuit module further includes a third subset of multi-qubit fixed quantum logic gates configured to create entanglement among the plurality of qubits. Each multi-qubit fixed quantum logic gate in the third subset includes a multi-qubit fixed Clifford quantum logic gate. Each multi-qubit fixed Clifford quantum logic gates in the third subset is one of a two-qubit iSWAP gate, a two-qubit Controlled-NOT gate, a two-qubit Controlled- Z gate, or a two-qubit Controlled-phase (CPHASE) gate. Each subcircuit module includes a plurality of Hadamard gates.

[0148] Implementations of the first example may include one or more of the following features. Each fixed non-Clifford quantum logic gate in the first subset includes a two-qubitAttorney Docket No.: RIGET-134WO1 fixed non-Clifford quantum logic gate. The two-qubit fixed non-Clifford quantum logic gate includes a two-qubit controlled-T gate.

[0149] Implementations of the first example may include one or more of the following features. Causing the quantum computing resource to execute the quantum logic circuit includes introducing magic by applying the first subset of fixed non-Clifford quantum logic gates on the plurality of qubits; and encoding the plurality of data points as parameters of the quantum logic circuit by applying the second subset of parametric quantum logic gates on the plurality of qubits. Each subcircuit module further includes a third subset of multi- qubit fixed Clifford quantum logic gates, and causing the quantum computing resource to execute the quantum logic circuit includes after introducing the magic and prior to encoding, spreading the created magic among the plurality of qubits by applying the third subset of multi-qubit fixed Clifford quantum logic gates to the plurality of qubits. The second subset of parametric quantum logic gate includes a plurality of single-qubit parametric quantum logic gate, and encoding the plurality of data points as parameters of the quantum logic circuit includes determining rotation angle values of the plurality of single-qubit parametric quantum logic gates according to the number of layers of the subcircuit modules and the number of qubits to which the quantum logic circuit is applied. The single-qubit parametric quantum logic gates in the second subset are single-qubit rotation gates with phase rotations along the Z axis of the Bloch sphere.

[0150] Implementations of the first example may include one or more of the following features. The quantum feature vector includes a quantum density operator, and determining the quantum feature vector includes determining estimated expectation values of a set of observables based on the obtained bitstrings; and determining the quantum density operator based on the estimated expectation values of a set of observables. The set of observables includes observables of first degree. The observables of first degree correspond to single-body Pauli operators. The set of observables includes observables of second degree. The observables of second degree correspond to two-body Pauli operators. The quantum density operator includes one of a density matrix; or a reduced form of a density matrix.Attorney Docket No.: RIGET-134WO1

[0151] Implementations of the first example may include one or more of the following features. The quantum computing resource includes at least one of a quantum simulator, a quantum computing system, or a hybrid computing system. Providing the quantum feature vector as an input to the optimization model includes performing a Chebyshev inequality- based anomaly classifier based on the quantum feature vector.

[0152] In a second example, a quantum computing system includes a quantum computing resource; and a classical computing resource configured to perform one or more operations in the first example.

[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 such separation 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

Attorney Docket No.: RIGET-134WO1 CLAIMS What is claimed is:

1. A method of processing an input dataset for an optimization model, the method comprising: obtaining the input dataset comprising a plurality of data points; causing a quantum computing resource to execute a quantum logic circuit, the quantum logic circuit comprising multiple layers of subcircuit modules applied on a plurality of qubits, each subcircuit module comprising a first subset of fixed non-Clifford quantum logic gates and a second subset of parametric quantum logic gates that are parameterized by the plurality of data points, obtaining bitstrings from the quantum computing resource based on output quantum states generated by executing the quantum logic circuit; determining a quantum feature vector associated with the input dataset based on the obtained bitstrings; and providing the quantum feature vector as an input to the optimization model.

2. The method of claim 1, wherein each fixed non-Clifford quantum logic gate in the first subset comprises a single-qubit T gate and each subcircuit module further comprises a third subset of multi-qubit fixed quantum logic gates configured to create entanglement among the plurality of qubits.

3. The method of claim 2, wherein each multi-qubit fixed quantum logic gate in the third subset comprises a multi-qubit fixed Clifford quantum logic gate.

4. The method of claim 3, wherein each multi-qubit fixed Clifford quantum logic gates in the third subset is one of: a two-qubit iSWAP gate, a two-qubit Controlled-NOT gate, a two-qubit Controlled-Z gate, or a two-qubit Controlled-phase (CPHASE) gate.

5. The method of claim 2, wherein each subcircuit module comprises a plurality of Hadamard gates.Attorney Docket No.: RIGET-134WO1 6. The method of claim 1, wherein each fixed non-Clifford quantum logic gate in the first subset comprises a two-qubit fixed non-Clifford quantum logic gate.

7. The method of claim 1, wherein the two-qubit fixed non-Clifford quantum logic gate comprises a two-qubit controlled-T gate.

8. The method of claim 1, wherein causing the quantum computing resource to execute the quantum logic circuit comprises: introducing magic by applying the first subset of fixed non-Clifford quantum logic gates on the plurality of qubits; and encoding the plurality of data points as parameters of the quantum logic circuit by applying the second subset of parametric quantum logic gates on the plurality of qubits.

9. The method of claim 8, wherein each subcircuit module further comprises a third subset of multi-qubit fixed Clifford quantum logic gates, and causing the quantum computing resource to execute the quantum logic circuit comprises: after introducing the magic and prior to encoding, spreading the created magic among the plurality of qubits by applying the third subset of multi-qubit fixed Clifford quantum logic gates to the plurality of qubits.

10. The method of claim 8, wherein the second subset of parametric quantum logic gate comprises a plurality of single-qubit parametric quantum logic gate, and encoding the plurality of data points as parameters of the quantum logic circuit comprises: determining rotation angle values of the plurality of single-qubit parametric quantum logic gates according to the number of layers of the subcircuit modules and the number of qubits to which the quantum logic circuit is applied.

11. The method of claim 10, wherein the single-qubit parametric quantum logic gates in the second subset are single-qubit rotation gates with phase rotations along the Z axis of the Bloch sphere.

12. The method of claim 1, wherein the quantum feature vector comprises a quantum density operator, and determining the quantum feature vector comprises: determining estimated expectation values of a set of observables based on theAttorney Docket No.: RIGET-134WO1 obtained bitstrings; and determining the quantum density operator based on the estimated expectation values of a set of observables.

13. The method of claim 12, wherein the set of observables comprises observables of first degree.

14. The method of claim 13, wherein the observables of first degree correspond to single-body Pauli operators.

15. The method of claim 12, wherein the set of observables comprises observables of second degree.

16. The method of claim 15, wherein the observables of second degree correspond to two-body Pauli operators.

17. The method of claim 12, wherein the quantum density operator comprises one of: a density matrix; or a reduced form of a density matrix.

18. The method of any one of claims 1 to 17, wherein the quantum computing resource comprises at least one of: a quantum simulator, a quantum computing system, or a hybrid computing system.

19. The method of any one of claims 1 to 17, wherein providing the quantum feature vector as an input to the optimization model comprises: performing a Chebyshev inequality-based anomaly classifier based on the quantum feature vector.

20. A quantum computing system comprising: a quantum computing resource; and a classical computing resource configured to: obtain the input dataset comprising a plurality of data points; cause a quantum computing resource to execute a quantum logic circuit, theAttorney Docket No.: RIGET-134WO1 quantum logic circuit comprising multiple layers of subcircuit modules applied on a plurality of qubits, each subcircuit module comprising a first subset of fixed non-Clifford quantum logic gates and a second subset of parametric quantum logic gates that are parameterized by the plurality of data points, obtain bitstrings from the quantum computing resource based on output quantum states generated by executing the quantum logic circuit; determine a quantum feature vector associated with the input dataset based on the obtained bitstrings; and provide the quantum feature vector as an input to the optimization model.

21. The system of claim 20, wherein each fixed non-Clifford quantum logic gate in the first subset comprise a single-qubit T gate and each subcircuit module further comprises a third subset of multi-qubit fixed quantum logic gates configured to create entanglement among the plurality of qubits.

22. The system of claim 21, wherein each multi-qubit fixed quantum logic gate in the third subset comprises a multi-qubit fixed Clifford quantum logic gate.

23. The system of claim 21, wherein each multi-qubit fixed Clifford quantum logic gates in the third subset is one of: a two-qubit iSWAP gate, a two-qubit Controlled-NOT gate, a two-qubit Controlled-Z gate, or a two-qubit Controlled-phase (CPHASE) gate.

24. The system of claim 21, wherein each subcircuit module comprises a plurality of Hadamard gates.

25. The system of claim 20, wherein each fixed non-Clifford quantum logic gate in the first subset comprises a two-qubit fixed non-Clifford quantum logic gate.

26. The system of claim 20, wherein the two-qubit fixed non-Clifford quantum logic gate comprises a two-qubit controlled-T gate.

27. The system of claim 20, wherein causing the quantum computing resource to execute the quantum logic circuit comprises:Attorney Docket No.: RIGET-134WO1 introducing magic by applying the first subset of fixed non-Clifford quantum logic gates on the plurality of qubits; and encoding the plurality of data points as parameters of the quantum logic circuit by applying the second subset of parametric quantum logic gates on the plurality of qubits.

28. The system of claim 27, wherein each subcircuit module further comprises a third subset of multi-qubit fixed Clifford quantum logic gates, and causing the quantum computing resource to execute the quantum logic circuit comprises: after introducing the magic and prior to encoding, spreading the created magic among the plurality of qubits by applying the third subset of multi-qubit fixed Clifford quantum logic gates to the plurality of qubits.

29. The system of claim 27, wherein the second subset of parametric quantum logic gate comprises a plurality of single-qubit parametric quantum logic gate, and encoding the plurality of data points as parameters of the quantum logic circuit comprises: determining rotation angle values of the plurality of single-qubit parametric quantum logic gates according to the number of layers of the subcircuit modules and the number of qubits to which the quantum logic circuit is applied.

30. The system of claim 29, wherein the single-qubit parametric quantum logic gates in the second subset are single-qubit rotation gates with phase rotations along the Z axis of the Bloch sphere.

31. The system of claim 20, wherein the quantum feature vector comprises a quantum density operator, and determining the quantum feature vector comprises: determining estimated expectation values of a set of observables based on the obtained bitstrings; and determining the quantum density operator based on the estimated expectation values of a set of observables.

32. The system of claim 31, wherein the set of observables comprises observables of first degree.Attorney Docket No.: RIGET-134WO1 33. The system of claim 32, wherein the observables of first degree correspond to single-body Pauli operators.

34. The system of claim 31, wherein the set of observables comprises observables of second degree.

35. The system of claim 34, wherein the observables of second degree correspond to two-body Pauli operators.

36. The system of claim 31, wherein the quantum density operator comprises one of: a density matrix; or a reduced form of a density matrix.

37. The system of any one of claims 20 to 36, wherein the quantum computing resource comprises at least one of: a quantum simulator, a quantum computing system, or a hybrid computing system.

38. The system of any one of claims 20 to 36, wherein providing the quantum feature vector as an input to the optimization model comprises: performing a Chebyshev inequality-based anomaly classifier based on the quantum feature vector.

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