Method and system for improving estimation of properties of quantum states

Abstract

A method for improving an estimate of a property of a quantum state may include (a) receiving, using a digital computer interface, indicia of (i) a property of the quantum state to be estimated, (ii) at least one quantum device, and (iii) at least one computational platform. The method may include obtaining a plurality of measurements of the quantum state using the at least one quantum device. The method may include constructing a neural network using the at least one computational platform and training the neural network using the plurality of measurements, the neural network including at least one trainable parameter, the neural network representing the quantum state. The method may include training at least one trainable parameter of the neural network using the at least one computational platform and the property of the quantum state to variationally improve the quantum state.

Description

[Technical Field]

[0001] Background technology New noisy intermediate-scale quantum (NISQ) devices are being developed, improved, and marketed. While these devices can perform a variety of tasks, such as optimization and stochastic sampling, they can lack precision. Furthermore, the physical wiring on the qubit chip can limit the connectivity between qubits. Summary of the Invention

[0002] At least some of these difficulties can be alleviated by using hybrid quantum-classical optimization algorithms, such as variational quantum computing. However, hybrid algorithms may not be tolerant to decoherence and gate errors, which may lead to inaccurate expectation value estimates. Furthermore, the ansatz in variational quantum computing are usually not general-purpose, and therefore variational quantum computing may result in an approximation to the target state. Another drawback of this method is that classical optimization of variational quantum computing parameters is a complex problem and may lead to suboptimal rather than optimal parameters.

[0003] It is recognized herein that there is a need for methods and systems that overcome the limitations associated with the precision of such devices and experiments.

[0004] The present disclosure provides methods and systems for improving estimation of properties of quantum states. In some cases, the methods and systems disclosed herein can be used to mitigate errors in neural network representations of quantum states of quantum systems. In some cases, the methods and systems disclosed herein can improve estimation of properties of quantum states. In some cases, the methods and systems disclosed herein can be applied to a variety of quantum devices. In some cases, the methods and systems disclosed herein can be applied to a variety of quantum experiments and quantum computations. In some cases, the methods and systems disclosed herein can utilize a variety of neural networks. In some cases, reconstructing a state using tomography with a neural network can enable preservation of states prepared by a quantum circuit. Creating a neural network wave function from imperfect measurements can enable long-lived states outside of experiments.

[0005] An advantage of the methods and systems disclosed herein is that they can be used to mitigate errors in neural network representations of quantum states of quantum systems.

[0006] Another advantage of the methods and systems disclosed herein is that they improve the estimation of properties of quantum states.

[0007] Another advantage of the methods and systems disclosed herein is that they can be applied to a variety of quantum devices.

[0008] Another advantage of the methods and systems disclosed herein is that they are applicable to a variety of quantum experiments and a variety of quantum computations.

[0009] Another advantage of the methods and systems disclosed herein is that they can utilize a variety of neural networks.

[0010] Another advantage of the methods and systems disclosed herein is that they allow for the preservation of states prepared by quantum circuits by using tomography with neural networks to reconstruct the states. The advantage of creating neural network wave functions from imperfect measurements allows for long-lived states outside of experiments.

[0011] Another advantage of the methods and systems disclosed herein is that in some embodiments, for example, where the properties are properties of quantum states of a parameterized Hamiltonian, the properties of the basis states can be estimated from the quantum states of the neural network for any value of the parameters, not just those used for training.

[0012] Another advantage of the methods and systems disclosed herein is that neural networks can be constructed that represent a continuous family of quantum states. The quantum states of a parameterized Hamiltonian can be represented using a limited number of parameter values, thereby allowing for the long-lived existence of an infinite number of related quantum states.

[0013] Aspects of the present disclosure provide a method for improving an estimate of a property of a quantum state, which may include: (a) receiving, using a digital computer interface, indications of (i) a property of a quantum state to be estimated, (ii) at least one quantum device, and (iii) at least one computational platform, using the at least one quantum device to obtain a plurality of measurements of the quantum state, and using the at least one computational platform to construct and train a neural network using the plurality of measurements, the neural network including at least one trainable parameter, the neural network representing the quantum state, (d) training the at least one trainable parameter of the neural network using the properties of the at least one computational platform and the quantum state to variationally improve the quantum state represented by the neural network, and (e) providing an estimate of the property of the quantum state at the interface.

[0014] In some embodiments, the method further includes repeating steps (a)-(d) until a stopping criterion is met. In some embodiments, step (a) further includes receiving an index of a set of measurement operators, and step (b) further includes (i) experimentally preparing an approximation of the quantum state using a quantum experiment, (ii) selecting a measurement operator from the set of measurement operators, and (iii) performing a measurement of the prepared quantum state using the operator selected from the set of measurement operators, until a stopping criterion is met. In some embodiments, (i) further includes applying at least one unitary transformation to the initial state.

[0015] In some embodiments, the neural network further includes a cost function. In some embodiments, step (c) includes (i) using the plurality of measurements to provide input to the neural network, (ii) calculating values ​​of a cost function for the neural network, (iii) calculating a gradient of the cost function with respect to the at least one trainable parameter of the neural network, (iv) using the calculated gradient and the calculated cost function to update the at least one trainable parameter of the neural network, and (v) repeating (i)-(v) multiple times. In some embodiments, a regularization term is added to the cost function.

[0016] In some embodiments, step (d) comprises (i) sampling at least one configuration using the neural network, (ii) using the sampled at least one configuration to estimate a variational energy of the wave function represented by an average of local energies, (iii) using the sampled at least one configuration to estimate a gradient of the variational energy with respect to the at least one parameter of the neural network, (iv) using the estimated variational energy and the estimated gradient of the variational energy to update at least one parameter of the neural network, and (v) repeating (i)-(iv) until a stopping criterion is met. In some embodiments, a regularization term is added to the variational energy of the wave function.

[0017] In some embodiments, the quantum experiment comprises quantum computing. In some embodiments, the quantum computing comprises at least one of circuit-model quantum computing, quantum annealing, measurement-based quantum computing, and adiabatic quantum computing. In some embodiments, the at least one quantum device comprises at least one of a quantum annealer, a trapped-ion quantum computer, an optical quantum computer, a photonic quantum computer, a spin quantum dot computer, and a superconductor quantum computer.

[0018] In some embodiments, the quantum state comprises a ground state of a Hamiltonian. In some embodiments, the quantum computation comprises solving an optimization problem, and further, the quantum state comprises a ground state of a Hamiltonian. In some embodiments, the Hamiltonian represents a classical optimization problem. In some embodiments, the ground state of the Hamiltonian represents an optimal solution to the optimization problem.

[0019] In some embodiments, step (b) comprises performing a variational quantum computing procedure, which comprises: (i) obtaining an initial state; (ii) using a quantum processor including a layer of parameterized quantum gates, preparing a multi-qubit quantum state by evolving the initial state through the layer of parameterized quantum gates; (iii) calculating a variational energy of the prepared multi-qubit quantum state; (iv) updating the parameters of the parameterized quantum gates using a classical optimization algorithm to minimize the variational energy; (v) repeating (i)-(v) a number of times; and (vi) providing the resulting quantum state.

[0020] In some embodiments, the quantum computation comprises a quantum chemistry simulation, and the quantum state is a quantum state of a Hamiltonian representing a quantum chemistry problem. In some embodiments, the Hamiltonian comprises an electronic structure Hamiltonian of one of a molecule and a material. In some embodiments, the property of the quantum state comprises an observable of the quantum state. In some embodiments, the observable of the quantum state is an expectation value of the energy of the quantum state.

[0021] In some embodiments, the neural network comprises at least one of an autoregressive model, a recurrent neural network, a transformer, an autoregressive generative model, an attention-based architecture, a dense deep neural network, a convolutional neural network, a variational autoencoder, a generative adversarial network, a restricted Boltzmann machine, a generalized Boltzmann machine, an energy-based model, a reversible neural network, and a flow-based generative model. In some embodiments, step (d) comprises using at least one of a tensor network assumption, a Jastrow-type wave function, and a Hartree-Fock-type wave function.

[0022] In some embodiments, step (c) includes using at least one of a tensor processing unit (TPU), a graphics processing unit (GPU), a field-programmable gate array (FPGA), and an application-specific integrated circuit (ASIC). In some embodiments, the quantum state is a quantum state of a parameterized Hamiltonian, and further, the parameterization of the parameterized Hamiltonian is continuous. In some embodiments, the neural network further receives parameter values ​​of the parameterization as input. In some embodiments, step (e) includes neural network inference to estimate properties of the quantum state of the parameterized Hamiltonian using parameter values ​​not used during training.

[0023] Another aspect of the present disclosure provides a system for improving estimation of properties of quantum states, the system comprising: (a) a digital computer including an interface, a memory including instructions, the digital computer configured to execute the instructions to receive at least (i) a property of a quantum state to be estimated, (ii) a set of measurement operators, (iii) at least one quantum device among a plurality of quantum devices, and (iv) an indication of at least one computational platform among a plurality of platforms, the digital computer further configured to provide an estimate of the property of the quantum state at the interface; and (b) at least one quantum device operably connected to the digital computer, the at least one quantum device including at least a quantum processor and a readout control system, the at least one quantum device configured to perform a quantum experiment and to perform a quantum experiment using the readout control system. and (c) the at least one computational platform operably connected to the digital computer, the at least one computational platform including at least one processor and a readout control system, the at least one computational platform configured to (i) receive from the digital computer a configuration of a neural network including at least one trainable parameter, the plurality of measurements, and the property of the quantum state, (ii) train the neural network representing the quantum state, and (iii) train the at least one trainable parameter of the neural network to variationally improve the quantum state represented by the neural network.

[0024] In some embodiments, the computing platform includes at least one component from the group consisting of a field-programmable gate array (FPGA), an application-specific integrated circuit (ASIC), a central processing unit (CPU), a graphics processing unit (GPU), a tensor processing unit (TPU), and a tensor streaming processor (TSP).

[0025] In another aspect, the present disclosure provides a method for mitigating errors in estimating properties of a quantum state. The method may include: (a) receiving a set of measurements of a quantum state from a quantum device; (b) preparing a representation of the quantum state using a computational platform and the set of measurements, the representation including a neural network including one or more tunable parameters; and (c) training the neural network using the computational platform by adjusting the one or more tunable parameters, the training including variational analysis, wherein the training mitigates errors in the estimation of the properties of the quantum state.

[0026] In some embodiments, the training comprises a variational Monte Carlo procedure. In some embodiments, the variational Monte Carlo procedure comprises a neural network representing a hypothetical ground state wave function. In some embodiments, the variational Monte Carlo procedure may comprise one or more of a tensor network hypothesis, a Jastrow-type wave function, or a Hartree-Fock-type wave function.

[0027] Another aspect of the present disclosure provides a system including one or more computer processors and a computer memory coupled thereto, the computer memory including machine-executable code that, when executed by the one or more computer processors, performs any of the methods described above or elsewhere herein.

[0028]

[0013] Further aspects and advantages of the present disclosure will become readily apparent to those skilled in the 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 its several details are capable of modifications in various obvious respects, all without departing from the present disclosure. Accordingly, the drawings and description are to be regarded as illustrative in nature, and not as restrictive.

[0029] Citation by reference All publications, patents, and patent applications mentioned herein are 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 that the publications, patents, or patent applications incorporated by reference conflict with the disclosure contained herein, the present specification is intended to supersede and / or supersede such conflicting material. [Brief explanation of the drawings]

[0030] 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 referred to herein as "Figure" and "FIG."). [Figure 1] 1 is a flowchart illustrating an example of a method for improving the estimation of properties of a quantum state. [Figure 2] 1 is a flow chart illustrating an example of a method for obtaining multiple measurements of a quantum state. [Figure 3] 1 is a flowchart illustrating an example of a method for constructing and training a neural network that includes at least one trainable parameter that represents a quantum state. [Figure 4]1 is a flowchart illustrating an example of a method for training at least one trainable parameter of a neural network to variationally improve the quantum state represented by the neural network. [Figure 5] 1 is a flowchart illustrating an example of a method for implementing a variational quantum computing procedure. [Figure 6] 1 is a flowchart illustrating an example of a method for preparing a multi-qubit quantum state and obtaining multiple measurements thereof. [Figure 7] FIG. 1 is a diagram of an example system for improving estimation of properties of quantum states. [Figure 8] We present error mitigation results using variational Monte Carlo for the spin N=8 lattice Schwinger model for mass values ​​m up to [-1.8,1.0]. DETAILED DESCRIPTION OF THE INVENTION

[0031] While various embodiments of the present invention have been shown and described herein, it will be apparent to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions will occur to those skilled in the art without departing from the invention. It is understood that various alternatives to the embodiments of the invention described herein may be employed.

[0032] 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. References to "or" herein are intended to encompass "and / or" unless stated otherwise.

[0033] The term "plurality" means "two or more" unless expressly specified otherwise.

[0034] The term "herein" means "in this application, including anything that may be incorporated by reference," unless expressly stated otherwise.

[0035] The term "eg" and similar terms mean "for example," and thus do not limit the term or phrase that it describes. For example, in the sentence "a computer sends data (e.g., instructions, data structures) over the Internet," the term "eg" explains that "instructions" are an example of "data" that a computer can send over the Internet, and that "data structures" are an example of "data" that a computer can send over the Internet. However, both "instructions" and "data structures" are merely examples of "data," and things other than "instructions" and "data structures" can also be "data."

[0036] When values ​​are stated as ranges, such disclosure includes disclosure of all possible subranges of such ranges and specific numerical values ​​falling within such ranges, whether or not a specific numerical value or specific subrange is explicitly stated.

[0037] In the following detailed description, reference is made to the accompanying drawings, which form a part of this specification. In the drawings, like symbols generally identify like elements, unless context dictates otherwise. The exemplary embodiments described in the detailed description, figures, and claims are not intended 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 aspects of the present disclosure, as generally described herein and illustrated in the figures, can be arranged, substituted, combined, separated, and designed in a variety of different configurations, all of which are expressly contemplated herein.

[0038] As used herein, the term "classical," when used in the context of computing or computation, generally refers to computation performed with binary values ​​of discrete bits, without quantum mechanical superposition or quantum mechanical entanglement. A classical computer may be a digital computer, such as a computer that employs discrete bits (e.g., 0 and 1), without quantum mechanical superposition or quantum mechanical entanglement.

[0039] As used herein, the term "non-classical," when used in the context of computing or computation, generally refers to any method or system for performing computational procedures outside the paradigm of classical computing.

[0040] As used herein, the term "quantum device" generally refers to any device or system that performs computations using any quantum mechanical phenomenon, such as quantum mechanical superposition or quantum mechanical entanglement.

[0041] As used herein, the terms "quantum computation," "quantum procedure," "quantum operation," and "quantum computer" generally refer to any method or system for performing computations using quantum mechanical operations (such as unitary transformations or completely positive trace-preserving (CPTP) maps on quantum channels) in a Hilbert space represented by quantum devices.

[0042] As used herein, the term "noisy intermediate-scale quantum device" (NISQ) generally refers to any quantum device that can perform tasks that exceed the capabilities of today's classical digital computers.

[0043] The present disclosure discloses methods and systems for using quantum devices to improve the estimation of properties of quantum states prepared using quantum experiments.

[0044] Neither the title nor the Abstract should be construed in any way as limiting the scope of the disclosed invention. The title of this application and the section headings within this application are provided merely for convenience and should not be construed as limiting the application in any way.

[0045] NISQ - Noisy Intermediate Scale Quantum Technology The term Noisy Intermediate-Scale Quantum (NISQ) was coined by John Preskill in "Quantum Computing in the NISQ era and beyond," arXiv:1801.00862, the entirety of which is incorporated herein by reference. Here, "noisy" implies imperfect control of qubits, and "intermediate-scale" refers to qubit counts ranging from 50 to several hundred. Several physical systems made from superconducting qubits, artificial atoms, and trapped ions have been proposed as viable candidates for building NISQ quantum devices and ultimately universal quantum computers.

[0046] quantum devices Any type of quantum computer may be suitable for the techniques disclosed herein. In accordance with the description herein, suitable quantum computers include, by way of non-limiting example, superconducting quantum computers (qubits implemented as small superconducting circuits—Josephson junctions) (Clarke, John, and Frank K. Wilhelm. “Superconducting quantum bits.” Nature 453.7198 (2008): 1031); trapped-ion quantum computers (qubits implemented as states of trapped ions) (Kielpinski, David, Chris Monroe, and David J. Wineland. “Architecture for a large-scale ion-trap quantum computer.” Nature 417.6890 (2002): 709); and optical lattice quantum computers (qubits implemented as states of neutral atoms trapped in an optical lattice) (Deutsch, Ivan H., Gavin K. Brennen, and Poul S. Jessen. “Quantum computing with neutral atoms in an optical lattice.” arXiv preprint quant-ph / 0003022 (2000)); spin-based quantum dot computers (qubits implemented as the spin states of trapped electrons) (Imamog, A., David D. Awschalom, Guido Burkard, David P. DiVincenzo, Daniel Loss, M. Sherwin, and A. Small. “Quantum information processing using quantum dot spins and cavity QED.” arXiv preprint quant-ph / 9904096 (1999)); space-based quantum dot computers (qubits implemented as the position of an electron in a double quantum dot) (Fedichkin, Leonid, Maxim Yanchenko, and KA Valiev.“Novel coherent quantum bit using spatial quantization levels in semiconductor quantum dot.” arXiv preprint quant-ph / 0006097 (2000)); coupled quantum wires (qubits implemented as paired quantum wires coupled by quantum point contacts) (Bertoni, A., Paolo Bordone, Rossella Brunetti, Carlo Jacoboni, and S. Reggiani. “Quantum logic gates based on coherent electron transport in quantum wires.” Physical Review Letters 84, no. 25 (2000): 5912.); nuclear magnetic resonance quantum computers (qubits implemented as atomic nuclear spins probed with radio waves) (Cory, David G., Mark D. Price, and Timothy F. Havel. “Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum computing.” arXiv preprint quant-ph / 9709001(1997)); solid-state NMR Kane quantum computer (qubits implemented as nuclear spin states of Lindner in silicon) (Kane, Bruce E. “A silicon-based nuclear spin quantum computer.” Nature 393, no. 6681 (1998): 133.); Electron-on-Helium quantum computer (qubits implemented as electron spin) (Lyon, Stephen Aplin. “Spin-based quantum computing using electrons on liquid helium.”” arXiv preprint cond-mat / 0301581 (2006)); quantum computers using cavity quantum electrodynamics (qubits are implemented as trapped atomic states coupled to high-finesse cavities) (Burell, Zachary. “An Introduction to Quantum Computing using Cavity QED concepts.” arXiv preprint arXiv:1210.6512 (2012)); quantum computers using molecular magnets (qubits are implemented as spin states) (Leuenberger, Michael N., and Daniel Loss. “Quantum computing in molecular magnets.” arXiv preprint cond-mat / 0011415 (2001)); ESR quantum computers using fullerenes (qubits are implemented as electron spins of atoms or molecules wrapped in fullerenes) (Harneit, Wolfgang. “Quantum Computing with Endohedral Fullerenes.” arXiv preprint arXiv:1708.09298 (2017).); linear optical quantum computers (qubits are implemented as processing states of light in different modes through linear optical elements such as mirrors, beam splitters, and phase shifters) (Knill, E., R. Laflamme, and G. Milburn. “Efficient linear optics quantum computation.” arXiv preprint quant-ph / 0006088 (2000).); diamond-based quantum computers (qubits are implemented as electron or nuclear spins in nitrogen-vacancy centers in diamond) (Nizovtsev, A. P., S. Ya Kilin, F. Jelezko, T. Gaebal, Iulian Popa, A. Gruber, and Jorg Wrachtrup.“A quantum computer based on NV centers in diamond: optically detected nutations of single electron and nuclear spins.” Optics and spectroscopy 99, no. 2 (2005): 233-244.); quantum computers using Bose-Einstein condensates (qubits implemented as two-component BECs) (Byrnes, Tim, Kai Wen, and Yoshihisa Yamamoto. “Macroscopic quantum computation using Bose-Einstein condensates.” arXiv preprint quantum-ph / 1103.5512 (2011)); quantum computers using transistors (qubits implemented as semiconductors coupled to nanophotonic cavities) (Sun, Shuo, Hyochul Kim, Zhouchen Luo, Glenn S. Solomon, and Edo Waks. “A single-photon switch and transistor enabled by a solid-state quantum memory.” arXiv preprint quant-ph / 1805.01964 (2018)); Quantum computer using rare-earth metal ion-doped inorganic crystals (qubits implemented as ground-state hyperfine levels of atoms in rare-earth ion-doped inorganic crystals) (Ohlsson, Nicklas, R. Krishna Mohan, and Stefan Kroll. “Quantum computer hardware based on rare-earth-ion-doped inorganic crystals.” Optics communications 201, no. 1-3 (2002): 71-77.Quantum computers using metallic carbon nanospheres (qubits are implemented as electron spins in conductive carbon nanospheres) (Nafradi, Balint, Mohammad Choucair, Klaus-Peter Dinse, and Laszlo Forro. “Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres.” arXiv preprint cond-mat / 1611.07690 (2016)); and D-Wave's quantum annealer (qubits are implemented as superconducting logic elements) (Johnson, Mark W., Mohammad HS Amin, Suzanne Gildert, Trevor Lanting, Firas Hamze, Neil Dickson, R. Harris et al. “Quantum annealing with manufactured spins.” Nature 473, no. 7346 (2011): 194-198.), the entire contents of each of which are incorporated herein by reference.

[0047] quantum annealer A quantum annealer is an example of a quantum mechanical system that can consist of multiple qubits.

[0048] Each qubit may be inductively coupled to a bias source, called a local field bias. In some cases, the bias source is an electromagnetic device used to pass a magnetic flux through the qubit, providing control of the qubit's state (see, for example, U.S. Patent Application No. 2006 / 0225165, the entire contents of which are incorporated herein by reference).

[0049] The local field biases at the qubits may be programmable and controllable. In some cases, a qubit control system including a digital processing device is connected to the system of qubits and can program and adjust the local field biases of the qubits.

[0050] The quantum annealer may further include a plurality of couplings between the plurality of pairs of qubits. In some cases, the coupling between two qubits is a device proximate to both qubits that passes magnetic flux to both qubits. In some cases, the coupling may include a superconducting circuit interrupted by a compound Josephson junction. A magnetic flux can be passed through the compound Josephson junction, thereby passing magnetic flux to both qubits (see, for example, U.S. Patent Application Publication No. 2006 / 0225165, the entire contents of which are incorporated herein by reference). The strength of this magnetic flux may contribute quadratically to the energy of a quantum Ising model with a transverse magnetic field. In some cases, the strength of the coupling is implemented by adjusting a coupling device proximate to both qubits.

[0051] The strength of the couplings may be controllable and programmable. In some cases, a quantum annealer control system including a digital processing device may be connected to multiple couplings. In some cases, a quantum annealer control system including a digital processing device may be capable of programming the strength of the couplings of the quantum annealer.

[0052] In some cases, the quantum annealer performs a transformation of the quantum Ising model with a transverse magnetic field from an initial configuration to a final configuration. In some cases, the initial and final configurations of the quantum Ising model with a transverse magnetic field provide a quantum system described by their corresponding initial and final Hamiltonians.

[0053] In some cases, quantum annealers may be used as heuristic optimizers of these energy functions. An example of such an analog processor is described in McGeoch, Catherine C. and Cong Wang, (2013), “Experimental Evaluation of an Adiabatic Quantum System for Combinatorial Optimization,” Computing Frontiers, May 14-16, 2013, and in U.S. Patent Application No. 2006 / 0225165, each of which is incorporated herein by reference in its entirety.

[0054] In some cases, a quantum annealer may also be used to provide samples from the Boltzmann distribution of the corresponding Ising model at finite temperatures. See, for example, Bian, Z., Chudak, F., Macready, W. G., and Rose, G. (2010), “The Ising model: teaching an old problem new tricks,” and Amin, M. H., Andriyash, E., Rolfe, J., Kulchytskyy, B., and Melko, R. (2016), “Quantum Boltzmann Machine” arXiv:1601.02036, the entire contents of which are incorporated herein by reference. This sampling method is called quantum sampling.

[0055] digital computer In some cases, the digital computer includes one or more hardware central processing units (CPUs) that perform the functions of the digital computer. In some cases, the digital computer further includes an operating system (OS) configured to execute the executable instructions. In some cases, the digital computer is connected to a computer network. In some cases, the digital computer is connected to the Internet to access the World Wide Web. In some cases, the digital computer is connected to a cloud computing infrastructure. In some cases, the digital computer is connected to an intranet. In some cases, the digital computer is connected to a data storage device.

[0056] As described herein, suitable digital computers may include, by way of non-limiting example, server computers, desktop computers, laptop computers, notebook computers, subnotebook 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, in some cases, be suitable for use with the methods and systems described herein. Select televisions, game consoles, and digital music players, which may, in some cases, have computer network connectivity, may be suitable for use with one or more variations, examples, or embodiments of the systems and methods described herein. Suitable tablet computers may include those with booklet, slate, and convertible configurations.

[0057] In some cases, the digital computer includes an operating system configured to execute executable instructions. An operating system may be software, including programs and data, that manages the device's hardware and provides services for running applications, for example. Suitable server operating systems include, by way of non-limiting example, 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 example, Microsoft® Windows®, Apple® Mac OS X®, UNIX®, and UNIX-like operating systems such as GNU / Linux®. In some cases, operating systems are provided for cloud computing. Suitable mobile smartphone operating systems may include, by way of non-limiting example, 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 example, 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 example, Sony® PS3®, Sony® PS4®, Microsoft® Xbox 360®, Microsoft Xbox One, Nintendo® Wii®, Nintendo® Wii U®, and Ouya®.

[0058] In some cases, the digital computer includes a storage device and / or memory device. In some cases, the storage device and / or memory device is one or more physical devices used to temporarily or permanently store data or programs. In some cases, the device includes volatile memory and requires power to maintain the stored information. In some cases, the device includes non-volatile memory and retains the stored information even when the digital computer is not powered. In some cases, the non-volatile memory includes flash memory. In some cases, the non-volatile memory includes dynamic random access memory (DRAM). In some cases, the non-volatile memory includes ferroelectric random access memory (FRAM). In some cases, the non-volatile memory includes phase change random access memory (PRAM). In some cases, the device includes a storage device, non-limiting examples of which include CD-ROMs, DVDs, flash memory devices, magnetic disk drives, magnetic tape drives, optical disk drives, and cloud computing-based storage devices. In some cases, the storage device and / or memory device is a combination of devices such as those disclosed herein.

[0059] In some cases, a digital computer includes a display used to provide visual information to a user. In some cases, the display includes a cathode ray tube (CRT). In some cases, the display includes a liquid crystal display (LCD). In some cases, the display includes a thin film transistor liquid crystal display (TFT-LCD). In some cases, the display includes an organic light emitting diode (OLED) display. In some cases, the OLED display includes a passive matrix OLED (PMOLED) or an active matrix OLED (AMOLED) display. In some cases, the display includes a plasma display. In some cases, the display includes a video projector. In some cases, the display includes a combination of devices such as those disclosed herein.

[0060] In some cases, a digital computer includes an input device that receives information from a user. In some cases, the input device includes a keyboard. In some cases, the input device includes a pointing device, including, by way of non-limiting example, a mouse, trackball, trackpad, joystick, game controller, or stylus. In some cases, the input device includes a touchscreen or multi-touchscreen. In some cases, the input device includes a microphone that captures voice or other audio input. In some cases, the input device includes a video camera or other sensor that captures motion or visual input. In some cases, the input device includes a Kinect, Leap Motion, etc. In some cases, the input device includes a combination of devices, such as those disclosed herein.

[0061] Neural networks representing quantum states In recent years, advances in machine learning have led to the creation of neural network-related models of quantum systems. In some cases, neural networks can learn and represent probability distributions. Thus, neural networks can be used as functional representations of wave functions that describe quantum states (see, for example, J. Carrasquilla, “Machine learning for quantum matter,” 2020, the entire contents of which are incorporated herein by reference). Quantum state tomography using neural networks may be one possible process for training the quantum state of a neural network.

[0062] Quantum state tomography (QST) involves reconstructing quantum states using measurements. QST is standard for validating and benchmarking quantum devices (see M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, “Efficient quantum state tomography,” Nature Communications 1 no. 1, (2010), the entire contents of which are incorporated herein by reference). The number of measurements and time required to reconstruct a state using QST may scale exponentially with the size of the system. Neural network-based tomography uses a set of measurements of a system to derive the wave function

[0063]

number

[0064] Variational Monte Carlo Variational Monte Carlo is a general set of algorithms that use a digital computer to iteratively improve the classical parameterization of a quantum state or set of quantum states according to a given criterion. There are a wide variety of algorithms that can be called VMC.

[0065] The VMC algorithm may be iterative, possibly alternating between computing a quantity related to a criterion and updating the parameters of the classical representation in small increments until a stopping criterion is met.

[0066] The criterion may include the expectation value of the quantum operator in the represented quantum state. In some cases, the expectation value may be estimated by expressing it as a so-called local operator probability expectation and using a Monte Carlo procedure.

[0067] A VMC algorithm may be applied to obtain a classical approximation of the ground state of the Hamiltonian. In some cases, the criterion is minimizing the expectation value of the Hamiltonian, which may be expressed as a probability expectation of the local energy. An estimate of the gradient of the expectation value of the Hamiltonian with respect to the parameters of the representation may be calculated. A gradient-based optimization procedure may be used to update the parameters.

[0068] Error Mitigation Using Variational Monte Carlo As described herein, there are many potential sources of error that can occur during the preparation of quantum states and / or in representing these states on a quantum computer. In some cases, ways to mitigate errors resulting from noisy and imperfect calculations from NISQ devices can be advantageous.

[0069] The methods disclosed herein can be used to mitigate errors in neural network representations of the ground states of a quantum system. For example, improvements to the quantum states of a neural network reconstructed using neural network quantum state tomography may be considered. In neural network tomography, information about the physical system may be present in the measurement data input to the neural network. A cost function, such as KL divergence, may be used to train the network according to the measurement data. In some cases, training is performed without direct knowledge of the system. While this can be a powerful method for reconstructing quantum states from laboratory data, it may be limited, at least in some cases, by the number of available measurements. Furthermore, at least in some cases, the method may be limited by noise in the measurement data. In the NISQ era, it may be advantageous not to assume that the ground state is perfectly prepared or that the measurements were noise-free.

[0070] One potential path to improving the approximation of ground states prepared using quantum devices may involve post-processing of tomographic states with neural networks using variational Monte Carlo. As described herein, variational Monte Carlo involves training a neural network on quantum states by minimizing the variational energy of the quantum states. In some cases, post-processing using variational Monte Carlo may be considered as fine-tuning the parameters of the neural network to improve the estimation of the ground states.

[0071] One potential bottleneck of variational Monte Carlo may be the expressivity of the selected wave function hypotheses. Another potential bottleneck may depend on how a large Hilbert space is sampled. Due at least in part to either of these potential limitations, although not restricted by theory, variational Monte Carlo may be sensitive to the initial hypotheses, and in some cases may get stuck in a local minimum or saddle point due at least in part to this sensitivity. In some cases, the wave function of a trained neural network may be used as the initial hypotheses for variational Monte Carlo. In some cases, the method may use the wave function

[0072]

number

[0073]

number

[0074] Computing Platform A computational platform as disclosed herein may include various types of hardware. Each type of hardware may be used alone or in combination with other hardware as part of a system to perform the method in whole or in part. In some cases, hardware may be used for various operations of the method disclosed herein, including one or more of the following: experimentally preparing an approximation of the quantum state; · Perform one or more measurements of the prepared quantum state. Calculate the value of the neural network's cost function. · Calculate the gradient of the cost function. ·Estimate the variational energy of the wave function. Generate random numbers. · Update the parameters of one or more neural networks. Updating one or more parameters of a parameterized quantum gate. ·Implement quantum development. · Perform one or more functions of the interface, including some or all of the above.

[0075] The computing platform may include a central processing unit (CPU). The CPU may be a low-latency integrated circuit chip that includes the main processor in a computer. The CPU may execute instructions provided by an algorithm. The CPU may include components configured to perform one or more of: performing arithmetic and logic operations; registering to store the results of those operations; and directing the operation of the former using a control device.

[0076] The computing platform may include a graphics processing unit (GPU), which may be a specialized electronic circuit optimized for high throughput and capable of performing the same set of operations in parallel on multiple blocks of data at once.

[0077] The computing platform may include a field-programmable gate array (FPGA), which may include an integrated circuit chip containing configurable logic blocks and programmable interconnects, and can be programmed after manufacture to execute custom algorithms.

[0078] The computing platform may include an application-specific integrated circuit (ASIC). An ASIC may be an integrated circuit chip customized to execute a particular algorithm. In some cases, the ASIC is not programmed after manufacture.

[0079] The computing platform may include a tensor processing unit (TPU), which may include a proprietary type of ASIC developed by Google Inc. for low-bit precision processing. See U.S. Patent Application No. 2016 / 0342891A1, the entirety of which is incorporated herein by reference for all purposes.

[0080] The computational platform may include a tensor streaming processor (TSP), which may be a domain-specific programmable integrated chip designed for linear algebra computations such as may be implemented in artificial intelligence applications (see, for example, https: / / groq.com / wp-content / uploads / 2020 / 01 / Groq-Rocks-NNs-Linley-Group-MPR-2020Jan06.pdf, the entirety of which is incorporated herein by reference for all purposes).

[0081] 1, a flowchart of an example method for improving estimation of properties of quantum states is shown. In some cases, the method can reduce errors in estimation of properties of quantum states.

[0082] In accordance with the processing operation (100), an indication of a property of the quantum state to be estimated is provided. In some cases, the property of the quantum state may be of various types. The property of the quantum state may include an observable of the quantum state. In some cases, the observable of the quantum state is an expectation value of the energy of the quantum state. In some cases, the indication of the property of the quantum state includes a Hamiltonian. In some cases, the quantum state may be a ground state of the Hamiltonian. In some cases, the quantum state may be an excited state of the Hamiltonian. In some cases, the Hamiltonian represents a classical optimization problem, and the ground state represents an optimal solution to the classical optimization problem.

[0083] In some cases, the Hamiltonian is a parameterized Hamiltonian representing a family of Hamiltonians. In some cases, the parameterization is continuous. Properties of the ground state of the Hamiltonian may be estimated for each possible value of the parameters. In some cases, the parameters may include multidimensional parameters. In some cases, each parameter value defines a Hamiltonian.

[0084] In accordance with the processing operation (102), an index of a set of measurement operators is provided. In some cases, the measurement operators may be of various types. In some cases, the measurement operators are any Pauli operators. In some cases, the set of measurement operators may include a set of tensor products of Pauli operators acting on qubits of a quantum device. In some cases, where the quantum state whose properties are to be estimated is a ground state of a Hamiltonian, the set of tensor products of Pauli operators is selected so that the Hamiltonian can be expressed as a weighted sum thereof. In some cases, the set of tensor product Pauli operators is selected to reduce non-computational measurements acting on qubits of the quantum device. For example, a tensor product set of Pauli operators may give a small weight to X and Y measurements, such as tensor product Pauli operators that include only one or two X and Y operators, and include Z operators everywhere else, such as ZZZZZX, ZZZZXX, ZZZZZY, and ZZZZYY.

[0085] In some cases, a set of measurement operators is selected such that the measurement results (optionally together with some knowledge of the properties of the prepared state) allow the prepared state to be proximally reconstructed using tomography.

[0086] In accordance with processing operation 104, multiple measurements of the quantum state are obtained. Optionally, the multiple measurements are obtained using a quantum experiment using a quantum device. Optionally, the quantum state is a quantum state of a parameterized Hamiltonian, multiple possible values ​​for a parameter are selected, and multiple measurements of the quantum state are obtained for each parameter value of the selected multiple possible values.

[0087] Referring now to FIG. 2, a flow chart of an example method for obtaining multiple measurements of a quantum state is shown.

[0088] According to the processing operation (200), a quantum state is experimentally prepared and a quantum experiment is performed using a quantum device. In some cases, the quantum experiment may be various types of quantum experiments, such as any of the quantum experiments disclosed herein. In some cases, performing the quantum experiment to experimentally prepare the quantum state includes applying at least one unitary transformation to the initial state of the qubits. In some cases, the quantum experiment includes quantum computing. In some cases, the quantum computing may include at least one component of the group consisting of circuit-model quantum computing, quantum annealing, measurement-based quantum computing, and adiabatic quantum computing. In some cases, the quantum computing may include a variational quantum computing procedure described below.

[0089] In some cases, the quantum computing includes solving an optimization problem. In some cases, the quantum computing includes quantum chemistry simulation. The Hamiltonian may include an electronic structure Hamiltonian of one of a molecule and a material, and the quantum state may be an eigenstate of the Hamiltonian.

[0090] In some cases, the quantum device may be of various types, such as any quantum device disclosed herein. The quantum device may be any suitable quantum device, such as any quantum device (704) described herein with respect to the system shown in FIG. 7. The quantum device may be of any type suitable for the methods disclosed herein. In some cases, the quantum device includes a NISQ device. In some cases, the quantum device includes a superconducting qubit. The quantum device may include at least one component of the group consisting of a quantum annealer, a trapped-ion quantum computer, an optical quantum computer, a photonic quantum computer, and a spin quantum dot computer.

[0091] 2, according to processing operation (202), a measurement operator may be selected from a set of measurement operators. In some cases, the selection criteria is based on the order of the measurement operators in the list. In some cases, the selection criteria is based on previously selected measurement operators. In some cases, the selection criteria is based on previously selected measurement operators and previously obtained measurement results.

[0092] According to processing operation (204), a measurement of the prepared quantum state is performed using a selected operator. In some cases, the measurement procedure varies depending on the nature of the quantum device. This may include further applying a unitary transformation, an experimental readout procedure, and post-processing to the prepared quantum state using electronics and / or a digital computer. The experimental readout procedure may be performed using a readout control system, such as the readout control system described herein with respect to the system shown in FIG. 7.

[0093] A stopping criterion may be verified according to processing operation (206). If the stopping criterion is met, a measurement of the quantum state may be provided according to processing operation (208), and if the stopping criterion is not met, processing operations (200, 202, and 204) may be repeated. In some cases, the stopping criterion may be of various types. In some cases, the stopping criterion is that the processing operations (200, 202, and 204) are repeated a predetermined number of times. In some cases, the stopping criterion is that a given function of a set of operators selected so far and measurements obtained so far exceeds a given value.

[0094] Referring back to FIG. 1 , according to processing operation (106), a neural network including at least one trainable parameter may be constructed and trained using at least one computing platform. In some cases, the neural network represents a quantum state. In some cases, quantum state tomography may be used to train the neural network. In some cases, the neural network is trained using multiple measurements. In some cases, the neural network may be of various types, including, but not limited to, autoregressive models, recurrent neural networks, transformers, autoregressive generative models, attention-based architectures, dense deep neural networks, convolutional neural networks, variational autoencoders, generative adversarial networks, restricted Boltzmann machines, generalized Boltzmann machines, energy-based models, reversible neural networks, and flow-based generative models.

[0095] In some cases, the computing platform may be of various types. The computing platform may be any suitable computing platform, such as any computing platform described herein with respect to the system shown in FIG. 7. In some cases, the computing platform includes at least one component from the group consisting of a field-programmable gate array (FPGA), an application-specific integrated circuit (ASIC), a central processing unit (CPU), a graphics processing unit (GPU), a tensor processing unit (TPU), and a tensor streaming processor (TSP).

[0096] 3, a flowchart of an example method for constructing and training a neural network including at least one trainable parameter representing a quantum state is shown. In some cases, constructing and training the neural network may include using at least one component from the group consisting of a tensor processing unit (TPU), a graphics processing unit (GPU), a field-programmable gate array (FPGA), a tensor streaming processor (TSP), and an application-specific integrated circuit (ASIC).

[0097] In accordance with processing operation 300, the plurality of measurements are used to provide input to the neural network. In some cases, the quantum state is a quantum state of a parameterized Hamiltonian, and the input to the neural network further includes selected parameter values ​​corresponding to the measurements. In some cases, the input data includes the plurality of measurements and the corresponding parameter values. In some cases, the plurality of measurements may be preprocessed prior to training. In some cases, the quantum state is a quantum state of a parameterized Hamiltonian, and the plurality of measurements are preprocessed along with the corresponding parameter values. In some cases, the input data is separated into training data and validation data. In some cases, the training data may be divided into batches. In some cases, the training procedure may depend on the particular type of neural network. For example, in some cases, the neural network is an energy-based model, and the training procedure is a contrastive divergence procedure. In some cases, the neural network is an autoregressive model, and the training procedure comprises maximizing the likelihood of training inputs.

[0098] According to processing operation (302), a cost function value is calculated for the neural network. In some cases, the cost function for the neural network may be of various types. Types of cost functions for the neural network may include, but are not limited to, the cross-entropy between the empirical distribution of measurement outcomes and the probabilities assigned to those outcomes by applying the Born rule to the quantum states represented by the neural network. For example, the cost function L may be given by the following equation:

[0099]

number

[0100]

number

[0101] In some cases, the type of cost function for a neural network may depend on the particular type of neural network, for example, in some cases the neural network represents non-normalized quantum states and the cost function may take normalization into account.

[0102] In some cases, a regularization term may be added to the cost function. The regularization term may be of various types, including, but not limited to, an L1 term, an L2 term, and an entropy term. A schedule may be used to control the contribution of the regularization term during the training process.

[0103] 3, according to a processing operation (304), a gradient of the cost function with respect to at least one trainable parameter of the neural network is calculated. In some cases, the calculation may depend on the particular type of neural network.

[0104] In accordance with processing operation 306, at least one trainable parameter is updated using the calculated cost function value and gradient. In some cases, the type of the at least one trainable parameter may depend on the particular type of neural network. In some cases, the neural network is an LSTM recurrent neural network, and the trainable parameters include weights and biases of one or more layers of cells or gates. In some cases, the neural network is a restricted Boltzmann machine, and the trainable parameters are weights associated with connections between each hidden unit and each visible unit.

[0105] According to process operation (308), if a stopping criterion is met, the training procedure is terminated, and if the stopping criterion is not met, process operations (300, 302, 304, and 306) are repeated. In some cases, the stopping criterion is that process operations (300, 302, 304, and 306) are repeated a predetermined number of times. In some cases, the stopping criterion is that the value of at least one trainable parameter converges.

[0106] Referring again to FIG. 1 , at least one trainable parameter of the neural network may be trained using properties of the quantum state to variationally improve the quantum state represented by the neural network, according to processing operation (108). In some cases, the training may be performed using at least one computing platform. The training may be performed using a variational Monte Carlo procedure. In some cases, the variational Monte Carlo procedure includes the neural network representing a hypothetical ground state wave function. In some cases, at least one trainable parameter of the neural network represents a set of variational degrees of freedom. The variational Monte Carlo procedure may be performed to improve the estimation of properties of the quantum state, for example, to reduce errors in the estimation. In some cases, performing the variational Monte Carlo procedure may include one or more of a tensor network hypothesis, a Jastrow-type wave function, or a Hartree-Fock-type wave function.

[0107] In some cases, the computing platform may be of various types. The computing platform may be any suitable computing platform, such as any computing platform described herein with respect to the system shown in FIG. 7. In some cases, the computing platform includes at least one component from the group consisting of a field-programmable gate array (FPGA), an application-specific integrated circuit (ASIC), a central processing unit (CPU), a graphics processing unit (GPU), a tensor processing unit (TPU), and a tensor streaming processor (TSP).

[0108] Referring now to FIG. 4, there is shown a flowchart of an example of a method for training at least one trainable parameter of a neural network to variationally improve the quantum state of the neural network.

[0109] In accordance with process operation 400, the trained neural network is used to sample at least one configuration. In some cases, the quantum state is a quantum state of a parameterized Hamiltonian, where multiple possible parameter values ​​are sampled, and the sampled multiple possible parameter values ​​are provided as inputs to the neural network, where at least one configuration is sampled from the neural network for each parameter value. For example, in some cases, the neural network is an autoregressive model, and at least one configuration is sampled by sampling from the conditional probabilities represented by the autoregressive model.

[0110] According to a processing operation (402), a variational energy of the wave function, represented by an average of the local energies, is estimated using at least one sampled configuration. In some cases, the variational energy of the wave function is estimated by the following equation:

[0111]

number

[0112]

number

[0113] In some cases, the quantum state is a quantum state of a parameterized Hamiltonian, and the variational energies of each sampled parameter value are combined into a loss function. In some cases, the loss function may include an average of the variational energies of the sampled parameter values, or a sum of the variational energies weighted by a function of the parameters.

[0114] In some cases, a regularization term may be added to the variational energy. The regularization term may be of various types. The regularization term may include, but is not limited to, an L1 term, an L2 term, and an entropy term. A schedule may be used to control the contribution of the regularization term during the training process.

[0115] In some cases, the underlying probability distribution of a quantum chemical system may peak sharply, resulting in a sparse ground state. In some cases, where the Hamiltonian is an electronic structure Hamiltonian, a regularization term may be added to the variational energy to overcome the ground state sparsity. The ground state in electronic structure theory may peak at a Hartree-Fock state. There may be one more common computational ground state and several less likely non-dominant states that characterize the ground state.

[0116] In some cases, the sampled configuration is likely to be the dominant Hartree-Fock state. Because we oversample this state during training, we may train the neural network to represent the dominant Hartree-Fock state. As a result, the neural network may not learn the topological structure because it represents the Hartree-Fock state and all other states have near-zero amplitudes. In some cases, the topological structure may be important for learning the ground states and navigating the optimization space. To prevent the wave function from collapsing into a Hartree-Fock state (a sparse solution) and failing to learn the topological structure, a regularization term may be added to the loss function represented by the variational energy. In some cases, a regularization term that discourages sparse solutions, such as L1 or entropy, may be added to the loss function. In early training iterations, the regularization term may stimulate the neural network to over-represent the amplitudes of all computational ground states, thereby allowing the neural network to learn the topological structure. In some cases, a schedule may be used to mitigate the contribution of the regularization term. The regularization term allows the network to learn the topological structure, allowing the optimization to navigate the optimization space more effectively and accurately represent the amplitudes of Hartree-Fock states and non-dominated states.

[0117] 4, according to processing operation (404), a gradient of variational energy with respect to at least one parameter of the neural network is estimated using at least one sampled configuration. In some cases, the gradient of variational energy is estimated by the following equation:

[0118]

number

[0119] 4, according to processing operation (406), at least one parameter of the neural network may be updated using the estimated variational energy and the estimated gradient of the variational energy.

[0120]

number

[0121]

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[0122]

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[0123] 4, according to process operation (408), if a stopping criterion is met, the training procedure is terminated, and if the stopping criterion is not met, process operations (400, 402, 404, and 406) are repeated. In some cases, the stopping criterion is that the fluctuation energy falls within a threshold, such as a threshold value, a number of iterations, or a degree of value decline.

[0124] Referring again to FIG. 1 , a stopping criterion is verified according to processing operation (110), and if the stopping criterion is met, a property of the quantum state is estimated and provided according to processing operation (112), and if the stopping criterion is not met, processing operations (102, 104, 106, and 108) are repeated. In some cases, the stopping criterion may be of various types. In some cases, the stopping criterion is that processing operations (102, 104, 106, and 108) have been repeated a predetermined number of times. In some cases, the stopping criterion is that the property estimate is of sufficient quality.

[0125] Referring now to FIG. 5, an example of a method for implementing a variational quantum computing procedure is shown. The variational quantum computing procedure involves applying a hybrid quantum-classical optimization algorithm using a quantum device including a quantum processor including a layer of parameterized quantum gates. The quantum device may be any quantum device including quantum gates, which may be parameterized. The quantum device may be any quantum device suitable for the technology, such as any quantum device disclosed herein, for example, as described with respect to the system shown in FIG. 7. In some cases, the quantum device is a trapped ion analog quantum simulator, such as IonQ™ or the trapped ion analog quantum simulator from Innsbruck University. In some cases, the quantum device is a superconducting circuit model quantum device, such as quantum devices from IBM™, Rigetti™, or Google™. The quantum device may be part of at least one of the group consisting of CV quantum computing by Xanadu™, cold atom quantum simulators such as quantum simulators from ColdQuanta™ and Atom Computing™, and annealers such as annealers from NTT™, D-Wave™ and QEO™.

[0126] According to the processing operation (500), an initial state and a set of measurement operators are obtained. In some cases, the initial state is a standard initial state for each iteration.

[0127]

number

[0128]

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[0129] According to a processing operation (502), a multi-qubit quantum state is prepared.

[0130] Referring now to FIG. 6, an example of a method for preparing a multi-qubit quantum state and obtaining multiple measurements thereof is shown.

[0131] According to a process operation (600), an initial state is set for the quantum device.

[0132] According to a processing operation (602), a multi-qubit quantum state is prepared. The preparation may include evolving an initial state through a layer of parameterized quantum gates using a quantum device including a quantum processor including a layer of parameterized quantum gates. In some cases, the quantum device is a trapped-ion analog quantum simulator, and the layer is a sequence that alternates between single-qubit rotation and time evolution with a long-range coupling Hamiltonian, the parameters being the rotation angle and the evolution time.

[0133] 6, according to processing operation (604), a measurement operator is selected from a set of measurement operators. In some cases, the selection criteria is based on the order of the measurement operators in the list. In some cases, the selection criteria is based on previously selected measurement operators. In some cases, the selection criteria is based on previously selected measurement operators and previously obtained measurement results.

[0134] According to a processing operation (606), a measurement of the prepared quantum state is performed using the selected operator. In some cases, the measurement procedure varies depending on the nature of the quantum device. This may include further application of unitary transformations, experimental readout procedures, and post-processing to the prepared quantum state using electronic devices and / or digital computers.

[0135] A stopping criterion is verified according to processing operation (608). For example, if the stopping criterion is met according to processing operation (308), a measurement result of the quantum state is provided according to processing operation (610); if the stopping criterion is not met, processing operations (600, 602, 604, and 606) are repeated. In some cases, the stopping criterion may be of various types. In some cases, the stopping criterion is that the processing (600, 602, 604, and 606) is repeated a predetermined number of times. In some cases, the stopping criterion is that a given function of a set of operators selected so far and measurements obtained so far exceeds a given value.

[0136] 5, according to processing operation (504), the variational energy of the prepared multi-qubit quantum state is calculated using the provided measurement results. In some cases, the Hamiltonian of the system is

[0137]

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[0138]

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[0139]

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[0140]

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[0141]

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[0142] According to processing operation 506, parameters of the parameterized quantum gate are updated to minimize the variational energy using a classical optimization algorithm. The classical optimization algorithm may be of various types. In some cases, the classical optimization algorithm is the Nelder-Mead algorithm. In some cases, the algorithm is the Adam algorithm, and the gradient of the variational energy is approximated using a shift law, or a finite difference gradient, or a combination of the two.

[0143] 5, according to processing operation (508), a stopping criterion is verified, and if the stopping criterion is met, the resulting quantum state is provided according to processing operation (510), and if the stopping criterion is not met, processing operations (500, 502, 504, and 506) are repeated. In some cases, the stopping criterion is that processing operations (500, 502, 504, and 506) are repeated a predetermined number of times. In some cases, the stopping criterion is that the parameters of the quantum gates converge.

[0144] Referring again to FIG. 1 , in some cases, all or part of the following steps may be performed together or separately: preparing a quantum state, training a neural network representing the quantum state, and performing a variational Monte Carlo procedure. In some cases, measurement results obtained from the prepared quantum state are used to train the neural network in the variational Monte Carlo procedure. In some cases, measurement results obtained from the prepared quantum state are used instead of operation (400) of FIG. 4. In some cases, the neural network is trained alternately with at least one training iteration described in processing operations (300, 302, 304, and 306) of FIG. 3 and at least one training iteration described in processing operations (400, 402, 404, and 406) of FIG. 4. In some cases where preparing a quantum state includes variational quantum computation, the neural network receives parameters of the variational quantum computation as additional inputs and is trained to represent multiple quantum states prepared with multiple parameter values ​​of the variational quantum computation. In some cases, the quantum state obtained from performing the processing operations of FIG. 3 or from performing the processing operations of FIG. 4 is used in processing operation (506) of FIG. 5 to update parameters of a quantum gate.

[0145] Referring now to Figure 7, a diagram of a system for improving estimation of properties of quantum states is shown. The system includes at least one processing device (706), a display device (708), an interface (710), a communication port (714), and a memory (712), where the memory (712) includes a computer program executable by the processing device to obtain indices of properties of the quantum state to be estimated, a set of measurement operators, at least one quantum device, and at least one computational platform, obtain a plurality of experimentally prepared measurements of the quantum state, and communicate with the quantum device (704) and the computational platform (702). In some cases, the digital computer (700) may be of various types, such as any of the digital computers disclosed herein.

[0146] The system further includes at least one computing platform (702). The computing platform (702) is operably connected to the digital computer (700). The computing platform (702) includes at least one processing device. In some cases, the at least one processing device (714) may be of various types, such as any of the processing devices disclosed herein. More precisely, the at least one processing device may include at least one component of a hardware group consisting of an FPGA, an ASIC, a GPU, a TSP, a CPU, and a TPU. The computing platform further includes a readout control system (718).

[0147] The system further includes at least one quantum device (704). The quantum device (704) includes at least a quantum processor (722) and a readout control system (720). The quantum device (704) may be of various types, such as any of the quantum processors disclosed herein. More precisely, the at least one quantum device may be at least one component of the group consisting of a superconducting quantum computer, a trapped-ion quantum computer, an optical lattice quantum computer, a spin quantum dot computer, a quantum dot computer in space, a coupled quantum wire, a nuclear magnetic resonance quantum computer, a solid-state NMR Kane quantum computer, an electron-on-helium quantum computer, a cavity quantum electrodynamics quantum computer, a molecular magnet quantum computer, a fullerene-based ESR quantum computer, a linear optical quantum computer, a diamond-based quantum computer, a Bose-Einstein condensate quantum computer, a transistor-based quantum computer, a rare-earth metal ion-doped inorganic crystal quantum computer, a metallic carbon nanosphere quantum computer, and a quantum annealer.

[0148] In some cases, each piece of hardware may be used alone or in combination with other hardware as part of a system to perform the entire method, or any portion thereof. In some cases, the hardware may be used to experimentally prepare an approximation of a quantum state, perform measurements of a prepared quantum state, calculate values ​​of a cost function of a neural network, calculate gradients of the cost function, estimate variational energy of a wave function, generate random numbers, update parameters of a neural network, update parameters of parameterized quantum gates, perform quantum evolution, and perform interface functions, including some or all of the above.

[0149] Schwinger model The lattice Schwinger model describes the interaction between scalar fermion fields and Abelian quantized electromagnetic fields in one dimension. Using the Kogut-Susskind encoding, open boundary conditions, and the Jordan-Wigner transformation, the lattice Schwinger Hamiltonian can be written as

[0150]

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[0151] The first term describes the creation or annihilation of pairs of fermions, including the spin-flop term w. The second term is a mass term, including the bare mass m. The last term is the electric field energy, including the bond g. If we set g=w=1, the behavior of the system can be studied as a function of the mass m. By Gauss's law, the electric field

[0152]

number

[0153]

number

[0154]

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[0155] The properties of interest are the ground state energy, the entanglement entropy, and the order parameter. Quantum phase transitions may be detected by calculating the order parameter O of the Hamiltonian. For the Schwinger model, the order parameter is:

[0156]

number

[0157] Variational quantum simulation (VQS) of the lattice Schwinger model has been shown to converge to a ground state (see C. Kokail, C. Maier, R. van Bijnen, T. Brydges, MK Joshi, P. Jurcevic, CA Muschik, P. Silvi, R. Blatt, CF Roos, and et al., “Self-verifying variational quantum simulation of lattice models,” Nature 569 no. 7756, (May 2019) 355-360, the entire contents of which are incorporated herein by reference). Variational quantum simulation is a quantum-classical optimization method used to find the ground state of a given Hamiltonian, such as the variational quantum procedure shown in FIG. 5. In this example, samples are obtained from an incomplete ground state prepared using VQS. Measurements are obtained by sampling the prepared state using a quantum device, in this case sampling the VQS state. Error mitigation procedures disclosed herein are used to find the ground state.

[0158]

number

[0159]

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[0160] Then, the neural network quantum state (NNQS) is trained on the measurement dataset D, and the neural network parameters

[0161]

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[0162] Referring now to Figure 8, we present results for error mitigation using variational Monte Carlo, also known as neural error mitigation (NEM), for the N = 8-site lattice Schwinger model for mass values ​​m up to [-1.8, 1.0]. Shown are estimates of the ground state energy (a), order parameter (b), entanglement entropy (c), and infidelity to the exact ground state (d). Each panel includes results for quantum states prepared by VQS (blue triangles), results for NNQS trained using neural quantum state tomography (NQST, green circles), final neural error mitigated NNQS results (NEM, red diamonds), and exact results, where applicable (solid black lines). In all panels, the median value over 10 runs is shown, with shaded regions containing three values ​​on either side of the median.

[0163] As shown in Figure 8, a simple VQS scheme may be constructed to approximately represent the ground state of the lattice Schwinger model. While the qualitative behavior of the exact ground state energy as a function of mass can be reproduced to some extent by VQS, the qualitative behavior of other physical properties (order parameter, entanglement entropy, and infidelity) may not be well reproduced, which may limit the use of VQS alone for the study of this model. During the first run of the error mitigation protocol, tomography can accurately reconstruct optimized VQS results using the selected measurement basis (see, for example, NQST results). The purpose of this run may be to extract information about the incomplete ground state approximation prepared using VQS from experimental measurements.

[0164] Analysis of the results of the error mitigation methods disclosed herein shows that the properties of the final NEM results are substantially improved over VQS. In particular, post-processing of the tomographic NNQS using variational Monte Carlo can significantly improve the estimation of the ground state wavefunction and ground state observables represented by the NNQS. The NEM states are characterized by an absolute energy error of 10 -2 and infidelity reaches 10 -3 Importantly, we show that by using the error mitigation methods disclosed herein, VQS results can be extended to low error and low infidelity.

[0165] While preferred embodiments of the present invention have been shown and described herein, it will be apparent to those skilled in the art that such embodiments are provided by way of example only. Numerous variations, changes, and substitutions will now occur to those skilled in the art without departing from the invention. It will be understood that various alternatives to the embodiments of the invention described herein may be employed in practicing the invention. The following claims define the scope of the invention, and it is intended that methods and structures within the scope of these claims and their equivalents be covered thereby.

Claims

1. 1. A method for mitigating errors in estimating properties of a quantum state, comprising: (a) receiving a plurality of measurements of a quantum state from a quantum device; (b) preparing, at a computational platform, a representation of the quantum state based at least in part on the plurality of measurements of the quantum state, the computational platform being a classical computational device communicatively coupled to the quantum device, the representation comprising a neural network including one or more tunable parameters; (c) training the neural network on the computing platform to variationally improve the representation of the quantum state by adjusting the one or more tunable parameters, wherein the training reduces errors in the estimation of the properties of the quantum state; and A method comprising:

2. The method of claim 1 , wherein step (c) comprises performing a variational Monte Carlo procedure.

3. 3. The method of claim 2, wherein the variational Monte Carlo procedure includes one or more neural networks representing ansatz ground state wave functions, tensor network hypotheses, Jastrow-type wave functions, or Hartree-Fock-type wave functions, respectively.

4. 10. The method of claim 1, further comprising, prior to step (a), receiving at an interface of a digital computer an indication of a property of a quantum state to be estimated, and after step (c), providing at the interface the estimate of the property of the quantum state.

5. The method of claim 1 , further comprising repeating steps (a) through (c) until a stopping criterion is met.

6. Prior to step (a), the method further includes receiving an index of a set of measurement operators, and step (a) continues until a stopping criterion is met: (i) experimentally preparing an approximation of said quantum state; (ii) selecting a measurement operator from the set of measurement operators; and (iii) performing a measurement of the approximation of the prepared quantum state using the measurement operator selected from the set of measurement operators; The method of claim 1 further comprising:

7. The neural network further comprises a cost function, and step (b) further comprises: (i) providing inputs to the neural network based on the plurality of measurements; (ii) calculating a value of a cost function of the neural network; (iii) calculating the gradient of the cost function with respect to the one or more adjustable parameters of the neural network; (iv) updating the one or more tunable parameters of the neural network using the calculated gradients and the calculated cost function; and (v) Repeating (i) to (iv) any number of times. The method of claim 1 , comprising:

8. Step (c) is (i) using said neural network to sample at least one configuration; (ii) estimating a variational energy of the wave function represented by an average of local energies based at least in part on the sampled at least one configuration; (iii) estimating a gradient of the variational energy with respect to the one or more adjustable parameters of the neural network based at least in part on the sampled at least one configuration; (iv) updating the one or more tunable parameters of the neural network based at least in part on the estimated variational energy and the estimated gradient of the variational energy; and (v) repeating (i) through (iv) until a stopping criterion is met. The method of claim 3 further comprising:

9. 7. The method of claim 6, wherein (i) includes one or more of quantum computing, circuit model quantum computing, quantum annealing measurement quantum computing, and adiabatic quantum computing.

10. The method of claim 1 , wherein the quantum state comprises a ground state of a Hamiltonian.

11. 10. The method of claim 9, wherein the quantum computation comprises solving an optimization problem, and further wherein the quantum state comprises a ground state of a Hamiltonian.

12. The method of claim 11 , wherein the Hamiltonian represents a classical optimization problem.

13. The method of claim 11 , wherein the ground state of the Hamiltonian represents an optimal solution to the optimization problem.

14. The method of claim 1 , wherein step (a) comprises performing a variational quantum computing procedure in the quantum device.

15. The method of claim 9 , wherein the quantum computation comprises a quantum chemistry simulation, and the quantum state is a quantum state of a Hamiltonian representing a quantum chemistry problem.

16. The method of claim 15 , wherein the Hamiltonian comprises an electronic structure Hamiltonian of one of a molecule and a material.

17. The method of claim 1 , wherein the property of the quantum state comprises an observable of the quantum state.

18. 18. The method of claim 17, wherein the observable of the quantum state is an expectation value of the energy of the quantum state.

19. 10. The method of claim 1, wherein the neural network comprises at least one of an autoregressive model, a recurrent neural network, a transformer, an autoregressive generative model, an attention-based architecture, a dense deep neural network, a convolutional neural network, a variational autoencoder, a generative adversarial network, a restricted Boltzmann machine, a generalized Boltzmann machine, an energy-based model, a reversible neural network, and a flow-based generative model.

20. The method of claim 1 , wherein the quantum state is a quantum state of a parametrized Hamiltonian, and further wherein the parametrization of the parametrized Hamiltonian is continuous.

21. The method of claim 20 , wherein the neural network is further configured to receive parameter values ​​of the parameterization as inputs.

22. 21. The method of claim 20, further comprising using neural network inference to estimate properties of quantum states of the parametrized Hamiltonian using a second parameter value not used during training, to provide an estimate of properties of the quantum states.

23. 1. A system for improving estimation of properties of a quantum state, comprising: (a) a digital computer including an interface and a memory containing instructions, said digital computer comprising at least: receiving a plurality of measurements of the quantum state; preparing, at a computational platform, a representation of the quantum state based at least in part on the plurality of measurements, the computational platform being a classical computational device communicatively coupled to a quantum device, the representation including a neural network including one or more tunable parameters; and training the neural network on the computing platform to variationally improve the representation of the quantum state by adjusting the one or more tunable parameters. a digital computer configured to execute the instructions to perform (b) at least one quantum device communicatively coupled to the digital computer, the at least one quantum device including at least a quantum processor and a readout control system, the at least one quantum device configured to perform a quantum experiment and obtain the plurality of measurements of the quantum state using the readout control system; and (c) at least one computing platform communicatively connected to the digital computer, the at least one computing platform including at least one processor and a readout control system, the at least one computing platform configured to (i) receive from the digital computer at least one tunable parameter and a configuration of a neural network including the plurality of measurements, and (ii) train the neural network representing the quantum state by adjusting the at least one tunable parameter of the neural network to variationally improve the quantum state. at least one computing platform, Including, the system.

24. 24. The system of claim 23, wherein the computing platform includes at least one component from the group consisting of a field-programmable gate array (FPGA), an application-specific integrated circuit (ASIC), a central processing unit (CPU), a graphics processing unit (GPU), a tensor processing unit (TPU), and a tensor streaming processor (TSP).

25. 24. The system of claim 23, wherein the at least one quantum device comprises at least one of a quantum annealer, a trapped ion quantum computer, an optical quantum computer, a photonic quantum computer, a spin quantum dot computer, and a superconductor quantum computer.

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