Quantum feature maps

Abstract

In a general aspect, a quantum feature-map transforms an input dataset with original features to a preprocessed dataset with quantum-enhanced features. In some cases, pre-processing an input dataset for a machine learning model includes obtaining the input dataset comprising a plurality of data points; encoding the plurality of data points as parameters of a quantum logic circuit; and executing the quantum logic circuit on a quantum computing resource. Expectation values of a set of observables are obtained based on an output quantum state generated by executing the quantum logic circuit. The set of observables includes observables of first degree and observables of second degree. A pre-processed dataset is generated based on the expectation values and provided as an input to the machine learning model.

Description

Attorney Docket No.: RIGET-118WO1 Quantum Feature Maps CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 497,959, filed April 24, 2023, entitled “Quantum-Enhanced Maps.” The above- referenced priority document is incorporated herein by reference in its entirety. TECHNICAL FIELD

[0002] The following description relates generally to quantum feature maps. 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 computing process.

[0006] FIG.3 is a block diagram showing aspects of an example quantum feature mapping process.

[0007] FIG.4 is a schematic diagram showing an example quantum logic circuit.

[0008] FIGS.5A-5B include a plot showing recession score as a function of time in months using a classical signature kernel and a plot showing recession score as a function of time in months using a quantum enhanced signature kernels for the single-shot prediction of recession for the United States economy.Attorney Docket No.: RIGET-118WO1 DETAILED DESCRIPTION

[0009] In some aspects of what is described here, input datasets are preprocessed by a quantum computing resource. In some examples, the input datasets are preprocessed to prepare data for a machine learning model. In some instances, a set of original data can be received, processed, and transformed by operation of a quantum feature map system to generate a new set of data. The new set of data compared to the set of original data can have enhanced features making the new set of data more informative, discriminative, or suitable for the machine learning model.

[0010] In some implementations, a quantum feature map generates quantum-enhanced features. In some examples, a quantum feature map can be configured to transform or otherwise map an input vector of real numbers into a quantum state. In some cases, the quantum computing system utilizes ^ qubits and the quantum state can be represented as a vector belonging to a linear space of dimension 2^. In some instances, a quantum feature map may be performed on a gate-based quantum computing system by encoding the inputs in parameters of parametric quantum logic gates of a quantum logic circuit. The quantum feature map then provides 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 a machine learning model.

[0011] In some implementations, quantum-enhanced features are generated by operation of a quantum computing resource that includes a quantum simulator, a quantum computing system, or a hybrid system. The quantum-enhanced features may be used, for example, as high-quality input to machine learning models. In some instances, the quantum-enhanced features can be used as an alternative to the features of the original dataset in a classical machine learning pipeline. For instance, with an opportune definition of the quantum logic circuit and of the observables, the features in the original dataset can be replaced with the quantum-enhanced features in a classical machine learning pipeline. In some instances, the quantum-enhanced features can be used to augment the features ofAttorney Docket No.: RIGET-118WO1 the original dataset. In certain examples, the quantum-enhanced features produced by the quantum feature map may be used in another manner.

[0012] In some instances, the quantum-enhanced features obtained from the techniques and processes presented here can improve the discriminant capability of features and increase performance of machine learning algorithms. The techniques and processes presented here may be implemented in a manner that is efficient and scalable in some systems. For example, signature kernels, a centerpiece of rough path theory, can be used to process data streams and capture essential information. Combining signature kernels with quantum feature maps has potential to improve performance of machine learning algorithms applied to data streams. The methods and techniques presented here may be implemented in a manner that is compatible with NISQ (Noisy Intermediate-Scale Quantum) era devices, with scalable and effective quantum error mitigation, combined with Clifford Data Regression or rescaling supported by randomized compilation. The techniques and processes may be embedded into classical machine learning pipelines. In some cases, a combination of these and potentially other advantages and improvements may be obtained.

[0013] 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 a computing environment may operate as described with respect to FIG.1 or in another manner.

[0014] 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.Attorney Docket No.: RIGET-118WO1

[0015] 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).

[0016] 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. For simplicity of discussion, the term “program,” as used herein, may refer to a ML program, a ML pipeline, a quantum feature map, and the like.

[0017] 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.

[0018] 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 communicationAttorney Docket No.: RIGET-118WO1 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.

[0019] 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.

[0020] The remote data connection in FIG.1 is provided by a wide area network 115, which may include, for example, the Internet or another type of wide area communication network. In some cases, remote user devices use another type of remote data connection (e.g., satellite-based connections, a cellular network, a virtual private network, etc.) to access the servers 108. The wide area network 115 may include one or more internet servers, firewalls, service hubs, base stations, or a combination of these and other types of remote networking elements. Generally, the computing environment 100 can be accessible to any number of remote user devices.

[0021] 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 userAttorney Docket No.: RIGET-118WO1 devices 110 based on output data from the computational tasks performed by the quantum computing systems 103A, 103B, and the other resources 107.

[0022] 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.

[0023] 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.

[0024] 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.

[0025] In some implementations, the servers 108 generate programs, identify appropriate computing resources (e.g., a QPU or QVM) in the computing system 101 toAttorney Docket No.: RIGET-118WO1 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.

[0026] 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.

[0027] In some cases, a program may be expressed in a hardware-independent format. For example, quantum machine instructions may be provided in a quantum instruction language such as Quil, described in the publication “A Practical Quantum Instruction Set Architecture,” arXiv:1608.03355v2, dated Feb.17, 2017, or another quantum instruction language. For instance, the quantum machine instructions may be written in a format that can be executed by a broad range of quantum processing units or simulators. In some cases, a program may be expressed in high-level terms of quantum logic gates or 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-Attorney Docket No.: RIGET-118WO1 quil-t-ea8945061e5b dated Dec.10, 2020, which is hereby incorporated by reference in the present disclosure.

[0028] 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 computing 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.

[0029] 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.

[0030] In some implementations, the servers 108 generate a schedule for executing programs (such as programs associated with the ML pipeline and / or quantum feature maps), 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.

[0031] 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 moreAttorney Docket No.: RIGET-118WO1 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 computing 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.

[0032] 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.

[0033] 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 fromAttorney Docket No.: RIGET-118WO1 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.

[0034] 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.

[0035] In some models, fault-tolerance can be achieved by applying a set of high-fidelity control and measurement operations to the qubits. For example, quantum error correcting codes can be deployed to achieve fault-tolerant quantum computation. Other computational regimes may be used; for example, quantum computing systems may operate in non-fault-tolerant regimes. In some implementations, a quantum computing system is constructed and operated according to a scalable quantum computing architecture. For example, in some cases, the architecture can be scaled to a large number of qubits to achieve large-scale general purpose coherent quantum computing. Other architectures may be used; for example, quantum computing systems may operate in small- scale or non-scalable architectures.

[0036] 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 controlsAttorney Docket No.: RIGET-118WO1 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.

[0037] 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.

[0038] In some instances, the quantum processing modules can include a superconducting quantum circuit that includes one or more quantum circuit devices. For instance, a superconducting quantum circuit may include qubit devices, readout resonator devices, Josephson junctions, or other quantum circuit devices. In some implementations, quantum circuit devices in a quantum processing unit can be collectively operated to define a single logical qubit. A logical qubit comprises 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.Attorney Docket No.: RIGET-118WO1

[0039] 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.

[0040] 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.

[0041] 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.Attorney Docket No.: RIGET-118WO1

[0042] 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.

[0043] 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.

[0044] In some instances, one or more components of the signal hardware 104A generate control signals, for example, based on control information from the controllers 106A. The control signals can be delivered to the quantum processing unit 102A during operation of the quantum computing system 103A. For instance, the signal hardware 104A may generate signals to implement quantum logic operations, readout operations, or other types of operations. As an example, the signal hardware 104A may include arbitrary waveform generators (AWGs) that generate electromagnetic waveforms (e.g., microwave or radio-frequency) 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.

[0045] 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 processingAttorney Docket No.: RIGET-118WO1 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.

[0046] 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.

[0047] 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 106AAttorney Docket No.: RIGET-118WO1 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.

[0048] 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.

[0049] 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.

[0050] 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.Attorney Docket No.: RIGET-118WO1

[0051] 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.

[0052] 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.

[0053] In some implementations, the quantum computing systems 103A, 103B are disparate systems that provide distinct modalities of quantum computation. For example, the computer system 101 may include both an adiabatic quantum computing system and a gate-based quantum computer system. As another example, the computer system 101 may include a superconducting circuit-based quantum computing system and an ion trap-based quantum computer system. In such cases, the computer system 101 may utilize each quantum computing system according to the type of quantum program that is being executed, according to availability or capacity, or based on other considerations.

[0054] In some instances, one or more components of the computing system 101 shown in FIG.1 are configured to perform the operations of the example processes shown in FIGS. 2-5, or another process. For example, a classical computing system (e.g., the classical processors 111 in the servers 108, the controllers 106 in the quantum computing systems 103, or another) can be configured to pre-process received datasets. In some cases, the classical computing system can determine a number of features and a range of featureAttorney Docket No.: RIGET-118WO1 values; design a quantum feature map by constructing 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 data points from the input dataset into quantum states; obtain and process measured quantum states; and determine quantum-enhanced features based on expectation values of observables of first and second degree. As another example, the classical computing system may include a machine learning system that receives the quantum-enhanced features and performs machine learning tasks. In some instances, the one or more components of the computing system 101 shown in FIG.1 may be configured to perform other operations.

[0055] FIG.2 is a flow chart showing aspects of an example computing process 200. The example process 200 can be used to construct a quantum feature map that encodes an input dataset into a quantum-enhanced dataset. The quantum-enhanced dataset may include expectation values of observables that can be used, for example, in supervised learning, unsupervised learning, or other machine learning applications. The example process 200 may include additional or different operations, and the operations may be performed in the order shown or in another order. In some cases, operations in the example process 200 can be combined, iterated or otherwise repeated, or performed in another manner.

[0056] 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 implementations, the features of the input dataset may be referred to as “original features”. In some implementations, the input dataset is received by a quantum feature map system, which may be implemented as the computing system 101 shown in FIG.1. In some instances, more than one input dataset may be received. In some instances, the input dataset is received by the computing system 101 from the user device 110 via the wide area network 115 as shown in FIG.1 or in another manner. As an example, let ^ be the input dataset. The cardinality of ^ can be indicated as [^]. Data points of theAttorney Docket No.: RIGET-118WO1 input datasets ^ ∈ ^ are of the form ^^: = (^^(1), … , ^^(^))^, where ^ is the number of features.

[0057] In some instances, the input dataset can be obtained from a classical machine learning problem with a classic formulation of the machine learning problem to make a prediction (e.g., a prediction of a recession or no recession in the economoy). In some examples, the machine learning problem may be a classification, a regression, or an anomaly detection task. The components of the original input dataset can be referred to as “features” in certain contexts. A feature can be or include a coordinate of a vector of numbers or a more qualitative element (such as a medical condition for example). In some instances, an “original feature” may go through a number of classical transformation steps before it is fed to the quantum feature map system for processing.

[0058] In certain instances, the input dataset with the original features may be obtained from the classical machine learning problem at any time. For example, the input dataset may include a training dataset or a test dataset which can be used to train and construct a machine learning model. For another example, the input dataset may be actual data that needs to be processed by operation of a trained machine learning model.

[0059] In some instances, the input dataset may be preprocessed. For example, features in the dataset can be calculated; and features can be standardized by determining a range of feature values (e.g., a maximum value and a minimum value). In some implementations, one or more attributes of the quantum logic circuit are determined by the determined number of features and the range of feature values. For example, the one or more attributes of the quantum logic circuit includes a number of elementary static blocks, a number of elementary variational blocks, a number of qubits to which the quantum logic circuit is applied, and other attributes of the quantum logic circuit. In some instances, the datasets can be preprocessed to ensure the drift impact of the quantum computing system is spread similarly across all data points in a dataset. In some instances, the dataset preprocessing may include resampling, reshuffling, reorganizing, and other operations.

[0060] At 204, a quantum feature map is obtained. as the quantum feature map is a composite function of a first map ^ defining a mapping from a subset ^ ∈ ^^to a set ofAttorney Docket No.: RIGET-118WO1 density matrices ^, and a second map ^ defining a mapping from the density matrices ^ to a subset ^ ∈ ^^. Accordingly, in some examples the quantum feature map may be defined as ^: ^^→ ^^, ^ ≡ ^ ∘ ^. In some instances, the first map ^ is achieved by executing a quantum logic circuit by associating to an element ^ ∈ ^ with a unitary transformation ^(^), starting from a fixed initial state!. In certain instances, the second map ^ can be realised by fixing ^ observables {#^}, % = 1, … , ^, and defining the %-th component of the second map ^ as &'[ #^], representing the trace of an observable operator #^in a quantum state . In some instances, an observable may be a Pauli operator or another operator. In the context of machine learning, the independent variables of the domain are normally called features. The quantum-enhanced features are independent variables of the codomain of ^. In other words, the quantum feature map ^ is a function from ^ ∈ ^^to ^ ∈ ^^that maps features into quantum-enhanced features. In some implementations, the execution of the first map ^ and then the second map ^ allows a transformation of the input dataset with original features to a preprocessed dataset with quantum-enhanced features. In some implementations, the quantum feature map is defined by a quantum logic circuit and measured observables. In some instances, the quantum feature map may be defined in another manner.

[0061] In some implementations, a quantum logic circuit is used to efficiently encode relatively low-dimensional classical data points into extremely high-dimensional quantum states. In this case all quantum states are initialized as 0, and the initial state is |)!^ = |0^ ⊗ ⋯ ⊗ |0^ ≡ |0^⊗^. The vector of classical features, ^, is mapped intoof gate parameters of the quantum logic circuit. In some instances, an angle encoding method is used, where the classical features are mapped to rotation angles of single-qubit quantum logic gates of the quantum logic circuit. In some instances, different schemes to map the classical features of the data to parameters of the quantum logic circuit may be used. For example, amplitude encoding, basis encoding, reuploading encoding, or other types of encoding schemes can be used to map classical features onto quantum states. In certain instances, classical features can be encoded on parameters of the multi-qubit quantum logic gates, parameters on both the single-qubit quantum logic gates and the multi-qubit quantum logic gates, or in another manner. The final state, |) / 〉, thus can encode theAttorney Docket No.: RIGET-118WO1 classical data points in a dataset and the subsequent sample classification can be performed in the high-dimensional Hilbert space. In some instances, the quantum logic circuit can provide more expressive power than equivalent classical models when only a polynomial number of parameters is allowed.

[0062] In some instances, a quantum logic circuit for quantum feature mapping may include one or more elementary static blocks. Each elementary static block includes one or more layers of single-qubit quantum logic gates for feature encoding, and one or more layers of multi-qubit quantum logic gates for feature entanglement. In some instances, each elementary static block includes parametric quantum logic gates with gate parameters configured to encode features of each data point. For example, a quantum logic circuit for quantum feature mapping may include multiple layers of single-qubit rotation gates with rotation angles as gate parameters for encoding the features of data points; and multiple layers of two-qubit quantum logic gates configured to create entanglement. In some instances, the quantum logic circuit may be implemented as the quantum logic circuit 400 shown in FIG.4 or in another manner.

[0063] In some instances, the quantum logic circuit may be constructed based on the dataset and the quantum computing hardware. In some instances, a quantum logic circuit may be determined by the depth and shape of the quantum logic circuit, the number of layers of the single-qubit quantum logic gates and multi-qubit quantum logic gates in the quantum logic circuit, the number of qubits on which the quantum logic circuit is applied, types of quantum logic gates, the ansatz (e.g., manner the quantum logic gates are connected together), measurement basis, features in the high-dimensional Hilbert space, and other parameters. In some instances, the quantum logic circuit is applied over a number (^) of qubits, which is determined by the number of features associated with the data points, e.g. one-to-one mapping between features and qubits. In some instances, the quantum logic circuit may be applied over a number of qubits that is more than the number of features, e.g., one-to-multiple mapping between features and qubits. When the quantum logic circuit includes two or more elementary static blocks, the layers of single-qubit quantum logic gates alternate with the layers of two-qubit quantum logic gates in the quantum logic circuit. In some instances, the range of rotation angles should be set atAttorney Docket No.: RIGET-118WO1 [0, 1 / 3], where 3 is the number of data encoding layers. In some instances, the quantum logic circuit is implemented as the quantum logic circuit 400 shown in FIG.4 or in another manner.

[0064] For example, the depth and number of layers and the gates used within the quantum logic circuit may be varied. For example, a multi-qubit quantum logic gate to obtain entanglement may include a controlled-Z (CZ) gate, an imaginary SWAP (iSWAP) gate, a controlled-NOT (CNOT) gate, a controlled-phase (CPHASE) gate, or another multi- qubit quantum logic gate. In some instances, circuit shape can be determined in order to best match the qubit topology of the quantum processing unit, e.g., connectivity of qubit devices. Features in the high-dimensional Hilbert space used may be Pauli operators of first order, or of first and second order. In some examples, other observables may be selected as features; or completely different features may also be utilized. In some instances, utilizing expectation values of Pauli operators 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 the number of qubits.

[0065] In certain examples, a number of features of the data points may be determined, e.g., by using feature extraction techniques or Random Fourier transforms. A range of feature values for each feature of the data points in the dataset is calculated. The quantum logic circuit is determined according to the number of features and the range of feature values. For example, a number of layers of the single-qubit quantum logic gates and a number of qubits to which the quantum logic circuit is applied are determined. In some instances, when the quantum logic circuit includes parametric single-qubit rotation gates, rotation angle values of the single-qubit rotation gates are determined according to the number of layers of the single-qubit quantum logic gates and the number of qubits to which the quantum logic circuit is applied. In some instances, features associated with data points may be encoded as parameters of the single-qubit quantum logic gates and parameters of the multi-qubit quantum logic gates.

[0066] In some instances, the quantum logic circuit and the measurement of quantum states can lead to an arbitrary choice of a system of reference. The arbitrariness in theAttorney Docket No.: RIGET-118WO1 choice of single-qubit gate operations can be removed, by breaking the directional asymmetry through the application of single-qubit quantum logic gates along random directions after state preparation. In some instances, directions along which the single- qubit quantum logic gates are applied in the same layer can be different for different qubits. In certain examples, any remaining directional asymmetry, namely measurement in the z basis, can be eliminated by the use of the approximate state formalism.

[0067] In some instances, the quantum logic circuit may include one or more elementary variational blocks (e.g., the operation 306 in FIG.3) each of which includes parametric quantum logic gates and enable the adjustability of the quantum feature map to the input dataset. In some instances, an elementary variational block can cause improvements to the quality of the quantum-enhanced features (measured in the results obtained with the target classical machine learning model). In some examples, an optimization objective function may be the kernel alignment. In some instances, the optimization function of a classical machine learning model can be used. In some instances, the optimization is executed with the approximate quantum state variational optimizer approach, which enables the computation of gradients, significantly containing the number of shots necessary. In some instances, each elementary variational block may include one layer of single-qubit quantum logic gates and one layer of two-qubit quantum logic gates. For example, directions along which the single-qubit quantum logic gates are applied may be parameters which may be varied and optimized using the expectation values of observables as an objective function. In this case, optimization may be performed based on the preprocessed dataset with the quantum-enhanced features. In some instances, the quantum logic circuit may not include an elementary variational block. In some instances, the number of elementary static and variational blocks in the quantum logic circuit may be varied; and may be configured to ensure an overall depth that avoids decoherence.

[0068] When two input datasets are received, data points in the two input datasets may be encoded in quantum states. In some instances, when data points from different datasets have different number of features, two different quantum logic circuits may be used. For example, the first and second quantum logic circuits for encoding the data points in the two input datasets may have different gate parameter values, different numbers of layers ofAttorney Docket No.: RIGET-118WO1 quantum logic gates, different number of qubits, etc. In some instances, the data points in the two input datasets may be encoded using the same quantum logic circuit, for example, when the two input datasets include the same number of features, and the features are within the same range of feature values. In the case of two input datasets are received, executing the quantum logic circuit can provide a transformation 4 that maps the data points ^^and 56in the two datasets into vectors 4(^^) and 4(56) of a Hilbert space 7. On the quantum computer, the transformation 4 is unitary transformation ^applied to the state |0^ ≡ |0^⊗^and |4^^^^ = = |^^. Note that if the transformation 4 is specified on 8 qubit devices, the dimensionality of the Hilbert space 7 is 2^.

[0069] At 206, the quantum feature map is performed. The quantum feature map may be performed, for example, by executing the quantum feature map to transform data points into quantum states, measuring the quantum states and determining traces of various observables of respective quantum states. In some implementations, a quantum feature map is performed by operation of a quantum computing resource (e.g., the computing system 101 in FIG.1). In some instances, the quantum computing resource may include a quantum simulator (e.g., the classical processors 111 of the servers 108), a quantum computing system with a quantum processing unit (e.g., the quantum processing unit 102 of the quantum computing system 103), or a hybrid computing system. As shown in FIG.2, the operation 206 includes two sub-operations 212, during which the quantum-enhance map generates output quantum states; and 214, during which the expectation values of observables are determined based on the output quantum states. Given the quantum feature map is a map from ^^to ^^, a sequence of quantum feature maps may be performed in a quantum feature map pipeline, for example, by iteratively performing the suboperations 212, 214 before a preprocessed dataset is sent to a machine learning system for processing.

[0070] At 212, the quantum-enhanced map is applied to input quantum states to generate output quantum states. By applying the quantum-enhanced map to an input quantum state, the original features of the input dataset are mapped to the output quantum states. The input dataset may be an input vector of real numbers; and the output quantumAttorney Docket No.: RIGET-118WO1 state domain may be a vector belonging to a linear space of dimension 2^for a system of 8 qubits. In this case, the input vector of real numbers is mapped to the vector belonging to a linear space of a higher dimension, for example, by operation of a quantum feature map system (e.g., the QPUs 102A, 102B of the quantum computing system 103A, 103B shown in FIG.1). In some implementations, the input dataset is transformed into a set of quantum states by passing through the original features through the quantum logic circuit.

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

[0072] In some instances, features may be reshuffled prior to executing the quantum logic circuit. For example, features associated with a data point may be randomly assigned to different qubits. In this case, features associated with a data point are encoded as parameters of the quantum logic circuit based on the reshuffled features. In some instances, orders of the data points in the input dataset may be randomly adjusted before being processed by the quantum logic circuit. Quantum computers require a calibration (re-tune) process to be run regularly in order to determine the control pulses needed to control the device, perform the numerous quantum operations, and to perform readout / measurement. After a re-tune, the performance can slowly degrade as the system gets impacted by changes in the environment. This drift can impact results when the time taken to perform many quantum calculations and obtain results is lengthy. In some implementations, the methods and techniques presented here can be used to reduce or minimize impact from drift to data points in the input dataset.

[0073] At 214, the expectation values of observables are determined based on the quantum states. After transforming the original features to the quantum states by executing the quantum logic circuit, the quantum states of qubits are measured byAttorney Docket No.: RIGET-118WO1 operation of the quantum processing unit. During the measurement, the quantum states of the qubits are collapsed to one of their respective basis states (|0^ or |1^). In some implementations, the observables on the quantum states are measured. In some implementations, the preprocessed dataset includes expectation values of different observables on the quantum state. 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. In some implementations, observables correspond to measurement along specific axis of the Bloch sphere, such as the Pauli Matrices. In some implementations, a preprocessed dataset includes the expectation values of observables obtained which are referred to as quantum-enhanced features.

[0074] In some instances, a common choice for the observables {#^} is tensor product of Pauli operators and identities. In some implementations, the expectation values of observables in the preprocessed dataset includes 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 near-neighbors qubits. In this case, the process 200 can avoid a quadratic or higher growth in the number of quantum-enhanced features. 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, an observable of degree 8 (where 8 > 1) means an operator acting non-trivially on a system with 8 qubits rather than on individual qubits.

[0075] In some instances, Pauli operators of first degree on all qubits can be determined. For example, for each data point % = 1, … , [^], the quantum logic circuit may be executed for ^ times or snapshoots; after each execution of the quantum logic circuit, Pauli :;is measured on all qubits, where < = 1, … , 8. An expectation value :;=^ is calculated as an average value of the recorded :;across all ^ times of quantum logic circuit executions.Attorney Docket No.: RIGET-118WO1 Similarly, an expectation value >;=^ is calculated as an average value of recorded Pauli >;measured on all qubits across all ^ times of quantum logic circuit executions; and an expectation value ?;=^ is calculated as an average value of recorded Pauli ?;measured on all qubits across all of quantum logic circuit executions. In some instances, Paulioperators of on nearest neighbor pairs of qubits can be also determined. For example, Pauli >;is measured on qubit <; and Pauli ?@is measured on qubit A. The expectation value of the Pauli operator of second degree on a pair of qubits < and A >;?@=^ is then calculated as average value of the product >;?@of recorded values of >;and ?@across all ^ times of the quantum logic circuit executions.

[0076] Quantum feature maps are computed through the approximate state formalism, which requires the fast execution of snapshots. For classical datasets with a number of data points in the order of 100,000, in order to keep QPU execution times within a few hours, if assuming the use of 1,000 snapshots per data point, snapshots need to be collected at a rate in the order of 0.1 milliseconds / snapshot.

[0077] In some instances, the preprocessed dataset with quantum-enhanced features may be further processed by the same quantum feature map by executing the same quantum logic circuit. In other words, the preprocessed dataset can be encoded by the same quantum logic circuit; and multiple iterations of the sub-operations 212, 214 may be performed chronologically. In some instances, the quantum logic circuit may be modified according to the quantum-enhanced features in the preprocessed dataset. The modified quantum logic circuit may be applied over a number of qubits different from the number of qubits that the initial quantum logic circuit is applied to. For another example, the modified quantum logic circuit may include a number of elementary static blocks different from the number of elementary static blocks in the initial quantum logic circuit. In some instances, the modified quantum logic circuit may produce a new preprocessed dataset with quantum-enhanced features of higher dimension. For example, the new preprocessed dataset includes quantum-enhanced features that represent more variables, dimensions, or more complex patterns added to the preprocessed dataset. In other words, feedback loops, e.g., feeding a preprocessed dataset back to the quantum-enhanced map as an inputAttorney Docket No.: RIGET-118WO1 dataset, can result in expanded quantum-enhanced features providing additional dimensions to the input space.

[0078] At 206, the preprocessed dataset with quantum-enhanced features is provided to a machine learning model. In some instances, a machine learning 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 has 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 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.Attorney Docket No.: RIGET-118WO1

[0079] In some instances, when the quantum-enhanced features are within the same domain as the original features, the quantum-enhanced features are compatible with systems and operations designed to receive the original features, e.g., classical machine learning models or systems. In this case, the quantum-enhanced 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, but could 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.

[0080] In certain instances, when the quantum-enhanced 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 quantum-enhanced 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.

[0081] In some instances, the machine learning 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.Attorney Docket No.: RIGET-118WO1

[0082] FIG.3 is a block diagram showing aspects of an example quantum feature mapping process 300. In some implementations, the example process 300 is used for performing a quantum feature map that encodes data points from an input dataset into quantum states by executing a quantum logic circuit on a quantum computing system (e.g., the quantum computing system 103 in FIG.1); and determining measured expectation values of observables in the quantum states. In some implementations, the example process 300 is used to transform the input dataset with original features into a preprocessed dataset with quantum-enhanced features. As shown in FIG.3, repetitions of the quantum feature map can generate quantum-enhanced features of higher degrees. The quantum feature map process 300 may be implemented, in whole or in part, by any of the computer systems shown in FIG.1, such as by the user devices 110, the classical processor 111, the control system 105, or any combination thereof. The example process 300 may include additional or different operations, and the operations may be performed in the order shown or in another order. In some cases, operations in the example process 300 can be combined, iterated or otherwise repeated, or performed in another manner.

[0083] At 302, at least one elementary static block in the quantum logic circuit is executed. In some instances, each elementary static block includes parametric quantum logic gates with gate parameters configured to encode features associated with each data point. For example, a quantum logic circuit for quantum feature mapping may include multiple elementary static blocks and thus multiple layers of single-qubit rotation gates with rotation angles as gate parameters for encoding the vectors of the features of data points; and multiple layers of multi-qubit quantum logic gates configured to create entanglement. In some instances, the quantum logic circuit may be implemented as the quantum logic circuit 400 shown in FIG.4 or in another manner. As shown in FIG.3, operation 302 includes suboperation 312, during which feature encoding is performed; and suboperation 314, during which future entangling is performed. In some instances, the operation 302 may include other suboperations.

[0084] At 312, a layer of quantum logic gates in an elementary static block of the quantum logic circuit is executed to efficiently encode relatively low-dimensional classical data points into extremely high-dimensional quantum states. The vector of classicalAttorney Docket No.: RIGET-118WO1 features, ^, is mapped into a set of gate parameters of the quantum logic circuit. In some implementations, the single layer of quantum logic gates for feature encoding includes single-qubit quantum logic gates. In some instances, an angle encoding method is used, where the classical features are mapped to rotation angles of the single-qubit quantum logic gates of the elementary static block. In some instances, different schemes to map the classical features of the data to parameters of the quantum logic circuit may be used. For example, amplitude encoding, basis encoding, reuploading encoding, or other types of encoding schemes can be used to map classical features onto quantum states. In certain instances, the single layer of quantum logic gates for feature encoding may include multi- qubit quantum logic gates, both single-qubit quantum logic gates and multi-qubit quantum logic gates, or in another manner.

[0085] In some implementations, prior to executing the elementary static block of a quantum logic circuit, initial states!are prepared. In some implementations, an initial state is given by the density matrix defined as the weighted sum with a uniform probability over all permutations of the features assignment to the qubits. The change in the computational model from a more standard pure state-based model to one with density matrices has the objective of avoiding arbitrariness in the correspondence between classical features and qubits. When executed on hardware, the approximate state would be utilized for the computation of expectation values, since the method works in an identical manner for pure and mixed states. Furthermore, on quantum hardware, the features-qubits assignment can be performed at preparation for every snapshot (or shadow, in the classical shadows language) with a control systems-level implementation.

[0086] In some instances, the single-layer of quantum logic gates in an elementary static block for feature mapping can be also configured to remove the z-asymmetry. For example, initial states as prepared are in a B basis (e.g., are eigenvectors of Pauli : operators); some quantum logic gates in the quantum logic circuit, e.g., CPHASE gates or other two-qubit quantum logic gates, are diagonal in the B basis; and measurement is in the B basis by default. This leads to an arbitrary choice of a system of reference. In some implementations, the arbitrariness in the choice of single-qubit gate operations can be removed, by breaking the directional asymmetry through the application of single-qubit rotation gates alongAttorney Docket No.: RIGET-118WO1 random directions after state preparation. The direction along which the single-qubit rotation gates are applied may be different for different qubits within the same elementary static block, and may be different for different elementary static blocks. In some instances, the directional asymmetry may remain, namely measurement in the B basis. In some implementations, this remaining direction asymmetry is eliminated by the use of the approximate state formalism.

[0087] At 314, a layer of quantum logic gates in an elementary static block of a quantum logic circuit is executed to efficiently create entanglement between features. In some implementations, the single layer of quantum logic gates for feature entangling includes multi-qubit quantum logic gates. For example, a multi-qubit quantum logic gate to obtain entanglement may include a controlled-Z (CZ) gate, an imaginary SWAP (iSWAP) gate, a controlled-NOT (CNOT) gate, a controlled-phase (CPHASE) gate, or another multi-qubit quantum logic gate. In some instances, circuit shape can be determined in order to best match the qubit topology of the quantum processing unit, e.g., connectivity of qubit devices.

[0088] In some implementations, executing the quantum logic circuit includes executing a set of elementary static blocks 302 as shown in FIG.3. After executing the set of elementary static blocks 302, the example process 300 continues with operation 304, during which a preprocessed dataset with quantum-enhanced features are determined based on the quantum states measured at the end of the set of elementary static blocks. In some implementations, the expectation values of the observables are measured based on the quantum states. In some instances, the operation 304 may be implemented as the operation 214 in the example process 200 shown in FIG.2 or in another manner. In some instances, the preprocessed dataset may be further encoded by the quantum logic circuit by executing the set of elementary static blocks multiple times. In some implementations, this process creates quantum-enhanced features of "higher degree". For example, a quantum-enhanced dataset output from a first quantum feature map may be received and processed by a second quantum feature map; the preprocessed dataset from the second quantum feature map can be received and processed by a third quantum feature map, etc.

[0089] In the case of degree 1, when the quantum-enhanced features are processed with a classical kernel, the result is equivalent to the use of a quantum kernel. In the case ofAttorney Docket No.: RIGET-118WO1 degrees higher than 1, processing of the quantum-enhanced features with a classical kernel, or with another classical machine learning model, does not trivially correspond to an existing quantum machine learning technique.

[0090] In some instances, the quantum logic circuit may include one or more elementary variational blocks, after the execution of the set of elementary static blocks, the example process 300 continues with operation 306, during which at least one elementary variational block configured to perform an approximate state optimization is executed. In some implementations, the at least one elementary variational block is parametric and enables the adjustability of the quantum feature map to the input dataset. In some instances, an elementary variational block includes a layer of single-qubit quantum logic gate, a layer of two-qubit quantum logic gates; or may be configured in another manner. In some instances, an optimization objective function of the elementary variational blocks may be the kernel alignment; and the optimization objective function of the target classical machine learning model may also be used. The optimization is executed with the approximate quantum state variational optimizer approach, which enables the explicit computation of gradients, significantly containing the number of shots necessary.

[0091] In some instances, the number of ansatz elementary static and variational blocks in the quantum logic circuit can be varied. In some implementations, the number of elementary static and variational blocks in a quantum logic circuit is configured to guarantee an overall depth that avoids decoherence. In some instances, the quality of the quantum-enhanced features (measured in the results obtained with the target classical machine learning model) may increase with the increasing number of elementary variational blocks in the quantum logic circuit.

[0092] In some instances, in a quantum fidelity kernel ^ for a system of ^ qubits, matrix elements ^^6of the quantum kernel may be expressed as ^^6= exp^−G||^−6||H)(1)where G is a scalar commonly as^a a %; and6is a quantum state of a qubit I.Attorney Docket No.: RIGET-118WO1

[0093] Instead of the exponential function, a different function of ||^−6||Hcould have been chosen for determining the element ^^6. With regard to the use of Frobenius norm and not just the trace &'[^ 6], the Frobenius norm or fidelity Kernel are equivalent are equivalent for pure states up to constants. In some instances, the norm distance can be preferable as it always maintains the correct meaning of distance between the points^and6for mixed states.

[0094] The argument of the exponential term of Eq. (1) can be expanded on the basis of the 4^Pauli operators K;= L;M⋯ L;N, with L;O∈ {P, >, ?, :}, ∀A: || ^ − 6||H=1 2^R(Tr[K;( ^ − 6)])H (2)where K is Pauli operator

[0095] For simplicity,use of the tensor product symbol between Pauli operators is omitted.@represents a reduced density matrix (RDM) to the A-th qubit;@M@Urepresents a reduced density matrix to the AV-th and AH-th qubits. The sum of Eq. (2) may be reorganized in partial sums, each including all terms with Pauli operators of the same degree. The different function of ||^− 6||Hcan be reorganized as: ^ X ( H 1 3) ⋯@⋯HAttorney Docket No.: RIGET-118WO1

[0096] New variables ^^V^^,@,;can be defined as ^^V^^,@,; ≡1 @ @(4)√2TrYL;^\

[0097] The first order term of^ X ^ X 1 (5) R R ^TrYL;@^^@−6@^\^H= R R ^^^V^^,@,;− ^^V^6,@,;^H= ||^^(V)− ^6(V)||Hwherestandard Euclidean diX^ ^V^stance in ^ . The 3^ features ^^represent the quantum-enhanced features of first degree. When used for example in a classical radial basis function (RBF) kernel ^^n6= exp o−Gp p^^ ^^V− ^^6V^p |Hq,(6)

[0098] The same resultkernel function using 1-RDMs, or, equivalently, the same result of Eq. (1), when the argument of the exponential is expanded and truncated at the first term as shown in Eq. (3).

[0099] In some instances, quantum-enhanced features of second degree or higher can be extracted. For example, quantum-enhanced features of first and second degrees can be obtained by defining new variables ^ ^V,H^ ^,@M,@U,;M,;U, with <V, <H∈ {0,1,2,3}, as ^^V,H^( ≡1TrYL@L@ @^M@U7) \.

[0100] The first andthe variables ^^V,H^^V,H^ ^ of components ^^,@M,@U,;M,;Uas1 ^ X X X 1 (8) whichAttorney Docket No.: RIGET-118WO1

[0101] In some implementations, all observables of first degree and only a few selected observables of the second degree, for example the ones where the indices AVand AHare limited to near neighbors in a lattice with limited connectivity are measured. The positivity of all terms in all the sums in Eq 3 means that every term maintained in the truncation contributes to getting closer to the exact value of Eq. (2). In other terms, a truncation always produces a value smaller than or equal to that of a truncation including a larger set of terms and, consequently, a value always smaller than or equal to that of Eq. (2). The value of Eq. (2) corresponds to the case of a fidelity quantum kernel. which, if using quantum feature maps, can be obtained with exponentially many (4^) quantum-enhanced features.

[0102] In some implementations, the methods and systems presented here can be efficient, as its processing time is linear in the number of samples and #^1^ in the number of features, assuming a number of observables {#^} that is linear in the number of features. In some implementations, the memory required is linear in the number of data points and in the number of features. The output of the quantum feature map is, in fact, a quantum- enhanced dataset with the same number of data points and a number of features linearly larger than the original dataset without any further data in memory. This is possible because of the method for the computation of expectation values in the approximate state: for each given data point, the expectation values of all observables {#^} can be updated after the collection of every snapshot and, therefore, the snapshots do not need to be stored. In some instances, the optimal solution is one where the updates to the expectation values are done on the control systems; and the measured values that form snapshots are not recorded.

[0103] In some instances, the quantum feature map pipeline may be conducted in parallel with a machine learning pipeline. For example, when the quantum logic circuit for quantum -feature mapping only includes elementary static blocks; and the target classical machine learning procedure admitting sequential processing of data points, the quantum feature map may run completely in parallel with a classical machine learning pipeline, adding fundamentally no extra processing time to the classical machine learning pipeline. In other words, a subset of data points with quantum-enhanced features when generated,Attorney Docket No.: RIGET-118WO1 can be communicated to the classical machine learning pipeline, while the remaining data points continue to be processed by the quantum feature map. In some instances, the quantum feature map pipeline may be conducted at least in part before a machine learning pipeline can be performed. For example, when the quantum logic circuit for quantum feature mapping also includes elementary variational blocks; and / or the classical machine learning pipeline needs to process multiple data points concurrently, multiple data points, from limited-size batches to the limit case of the full dataset, will need to be processed, possibly more than once, by quantum feature maps before they can be sent to the classical machines learning pipeline. In some instances, if the quantum-enhanced dataset is processed with a classical kernel, the classical kernel may continue to exhibit quadratic scaling requirements. In some instances, this quadratic scaling requirement may not be needed if a different classical model is used.

[0104] Similarly to the hardware accelerations before readout, an optimal implementation of quantum feature maps will make use of control systems-level acceleration for features reshuffling and real-time expectation values computation, and for randomized compilation in support of quantum error mitigations. In some instances, this would be the sampling / randomization of the permutation of the original features that would be performed at the FPGA level.

[0105] Additionally to the importance of speed, another important characteristic of the quantum system to optimally support quantum feature maps is the number of qubits. In fact, a higher number of qubits enables the use of more qubits per feature and higher degree quantum-enhanced features (at every repetition of a quantum feature map, even just with Pauli operators of first degree, the number of quantum-enhanced features is larger than the number of original features in the input dataset).

[0106] 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 hybrid computing program for performing a quantum two-sample test or another computing 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 circuitAttorney Docket No.: RIGET-118WO1 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 each data point from a classical dataset into corresponding quantum states.

[0107] In some implementations, the quantum logic circuit 400 may include single- qubit quantum logic gates, two-qubit quantum logic gates, or other multi-qubit quantum logic gates applied on a number of qubits. In some implementations, layers of the single qubit quantum logic gates in the quantum logic circuit 400 are data encoding layers; and layers of the multi-qubit quantum logic gates are entanglement-creating layers. In some implementations, the number of qubits where the quantum logic circuit 400 is applied is determined by the number of features that represent different aspects or attributes of the data points, or dimensionality of the data points in the dataset. In some instances, the number of features of data points in the dataset can be determined by implementing feature selection and extraction techniques. In some instances, the number of features may be equal to the number (8) of qubit devices (e.g., quantum registers) where the quantum logic circuit for performing quantum feature mapping is applied. In other words, each feature of data points in a dataset is encoded to a quantum state of a qubit device. In certain instances, the number of features may be different from the number of qubit devices. In this case, two or more features of data points may be encoded in a quantum state of a single qubit device; or quantum states of two or more qubit devices may be used to encode a single feature.

[0108] In some implementations, the quantum logic circuit 400 is configured to efficiently encode a relatively low-dimensional classical data point into extremely high- dimensional quantum states, thus realizing a quantum feature map. For example, all quantum states may be initialized as |0^, and the initial state is |)!^ = |0^ ⊗ ⋯ ⊗ |0^ ≡ |0^⊗^. The vector of features, ^, can be mapped into a set of gate parameters of the quantum logic circuit. For example, when the single-qubit quantum logic gates in the quantum logic circuit are parametric single-qubit rotation gates, the vectors of features canAttorney Docket No.: RIGET-118WO1 be mapped onto rotation angles of the parametric single-qubit rotation gates, e.g., an angle encoding method. The final state, |) / 〉, thus encodes the classical data point and the subsequent sample performed in the high-dimensional Hilbert space.

[0109] In some instances, the number of layers of the single-qubit quantum logic gates and multi-qubit logic gates can be defined by the amount of entanglement required, the fidelity of the single-qubit and multi-qubit gates used in the circuit, the topology of the quantum device, the native gates available on the quantum device, and whether data re- upload techniques are required, or in another manner.

[0110] As shown in FIG.4, the example quantum logic circuit 400 includes a sequence of alternating sets of single-qubit rotation gates 402 and two-qubit iSWAP gates 404 applied on 4 qubits 412. Specifically, the quantum logic circuit 400 includes a first set of single- qubit rotation gates 402A on the X basis, ^s^t^, applied on the 4 qubits at a first time step uV, each of which is configured to apply phase rotation to a qubit around the X axisof the Bloch sphere. The quantum logic circuit 400 includes a first set of two-qubit iSWAP gates 404A at a second time step uH, creating entanglement 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 set of single-qubit rotation gates 402B on the Z basis, ^v^t^, applied on the 4 qubits at a third time step uX, each of which is configured to apply a specific phase rotation to a qubit around the Z axis of the Bloch sphere. The quantum logic circuit 400 includes a second set of two-qubit iSWAP gates 404B at a fourth time step uw, creating entanglement 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 single-qubit rotation gates 402C on the Y basis, ^x^t^, applied on the 4 qubits at a fifth time step uy, each of which is configured to apply a specific phase rotation to a qubit around the Y axis of the Bloch sphere. The quantum logic circuit 400 includes a third set of two-qubit iSWAP gates 404C at a sixth time step uz, creating entanglement between qubit 412-1 and qubit 412-2 and between qubit 412-3 and qubit 412-4. The quantum logic circuit 400 includes a fourth set of single- qubit rotation gates 402D on the Z basis, ^v^t^, applied on the 4 qubits at a seventh time step u{, each of which is configured to apply a specific phase rotation to a qubit around the Z axis of the Bloch sphere. The quantum logic circuit 400 includes a fourth set of two-qubitAttorney Docket No.: RIGET-118WO1 iSWAP gates 404D at an eighth time step u|, creating entanglement between qubit 412-3 and qubit 412-2 and between qubit 412-1 and qubit 412-4. The quantum logic circuit 400 includes a fifth set of single-qubit rotation gates 402E on the X basis, ^s^t^, applied on the 4 qubits at a ninth time step u}, each of which is configured to apply a specific phase rotation to a qubit around the X axis of the Bloch sphere. The quantum logic circuit 400 includes a fifth set of two-qubit iSWAP gates 404E at a tenth time step uV!, creating entanglement between qubit 412-1 and qubit 412-2 and between qubit 412-3 and qubit 412-4. The quantum logic circuit 400 includes a sixth set of single-qubit rotation gates 402F on the Z basis, ^v^t^, applied on the 4 qubits at an eleventh time step uVV, each of which is configured to apply a specific phase rotation to a qubit around the Z axis of the Bloch sphere. The quantum logic circuit 400 includes a sixth set of two-qubit iSWAP gates 404F at a twelfth time step uVH, creating entanglement between qubit 412-3 and qubit 412- 2 and between qubit 412-1 and qubit 412-4. The quantum logic circuit 400 includes a seventh set of single-qubit rotation gates 402G on the Y basis, ^x^t^, applied on the 4 qubits at a thirteenth time step uVX, each of which is configured to apply a specific phase rotation to a qubit around the Y axis of the Bloch sphere.

[0111] As shown in FIG.3, a data point in a dataset can be encoded through one-qubit rotation operations which are rotations of the individual quantum states around the X, Y, or Z axes. Each data point can be represented by a vector of four features, and each feature is mapped into the corresponding rotation angle. The circuit consists of four quantum registers and seven data encoding layers. This specifies the feature mapping scheme where each qubit device encodes one feature (e.g., one-to-one mapping between feature and qubit) and all rotation angles are defined on the interval [0, 1 / 7]. To be more specific, the proposed angle encoding scheme is given by the following expression: ^^^I^ − min^^^I^^ =1^ (9) where ^%8^^^I^^ andacross all data points, ^ is the total number of data points in a dataset.Attorney Docket No.: RIGET-118WO1

[0112] The final quantum state, |) / ^, is obtained after application of the 8-qubit quantum logic circuit – a logic gates (linear unitary operators ^) controlled by parameters ^V, … , – the initial quantum state, |)!^: |) / ^ = ^^(Θ^) … ^2(Θ2)^1(Θ1)|)!^(10)

[0113] In somemeasured. 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 is a data point from the probability distribution encoded in the final quantum state (each quantum state encodes a probability distribution).

[0114] As shown in FIG.4, the quantum logic circuit 400 includes six layers of two-qubit iSWAP gates 404 creating entanglement; alternating with seven layers of single-qubit ^s, ^xand ^vgates 402 performing repeated data encoding. The quantum logic circuit 400 is configured to ensure that data points in datasets are not concentrated in any one region of the system’s Hilbert space. The quantum logic gates in the quantum logic circuit may be organized as the example quantum logic circuit 400 shown in FIG.4 or in another manner. For example, a quantum logic circuit may include a different number of layers of single- qubit rotation gates and two-qubit iSWAP gates. For another example, the seven layers of the single-qubit rotation gates may be organized in another order. In some instances, the quantum logic circuit may include other types of parametric quantum logic gates and the features can be encoded into gate parameters of the quantum logic circuits in another manner. In some instances, other multi-qubit gates different from the iSWAP gate may be used.

[0115] Table 1 presents the experimental results obtained with the probit model and signature kernel models (classical and quantum-enhanced) for recession prediction in the backtesting period. The separation metric is used to report results. It can be seen that the probit model obtained a separation of 77.5%. Classic signature kernels improved the result to 79.2% and the quantum-enhanced signature kernel outperformed both models with aAttorney Docket No.: RIGET-118WO1 separation of 85.8%, showing how quantum-enhanced features can improve the performance of a classical machine learning pipeline. The column Separation shows the separation metric obtained by the different models tested over the backtesting period [January 2000, January 2010].

[0116] The actual recession score and the actual recession label are provided by Moody’s. The dataset consisted of a set of eight economic variables (e.g., the number of features is ^=8) that are deemed to be relevant indicators / predictors of a future recession (examples are transformations such as moving average, moving standard deviation and / or the difference between two time series of well-known and publicly available economic indicators such as the US yield curve, the CPI, US employment numbers, etc.). Target labels consisted of a binary variable indicating for each month whether the US economy was in recession, and a 12m-forward-looking recession label, equal to 1 if in the next 12 months there was a recession and 0 otherwise.

[0117] Performance of the model was measured using the separation indicator, defined as the proportion of months over the backtesting period when the estimated probability of recession is less than 1 / 3 when the forward looking recession label is 0, and when the estimated probability of recession is greater than 2 / 3 when the forward looking recession label is equal to 1.

[0118] The forecasts generated by a regression model (e.g., a probit model) is used as the reference baseline to calculate probabilities of a recession. The ability of the model to forecast the probability of a recession in the future was evaluated through a backtesting exercise over a ten year period, which included two recession events. For every (monthly) date u between 01 / 2000 and 01 / 2010, the probit model provided the estimated probability of a recession in the following 12 months. The parameters of the probit model were estimated on the data between June 1978 and 12 months prior to u. To demonstrate the computational power of the method, separating the effects of the quantum computer’s errors from the algorithmic results, the quantum feature map is simulated on classical hardware. Table.1Attorney Docket No.: RIGET-118WO1 Model Separation

[0119] FIGS.5A-5Be as a function of time in months using a classical signature kernel and a plot 510 showing recession score as a function of time in months using a quantum-enhanced signature kernels for the single-shot prediction of recession for the United States economy. A different set of economic variables including SP500 monthly moves, CMS spread 10Y2Y, Payrolls, etc., considered to have lower explanatory power. The time interval considered is 1960-2019. The green lines 502, 512 represent the target recession indicator computed from the binary recession indicator, which are represented by the respective blue lines 504, 514. The red broken lines 506, 516 represent predicted recession scores by the classical signature kernel and the quantum- enhanced signature kernel. The signature kernels are trained to match the target recession indicator on the data of the period in the white part of the figure and tested on the period in grey in the figure. Differently from the backtesting approach, the problem in this case was formulated as a regression problem, where the target variable was a continuous recession score, defined as a weighted sum of the recession label over 18 future months for each point in time. As shown in FIG.5B, the use of the quantum-enhanced features improves the performance of the classical signature kernel model. The quantum-enhanced signature kernel, in fact, outperformed the classical signature kernel in the out-of-sample prediction, as visually evident in the smaller difference between the green and red curves in the grey areas of FIG.5B comparing to those in FIG.5A.

[0120] 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 matterAttorney Docket No.: RIGET-118WO1 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 generated propagated signal. The computer storage medium can also be, or be included in, one or more separate physical components or media.

[0121] 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.

[0122] In a general aspect, a quantum feature-map to transform an input dataset with original features to a preprocessed dataset with quantum-enhanced features is presented.

[0123] In a first example, a method of pre-processing an input dataset for a machine learning model includes obtaining the input dataset comprising a plurality of data points; encoding the plurality of data points as parameters of a quantum logic circuit; causing a quantum computing resource to execute the quantum logic circuit; obtaining expectation values of a set of observables based on an output quantum state generated by executing the quantum logic circuit, the set of observables including observables of first degree and observables of second degree; generating a pre-processed dataset based on the expectation values; and providing the pre-processed dataset as an input to the machine learning model.

[0124] Implementations of the first example may include one or more of the following features. The quantum logic circuit includes one or more elementary static blocks; and each elementary static block includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates. Encoding the plurality of data points as parameters of the quantum logic circuit includes encoding features of the input datasetAttorney Docket No.: RIGET-118WO1 as a first set of parameters of the single-qubit quantum logic gates and a second set of parameters of the multi-qubit quantum logic gates.

[0125] Implementations of the first example may include one or more of the following features. The quantum logic circuit includes one or more elementary variational blocks configured to perform approximate quantum state optimization. Encoding the plurality of data points as parameters of the quantum logic circuit includes calculating a number of features of the data points and a range of feature values; and determining one or more attributes of the quantum logic circuit according to the range of feature values and the number of features. Determining one or more attributes of the quantum logic circuit comprises at least one of: determining a number of elementary static blocks; determining a number of elementary variational blocks; and determining a number of qubits to which the quantum logic circuit is applied. Each elementary static block includes single-qubit quantum logic gates, and determining one or more attributes of the quantum logic circuit includes determining rotation angle values of the single-qubit quantum logic gates in respective elementary static blocks according to the number of elementary static blocks and the number of qubits to which the quantum logic circuit is applied.

[0126] Implementations of the first example may include one or more of the following features. The expectation values of observables include multiple subsets of expectation values associated with distinct measurement bases for the plurality of qubits in the quantum logic circuit. The plurality of data points is a first plurality of data points; the pre- processed dataset is a first pre-processed dataset including a second plurality of data points; the quantum logic circuit is a first quantum logic circuit; the expectation values are first expectation values; and the method includes prior to providing the pre-processed dataset to the machine learning model, encoding the second plurality of data points as parameters of a second quantum logic circuit; by operation of the quantum processing unit, executing the second quantum logic circuit; and measuring second expectation values of the set of observables based on an output quantum state generated by executing the second quantum logic circuit; generating a second pre-processed dataset based on the expectation values; and providing the second pre-processed dataset as an input to the machine learning model.Attorney Docket No.: RIGET-118WO1

[0127] Implementations of the first example may include one or more of the following features. The method includes shuffling features associated with the plurality of data points; and encoding the plurality of data points as parameters of the quantum logic circuit based on the shuffled features. The machine learning model includes a classical kernel, a fidelity quantum kernel, or a projected quantum kernel. The observables of first degree correspond to single-body Pauli operators; and the observables of second degree correspond to two-body Pauli operators. The set of observables includes first observables of first degree for respective qubits in the plurality of qubits; and second observables of second degree for respective nearest neighbor pairs of qubits in the plurality of qubits.

[0128] In a second example, a computer system includes a communication interface, and classical computing resources. The classical computing resources include one or more classical processing units; and memory storing instructions that, when executed by the one or more classical processing units, cause the one or more classical processing units to obtain the input dataset comprising a plurality of data points; encode the plurality of data points as parameters of a quantum logic circuit; cause, via the communication interface, a quantum computing resource to execute the quantum logic circuit; obtain expectation values of a set of observables based on an output quantum state generated by executing the quantum logic circuit, the set of observables comprising observables of first degree and observables of second degree; generate a pre-processed dataset based on the obtained expectation values; and provide the pre-processed dataset as an input to the machine learning model.

[0129] Implementations of the second example may include one or more of the following features. The computer system includes the quantum computing resource communicably coupled to the classical computing resource via the communication interface. The quantum computing resource includes at least one of a quantum simulator, a quantum computing system, or a hybrid computing system. The quantum logic circuit includes one or more elementary static blocks. Each elementary static block includes at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates. Encoding the plurality of data points as parameters of the quantum logic circuit includes encoding features of the input dataset as a first set of parameters ofAttorney Docket No.: RIGET-118WO1 the single-qubit quantum logic gates and a second set of parameters of the multi-qubit quantum logic gates.

[0130] Implementations of the second example may include one or more of the following features. The quantum logic circuit includes one or more elementary variational blocks configured to perform approximate quantum state optimization. Encoding the plurality of data points as parameters of the quantum logic circuit includes calculating a number of features of the data points and a range of feature values; and determining one or more attributes of the quantum logic circuit according to the range of feature values and the number of features. Determining one or more attributes of the quantum logic circuit includes at least one of determining a number of elementary static blocks; determining a number of elementary variational blocks; and determining a number of qubits to which the quantum logic circuit is applied. Each elementary static block includes single-qubit quantum logic gates, and determining one or more attributes of the quantum logic circuit includes determining rotation angle values of the single-qubit quantum logic gates in respective elementary static blocks according to the number of elementary static blocks and the number of qubits to which the quantum logic circuit is applied.

[0131] Implementations of the second example may include one or more of the following features. The expectation values of observables include multiple subsets of expectation values associated with distinct measurement bases for the plurality of qubits in the quantum logic circuit. The plurality of data points is a first plurality of data points; the pre-processed dataset is a first pre-processed dataset comprising a second plurality of data points, the quantum logic circuit is a first quantum logic circuit; the expectation values are first expectation values. The memory stores instructions that, when executed by the one or more classical processing units, cause the one or more classical processing units to prior to providing the pre-processed dataset to the machine learning model, encode the second plurality of data points as parameters of a second quantum logic circuit; cause the quantum computing resource to execute the second quantum logic circuit; obtain second expectation values of the set of observables based on an output quantum state generated by executing the second quantum logic circuit; generate a second pre-processed dataset based on theAttorney Docket No.: RIGET-118WO1 expectation values; and provide the second pre-processed dataset as an input to the machine learning model.

[0132] Implementations of the second example may include one or more of the following features. The memory stores instructions that, when executed by the one or more classical processing units, cause the one or more classical processing units to shuffle features associated with the plurality of data points; and to encode the plurality of data points as parameters of the quantum logic circuit based on the shuffled features. The machine learning model includes a classical kernel, a fidelity quantum kernel, or a projected quantum kernel. The observables of first degree correspond to single-body Pauli operators; and the observables of second degree correspond to two-body Pauli operators. The set of observables includes first observables of first degree for respective qubits in the plurality of qubits; and second observables of second degree for respective nearest neighbor pairs of qubits in the plurality of qubits.

[0133] 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.

[0134] 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.Attorney Docket No.: RIGET-118WO1

[0135] 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-118WO1 CLAIMS What is claimed is:

1. A method of pre-processing an input dataset for a machine learning model, the method comprising: obtaining the input dataset comprising a plurality of data points; encoding the plurality of data points as parameters of a quantum logic circuit; causing a quantum computing resource to execute the quantum logic circuit; obtaining expectation values of a set of observables based on an output quantum state generated by executing the quantum logic circuit, the set of observables comprising observables of first degree and observables of second degree; generating a pre-processed dataset based on the expectation values; and providing the pre-processed dataset as an input to the machine learning model.

2. The method of claim 1, wherein: the quantum logic circuit comprises one or more elementary static blocks, each elementary static block comprising at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates.

3. The method of claim 2, wherein encoding the plurality of data points as parameters of the quantum logic circuit comprises encoding features of the input dataset as a first set of parameters of the single-qubit quantum logic gates and a second set of parameters of the multi-qubit quantum logic gates.

4. The method of claim 2, wherein the quantum logic circuit comprises one or more elementary variational blocks configured to perform approximate quantum state optimization.

5. The method of claim 4, wherein encoding the plurality of data points as parameters of the quantum logic circuit comprises: calculating a number of features of the data points and a range of feature values; and determining one or more attributes of the quantum logic circuit according to the range of feature values and the number of features.Attorney Docket No.: RIGET-118WO1 6. The method of claim 5, wherein determining one or more attributes of the quantum logic circuit comprises at least one of: determining a number of elementary static blocks; determining a number of elementary variational blocks; and determining a number of qubits to which the quantum logic circuit is applied.

7. The method of claim 5, wherein each elementary static block comprises single-qubit quantum logic gates, and determining one or more attributes of the quantum logic circuit comprises: determining rotation angle values of the single-qubit quantum logic gates in respective elementary static blocks according to the number of elementary static blocks and the number of qubits to which the quantum logic circuit is applied.

8. The method of claim 1, wherein the expectation values of observables comprise multiple subsets of expectation values associated with distinct measurement bases for the plurality of qubits in the quantum logic circuit.

9. The method of claim 1, wherein the plurality of data points is a first plurality of data points, the pre-processed dataset is a first pre-processed dataset comprising a second plurality of data points, the quantum logic circuit is a first quantum logic circuit, the expectation values are first expectation values, and the method comprises, prior to providing the pre-processed dataset to the machine learning model: encoding the second plurality of data points as parameters of a second quantum logic circuit; causing the quantum processing unit to execute the second quantum logic circuit; and obtaining second expectation values of the set of observables based on an output quantum state generated by executing the second quantum logic circuit; generating a second pre-processed dataset based on the expectation values; and providing the second pre-processed dataset as an input to the machine learning model.Attorney Docket No.: RIGET-118WO1 10. The method of claim 1, comprising: shuffling features associated with the plurality of data points; and encoding the plurality of data points as parameters of the quantum logic circuit based on the shuffled features.

11. The method of claim 1, wherein the machine learning model comprises a classical kernel, a fidelity quantum kernel, or a projected quantum kernel.

12. The method of claim 1, wherein: the observables of first degree correspond to single-body Pauli operators; and the observables of second degree correspond to two-body Pauli operators.

13. The method of claim 1, wherein the set of observables comprises: first observables of first degree for respective qubits in the plurality of qubits; and second observables of second degree for respective nearest neighbor pairs of qubits in the plurality of qubits.

14. The method of claim 1, wherein the quantum computing resource comprises at least one of a quantum simulator, a quantum computing system, or a hybrid computing system.

15. A computer system comprising: a communication interface; and classical computing resources comprising: one or more classical processing units; and memory storing instructions that, when executed by the one or more classical processing units, cause the one or more classical processing units to: obtain the input dataset comprising a plurality of data points; encode the plurality of data points as parameters of a quantum logic circuit; cause, via the communication interface, a quantum computing resource to execute the quantum logic circuit; obtain expectation values of a set of observables based on an output quantum state generated by executing the quantum logic circuit, the set of observables comprising observables of first degree and observables of second degree;Attorney Docket No.: RIGET-118WO1 generate a pre-processed dataset based on the received expectation values; and provide the pre-processed dataset as an input to the machine learning model.

16. The computer system of claim 15, comprising the quantum computing resource, wherein the quantum computing resource comprises at least one of a quantum simulator, a quantum computing system, or a hybrid computing system.

17. The computer system of claim 15, wherein the quantum logic circuit comprises one or more elementary static blocks, each elementary static block comprising at least one layer of single-qubit quantum logic gates and at least one layer of multi-qubit quantum logic gates.

18. The computer system of claim 17, wherein encoding the plurality of data points as parameters of the quantum logic circuit comprises encoding features of the input dataset as a first set of parameters of the single-qubit quantum logic gates and a second set of parameters of the multi-qubit quantum logic gates.

19. The computer system of claim 17, wherein the quantum logic circuit comprises one or more elementary variational blocks configured to perform approximate quantum state optimization.

20. The computer system of claim 19, wherein encoding the plurality of data points as parameters of the quantum logic circuit comprises: calculating a number of features of the data points and a range of feature values; and determining one or more attributes of the quantum logic circuit according to the range of feature values and the number of features.

21. The computer system of claim 20, wherein determining one or more attributes of the quantum logic circuit comprises at least one of: determining a number of elementary static blocks; determining a number of elementary variational blocks; and determining a number of qubits to which the quantum logic circuit is applied.Attorney Docket No.: RIGET-118WO1 22. The computer system of claim 20, wherein each elementary static block comprises single-qubit quantum logic gates, and determining one or more attributes of the quantum logic circuit comprises: determining rotation angle values of the single-qubit quantum logic gates in respective elementary static blocks according to the number of elementary static blocks and the number of qubits to which the quantum logic circuit is applied.

23. The computer system of claim 15, wherein the expectation values of observables comprise multiple subsets of expectation values associated with distinct measurement bases for the plurality of qubits in the quantum logic circuit.

24. The computer system of claim 15, wherein the plurality of data points is a first plurality of data points, the pre-processed dataset is a first pre-processed dataset comprising a second plurality of data points, the quantum logic circuit is a first quantum logic circuit, the expectation values are first expectation values, and the memory stores instructions that, when executed by the one or more classical processing units, cause the one or more classical processing units to, prior to providing the pre-processed dataset to the machine learning model: encode the second plurality of data points as parameters of a second quantum logic circuit; cause the quantum computing resource to execute the second quantum logic circuit; and obtain second expectation values of the set of observables based on an output quantum state generated by executing the second quantum logic circuit; generate a second pre-processed dataset based on the expectation values; and provide the second pre-processed dataset as an input to the machine learning model.

25. The computer system of claim 15, wherein the memory stores instructions that, when executed by the one or more classical processing units, cause the one or more classical processing units to: shuffle features associated with the plurality of data points; andAttorney Docket No.: RIGET-118WO1 encode the plurality of data points as parameters of the quantum logic circuit based on the shuffled features.

26. The computer system of claim 15, wherein the machine learning model comprises a classical kernel, a fidelity quantum kernel, or a projected quantum kernel.

27. The computer system of claim 15, wherein: the observables of first degree correspond to single-body Pauli operators; and the observables of second degree correspond to two-body Pauli operators.

28. The computer system of claim 15, wherein the set of observables comprises: first observables of first degree for respective qubits in the plurality of qubits; and second observables of second degree for respective nearest neighbor pairs of qubits in the plurality of qubits.

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