Methods and systems for quantum state reconstruction using neural shadow quantum state tomography

Abstract

The present disclosure provides methods and systems for a quantum state reconstruction. The methods may comprise: obtaining one or more quantum measurements; using the one or more quantum measurements to construct a classical shadow state using a classical shadow technique; using the classical shadow state to construct a loss function of a machine learning (ML) model representative of a parameterized quantum state, wherein the loss function comprises an overlap between a snapshot state representative of the one or more quantum measurements and an ansatz state provided by the ML model; using samples from the classical shadow state to estimate the gradient of the loss function; and using the gradient of the loss function to train the ML model. The systems may comprise a hybrid system comprising a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system, and a high-performance computing platform.

Description

METHODS AND SYSTEMS FOR QUANTUM STATE RECONSTRUCTION USING NEURAL SHADOW QUANTUM STATE TOMOGRAPHY CROSS-REFERENCE

[0001] This application claims the benefit of U.S. Provisional Application No. 63 / 587,707, filed October 3, 2023, and U.S. Provisional Application No.63 / 574,832, filed April 4, 2024, which are incorporated herein by reference in their entireties. BACKGROUND

[0002] Quantum computing holds the promise to surpass classical computing in terms of performance. One key advantage of quantum computing is its potential to represent an exponentially large amount of data using a polynomial number of quantum bits, or “qubits”. This unique capability would allow for the simultaneous processing of quantum information on a massive scale, resulting in significantly faster computations compared to classical computers. However, a crucial drawback of quantum computing is introduced through the act of measurement, which may lead to the corruption of a quantum state. The measurement process results in the loss of a substantial amount of information contained within the quantum state. Consequently, the advantages of quantum computing may be nullified unless efficient methods for extracting quantum information are employed.

[0003] The accessing of experimentally prepared quantum states may be obstructed by the limited rate at which classical information can be extracted from a quantum state through quantum measurements, which may be costly. Moreover, in near-term quantum devices, various sources of physical noise may introduce additional systemic errors, corrupting the measurements. Even when preparing quantum states fault tolerantly, accessing the logical states’ observables of interest may be susceptible to statistical errors resulting from the finite number of executions of a fault-tolerant algorithm. For these reasons, it may be useful to treat the quantum data collected from quantum experiments as scarce, highly valuable information, and to maximize the utility of the available data.SUMMARY

[0004] The quantum data may need to be stored in a classical data structure as a proxy for access to information about a given quantum state. In an example, this data structure is the list of the measurements performed (e.g., a tomographically complete set of measurements such as those used in the classical shadow technique in Huang et al., “Predicting Many Properties of a Quantum System from Very Few Measurements,” Nature Physics 16, 1050–1057, 2020, which is incorporated herein by reference in its entirety). In principle, such a raw dataset already includes the information that has been collected from a quantum state. This raises the question as to what the role of a neural quantum state may be when it cannot learn more about the physics of the state than what it is provided in the training data. However: (a) the amount of memory required to store a tomographically complete set of measurements may grow exponentially with the size of a quantum system; (b) using this raw data to estimate various observables of the system may result in highly erroneous and often unphysical estimations; and (c) every such estimation may involve sweeping over the exponentially large list at least once.

[0005] While quantum state tomography techniques that solve inverse problems for various (partial or complete) approximations of the density matrix may overcome the last issue by providing a proxy to the raw data, they may not resolve the other issues.

[0006] Maximum likelihood estimation may offer a path to alleviating at least some of these challenges using memory-efficient parameterized models (e.g., an ansatz). Model parameters may variationally be optimized in update directions that may make the training data the most likely to be generated by the learned ansatz. Such variational ansatz may flexibly be used to impose physicality on the reconstructed states. Among various approaches, tensor networks and neural networks may provide flexible architectures that may be capable of capturing relevant regions within the Hilbert space. Moreover, when these networks possess an autoregressive property, they may be able to efficiently provide independent samples for subsequent tasks downstream as generative models for the learned quantum states.

[0007] In another example, an approach for training a neural quantum state using classical shadows may combine the advantages of variational ansatz and classical shadow measurements. However, these methods may employ non physical states and may have infidelities that are not constrained to physical ranges. Further, the method’s success may depend on the informational content of the underlying quantum state in the computational basis rendering it non-universal. The present disclosure provides methods and systems for a quantum state reconstruction using various measurement procedures, sampling methods, and loss functions to enhance state reconstruction efficiency.

[0008] Recognized herein is the need for improved methods and systems that may overcome at least one of the identified drawbacks.

[0009] Disclosed herein is an estimator for optimizing the cross-entropy loss function. The disclosed approach capitalizes on the classical shadow state in order to bootstrap the dataset for construction of an unbiased estimator. Moreover, during training, the utilization of stabilizer samples, instead of relying on samples generated from a generative ansatz, may be adopted. This not only reduces the necessity for pretraining but also provides a smoother training gradient. Among various non-limiting advantages, the present application provides: an improved loss function, an importance sampling strategy, supervision of a physically valid mixed-state autoregressive model on measurement data, and improved prediction of high-weight observables compared to raw classical shadow estimations.

[0010] Regarding the loss function, the present application provides an estimator for the cross-entropy loss function using classical shadows. This loss function may be viable for training mixed states. Regarding the importance sampling strategy, the present application provides stabilizer-based sampling for estimating the overlap between classical shadows and a neural quantum state. This may reduce or entirely eliminate the need for pretraining and may significantly reduce the variance of gradient estimations during training. Regarding the supervision of a physically valid mixed-state autoregressive model on measurement data, the present application provides methods and systems which use the loss function to train a neural network representing a purification of the mixed quantum state from classical shadow training data. Regarding the improved prediction of high- weight observables compared to raw classical shadow estimations, systems and methodsdisclosed herein may implement physicality constraints natively imposed by the explicit access of autoregressive models to conditional probability densities (in the autoregressive expansion), which improve the accuracy of predictions extracted from classical shadow data.

[0011] In an aspect, the present disclosure provides a method for a quantum state reconstruction. The method may comprise: (a) obtaining one or more quantum measurements; (b) constructing a classical shadow state based at least in part on said one or more quantum measurements and using a classical shadow technique; (c) constructing a loss function of a machine learning (ML) model representative of a parameterized quantum state based at least in part on said classical shadow state, wherein said loss function comprises an overlap between a snapshot state representative of said one or more quantum measurements and an ansatz state provided by said ML model; (d) estimating a gradient of said loss function based at least in part on samples from said classical shadow state; and (e) training said ML model based at least in part on using said gradient of said loss function to update one or more parameters of said ML model.

[0012] In some embodiments, said one or more quantum measurements comprise a tomographically complete set of quantum measurements. In some embodiments, said loss function comprises cross-entropy and (c) comprises computing a classical shadow data distribution which is used to construct said loss function. In some embodiments, said loss function of said ML model representative of said parameterized quantum state comprises the Kullback–Leibler (KL) divergence between the distribution corresponding to said one or more quantum measurements in a random basis and the distribution corresponding to said ansatz state; or KL divergence between the distribution corresponding to said ansatz state and the distribution corresponding to said one or more quantum measurements in a random basis. In some embodiments, said loss function of said ML model comprises an indication of infidelity between said ansatz state and said classical shadow state. In some embodiments, (d) comprises Monte Carlo (MC) sampling from said classical shadow state.

[0013] In some embodiments, said snapshot state comprises a stabilizer state. In some embodiments, (c) and (d) comprise evaluating said overlap between said stabilizer state and said ansatz state; wherein said overlap is evaluated based at least in part on samplesfrom said stabilizer state. In some embodiments, a validation dataset is used to prevent overfitting in training. In some embodiments, a training dataset is separated into mini- batches to calculate gradients. In some embodiments, said loss function comprises cross- entropy that is estimated using an empirical data distribution. In some embodiments, said classical shadow data distribution comprises probabilistic weights of said one or more quantum measurements computed based at least in part on said classical shadow state constructed using said classical shadow technique. In some embodiments, said training comprises pretraining using synthetic data.

[0014] In some embodiments, the method further comprises characterizing a quantum state based at least in part on said ML model. In some embodiments, the method further comprises characterizing a quantum channel based at least in part on said ML model. In some embodiments, said ML model comprises one or more members of the group consisting of: an autoregressive model, a neural network, a tensor network, and a generative model. In some embodiments, said one or more quantum measurements comprises a Pauli measurement or a Clifford measurement. In some embodiments, said quantum state comprises a pure quantum state or a mixed quantum state. In some embodiments, said mixed quantum state is purified into a pure state using auxiliary qubits.

[0015] In another aspect, the present disclosure provides a method for a quantum state reconstruction. The method may comprise: (a) obtaining one or more quantum measurements; (b) constructing a classical shadow state from a classical shadow technique based at least in part on said one or more quantum measurements; (c) constructing a loss function of a machine learning (ML) model representative of a parameterized quantum state based at least in part on said classical shadow state, wherein said loss function comprises an overlap between a snapshot state representative of said one or more quantum measurements and an ansatz state provided by said ML model; (d) calculating a gradient of said loss function based at least in part on samples from said classical shadow state; and (e) training said ML model, wherein said training comprises using said gradient of said loss function to update one or more parameters of said ML model.

[0016] In another aspect, the present disclosure provides a hybrid system for a quantum state reconstruction. The system may comprise (a) a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system; and (b) a high- performance computing platform operatively coupled to said quantum computer, said high-performance computing platform comprising one or more computing components capable of performing computational tasks in parallel, wherein said high-performance computing platform is configured to at least: (i) obtain one or more quantum measurements from said quantum computer; (ii) construct a classical shadow state using a classical shadow technique based at least in part on said one or more quantum measurements; (iii) construct a loss function of an ML model representative of a parameterized quantum state; (iv) estimate the gradient of said loss function; and (v) train said ML model by updating one or more parameters of said ML model.

[0017] In some embodiments, said high-performance computing platform is operatively coupled to said quantum computer using a digital computer. In some embodiments, the system further comprises a cloud-based computing platform. In some embodiments, said hybrid system provides computational services on the cloud. In some embodiments, said hybrid system is configured to operate in a distributed computing environment.

[0018] In another aspect, the present disclosure provides a system for a quantum state reconstruction. The system may comprise a high-performance computing platform operatively coupled to a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system, said high-performance computing platform comprising one or more computing components capable of performing computational tasks in parallel, wherein said high-performance computing platform is configured to at least: (i) obtain one or more quantum measurements from said quantum computer; (ii) construct a classical shadow state using a classical shadow technique based at least in part on said one or more quantum measurements; (iii) construct a loss function of an ML model representative of a parameterized quantum state; (iv) estimate the gradient of said loss function; and (v) train said ML model by updating one or more parameters of said ML model.

[0019] In some embodiments, said high-performance computing platform is operatively coupled to said quantum computer using a digital computer. In some embodiments, the system further comprises a cloud-based computing platform. In some embodiments, said hybrid system provides computational services on the cloud. In some embodiments, said hybrid system is configured to operate in a distributed computing environment.

[0020] In another aspect, the present disclosure provides a system for a quantum state reconstruction. The system may comprise (a) a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system, wherein said quantum computer is operatively coupled to a high-performance computing platform, said high- performance computing platform comprising one or more computing components capable of performing computational tasks in parallel, wherein said high-performance computing platform is configured to at least: (i) obtain one or more quantum measurements from said quantum computer; (ii) construct a classical shadow state using a classical shadow technique based at least in part on said one or more quantum measurements; (iii) construct a loss function of an ML model representative of a parameterized quantum state; (iv) estimate the gradient of said loss function; and (v) train said ML model by updating one or more parameters of said ML model.

[0021] In some cases, said high-performance computing platform is operatively coupled to said quantum computer using a digital computer. In some embodiments, the system further comprises a cloud-based computing platform. In some embodiments, said hybrid system provides computational services on the cloud. In some embodiments, said hybrid system is configured to operate in a distributed computing environment.

[0022] Another aspect of the present disclosure provides a system comprising one or more computer processors and computer memory coupled thereto. The computer memory comprises machine executable code that, upon execution by the one or more computer processors, implements any of the methods above or elsewhere herein.

[0023] Additional aspects and advantages of the present disclosure will become readily apparent to those skilled in this art from the following detailed description, wherein only illustrative embodiments of the present disclosure are shown and described. As will be realized, the present disclosure is capable of other and different embodiments, and itsseveral details are capable of modifications in various obvious respects, all without departing from the disclosure. Accordingly, the drawings and description are to be regarded as illustrative in nature, and not as restrictive. INCORPORATION BY REFERENCE

[0024] All publications, patents, and patent applications mentioned in this specification are herein incorporated by reference to the same extent as if each individual publication, patent, or patent application was specifically and individually indicated to be incorporated by reference. To the extent publications and patents or patent applications incorporated by reference contradict the disclosure contained in the specification, the specification is intended to supersede and / or take precedence over any such contradictory material. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] The novel features of the invention are set forth with particularity in the appended claims. A better understanding of the features and advantages of the present invention will be obtained by reference to the following detailed description that sets forth illustrative embodiments, in which the principles of the invention are utilized, and the accompanying drawings (also “Figure” and “FIG.” herein), of which:

[0026] FIG.1 is a schematic of an example hybrid computing system for a quantum state reconstruction, in accordance with some embodiments.

[0027] FIG.2 is a flowchart of an example method for a quantum state reconstruction, in accordance with some embodiments.

[0028] FIG.3A is a diagram of an example Clifford measurement framework: randomly selecting a unitary U from the n-qubit Clifford group.

[0029] FIG. 3B is a diagram of an example Pauli measurement framework: randomly selecting a unitary U by taking an n-qubit tensor product of single-qubit unitaries independently sampled from the single-qubit Clifford group, in accordance with some embodiments.

[0030] FIGS.4A-4F are a group of graphs of the infidelity and infidelity loss, along with the cross-entropy loss, between a reconstructed state and the actual state for four methods in reconstructing both six- and eight-qubit Greenberger–Horne–Zeilinger (GHZ) states using 1000 classical shadows based on Pauli or Clifford measurements, in accordance with some embodiments.

[0031] FIG.5A and FIG.5B are a pair of graphs of the predictive capabilities based on neural quantum states, classical shadows, and the simplex projection of the latter, for Pauli observables of a mixed GHZ state prepared using a series of noisy gates with noise levels p ranging from 0.0 to 0.5, in accordance with some embodiments.

[0032] FIGS.6A-6E are a group of graphs of a performance comparison of reconstructed six-qubit GHZ mixed states using classical shadows and neural quantum states collected from 5000 random Pauli measurements, in accordance with some embodiments.

[0033] FIG.7A and FIG.7B are a pair of graphs depicting the discrepancy in angle for the Monte Carlo-estimated gradients compared to the exact gradient throughout the training process, in accordance with some embodiments.

[0034] FIG.8A and FIG.8B are a pair of graphs of the Kullback–Leibler (KL) divergence which compares the proxy for a typically unknown data distribution with empirical and shadow-weight distributions, both accessible from measurement data, in accordance with some embodiments. DETAILED DESCRIPTION

[0035] While various embodiments of the invention are shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions may occur to those skilled in the art without departing from the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed.

[0036] Neither the Title nor the Abstract is to be taken as limiting in any way the scope of the disclosed invention(s). The title of the present application and headings of sections provided in the present application are for convenience only and are not to be taken aslimiting the disclosure in any way.

[0037] Unless otherwise defined, all technical terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise. Any reference to “or” herein is intended to encompass “and / or” unless otherwise stated.

[0038] The term “plurality” generally refers to “two or more,” unless expressly specified otherwise.

[0039] The term “e.g.” and like terms mean “for example,” and thus do not limit the terms or phrases they explain. For example, in a sentence “the computer sends data (e.g., instructions, a data structure) over the Internet,” the term “e.g.” explains that “instructions” are an example of “data” that the computer may send over the Internet, and also explains that “a data structure” is an example of “data” that the computer may send over the Internet. However, both “instructions” and “a data structure” are merely examples of “data,” and other things besides “instructions” and “a data structure” can be “data.”

[0040] Whenever the term “at least,” “greater than,” or “greater than or equal to” precedes the first numerical value in a series of two or more numerical values, the term “at least,” “greater than,” or “greater than or equal to” applies to each of the numerical values in that series of numerical values. For example, greater than or equal to 1, 2, or 3 is equivalent to greater than or equal to 1, greater than or equal to 2, or greater than or equal to 3.

[0041] Whenever the term “no more than,” “less than,” or “less than or equal to” precedes the first numerical value in a series of two or more numerical values, the term “no more than,” “less than,” or “less than or equal to” applies to each of the numerical values in that series of numerical values. For example, less than or equal to 3, 2, or 1 is equivalent to less than or equal to 3, less than or equal to 2, or less than or equal to 1.

[0042] Where values are described as ranges, the disclosure includes the disclosure of all possible sub-ranges within such ranges, as well as specific numerical values that fall within such ranges irrespective of whether a specific numerical value or specific sub-range is expressly stated.

[0043] Certain inventive embodiments herein contemplate numerical ranges. When ranges are present, the ranges include the range endpoints. Additionally, every sub-range and value within the range is present as if explicitly written out.

[0044] The term “about” or “approximately” may mean within an acceptable error range for the particular value, which will depend in part on how the value is measured or determined, e.g., the limitations of the measurement system. For example, “about” may mean within 1 or more than 1 standard deviation, per the practice in the art. Alternatively, “about” may mean a range of up to 20%, up to 10%, up to 5%, or up to 1% of a given value. Where particular values are described in the application and claims, unless otherwise stated the term “about” meaning within an acceptable error range for the particular value may be assumed. Quantum Computer / Quantum Hardware

[0045] Any type of quantum hardware may be suitable for the technologies disclosed herein. A quantum processor or quantum computer may comprise one or more quantum gate arrays, one-way quantum computers, topological quantum computers, superconductor-based quantum computers, trapped ion quantum computers, trapped atom quantum computers, optical lattices, quantum dot computers, spin-based quantum computers, spatial-based quantum computers, Loss–DiVincenzo quantum computers, nuclear magnetic resonance (NMR) based quantum computers, solution-state NMR quantum computers, solid-state NMR quantum computers, solid-state NMR Kane quantum computers, electrons-on-helium quantum computers, cavity-quantum- electrodynamics based quantum computers, molecular magnet quantum computers, fullerene-based quantum computers, linear optical quantum computers, diamond-based quantum computers, nitrogen-vacancy (NV) diamond-based quantum computers, Bose– Einstein condensate-based quantum computers, transistor-based quantum computers, and rare-earth-metal-ion-doped inorganic crystal based quantum computers. The quantum processor or quantum computer may comprise one or more of: quantum annealers, Ising solvers, optical parametric oscillators (OPO), and gate model quantum computers.

[0046] A quantum processor or quantum computer may comprise one or more qubits. The one or more qubits may comprise superconducting qubits, trapped ion qubits, trapped atom qubits, photon qubits, quantum dot qubits, electron spin-based qubits, nuclear spin- based qubits, molecular magnet qubits, fullerene-based qubits, diamond-based qubits, nitrogen-vacancy (NV) diamond-based qubits, Bose–Einstein condensate-based qubits, transistor-based qubits, or rare-earth-metal-ion-doped inorganic crystal-based qubits.

[0047] In some cases, a quantum device with a limitation of a two-dimensional structure of a quantum chip or a limitation on how many neighboring qubits each qubit is connected to may benefit from methods and systems disclosed herein. In accordance with the description herein, suitable quantum computers may include, by way of non-limiting examples, including the associated references, each of which is incorporated by reference herein in its entirety: superconducting quantum computers (qubits implemented as small superconducting circuits—Josephson junctions) (Clarke et al., “Superconducting quantum bits”, Nature 453, no. 7198, pp. 1031–1042, 2008); trapped-ion quantum computers (qubits implemented as states of trapped ions) (Kielpinski et al., “Architecture for a large-scale ion-trap quantum computer”, Nature 417, no.6890, pp.709–711, 2002); optical lattice quantum computers (qubits implemented as states of neutral atoms trapped in an optical lattice) (Deutsch et al., “Quantum computing with neutral atoms in an optical lattice”, Fortschritte der Physik: Progress of Physics 48, no. 9–11, pp. 925–943, 2000); spin-based quantum dot computers (qubits implemented as the spin states of trapped electrons) (Imamoḡlu et al., “Quantum information processing using quantum dot spins and cavity QED”, Physical Review Letters 83, no. 20, p. 4204, 1999); spatial-based quantum dot computers (qubits implemented as electron positions in a double quantum dot) (Fedichkin et al., “Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot”, arXiv:quant-ph / 0006097, 2000); coupled quantum wires (qubits implemented as pairs of quantum wires coupled by quantum point contact) (Bertoni et al., “Quantum logic gates based on coherent electron transport in quantum wires”, Physical Review Letters 84, no.25, p.5912, 2000); nuclear magnetic resonance quantum computers (qubits implemented as nuclear spins and probed by radio waves) (Cory et al., “Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing”, arXiv: quant-ph / 9709001, 1997); solid-state NMR Kane quantum computers (qubits implemented as the nuclear spin states of phosphorusdonors in silicon) (Kane, “A silicon-based nuclear spin quantum computer”, Nature 393, no. 6681, pp. 133–137, 1998); electrons-on-helium quantum computers (qubits implemented as electron spins) (Lyon, “Spin-based quantum computing using electrons on liquid helium”, arXiv:cond-mat / 0301581, 2006); cavity quantum electrodynamics- based quantum computers (qubits implemented as states of trapped atoms coupled to high-finesse cavities) (Burell, “An Introduction to Quantum Computing using Cavity QED concepts,” arXiv:1210.6512, 2012); molecular magnet-based quantum computers (qubits implemented as spin states) (Leuenberger et al., “Quantum Computing in Molecular Magnets”, arXiv:cond-mat / 0011415, 2001); fullerene-based electron spin resonance (ESR) quantum computers (qubits implemented as electronic spins of atoms or molecules encased in fullerenes) (Harneit, “Spin Quantum Computing with Endohedral Fullerenes”, arXiv:1708.09298, 2017); linear optical quantum computers (qubits implemented as processing states of different modes of light through linear optical elements such as mirrors, beam splitters and phase shifters) (Knill et al. “Efficient linear optics quantum computation”, arXiv:quant-ph / 0006088, 2000); diamond-based quantum computers (qubits implemented as electronic or nuclear spins of nitrogen-vacancy (NV) centres in diamond) (Nizovtsev et al., “A quantum computer based on NV centers in diamond: optically detected nutations of single electron and nuclear spins”, Optics and spectroscopy 99, no. 2, pp. 233–244, 2005); Bose–Einstein condensate-based quantum computers (qubits implemented as two-component Bose–Einstein condensates) (Byrnes et al., “Macroscopic quantum computation using Bose–Einstein condensates”, arXiv:quantum-ph / 1103.5512, 2011); transistor-based quantum computers (qubits implemented as semiconductors coupled to nanophotonic cavities) (Sun et al., “A single- photon switch and transistor enabled by a solid-state quantum memory”, arXiv:quant- ph / 1805.01964, 2018); rare-earth-metal-ion-doped inorganic crystal-based quantum computers (qubits implemented as atomic ground state hyperfine levels in rare-earth-ion- doped inorganic crystals) (Ohlsson et al. “Quantum computer hardware based on rare- earth-ion-doped inorganic crystals”, Optics Communications 201, no. 1–3, pp. 71–77, 2002); metal-like carbon nanospheres based quantum computers (qubits implemented as electron spins in conducting carbon nanospheres) (Náfrádi et al., “Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres”, arXiv:cond- mat / 1611.07690, 2016); topological quantum computers (qubits implemented as non-Abelian anyons) (Nayak et al., “Non-Abelian Anyons and Topological Quantum Computation,” arXiv:0707.1889, 2007); photonic continuous-variable quantum computing hardware (quantum variables represented by the quadrature operators of the quantum harmonic oscillators in a quantum optical mode) (Arrazola et al., “Quantum circuits with many photons on a programmable nanophotonic chip,” Nature 591, pp.54– 60, 2021); photonic qubit-based quantum hardware (qubits implemented on pairs of optical paths) (O’Brien et al., “Photonic quantum technologies,” Nature Photonics 3, pp. 687–695, 2009); quantum computing hardware based on bosonic codes (error-protected qubits are formed by embedding a finite-dimensional code space within the infinite- dimensional Fock space associated with a bosonic quantum field mode; examples include the Gottesman–Kitaev–Preskill (GKP) code, cat codes, and binomial codes, respectively) (Gottesman et al., “Encoding a qubit in an oscillator,” Physical Review A 64, 012310, 2001; Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes,” PRX Quantum 3, 010329, 2022; Michael et al., “New Class of Quantum Error-Correcting Codes for a Bosonic Mode,” Physical Review X 6, 031006, 2016); quantum hardware based on coherent network computing (operating by sampling low-energy eigenstates of an Ising Hamiltonian by encoding the spins in a network of optical parametric oscillators with all-to-all connectivity; and future architectures may exploit quantum entanglement for computation) (Inui et al., “Entanglement and quantum discord in optically coupled coherent Ising machines,” Physical Review A 102, 062419, 2020; and Yanagimoto et al., “Embedding entanglement generation within a measurement-feedback coherent Ising machine,” arXiv:1906.04902, 2019). High-Performance Computing Device

[0048] A computing device, which may interact with a quantum computing device disclosed herein and enhance a special-purpose computing platform disclosed herein, may be a high-performance computing (HPC) device. An HPC device may comprise one or more of a graphics processing unit (GPU), a tensor processing unit (TPU), a field- programmable gate array (FPGA), an application-specific integrated circuit (ASIC), and a tensor streaming processor (TSP). Any other suitable processing unit that is capable of performing matrix multiplication may be used. Certain computing devices may be more efficient at operations such as matrix multiplication. These computing devices mayprovide additional efficiency gains over other computing systems. For example, a matrix multiplication device may be a GPU. A GPU may be a specialized electronic circuit optimized for high throughput, which can perform the same set of operations in parallel on many data blocks at a time. For example, a matrix multiplication device may be a TPU. A TPU may be a type of ASIC developed for low bit precision processing by Google® Inc. For example, a matrix multiplication device may be an FPGA. An FPGA may be an integrated circuit chip that comprises configurable logic blocks and programmable interconnects. It can be programmed after manufacturing to execute custom algorithms. For example, a matrix multiplication device may be an ASIC. An ASIC may be an integrated circuit chip that is customized to run a specific algorithm. In some case, an ASIC cannot be programmed after manufacturing. For example, a matrix multiplication device may be a TSP. A TSP may be a domain-specific programmable integrated chip that is designed for linear algebra computations as they may be performed in artificial intelligence applications, an example of which may be found in Gwennap, “Groq Rocks Neural Networks,” Microprocessor Report, Tech. Rep., Jan., 2020, which is incorporated by reference herein in its entirety. Digital Computer

[0049] In some cases, a digital computer comprises one or more hardware central processing units (CPU) that carry out a classical computer’s functions. In some cases, the classical computer further comprises an operating system (OS) configured to perform executable instructions. In some cases, the classical computer is connected to a computer network.

[0050] In some cases, the classical computer is connected to a computer network. In some cases, the classical computer is connected to the Internet such that it accesses the World Wide Web. In some cases, the classical computer is connected to one or more computer servers, which can enable distributed computing, such as a cloud computing infrastructure. In some cases, the classical computer is connected to an intranet and / or extranet or an intranet and / or extranet that is in communication with the Internet. In some cases, the classical computer is connected to a data storage device. In some cases, the network is a telecommunication and / or data network. In some cases, the network is a peer- to-peer network, which may enable devices coupled to the computer system to behave asa client or a server.

[0051] In accordance with the description herein, suitable classical computers may include, by way of non-limiting examples, server computers, desktop computers, laptop computers, notebook computers, sub-notebook computers, netbook computers, netpad computers, set-top computers, media streaming devices, handheld computers, Internet appliances, mobile smartphones, tablet computers, personal digital assistants, video game consoles, and vehicles. Smartphones may be suitable for use with methods and systems described herein. Select televisions, video players, and digital music players, in some cases, with computer network connectivity, may be suitable for use in the systems and methods described herein. Suitable tablet computers may include those with booklet, slate, and convertible configurations.

[0052] In some cases, the classical computer includes an operating system configured to perform executable instructions. The operating system may be, for example, software, including programs and data, which manages the device’s hardware and provides services for execution of applications. Suitable server operating systems include, by way of non- limiting examples, FreeBSD, OpenBSD, NetBSD®, Linux®, Apple® Mac OS X Server®, Oracle® Solaris®, Windows Server®, and Novell® NetWare®. Suitable personal computer operating systems may include, by way of non-limiting examples, Microsoft® Windows®, Apple® Mac OS X®, Apple® macOS®, UNIX®, and UNIX- like operating systems such as GNU / Linux®. In some cases, the operating system is provided by cloud computing. Suitable mobile smart phone operating systems may include, by way of non-limiting examples, Nokia® Symbian® OS, Apple® iOS®, Research In Motion® BlackBerry OS®, Google® Android®, Microsoft® Windows Phone® OS, Microsoft® Windows Mobile® OS, Linux®, and Palm® WebOS®. Suitable media streaming device operating systems may include, by way of non-limiting examples, Apple TV®, Roku®, Boxee®, Google TV®, Google Chromecast®, Amazon Fire®, and Samsung® HomeSync®. Suitable video game console operating systems may include, by way of non-limiting examples, Sony® PS3®, Sony® PS4®, Microsoft® Xbox 360®, Microsoft® Xbox One®, Nintendo® Wii®, Nintendo® Wii U®, and Ouya®.

[0053] In some cases, the classical computer includes a storage and / or memory device. In some cases, the storage and / or memory device is one or more physical apparatuses used to store data or programs on a temporary or permanent basis. In some cases, the storage and / or memory device may have one or more additional data storage units that are external to the classical computer, for example, being located on a remote server that is in communication with the classical computer through an intranet or the Internet. In some cases, the device comprises a volatile memory and requires power to maintain stored information. In some cases, the device comprises non-volatile memory and retains stored information when the classical computer is not powered. In some cases, the non-volatile memory comprises flash memory. In some cases, the non-volatile memory comprises dynamic random-access memory (DRAM). In some cases, the non-volatile memory comprises ferroelectric random-access memory (FRAM). In some cases, the non-volatile memory comprises phase-change random-access memory (PRAM). In some cases, the non-volatile memory comprises resistive random-access memory (RRAM). In some cases, the device comprises a storage device including, by way of non-limiting examples, CD-ROMs, DVDs, flash memory devices, magnetic disk drives, magnetic tapes drives, optical disk drives, and cloud computing-based storage. In some cases, the storage and / or memory device comprises a combination of devices such as those disclosed herein.

[0054] In some cases, the classical computer includes a display to send visual information to a user. In some cases, the display is a cathode ray tube (CRT). In some cases, the display is a liquid crystal display (LCD). In some cases, the display is a thin film transistor liquid crystal display (TFT-LCD). In some cases, the display is an organic light emitting diode (OLED) display. In some cases, on OLED display is a passive-matrix OLED (PMOLED) or active-matrix OLED (AMOLED) display. In some cases, the display is a plasma display. In some cases, the display is a video projector. In some cases, the display is a combination of devices such as those disclosed herein.

[0055] In some cases, the classical computer includes an input device to receive information from a user. In some cases, the input device is a keyboard. In some cases, the input device is a pointing device including, by way of non-limiting examples, a mouse, trackball, track pad, joystick, game controller, or stylus. In some cases, the input device is a touch screen or a multi-touch screen. In some cases, the input device is a microphoneto capture voice or other sound input. In some cases, the input device is a video camera or other sensor to capture motion or visual input. In some cases, the input device is a Kinect®, Leap Motion®, or the like. In some cases, the input device is a combination of devices such as those disclosed herein.

[0056] In the following detailed description, reference is made to the accompanying figures, which form a part hereof. In the figures, similar symbols typically identify similar components, unless context dictates otherwise. The illustrative embodiments described in the detailed description, figures, and claims are not meant to be limiting. Other embodiments may be used, and other changes may be made, without departing from the scope of the subject matter presented herein. It will be readily understood that the aspects of the present disclosure, as generally described herein, and illustrated in the figures, can be arranged, substituted, combined, separated, and designed in a wide variety of different configurations, all of which are explicitly contemplated herein. Classical Shadows

[0057] In some cases, the classical shadow formalism may provide an efficient method for predicting various properties using relatively few measurements. An example of a description of the classical shadow formalism is provided at, for example, Huang et al., “Predicting Many Properties of a Quantum System from Very Few Measurements,” Nature Physics 16, 1050–1057, 2020, which is incorporated herein by reference in its entirety. By randomly measuring quantum states, exponentially many observables may be predicted.

[0058] In some cases, a classical shadow may be created by repeatedly performing the following procedure: at every iteration ^^^^, a random unitary transformation ^^^^ ↦ ^^^^ ^^^^ ^†^^^^^^^^^^^is applied, after which all the qubits in the computational basis are measured. The pairs ( ^^^^^^^^, ^^^^^^^^) may be recorded in classical memory. Let ^^^^^^^^=^^^^ and ^^^^^^^^= ^^^^| ^^^^^^^^^ be the associated density matrix and stabilizer state, respectively. The mapping ℳ: ^^^^ ↦ ^^^^[^^^^^^^^]is invertible if the measurement protocol is tomographically complete. The set ^^^^ ^^^^ ={ℳ−1( ^^^^ ^^^^)} ^^^^^^^^=1 is a classical shadow of ^^^^ of size ^^^^, and can be used to construct anunbiased estimator ^^�^^ = ^^^^[ℳ−1( ^^^^^^^^)] of ^^^^ called a classical shadow state.

[0059] The classical shadow ^^^^^^^^suffices to predict ^^^^ arbitrary linear and nonlinear log(^^^^)max‖ ^^ ‖2 functions of the state ^^^^ up to an additive error ^ if it is of size ^^^^^^ ^^^^shadowThe shadow norm may depend on the ensemble from which the unitaries ^^^^^^^^are drawn for creating the classical shadow and has efficient bounds. In some cases, a shadow norm may have efficient bounds if the ensemble satisfies a unitary 3-design property. For example, this is the case if the unitaries are drawn from the Clifford set as in Huang et al., “Predicting Many Properties of a Quantum System from Very Few Measurements,” Nature Physics 16, 1050–1057, 2020, or using analog evolutions of different lengths of time as in Liu et al., “Predicting Arbitrary State Properties from Single Hamiltonian Quench Dynamics,” arXiv:2311.00695, 2023, each of which is incorporated herein by reference in its entirety.

[0060] Two example ensembles include the Pauli and Clifford groups. In some cases, these ensembles may be computationally favourable because of their efficient classical simulations, according to the Gottesman–Knill theorem. The inverse twirling maps for the n-qubit Clifford and Pauli ensembles take the forms= (2^^^^+ 1) ^^^^ − ^^^^ and ℳ1−1, respectively. The shadow norm scales well for the Clifford ensemble, while the Pauli ensemble is more experimentally friendly. In some cases, tor the Pauli ensemble, the shadow norm may scale exponentially poorly with respect to the locality of an observable, rendering the procedure inefficient at estimating high-weight observables. A middle-ground solution is to use shallow subsets of the Clifford group, examples of which is described in Ippoliti et al., “Operator Relaxation and the Optimal Depth of Classical Shadows, Physical Review Letters 130, 230403, 2023; Rozon, et al., “Optimal twirling depths for shadow tomography in the presence of noise,” arXiv:2311.10137, 2023; and Hu, et al., “Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows,” arXiv:2402.17911, 2024, each of which is incorporated herein by reference in its entirety.

[0061] Now referring to FIG. 3A and FIG. 3B, FIG. 3A illustrates a diagram of an example Clifford measurement framework: randomly selecting a unitary U from the n- qubit Clifford group. FIG. 3B illustrates a diagram of an example Pauli measurement framework: randomly selecting a unitary U by taking an n-qubit tensor product of single-qubit unitaries independently sampled from the single-qubit Clifford group.

[0062] The Clifford measurement may comprise randomly selecting a unitary transformation U from the n-qubit Clifford group. The Pauli measurement may comprise randomly selecting a unitary transformation U by taking an n-qubit tensor product of single-qubit unitaries independently sampled from the single-qubit Clifford group.

[0063] The Clifford measurement may use Clifford circuits generated by CNOT, Hadamard, or phase gates. The associated classical snapshot may be written aswherein ^^^^^^^^is the measured bitstring resulting from the application of the unitaries ^^^^^^^^, and ^^^^ is the identity operator. suffices topredict M linear functions. This technique may be more powerful when the target has aconstant tr(O2^^^^ ) value. Examples include estimating quantum fidelities for whichtr(O2^^^^ )=1.

[0065] The Pauli measurement may use single-qubit Clifford gates, which may be decomposed into tensor products. The classical snapshot may be written aswhere the external index ^^^^ refers to the measurement index while the internal index ^^^^ refers to the qubit index.the locality of ^^^^^^^^, suffices to predict M linear functions. For observables that are tensor products of kThis type of measurement may be more powerful when the target functions are local. Examples include k-point correlators and Hamiltonian terms associated with many-body quantum states.

[0067] In some cases, Pauli measurements may be used to generate a classical shadow. It may require a shorter quantum circuit and hence may be less sensitive to quantum device errors.

[0068] While classical shadows may efficiently predict observables, achieving a complete and accurately reconstructed quantum state may remain a challenging task. The neural shadow quantum state tomography (NSQST) method addresses this limitation by building upon neural network quantum state tomography (NNQST) and incorporating infidelity (observable prediction from the classical shadow) as a loss function. An example of the NSQST method is provided in Wei et al., “Neural-Shadow Quantum State Tomography,” arXiv:2305.01078, 2023, which is incorporated herein by reference in its entirety.

[0069] The NSQST approach for training a neural quantum state using classical shadows may combine the advantages of variational ansatz and classical shadow measurements. In the NSQST approach the infidelity between the classical shadow state and the ansatz may be introduced as a loss function, enabling the training of the ansatz on Clifford measurements. Unlike the traditionally used cross-entropy loss function, infidelity may offer a better optimization landscape due to its metric properties and bounded range. However, in the case of an unphysical state like the classical shadow state, the infidelity may not be constrained within the physically valid [0, 1] range. Additionally, infidelity cannot guarantee bounded errors in approximating many quantities of interest from the trained model. Finally, the loss may not generalize to ansatz for mixed states.

[0070] Moreover, the NSQST approach may face many practical challenges in its training protocol. The trainability of the ansatz may use an ad hoc initialization step, without which the infidelity-based optimization stalls. In some cases, it may be possible o overcome this barren plateau by pretraining the model using the cross-entropy loss function on the classical dataset with measurements in the computational basis. However, this pretraining requires an additional set of measurements beyond the one used in the classical shadow technique. Furthermore, its success may depend on the informational content of the underlying quantum state in the computational basis, rendering it non-universal.

[0071] Methods and systems disclosed herein may improve upon NSQST and may provide an alternative approach to improving the predictive power of Pauli ensembles (e.g., the shallowest possible shadows) by using a neural network (or other parameterized ansatz). Whereas the raw classical shadow is powerful at predicting many observables efficiently, it may not always be sample efficient due to the unphysical nature of the state it provides. This is because, while shadow tomography yields a trace-one quantum state, not all eigenvalues are guaranteed to be positive. However, neural quantum states may allow for natively imposing physicality constraints on a reconstructed state.

[0072] Another advantage of the unitary design property of the Clifford ensemble is that it may allow for the efficient prediction of nonlinear functions of a quantum state. This may be particularly useful in estimating entropy-based quantities such as the Rényi entropies, since they involve two copies of a state, and thus quadratic functions of it. Neural networks may also improve the predictive power of such nonlinear quantities when combined with a purification technique. Parameterized Quantum States

[0073] In some examples, it may be possible to efficiently parameterize a pure quantum state |ψ^ of multiple qubits with a machine learning (ML) model parameter vector λ aswhere ^^^^ ∈ ℬ^^^^={0, 1} ^^^^is a binary vector in the computational basis. Similarly, the larger state space of density matrices may be parameterized with an ML model. Among competing approaches, one based on the idea of purification in Torlai et al., “Latent Space Purification via Neural Density Operators,” Physical Review Letters 120, 240503, 2018, which is incorporated herein by reference in its entirety,may be useful, as the resulting density matrix is physical. Here, auxiliary input variables ^^^^ represent the extended dimensions of the Hilbert space of the purified state. Therefore, their number ^^^^^^^^constrains the entropy of ^^^^^^^^, with ^^^^^^^^being a theoretically proven sufficient upper bound for capturing any density matrix over the physical input variables^^^^. ^^^^^^^^may be treated as a hyperparameter on the same footing as other parameters affecting the neural network’s architecture and hence its expressive power.

[0074] In some cases, the ML model may comprise one or more members of the group consisting of: an autoregressive model, a neural network, a tensor network, and a generative model.

[0075] When the ML model parameters λ in the ansatz equations disclosed herein are embedded within a generative model based on an autoregressive architecture, the resulting model may be very expressive and amenable to an efficient extraction of information about the underlying wave function as long as the quantity of interest can be cast as an unbiased estimator over samples from the model. For example, the purity of ^^^^^^^^may be estimated by evaluating the Swap operator between the physical and auxiliary qubits. Examples of evaluating the Swap operator are provided at: Hastings et al., “Measuring Rényi Entanglement Entropy in Quantum Monte Carlo Simulations,” Physical Review Letters 104, 157201, 2010; Kulchytskyy, “Probing universality with entanglement entropy via quantum Monte Carlo,” Ph.D. thesis, University of Waterloo, 2019; and Beach et al., “QuCumber: wavefunction reconstruction with neural networks,” SciPost Physics 7, 009, 2019, each of which is incorporated herein by reference in its entirety. Neural Shadow Quantum State Tomography

[0076] Neural network quantum state tomography (NNQST) is a technique that may be used for harnessing the capabilities of artificial neural networks to effectively extract essential information about quantum states from experimental measurements. This approach may achieve its goal by training a neural network to recognize underlying patterns within measurement data. NNQST may approximate a neural network wave function that closely matches some class of target states. However, reconstructing a general unknown quantum state may still demand an exponentially large number of measurements (Torlai et al., “Many-body quantum state tomography with neural networks,” arXiv:1703.05334, 2017; and Bennewitz et al., “Neural Error Mitigation of Near-Term Quantum Simulations,” arXiv:2105.08086, 2021, each of which is incorporated herein by reference in its entirety). An example is the phase-shifted multi-qubit GHZ state, where the relative phase is associated with non-local correlations.

[0077] In NSQST, a classical shadow may be used to provide information about the target state instead of traditional measurement data obtained from few qubits. By leveraging the predictive capability of the classical shadow technique, methods and system disclosed herein may construct a loss function of the neural network wave function based on predicted observables, e.g., the infidelity between the ansatz and the classical shadow state, and then train the ansatz via, e.g., variational Monte Carlo (VMC) to provide an optimal set of parameters.

[0078] Now referring to FIG. 1, there is shown a schematic of an example hybrid computing system for a quantum state reconstruction. In some cases, the system may comprise a quantum computer 126 and a high-performance computing (HPC) platform 100. In some cases, the system may comprise a digital computer 114. In some cases, the quantum computer 126 and the HPC platform 100 may be operatively coupled using the digital computer 114. In some cases, the HPC platform 100 may comprise one or more computing components capable of performing computational tasks in parallel. The quantum computer 126 may be of various types, such as any quantum computer disclosed elsewhere herein. In some cases, the quantum computer 126 may comprise a quantum chip 128 and a control system 130. In some cases, the quantum computer 126 may comprise a fault-tolerant quantum computer. The fault-tolerant quantum computer may be of various types such as any fault-tolerant quantum computer disclosed elsewhere herein.

[0079] In some cases, the digital computer 114 may comprise at least one processing device 120, a display device 122, an input device 116, communication ports 124 and a memory 118 comprising a computer program executable by the at least one processing device 120. The memory 118 may be of various types, such as any type disclosed elsewhere herein. The digital computer 114 may be of various types, such as any digital computer disclosed elsewhere herein.

[0080] In some cases, the HPC platform 100 may comprise one or more HPC nodes 102, 110, 112. The HPC platform 100 may be of various types, such as any HPC platform disclosed elsewhere herein. In some cases, the HPC platform 100 may comprise a special-purpose computing platform for a quantum state reconstruction. In some cases, the HPC platform 100 may be operatively coupled to the quantum computer 126. In some cases, the HPC platform 100 may be operatively coupled to the quantum computer 126 using the digital computer 114. In some cases, the HPC platform 100 may comprise at least one member of the group consisting of: a central processing unit (CPU), a field-programmable gate array (FPGA), an application-specific integrated circuit (ASIC), a graphics processing unit (GPU), a tensor processing unit (TPU), and a tensor streaming processor (TSP).

[0081] In some cases, the one or more HPC nodes 102, 110, 112 may comprise one or more computing components. In some cases, the one or more HPC nodes 102, 110, 112 may perform computations in parallel. In some cases, the one or more computing components may perform computations in parallel. In general, each node corresponds to a classical computing instance that can perform a specific individual task, with the digital computer 114 combining the results from the HPC nodes. In some cases, the HPC platform 100 may be configured to at least (i) obtain one or more quantum measurements from the quantum computer 126; (ii) construct a classical shadow state using a classical shadow technique based at least in part on the one or more quantum measurements; (iii) use the classical shadow state to construct a loss function of an ML model representative of a parameterized quantum state; (iv) use samples from the classical shadow state to estimate the gradient of the loss function; and (v) use the gradient of the loss function to train the ML model by updating its parameters. The execution of tasks (ii) to (v) may be parallelized over the HPC nodes.

[0082] In some cases, the one or more HPC nodes 102, 110, 112 may comprise a processing device 106, communication ports 108, and a memory 104. In some cases, the communication ports 108 may be necessary for the individual components to interact with each other and the central computer that delegates tasks to each individual component. In some cases, the memory 104 may be required in case one step needs to exchange information with the next and may comprise RAM or a hard drive.

[0083] In some cases, the hybrid computing system may comprise a cloud-based computing platform. In such cases, the interactions between the digital computer 114 and the HPC nodes may be conducted through the internet, but the basic structure may remainthe same. The HPC platform is generally located in a separate space from the digital computer.

[0084] In some cases, the hybrid computing system may be used for providing computational services on or through the cloud. In such cases, the user may provide the digital computer 114, the HPC resources 100 may be provided by the one or more cloud services, and the communication ports 108 may interact through both the internet and a fast local interconnect.

[0085] In some cases, the hybrid computing system may be configured to operate in a distributed computing environment. In such cases, the different one or more HPC nodes 102, 110, 112 may be in different locations, so long as the communication ports 108 allow them all to interact with each other and with the digital computer 114.

[0086] Now referring to FIG.2, there is shown a flowchart of an example method for a quantum state reconstruction. In some cases, the quantum state may comprise a pure quantum state or a mixed quantum state. In some cases, the mixed quantum state may correspond to a state prepared via a noisy quantum circuit. In some cases, the mixed quantum state may correspond to a subsystem embedded into a larger pure quantum state. In some cases, the mixed quantum state may comprise an ensemble of pure quantum states prepared via repeated execution of a quantum circuit containing measurement gates.

[0087] According to processing step 202, one or more quantum measurements are obtained. In some cases, the one or more quantum measurements may comprise a tomographically complete set of quantum measurements that permit a complete reconstruction of the measured quantum state. In some cases, the one or more quantum measurements may comprise Pauli measurements or Clifford measurements. In some case, the measurements may be randomized by probabilistic sampling from the aforementioned ensembles.

[0088] According to processing step 204, a classical shadow state is constructed using the one or more quantum measurements using a classical shadow technique. In some cases, the classical shadow state may be created using a hybrid computing system comprising an HPC platform, such as the hybrid computing system disclosed herein with respect toFIG.1.

[0089] In some cases, Pauli measurements may be used to generate a classical shadow. It may require a shorter quantum circuit and hence may be less sensitive to quantum device errors.

[0090] According to processing step 206, a loss function of a machine learning (ML) model representative of a parameterized quantum state is constructed using the classical shadow state. In some cases, the loss function may comprise an overlap between a snapshot state representative of the one or more quantum measurements and an ansatz state provided by the ML model; the ansatz may also be referred to as a neural shadow. In some cases, the construction of the loss function may comprise evaluating the overlap between the snapshot state and the ansatz state. In some cases, the overlap may be evaluated using samples from the snapshot state. In some cases, the snapshot state may comprise a stabilizer state. In some cases, the loss function of an ML model may be constructed using a hybrid computing system comprising an HPC platform, such as the hybrid computing system disclosed herein with respect to FIG.1.

[0091] According to processing step 208, the gradient of the loss function is estimated using samples from the classical shadow state. In some cases, the classical shadow state may comprise the stabilizer state, the snapshot state, or the sum of the snapshot or stabilizer states. In some cases, the estimation may comprise Monte Carlo (MC) sampling from the classical shadow state. In some cases, the estimation may comprise MC sampling of stabilizer states from the classical shadow state. In some cases, the estimation may comprise MC sampling from the stabilizer state. In some cases, the estimation may comprise MC sampling of computational basis measurements from the stabilizer state. In some cases, the gradient of the loss function may be estimated using a hybrid computing system comprising an HPC platform, such as the hybrid computing system disclosed herein with respect to FIG.1.

[0092] In some cases, it may be useful to evaluate the overlap between a stabilizer state^^^^ and the neural stateto improve the computationaltractability of the considered loss functions and their gradients.

[0093] In some cases, this sum may be exponentially large in the computational basis with respect to the number of qubits and may be intractable. However, it may be estimated using Monte Carlo sampling in at least two ways. ^^^^ ∈ ℬ may be sampled according to

[0094] Alternatively, ^^^^ ∈ ℬ may be sampled with a probability proportionate to itsoverlap with the stabilizer state ^^^^, ^^^^ ^^^^(^^^^)=‖^^^^(^^^^)‖2, to obtain the reformulated estimator

[0095] Both estimators are efficient to compute, as the amplitudes and samples may be obtained in polynomial time for both the stabilizer state ^^^^ and an autoregressive generative model representing ^^^^^^^^. The first sampling strategy is akin to on-policy reinforcement learning, wherein an agent ^^^^^^^^may attempt to optimize its actions ^^^^ in order to optimize a reward. Hence, just like ordinary policy-gradient methods based on the REINFORCE technique, training is prone to suffering from a lack of diversity and therefore highly erroneous gradients.

[0096] The second estimator may keep the training process more analogous to supervised learning. In this case, the stable statistics collected from ^^^^ may provide a steady targetduring training. The gradient for^^^^^ ^^^^|^^^^^= may take the simpler form

[0097] Now referring to FIG.7A and FIG.7B, there are shown a pair of graphs depicting the discrepancy in angle for the Monte Carlo-estimated gradients compared to the exact gradient throughout the training process. The dataset consists of 1000 Clifford measurements performed on a three-qubit GHZ state. Estimation using neural state-based sampling (NS) and stabilizer-based sampling (SS) are depicted with circular and square markers, respectively.

[0098] Each estimation uses 500 Monte Carlo samples. The gradient errors are evaluated over (FIG. 7A) mini-batches and (FIG. 7B) a full batch. In the case of the mini-batch gradient error, the error bars correspond to the standard deviation in the error over all mini-batches. The graphs depict the error in the gradient direction as quantified by the angle between the gradient evaluated exactly and the gradient estimated with estimators using NS and SS. The errors are evaluated on samples from mini-batches and a full batch, as shown in FIG.7A and FIG.7B, respectively. The mini-batch plot shows the stochastic noise in the gradient estimations while the full-batch plot shows their systematic bias. In the numerical experiment for the three-qubit state GHZ state, the method relying on the stabilizer-based samples reduces both the variance and bias during the entire training process the progress of which is indicated by the training epoch index specified on the x- axis.

[0099] In some cases, the loss function of an ML model may comprise infidelity between the ansatz state and the classical shadow state.

[0100] The infidelity between the ansatz state and the classical shadow state from Cliffordensembles, 1 − ^^^^ ^^^^ = 1 −^^^^^ ^^^^|^^�^^|^^^^ ^^^^^, may be used to derive the loss functionℒCinf(^^^^)= −∑ ^^^^^^=^^1^^^^ ^^^^(^^^^ ^^^^),where ^^^^^^^^( ^^^^^^^^) =^^^^^^^^^|2is the probability of measuring the stabilizer ^^^^^^^^= ^^^^^^^^| ^^^^^^^^^.

[0101] In some cases, the same loss function may also be applied for the case of Pauliensembles as follows. For each Pauli ^^^^ ^^^^ = ^^^^ ^^^^,1 ⊗ ⋯⊗ ^^^^ ^^^^, ^^^^ and any bitstring ^^^^ ∈ ℬ ^^^^ ≡{0, 1} ^^^^, let | ^^^^| = ^^^^1 ^^^^ ^^^^be the number of ones in ^^^^. Let ^^^^ ^^^^, ^^^^ = ^^^^ ^^^^,1 ⊗ ⋯⊗ ^^^^ ^^^^, ^^^^ be theoperator that has an identity factor for each zero index in ^^^^. Let the stabilizer state bewhere the computational basisrepeats the bit in ^^^^^^^^for locations that are hot in the bitstring ^^^^ and uses the bit in bitstring ^^^^ in every other location. The resulting training loss function is

[0102] Unfortunately, the loss functions based on infidelity may not be generalizable to mixed states. To circumvent this drawback, the Kullback–Leibler (KL) divergence may be considered.

[0103] In some cases, the loss function of the ML model representative of the parameterized quantum state may comprise the KL divergence between the distribution corresponding to the one or more quantum measurements in a random basis and the distribution corresponding to the ansatz state. In some cases, the loss function of the ML model representative of the parameterized quantum state may comprise the KL divergence between the distribution corresponding to the ansatz state and the distribution corresponding to the one or more quantum measurements in a random basis.

[0104] The superscripts “P” and “C” in a given loss function standing for “Clifford” and “Pauli”, respectively, may be omitted, as the discussion that follows applies to both training datasets.

[0105] The Pauli or Clifford datasets may be viewed as samples drawn from a target probability measure ^^^^data: Stab → ℝ on the space of stabilizer states. For Pauli measurements, this is the product of single-qubit stabilizer states Stab⊗ ^^^^1 , and for Clifford measurements it is Stab^^^^, the full set of all n-qubit stabilizer states. The ML model (the neural network, the autoregressive model, or the like) representing the quantum state is a priori a generative model with explicit access to a probability distribution{0, 1}^^^^→ ℝ on the computational basis states. However, the efficiency of stabilizer state tableau representation allows it to be extended to a measureStab → ℝ on the stabilizer states.The KL divergence between dataand ^^^^ can be written as

[0106] In some cases, the entropy of the underlying data distribution may be constant with respect to the variational parameters ^^^^. Consequently, the loss function reduces tothe cross-entropy term

[0107] In some cases, the loss function may comprise cross-entropy. In some cases, the cross-entropy may be estimated using empirical data distribution. In some cases, the construction of the loss function may comprise computing a classical shadow data distribution. In some cases, the classical shadow data distribution may comprise probabilistic weights of the one or more quantum measurements computed based at least in part on the classical shadow state constructed using the classical shadow technique.

[0108] In some cases, ^^^^ ⊆ Stab may denote an independent identically distributed family of samples (allowing repetitions) drawn from ^^^^data, where ^�^^^ may denote the set of elements in ^^^^ (removing repetitions). The cross-entropy loss function may be approximated by the log-likelihood of ^^^^,which may be referred to as the empirical cross-entropy (ECE) loss function.

[0109] This is consistent with the notion that the optimal model should treat the observed samples as the most probable ones. In some cases, the loss function ℒECE(^^^^)may be viewed in light of the empirical distributionis an indicator function. Since ^^^^emp datais an unbiased estimator for ^^^^ , theloss function ℒECE(^^^^)may be rewritten as

[0110] The new summation runs over distinct elements of the training data without repetition. This reformulation helps alternative unbiased estimators to be considered for the data distribution. An approach based on the classical shadow state ^^�^^ may be considered. For a stabilizer state ^^^^, its shadow weight may be defined as ^^^^(^^^^)=|^^^^^|^^�^^|^^^^^|,where the absolute values are taken because ^ ^^^^| ^^�^^| ^^^^^ can take negative values, which prevents its being interpreted as a probability distribution. The normalized shadow weightsmay now replace the empirical averaging in the cross-entropy approximation, justifying the new loss functionwhich is referred to as the shadow-based cross-entropy (SCE) loss function. In contrast to the empirical cross-entropy loss function, which asserts equal weights on the contributions of each stabilizer state ^^^^, the new loss function for ℒSCE(^^^^)may leverage the classical shadow state to inject further signals useful for the training dynamics.

[0111] The loss function comprising the cross-entropy may be derived using the estimator ,

[0112] Now referring back to FIG.2 and according to processing step 210, the ML model is trained by updating its parameters using the gradient of the loss function. In some cases, the training may comprise pretraining using synthetic data.

[0113] In some cases, a fixed training dataset may be used for the ML model’s training.

[0114] In some cases, a validation dataset may be used to prevent overfitting in training. In some cases, a fraction of available snapshot states representative of the one or more quantum measurements are not used in the training to evaluate the gradient. Instead, they may be held off in a separate validation dataset. After each training step, the loss functionmay be evaluated using the validation dataset. The training may be considered complete once this value consistently increases (beyond statistical fluctuations).

[0115] In some cases, a training dataset is separated into mini-batches to calculate gradients.

[0116] The dataset is separated into mini-batches to calculate gradients instead of running through the entire dataset.

[0117] In some cases, the method disclosed with respect to FIG. 2 may be used for quantum state characterization. In some cases, the process of data collection may involve repeated preparation and measurement of a quantum state prepared on quantum hardware. In some cases, prior knowledge about this quantum state may be used to constrain the architecture of an ML model. The ML model may be trained according to the disclosed methods. In some cases, the ML model may be used to extract observables that may characterize the underlying quantum state.

[0118] In some cases, the method disclosed with respect to FIG. 2 may be used for quantum channel characterization. In some cases, quantum channel characterization may rely on quantum state characterization as a subroutine.

[0119] Obtaining accurate quantum states may be useful for quantum channel characterization. In some cases, after running quantum circuits, information about resultant quantum states may be obtained through projective measurements or positive operator-valued measurements. In some cases, measurements may be sampled from random ensembles. In some cases, these data may be used to train and validate neural shadow quantum states to obtain accurate quantum states. In some cases, neural quantum states may be used to compute quantities for quantum channel characterization. One example of a quantum channel characterization method is proposed in WO2024 / 150142. It is designed to detect and quantify crosstalk errors by using properties such as quantum discord, quantum correlation, entanglement, and other physical properties. In some cases, NSQST may provide accurate quantum states for computing these quantities. Another example of a quantum channel characterization method is proposed in PCT / US2024 / 018752. It is designed to accurately measure parameters in gate operations.

[0120] In some cases, the method disclosed with respect to FIG. 2 may be used for quantum error correction (QEC) or fault-tolerant quantum computation (FTQC). In some cases, methods of quantum state and channel characterization may be applied to QEC and FTQC. As part of their application, a quantum state of interest may be prepared by a quantum circuit relating to a QEC or FTQC protocol. In some cases, the quantum circuit may comprise magic-state preparation, fault-tolerant preparation of a logical state, fault- tolerant logical gates, or lattice surgery. In some cases, the information about these states may be characterized with the help of methods disclosed elsewhere herein. Scaling and Variance Study

[0121] Prior to studying the training of neural quantum states with the methods disclosed herein, the benefits of using shadow weights and stabilizer-based importance sampling were explored. To this end, a focus was placed on the key training metrics: the informational content and the statistical properties of the gradient estimator.

[0122] Now referring to FIG. 8A and FIG. 8B, there is shown a pair of graphs of the Kullback–Leibler (KL) divergence which compares the proxy for a typically unknown data distribution with empirical and shadow-weight distributions, both accessible from measurement data. The KL divergence is examined as a function of (FIG.8A) the system size for a dataset of 1000 shadows and (FIG.8B) the number of shadows for a six-qubit system. The measurement type is specified by the marker type: square for Clifford measurements and circle for Pauli measurements. The target distribution type is specified by the line styles: dashed and solid curves for the empirical distribution and shadow weights, respectively. The findings are averaged over 32 instances of pure states produced using random Clifford circuits, with error bars showing the standard deviation.

[0123] To facilitate an analysis, an empirical proxy for the data distribution ^^^^edmatpa(^^^^)∝ ^^^^data( ^^^^) ^^^^^^^^∈ ^�^^^was considered, whereis the indicator function that excludes unobserved stabilizers. This distribution represents the most-accurate reconstruction of the probability measure ^^^^datagiven a dataset ^^^^, assuming no prior knowledge about^^^^data. To assess the informational content in the shadow weights, the KL divergenceKL� ^^^^data targetemp� ^^^^ � was considered between this typically unknown distribution and^^^^target, which denotes either of the two empirically accessible distributions used fortraining: ^^^^empis based on the equation ^^^^empwhereas ^^^^shisspecified by the equation ^^^^shSpecifically, the dependence was studiedon the size of the dataset relative to the size of the system by scaling the number of qubits given a fixed dataset, as disclosed herein with respect to FIG.8A, and scaling the number of measurements in the dataset for a fixed number of qubits, as disclosed herein with respect to FIG.8B.

[0124] As disclosed herein with respect to FIG.8A and FIG.8B, the advantage of shadow weights in terms of estimating ^^^^edmatpais evident for smaller systems, as disclosed herein with respect to FIG. 8A, though it gradually diminishes as the systems grow larger. In contrast, FIG.8B indicates that this benefit can be reclaimed by increasing the number of measurements performed. A crucial factor in interpreting the plots disclosed herein with respect to FIG.8A and FIG.8B may be in considering the difference in the configuration space dimensions of ^^^^edmatpaassociated with Pauli and Clifford measurements. Notably, Pauli measurements can reconstruct 4^^^^distinct stabilizers, whereas Clifford measurements can reconstruct 20.5+ ^^^^(1) ^^^^2distinct stabilizers (S. Aaronson and et. al., “Improved simulation of stabilizer circuits” Physical Review A—Atomic, Molecular, and Optical Physics 70, 052328 (2004)). For n = 6 qubits, the dataset sizes considered in FIG. 8B were compared to the dimension of the stabilizer space. For the Pauli distribution, the dataset ranges from 5% to 500% of the total set size of distinct stabilizers. However, for the Clifford distribution, the maximum number of measurements is less than 0.001% of the corresponding stabilizer set size. This significant difference may explain the slight scaling advantage of the ^^^^empPauli measurements over the ^^^^empClifford measurements, as the former have a higher likelihood of obtaining repeated stabilizers, while the latter degenerate to a uniform distribution. For shadow weights, a scaling advantage over empirical distributions in both regimes is evident. However, unlike in the case of the empirical distributions, the Clifford distribution shows a slight improvement over the Pauli distribution. Therefore, given a sufficiently large measurement dataset, training with shadow weights-based cross-entropy may be expected to produce neural quantum states that better capture the underlying quantum state.

[0125] The advantages of the proposed stabilizer-based method represented by the equation ^ ^^^^^^^^| ^^^^^ = 〈^^^^∗^^^^ ( ^^^^)^^^^∗〉^^^^~ ^^^neural state-based method represented by the ^( ^^^^) ^ ^^^^equationevaluated for Monte Carlo estimation of the overlap,which was used to compute the training gradient. For benchmarking purposes, the errors in the gradient estimates relative to the true gradient, obtained through an exhaustiveevaluation of the inner product of the equation^^^^^ ^^^^|^^^^^wereanalyzed. Specifically, focus was placed on the error in the gradient direction and distinguishment made between errors calculated over a mini-batch and a full batch of measurements.

[0126] Shown herein with respect to FIGS. 7A and FIG. 7B are illustrations of the evolution of these errors as the neural quantum state was trained using the true gradient. FIG. 7A shows that both estimators struggle most with capturing the correct gradient direction during the early stages of training, where the neural quantum state is likely the most different from the target state. However, after several epochs, as the neural state becomes more similar to the target, the stabilizer-based estimation significantly improves, while the neural state estimation remains relatively high and inconsistent. Throughout the entire training process, the stabilizer-based estimator consistently provides estimates with much lower variance, which is essential for training stability. Additionally, the estimator proposed herein is consistently more accurate than the neural-based estimator. Being an unbiased estimator, the stochastic error due to mini-batching is expected to be cancelled out when averaged over the mini-batches and thus can be lowered using smaller learning rates, albeit at the cost of slower training dynamics. FIG.7B shows whether this expected behaviour holds for the given estimators. There is a clear difference between them: the stabilizer-based estimator functions as an unbiased estimator, reducing gradient errors by averaging over mini-batches. In contrast, the neural state-based estimator exhibits a systematic bias throughout the entire training duration, resulting in an extreme accumulation of errors from the true gradient. To eliminate this finite-sample bias, a significant increase in the samples generated by the neural state may be necessary. Therefore, given a fixed computational resource budget, the benchmarking results disclosed herein demonstrate that the stabilizer-based gradient estimation has a significant advantage over the neural state-based gradient estimation, highlighting its potential foroptimization. Benchmarking Loss Functions on Pure GHZ States

[0127] In some cases, the target quantum state to be reconstructed may be a pure Greenberger–Horne–Zeilinger (GHZ) state, wherein the reconstruction may be from Pauli or Clifford measurements. In some cases, the shadow-based cross-entropy (ℒSCE), the empirical cross-entropy (ℒECE), and the infidelity (ℒinf) may be used as the loss functions.In some cases, the estimatorsmay beemployed in evaluating the overlap between the neural network (ansatz state) and stabilizer states. The superscripted abbreviations “NS” (for neural state-based samples) and “SS” (for stabilizer-based samples) appear with the corresponding loss function ℒ to distinguish between the two estimators.

[0128] Now referring to FIG. 4A-FIG. 4F, there are shown a group of graphs of the infidelity and infidelity loss, along with the cross-entropy loss, between a reconstructed state and the actual state for four methods in reconstructing both six- and eight-qubit GHZ states. The results are averaged over five training trials on the same dataset trained over 50 epochs. Gradient estimation is based on 500 Monte Carlo samples. The final states for each loss function and trial are represented by markers. Triangle and square markers represent the samples from the last epoch for the stabilizer- and neural state-based infidelity loss functions, respectively. Circle and “x” markers represent the samples from the last epoch for the empirical and shadow based cross-entropy loss functions, respectively. The upper row displays the loss functions while the lower row shows the associated infidelity with respect to the true state as a measure of the generalization power of the neural quantum state.

[0129] The experiments were conducted using 1000 classical shadows based on Pauli or Clifford measurements (as specified in the titles above the plots) and averaging the results over five training trials on the same dataset trained over 50 epochs. Each estimate of the overlap between a neural quantum state and a stabilizer is based on 500 newly generated samples. The final states for each loss function and trial are represented by markers. Square and triangle markers represent the samples for the stabilizer- and neural state-based infidelity loss functions, respectively. Circle and “x” markers represent the samples for the empirical and shadow-based entropy loss functions, respectively. The upper row displays the infidelity loss (left y-axis) and the cross-entropy loss (right y-axis) for the loss functions, while the lower row shows the associated infidelity with respect to the true state as a measure of the generalization power of the neural quantum state. The cross- entropy loss is normalized to be below one in order to facilitate a visual comparison with the infidelity loss.

[0130] FIG. 4A – FIG. 4F show the trainability and the generalization power of these training methods via the optimization progress curves of loss functions evaluated on the training set (upper row) and infidelity with respect to the true states (lower row), over 50 epochs. For Pauli measurements, shown in FIG.4A and FIG.4B, the superiority of cross- entropy-based training over the infidelity-based approach is apparent. The latter struggles to achieve significant progress in approximating the underlying quantum state. Moreover, the benefits of the cross-entropy-based loss function become even more pronounced when considering computational costs. Both loss functions rely on computing the overlap between the neural quantum state and stabilizers. For the infidelity loss function, each Pauli measurement leads to a proliferation of stabilizers that grows super-exponentially with the number of qubits. Using a Monte Carlo approach to address this prohibitive scaling of the infidelity-based approach through sampling of stabilizers from a classical shadow state leads to unstable training. In contrast, this prohibitive scaling is entirely absent for both cross-entropy-based loss functions, as each Pauli measurement contributes a single stabilizer. Additionally, stabilizer-based sampling improves in-training performance across all loss functions (the results for neural state-based cross-entropy loss functions have been omitted for visual clarity). Moreover, these improvements during training translate to enhanced generalization only for cross-entropy loss functions. In fact, for the cross-entropy loss function, stabilizer-based training proves to be crucial.

[0131] For Clifford measurements, the benchmarking performed on the six-qubit GHZ state is shown in FIG. 4C and FIG. 4D. Observations regarding the advantages of stabilizer-based sampling generalize to this scenario and, in fact, are applicable to the infidelity loss function as well. Indeed, the loss function ℒiSnSf leads to an improved generalization with the final infidelity of roughly an order of magnitude compared to thatof the loss function as shown in FIG. 4D. This improvement in generalization is achieved despite their corresponding loss functions yielding similar values at the end of training. Further examination of FIG.4C and FIG.4D reveals that the loss function ℒESSCEgreatly outperforms the previously proposed shadow-based loss functionin both the convergence rate and generalization. In addition, a comparison between the loss functions ℒSSCSE and ℒESSCEreveals that the cross-entropy estimator leads to a superior convergence rate and lower infidelity, illustrating the benefits of shadow weights extracted from theclassical shadow state ( ^^^^sh(^^^^)=Overall, both of the loss functionsandℒiSnSf demonstrate the best convergence rates. The loss functions ℒSSCSE and ℒESSCEexhibit the greatest generalization, approaching an infidelity of 0.01, while the estimator disclosed herein shows the greatest stability in reaching states in proximity to the true states. An even more drastic difference in performance is observed for the eight-qubit GHZ state, shown in FIG.4E and FIG.4F. In this case, the loss functionsSS SSℒinf, and ℒECEstruggle to escape local minima. While all the loss functions result in infidelities that remain stuck at 0.99, only the loss function ℒSSCSE stands out, achieving an infidelity as low as 0.05. The benchmarking results indicate that the methods disclosed herein provide the best training objective for neural networks trained to approximate an underlying quantum state from limited measurements. Applications on Mixed GHZ States

[0132] In some cases, the target quantum state to be reconstructed may be a mixed quantum state generated by noisy circuits, wherein the reconstruction is from Pauli or Clifford measurements. A circuit that prepares a six-qubit GHZ state beginning from the all-zeroes state|0^⊗6may be considered. The circuit may comprise a Hadamard gate applied to the first qubit, followed by a series of CNOT gates applied to the consecutive pairs of qubits. For the noise model, two-qubit depolarization noise may be assumed on the CNOT gates. The depolarizing noise channel ^^^^ ↦�1 −^^^^is parameterized via a noise strength parameter ^^^^. In some cases, Pauli measurement datasets, which are less noisy than Clifford measurements, may be considered. In some cases, as the ansatz for the density matrix, the purification ansatz ^^^^^^^^( ^^^^1, ^^^^2) =with a transformer neural network used to parameterize ^^^^ ^^^^(^^^^, ^̅^^^)may be employed. This approach allows physical constraints to be imposed on the ansatz. Pauli Observables

[0133] Although classical shadows based on Pauli measurements predict local observables with high accuracy, their predictive accuracy may worsen in the case of many-body observables. The benefits of the neural network approach were demonstrated over classical shadows in predicting Pauli observables. To do so, a set of 5000 random Pauli strings was generated by sampling each single-qubit Pauli operator from a uniform distribution.

[0134] Now referring to FIG. 5A and FIG. 5B, there are shown a pair of graphs of the predictive capabilities based on neural quantum states, classical shadows, and the simplex projection of the latter, for noisy observables of a mixed GHZ state prepared using a series of noisy gates with noise levels p ranging from 0.0 to 0.5.

[0135] The dataset used for reconstructions contains 5000 random Pauli measurements. FIG. 5A shows absolute error, ^, in predicting random Pauli observables with standard deviations represented by error bars. FIG.5A shows accuracy, defined as the average ofthe absolute error over the standard deviation,�^^^�^ / ^�^^^. The results are differentiated bymarker style, with neural quantum states shown using squares, classical shadows using circles, and the simplex projections of classical shadows using triangles.

[0136] The absolute error in predicting Pauli observables from neural networks and classical shadows is disclosed herein with respect to FIG.5A. The precision and accuracy of estimates extracted from the classical shadow technique become larger as the Pauli weight increases. This poor performance in predicting high-weight observables may be alleviated by projecting a classical shadow state to a physical state on the probability simplex. Despite this adjustment, the neural state ansatz consistently demonstrates superior performance across all noise levels. This observation is further supported when focusing solely on prediction accuracy, as shown herein with respect to FIG.5B, which comprises the same dataset, now normalized by variance and averaged across all noise levels.Purity and Trace Distance

[0137] Now referring to FIG. 6A-FIG. 6E, there are shown a group of graphs of a performance comparison of reconstructed six-qubit GHZ mixed states using classical shadows and neural quantum states collected from 5000 random Pauli measurements. FIG.6A shows theoretical values of purity for the ideal state (shown using a solid curve), along with the reconstructed states from neural quantum states (shown using square markers), classical shadows (shown using circles), and the simplex projections of classical shadows (shown using triangles). FIG.6B shows trace distance with respect to the ideal state. FIG. 6C-FIG. 6E show values of real components of the density matrix elements for the noise level ^^^^ = 0.5, comparing the ideal case (FIG.6C), the neural quantum state (FIG. 6D), and the classical shadow state (FIG. 6E). The grey-scale bar as well as the height of each data point indicates the magnitude of each component of the density matrix.

[0138] Assessments of purity and trace distance, as shown herein with respect to in FIG. 6A and FIG.6B, respectively, may be considered. In contrast to the behavior observed in the case of Pauli observables, projecting the classical shadow state onto the probability simplex leads to a degradation in purity estimation, as shown in FIG.6A. However, both neural quantum states and classical shadow states are able to accurately track theoretically predicted results. The distinguishing feature of the physically constrained neural quantum state lies in its ability to consistently provide physical estimates, whereas the classical shadow state’s estimate falls below zero for the least pure target state considered. The observed trends of the trace distance shown in FIG.6B align with the results of the Pauli observables: the simplex projection demonstrates successful improvement over the classical shadow state, while the neural quantum state exhibits the lowest error for all noise levels.

[0139] Relatively weaker performance of the neural quantum states in the low noise regime, specifically ^^^^ = 0.1 and 0.2, was observed. This behavior was investigated further through the production of visualizations of density matrices, as shown in FIG.6C, FIG.6D, and FIG.6E for ^^^^ = 0.5. For such high noise levels, all non-zero components in the theoretical density matrix are substantial. In this case, the neural quantum state can successfully capture the full structure of the theoretical density matrix, whereas the classical shadow state is extremely noisy. However, at lower noise regimes, most of thediagonal elements of the density matrix decrease, and only the corner components remain large. The neural quantum state seems to struggle with capturing signals from very small components, possibly due to the Monte Carlo sampling noise present during training, leading to the relative weakening of its performance.

[0140] While preferred embodiments of the present invention have been shown and described herein, it will be obvious to those skilled in the art that such embodiments are provided by way of example only. It is not intended that the invention be limited by the specific examples provided within the specification. While the invention has been described with reference to the aforementioned specification, the descriptions and illustrations of the embodiments herein are not meant to be construed in a limiting sense. Numerous variations, changes, and substitutions will now occur to those skilled in the art without departing from the invention. Furthermore, it shall be understood that all aspects of the invention are not limited to the specific depictions, configurations, or relative proportions set forth herein which depend upon a variety of conditions and variables. It should be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. It is therefore contemplated that the invention shall also cover any such alternatives, modifications, variations, or equivalents. It is intended that the following claims define the scope of the invention and that methods and structures within the scope of these claims and their equivalents be covered thereby.

Claims

CLAIMS:

1. A method for a quantum state reconstruction, said method comprising: (a) obtaining one or more quantum measurements; (b) constructing a classical shadow state based at least in part on said one or more quantum measurements and using a classical shadow technique; (c) constructing a loss function of a machine learning (ML) model representative of a parameterized quantum state based at least in part on said classical shadow state, wherein said loss function comprises an overlap between a snapshot state representative of said one or more quantum measurements and an ansatz state provided by said ML model; (d) estimating a gradient of said loss function based at least in part on samples from said classical shadow state; and (e) training said ML model based at least in part on using said gradient of said loss function to update one or more parameters of said ML model.

2. The method of claim 1, wherein said one or more quantum measurements comprise a tomographically complete set of quantum measurements.

3. The method of claim 1, wherein said loss function comprises cross-entropy and (c) comprises computing a classical shadow data distribution which is used to construct said loss function.

4. The method of claim 1, wherein said loss function of said ML model representative of said parameterized quantum state comprises the Kullback–Leibler (KL) divergence between the distribution corresponding to said one or more quantum measurements in a random basis and the distribution corresponding to said ansatz state; or KL divergence between the distribution corresponding to said ansatz state and the distribution corresponding to said one or more quantum measurements in a random basis.

5. The method of claim 1, wherein said loss function of said ML model comprises an indication of infidelity between said ansatz state and said classical shadow state.

6. The method of claim 1, wherein (d) comprises Monte Carlo (MC) sampling from said classical shadow state.

7. The method of claim 1, wherein said snapshot state comprises a stabilizer state.

8. The method of claim 7, wherein (c) and (d) comprise evaluating said overlap between said stabilizer state and said ansatz state; wherein said overlap is evaluated based at least in part on samples from said stabilizer state.

9. The method of claim 1, wherein a validation dataset is used to prevent overfitting in training.

10. The method of claim 1, wherein a training dataset is separated into mini-batches to calculate gradients.

11. The method of claim 1, wherein said loss function comprises cross-entropy that is estimated using an empirical data distribution.

12. The method of claim 3, wherein said classical shadow data distribution comprises probabilistic weights of said one or more quantum measurements computed based at least in part on said classical shadow state constructed using said classical shadow technique.

13. The method of claim 1, wherein said training comprises pretraining using synthetic data.

14. The method of claim 1, further comprising characterizing a quantum state based at least in part on said ML model.

15. The method of claim 1, further comprising characterizing a quantum channel based at least in part on said ML model.

16. The method of claim 1, wherein said ML model comprises one or more members of the group consisting of: an autoregressive model, a neural network, a tensor network, and a generative model.

17. The method of claim 1, wherein said one or more quantum measurements comprises a Pauli measurement or a Clifford measurement.

18. The method of claim 1, wherein said quantum state comprises a pure quantum state or a mixed quantum state.

19. The method of claim 18, wherein said mixed quantum state is purified into a pure state using auxiliary qubits.

20. A hybrid system for a quantum state reconstruction, said system comprising: (a) a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system; and (b) a high-performance computing platform operatively coupled to said quantum computer, said high-performance computing platform comprising one or more computing components capable of performing computational tasks in parallel, wherein said high-performance computing platform is configured to at least: (i) obtain one or more quantum measurements from said quantum computer; (ii) construct a classical shadow state using a classical shadow technique based at least in part on said one or more quantum measurements; (iii) construct a loss function of an ML model representative of a parameterized quantum state; (iv) estimate the gradient of said loss function; and (v) train said ML model by updating one or more parameters of said ML model.

21. The hybrid system of claim 20, wherein said high-performance computing platform is operatively coupled to said quantum computer using a digital computer.

22. The hybrid system of claim 20, further comprising a cloud-based computing platform.

23. The hybrid system of claim 22, wherein said hybrid system provides computational services on the cloud.

24. The hybrid system of claim 20, wherein said hybrid system is configured to operate in a distributed computing environment.

25. A system for a quantum state reconstruction, said system comprising: (a) a high-performance computing platform operatively coupled to a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system, said high-performance computing platform comprising one or more computing components capable of performing computational tasks in parallel, wherein said high-performance computing platform is configured to at least: (i) obtain one or more quantum measurements from said quantum computer; (ii) construct a classical shadow state using a classical shadow technique based at least in part on said one or more quantum measurements; (iii) construct a loss function of an ML model representative of a parameterized quantum state; (iv) estimate the gradient of said loss function; and (v) train said ML model by updating one or more parameters of said ML model.

26. The system of claim 25, wherein said high-performance computing platform is operatively coupled to said quantum computer using a digital computer.

27. The system of claim 25, further comprising a cloud-based computing platform.

28. The system of claim 27, wherein said hybrid system provides computational services on the cloud.

29. The system of claim 25, wherein said hybrid system is configured to operate in a distributed computing environment.

30. A system for a quantum state reconstruction, said system comprising: (a) a quantum computer comprising a quantum chip comprising a plurality of qubits and a control system, wherein said quantum computer is operatively coupled to a high-performance computing platform, said high-performance computing platform comprising one or more computing components capable of performing computational tasks in parallel, wherein said high-performance computing platform is configured to at least: (i) obtain one or more quantum measurements from said quantum computer; (ii) construct a classical shadow state using a classical shadow technique based at least in part on said one or more quantum measurements; (iii) construct a loss function of an ML model representative of a parameterized quantum state; (iv) estimate the gradient of said loss function; and (v) train said ML model by updating one or more parameters of said ML model.

31. The system of claim 30, wherein said high-performance computing platform is operatively coupled to said quantum computer using a digital computer.

32. The system of claim 30, further comprising a cloud-based computing platform.

33. The system of claim 32, wherein said hybrid system provides computational services on the cloud.

34. The system of claim 30, wherein said hybrid system is configured to operate in a distributed computing environment.

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