Quantum systems and methods for windowed estimation of expectation values of arbitrary observables on quantum states

Abstract

Systems, methods, and quantum circuits for estimating an expectation value of an observable of a physical quantum system. An outer quantum phase register, an inner quantum register, and a state simulation register are received. The state simulation register is prepared in a reference eigenstate of a Hermitian operator. The inner quantum register includes an inner phase register, a Hermitian operator register, and an observable operator register. A first unitary operation is performed that extracts phase information related to a reference eigenvalue of a first operator corresponding to the observable. A window function is applied to the inner phase register. The outer quantum phase register is measured to obtain classical measurement results that are used to determine the expectation value of the observable for the reference eigenstate.

Description

Attorney Docket No. 7246-02601PsiQ-600W01Quantum Systems and Methods for Windowed Estimation of Expectation Values of Arbitrary Observables on Quantum StatesTechnical Field

[0001] Embodiments herein relate generally to quantum computational methods, systems and devices, for emulating physical systems.Background

[0002] Calculating aspects of quantum systems such as molecular energies is an important potential application of fault-tolerant quantum computing in quantum chemistry. This problem has been studied and refined thoroughly, as resource estimates for computing eigenenergies of systems with ~ 100 orbitals up to chemical accuracy have improved from O(1016) non-Clifford gates to Q(1010) non-Clifford gates over the years. However, while calculating molecular energies is useful in some applications, it may also be desirable to learn other expectation values of a wavefunction in the ground state or in another state.Accordingly, improvements in the field of sy stems and methods used to calculate expectation values of observables are desirable.Summary

[0003] Some embodiments described herein include quantum computing devices, sy stems, quantum circuits and methods for estimating an expectation value of an observable of a physical quantum system.

[0004] In some embodiments, a plurality of qubits is received, including an outer phase register and an inner quantum register. The quantum registers may be prepared in a first initial state such as a null state. The inner quantum register may include an inner phase register, a Hermitian operator register and an observable operator register, in some embodiments. A state simulation register may also be received that is prepared in a reference eigenstate of a Hermitian operator.

[0005] A first unitary’ operation may be performed that receives as input the inner quantum register qubits and the state simulation register and outputs phase information based at least in part on the input.

[0006] Performing the first unitary’ operation may include performing a block encoding of the first operator on the state simulation register and the observable operator register, performingAttorney Docket No. 7246-02601PsiQ-600W01a quantum phase estimation (QPE) operation of the Hermitian operator on the state simulation register and the Hermitian operator register, reflecting the reference eigenenergy in the inner phase register, and performing an inverse QPE operation that reverts the inner phase register toward the first initial state. In some embodiments, performing the QPE operation outputs eigenvalues of the Hermitian operator into the inner phase register, where the output eigenvalues are entangled with respective Hermitian operator eigenstates of the state simulation register.

[0007] In some embodiments, performing the QPE operation and the inverse QPE operation includes applying a window function, such as a Kaiser window, to the inner phase register.

[0008] In some embodiments, the phase information is related to the reference eigenvalue and a plurality of eigenvalues of the first operator, where the first operator corresponds to an observable. The phase information is based on a summation over the plurality of eigenvalues of the first operator of a product of the reference eigenstate, respective ones of the plurality of eigenvalues of the first operator, and respective ones of a plurality of eigenstates of tire first operator. The unitary circuit may be performed conditionally on a state of a first subset of the outer phase register qubits. Conditionally outputting the phase information may transfer the phase information from the inner quantum register to the subset of the outer phase register qubits.

[0009] In some embodiments, the outer phase register qubits may be measured with measurement circuitry’ to obtain classical measurement results. In some embodiments, a classical processor determines an expectation value of the observable for the reference eigenstate based at least in part on the first classical measurement results.

[0010] The techniques described herein may be implemented in and / or used with a number of different types of devices, including but not limited to photonic quantum computing devices and / or systems, hybrid quantum / classical computing systems, and any of various other quantum computing systems.

[0011] This Summary is intended to provide a brief overview of some of the subject matter described in this document. Accordingly, it will be appreciated that the above-described features are merely examples and should not be construed to narrow the scope or spirit of the subject matter described herein in any way. Other features, aspects, and advantages of the subject matter described herein will become apparent from the following Detailed Description, Figures, and Claims.Attorney Docket No. 7246-02601PsiQ-600W01Brief Description of the Drawings

[0012] For a better understanding of the various described embodiments, reference should be made to the Detailed Description below, in conjunction with the following drawings in which like reference numerals refer to corresponding parts throughout the Figures.

[0013] Figure 1 is a system diagram illustrating a classical and quantum computing system, according to some embodiments;

[0014] Figures 2A-C are quantum circuit diagrams illustrating high-level subroutines for a method to estimate an expectation value of an observable, according to some embodiments;

[0015] Figure 3 is a quantum circuit diagram illustrating in more detail a method for estimating an expectation value of an observable, according to some embodiments;

[0016] Figure 4 is a flowchart illustrating a method for determining an expectation value of an observable, according to some embodiments;

[0017] Figures 5A-5B are quantum circuit diagrams illustrating examples implementations of the ℛπand ℛτsubroutines, according to some embodiments;

[0018] Figure 6 is a quantum circuit diagram illustrating a collection of multi-qubit Toffoli gates to implement a reflect operator, according to some embodiments;

[0019] Figure 7 is a quantum circuit diagram illustrating a single expectation value estimation (SEVE) implementation of an inner quantum phase estimation (1QPE) operation, according to some embodiments;

[0020] Figures 8A-B are quantum circuit diagrams illustrating example circuits to implement a Vxand a Qubitization (Q) operation, respectively, according to some embodiments;

[0021] Figure 9 is a quantum circuit diagram illustrating an example circuit to implement an inverse quantum Fourier transform (QFT†), according to some embodiments;

[0022] Figure 10 is a quantum circuit diagram configured to estimate an expectation value of an observable that utilizes a window function, according to some embodiments; and

[0023] Figure 11 is a plot of a Kaiser window with various values for the parameter, according to some embodiments.

[0024] While the features described herein may be susceptible to various modifications and alternative forms, specific embodiments thereof are shown by way of example in the drawings and are herein described in detail. It should be understood, however, that the drawings and detailed description thereto are not intended to be limiting to the particular form disclosed, but on the contrary’, the intention is to cover all modifications, equivalents and alternatives falling within the spirit and scope of the subject matter as defined by the appended claims.Attorney Docket No. 7246-02601PsiQ-600W01DETAILED DESCRIPTION

[0025] Disclosed herein are examples (also referred to as ‘‘embodiments”) of quantum systems and methods for estimating expectation values of arbitrary observables in arbitrary quantum states.

[0026] Although embodiments are described with specific detail to facilitate understanding, those skilled in the art with access to this disclosure will appreciate that the claimed invention may be practiced without these details. Reference will now be made in detail to embodiments, examples of which are illustrated in the accompanying drawings. In other instances, well-known methods, procedures, components, circuits, and networks have not been described in detail so as not to unnecessarily obscure aspects of the embodiments.Overview of Quantum Computing

[0027] To facilitate understanding of the disclosure, an overview of relevant concepts and terminology is provided in the following paragraphs.

[0028] Quantum computing relies on the dynamics of quantum objects, e.g,, photons, electrons, atoms, ions, molecules, nanostructures, and the like, which follow the rules of quantum theory. In quantum theory, the quantum state of a quantum object is described by a set of physical properties, the complete set of which is referred to as a mode. In some embodiments, a mode is defined by specifying the value (or distribution of values) of one or more properties of tire quantum object. For example, in the case w'here the quantum object is a photon, modes may be defined by the frequency of the photon, the position in space of the photon (e.g., which waveguide or superposition of waveguides the photon is propagating within), the associated direction of propagation (e.g., the ^-vector for a photon in free space), the polarization state of the photon (e.g., the direction (horizontal or vertical) of the photon’s electric and / or magnetic fields), a time window in which the photon is propagating, the orbital angular momentum state of the photon, and the like,

[0029] Persons of ordinary skill in the art will be able to implement examples using any of a variety of types of quantum systems, including but not limited to photonic systems, solid state system, topological quantum computing systems, hybrid quantum computing systems, and superconducting systems, among other possibilities.

[0030] As used herein, a “qubit” (or quantum bit) is a quantum system with an associated quantum state that may be used to encode information. A quantum state may be used toAttorney Docket No. 7246-02601PsiQ-600W01encode one bit of information if the quantum state space can be modeled as a (complex) two- dimensional vector space, with one dimension in the vector space being mapped to logical value 0 and the other to logical value 1. In contrast to classical bits, a qubit may have a state that is a superposition of logical values 0 and 1. More generally, a “qudit” describes any quantum system having a quantum state space that may be modeled as a (complex) n-dimensional vector space (for any integer n), which may be used to encode log2(n) bits of information. For the sake of clarity of description, the term “qubit” is used herein, although in some embodiments the system may also employ quantum information carriers that encode information in a manner that is not necessarily associated with a binary’ bit, such as a qudit or a plurality of qubits encoded to form an error-corrected logical qubit. For example, embodiments herein for quantum computational methods and circuits that utilize fault-tolerant quantum computing schemes use the term “qubit” to refer to an error-corrected logical qubit that contains a plurality of physical qubits entangled together in an errorcorrecting code.

[0031] Embodiments herein use the term “quantum register” (and more simply, “register”) to refer to a set of one or more logical qubits used in a quantum computational method.Typically, distinct quantum registers are separated at the logical layer, i.e., different quantum registers serve distinct logical purposes in a quantum computing method or circuit.

[0032] Qubits (or qudits) may be implemented in a variety of quantum systems. Examples of qubits include: polarization states of photons; presence of photons in waveguides; or energy states of molecules, atoms, ions, nuclei, or photons. Other examples include other engineered quantum systems such as flux qubits, phase qubits, or charge qubits (e.g., formed from a superconducting Josephson junction); topological qubits (e.g., Majorana fermions); or spin qubits formed from vacancy centers (e.g., nitrogen vacancies in diamond).Figure 1 - Quantum Computing System

[0033] Figure 1 is a system diagram of a quantum computing system 101 that may be utilized to implement method steps of embodiments described herein. As illustrated, the system includes a classical computing system 103 coupled to a quantum processing unit (QPU) 105 over a classical channel 112. The classical channel may relay classical information between the classical computing system and the QPU,

[0034] In some embodiments, the classical computing system 103 includes one or more non- transitory computer-readable memory media 104, one or more central processing unitsAttorney Docket No. 7246-02601PsiQ-600W01(CPUs) or processor(s) 102, a power supply, an input / output (I / O) subsystem, and a communication bus interconnecting these components. The processor(s) 102 may execute modules, programs, and / or instructions stored in memory 104 and thereby perform processing operations. The processor(s) may additionally or alternatively perform operations based on information and / or instructions received from the QPU 105 over tire channel 112. The processor may comprise a dedicated processor, or it may be a field programmable gate arrays (FPGA), an application specific integrated circuit (ASIC), or a “system on a chip” that includes classical processors and memory, among other possibilities. In some embodiments, memory 104 stores one or more programs (e.g., sets of instructions) and / or data structures and is coupled to the processor(s).

[0035] The classical computing system may be classical in the sense that it operates computer code represented as a plurality of classical bits that may take a value of 1 or 0. Programs may be written in the form of ordered lists of instructions and stored within the classical (e.g., digital) memory 104 and executed by the classical (e.g., digital) processor 102 of the classical computer. The memory 104 is classical in the sense that it stores data and / or program instructions in a non-transitory storage medium in the form of bits, which have a single definite binary state at any point in time. The processor may read instructions from the computer program in the memory 104 and / or write data into memory, and may optionally receive input data from a source external to the classical computing system 103, such as from a user input device such as a mouse, keyboard, or any other input device. The processor 102 may execute program instructions that have been read from the memory 104 to perform computations on data read from the memory 104 and / or input from the QPU, and generate output from those instructions. The processor 102 may store that output back into the memory 104 and / or provide the output to the QPU over the channel 112.

[0036] The QPU 105 may include a plurality of qubits 110 and a controller 106 configured to interface with the plurality of qubits. In some embodiments, the qubits are divided into one or more independent qubit modules, where each qubit module includes a self-contained plurality of fault-tolerant qubits, and different qubit modules may be interchangeably used for various steps within a quantum computation. The controller 106 may include physical hardware to interact with and / or perform operations on the qubits, e.g., to apply quantum gates or perform other operations. In some embodiments, the controller further includes a classical processor, potentially coupled to its own dedicated non-transitory (classical) memory, that is configured to direct the physical hardware to interact with the qubits and communicate with theAttorney Docket No. 7246-02601PsiQ-600W01processor of the classical computing system 103 over the channel 112. Alternatively, the classical processor of the classical computing system 103 may directly communicate with the hardware of the controller to provide instructions for interacting with and manipulating the qubits. The qubits may be configured to evolve in time under the directed influence of the controller, and a measurement system 108 may at times perform quantum measurements on all or a subset of the qubits to obtain quantum measurement results in the form of classical data bits (e.g., ones and zeros). The classical data from the measurement results may be intermediate results that inform behavior of the classical computing system and / or the controller 106 during a quantum computation, and they may additionally include classical results of the quantum computation. In some embodiments, the QPU further includes one or more decoders configured to receive and decode the classical measurement results, and the decoded measurement results may be provided to the classical computing system for processing. The measurement results may be communicated to the classical computing system and / or the controller 106, and further the classical computing system may provide directions and / or instructions to the controller 106 and the measurement system 108 to guide the behavior of the QPU while performing a quantum computation.Calculating an Expectation Value of an Observable of a Quantum System

[0037] In some quantum computing applications, it may be desirable to compute expectation values on the energy eigenstate wave function once the eigenenergy has been found. This could include expectation values of the nuclear gradients in geometry optimization and molecular dynamics, the electrical multipole moments of a molecule to describe the charge distribution, or the separate parts of the Hamiltonian operator like the kinetic energy and potential energy. For many classical methods, once the (approximate) solution to the Schrodinger equation has been found, obtaining expectation values has a low cost. However, estimating expectation values on a quantum computer is more involved and may be even harder than estimating eigenenergies.

[0038] A simple approach to solving this problem is to repeatedly prepare the reference state and measure the components of the observable. The expectation value of observable is then reconstructed from tire recorded measurement outcomes. This approach has several drawbacks. For example, since only the reference state eigenenergy is known, the state itself would be identified with (an equivalent of) quantum phase estimation, and one would typically either increase the number of samples through a “repeat until success” stateAttorney Docket No. 7246-02601PsiQ-600W01preparation or increase tire complexity of each sample using a method for eigenvalue filtering. Note that phase estimation with respect to the observable will not solve the expectation value problem any faster: one would only estimate the eigenvalues of the observable, not its expectation values, with respect to the Hamiltonian eigenstate.

[0039] Embodiments herein describe methods, quantum circuits, and quantum computing systems for calculating an expectation value of an observable for a quantum system in an eigenstate of a Hamiltonian. Each eigenstate of the Hamiltonian has a corresponding eigenenergy, which is an eigenvalue of the Hamiltonian. The eigenenergies may be obtained using quantum phase estimation. For expectation values of a general observable corresponding to a first operator, the situation is much different as the state of the quantum system, referred to herein as the ‘"reference state”, is not typically an eigenstate of the observable. In general, the reference state is an eigenstate state of a given Hamiltonian without a specified relationship to the observable or the first operator corresponding to the observable. Still, by reframing the expectation value problem as an eigenvalue problem for a related operator, embodiments herein calculate expectation values with techniques based on quantum phase estimation. The expectation values may often be calculated for the ground state of the Hamiltonian in many applications, but more generally the methods described herein may be applied to a quantum system in an arbitrary eigenstate of the Hamiltonian.

[0040] Embodiments herein perform an expectation value calculation using quantum phase estimation. An expectation value computation is formulated that utilizes the fact that the observable, as well as the Hamiltonian, may be block-encoded. Advantageously, large-sized optimization problems are avoided by replacing quantum singular value transformation (QSVT)-heavy subroutines within the computation by quantum phase estimation routines which are less complex in terms of QSVT, While this utilizes marginally more qubits, this cost is compensated for by finding the optimal phase factors much more easily.Figures 2-3 - Method for Calculating Expectation Value of an Observable

[0041] Figures 2A-C are quantum circuit diagrams illustrating subroutines for estimating an expectation value of an observable, according to some embodiments. Given an observable in the form of a Hermitian operator F, as well as a Hamiltonian H, it may be desirable to determine the expectation value of F with respect to an eigenstate of the Hamiltonian|

[042] ^E\F\yfE), (1)Attorney Docket No. 7246-02601PsiQ-600W01

[0043] where H \ yrs) = E i ips), and only the eigenenergy E corresponding to the eigenstate 11 / / .?) is known. In some embodiments, the expectation value represented by Equation 1 may be computed by repeatedly querying block encodings of F and H, Figures 2A, 2B and 2C illustrate the quantum circuit at different levels of detail. At a high level of abstraction of the method, shown in Figure 2A, outer phase register qubits 204a-d are received and initialized into an initial state at gate 202, and simulation qubits205 are received and prepared in an ansatz state using ansatz state preparation (ASP) at gate 203. Note that as used herein, a “gate” refers to a quantum circuit configured to perform a particular operation or subroutine on one or more qubits. Gates may exist hierarchically, e.g., a gate may itself contain one or more sub-gates, as illustrated in Figure 2C for the gate U. In some embodiments, the simulation qubits may include inner register qubits that serve as a “workspace” to store working information related to the encoded Hamiltonian (enc[HJ), the block encoded operator F (enc[F]), and phase information (inner phase during the computation, and state simulation qubits |sim) to simulate the state of the quantum system. Tire simulation qubits may be processed within the unitary operators Un. A quantum phase estimation (QPE) operation 201 is performed on the qubits, and the outer phase register qubits are then measured at 224 to produce classical measurement results to calculate the desired expectation value. Measuring the qubits at 224 produces, with high probability, the measurement outcome as the binary representation of a fixed-point number:

[0044] M = - ( 1 ± - arccos — (2)

[0045] where ± indicates that two solutions are possible, and F is normalized to ||Fi| < 1 by the block encoding of F. The expectation value may then be extracted from Eq. (2), for example, (I / EF\I / / E) = 2cos(±rr(2M — 1)) — 1. Advantageously, described embodiments may be able to distinguish which of the two solutions are obtained, to accurately extract the desired expectation value. Said another way, unlike some other estimation computations, the expectation value estimation is unambiguous in the sign of the expectation value, meaning that the sign ± in Eq. (2) does not obfuscate the sign of (^siF|The first unitary operation U 206 shown in Figure 2C may therefore be implemented without additional controls that are characteristic of some overlap estimation routines. Also, in some embodiments the computation may allow for a phase estimation with a larger response to (< E\F\! / E) - enabling estimation of the angles ±arccos(^i F| i s) and reducing the complexity' by a factor of two.Attorney Docket No. 7246-02601PsiQ-600W01

[0046] Figure 2B illustrates in greater detail the circuit operations involved in the QPE operation 201. As illustrated, Hadamard gates are depicted with a non-italicized capital H (to distinguish from the Hamiltonian, which is depicted as an italicized capital H) and are applied to the outer phase register qubits 204a-d to place them into the i+) state. A sequence of U2nunitary operators for n = {1,2, 3, 4} are then applied to the PFE,-,., qubits 205, controlled on the state of distinct subsets of the outer phase register qubits (as illustrated in Figure 2B, the control is based on the 11 } state, illustrated as a black dot, although other controls are also possible). In various embodiments, larger or smaller values for nmax may be utilized (e.g., rimax=4 in the example illustrated in Figure 2B) to increase or decrease the numerical precision of the estimated expectation value with a longer or shorter computation, respectively. Note that U2refers to a sequential application of two instances of the U operation 206, U4performs four such applications, etc.

[0047] Figure 2B illustrates the control 216 as a black dot, indicating a control over the | 1) state of a subset of the outer phase register qubits, but it is within the scope of the described embodiments to control the unitary operators (e.g., the first unitary’ operation 206) based on the |0) state, or more generally based on any state of the outer phase register qubits.Controlling the operators U” on the state of the outer phase register in this manner performs phase kickback to transfer phase information from the inner register qubits to the outer phase register qubits. Accordingly, the desired information used to determine the expectation value of the observable may be obtained by measuring the outer phase register qubits at gate 224 after performing an inverse quantum Fourier transform QFT†at gate 217. Subsequent terms in the sequence of U2nare utilized to transmit subsequent significant digits (in binary representation) of the information related to the expectation value of the observable, which are stored in respective subsets of the outer phase register qubits.

[0048] Figure 2C illustrates the quantum circuit elements of the unitary operator U in greater detail, according to some embodiments. As illustrated, the unitary operator U is divided into two rotation operators, ℛπ224 and ℛτ226. The ℛπoperation consists of an inner quantum phase estimation (iQPE, 210), a reflect operator (Refl, 212), and an inverse iQPE (iQPE†, 214). Note that “inner” QPE is simply an identifier to distinguish the QPE subroutine that occurs within the unitary operator U from the QPE operation 201 that performs the overall estimation of the quantum phase. A similar distinction is used between the inner phase register qubits that are operated on within the unitary operator U and the outer phase register qubits which are measured at step 224. The operator 7?^ acts upon the state simulation qubits,- 10 -Attorney Docket No. 7246-02601PsiQ-600W01the encoded Hamiltonian register qubits, and the inner phase register qubits, where the REFLECT operator (labelled Refl 212) is controlled on a state of the encoded F operator register qubits (controlled on the null |0) state as illustrated by an open circle 220 in Figure 2C, although the control could be on another state of the encoded F operator register qubits). The Rτoperation 226 consists of a block encoding of the F operator, which acts on the state simulation qubits and the encoded F operator register qubits, controlled on a state of the encoded Hamiltonian register qubits (controlled on the null |0) state as illustrated by an open circle 222 in Figure 2C, although the control could be on another state of the encoded Hamiltonian register qubits).

[0049] Figure 3 is a circuit diagram that is similar in certain respects to the diagrams shown in Figures 2A-C, although Figure 3 illustrates further aspects of the described methods and describes the subroutines with a greater level of detail, according to some embodiments. For example, block 301 illustrates how, at a high level, the single expectation value estimation (SEVE) method may involve an initial estimation of the eigenenergy of the reference state and an energy gap at step 302, if called for by the particular application. As one example, when the reference eigenstate is the ground state of the Hamiltonian, the energy gap between the ground state energy and the first excited state energy may be estimated to determine whether the quantum system is in a thermal regime that will stably exist in the ground state, such that an expectation value of the observable while the system is in the ground state is a meaningful quantity. Additionally or alternatively, a single instance SEVEi of the expectation value estimation method may be iteratively repeated a plurality of times at step 304, where the subscript i is an index to denote each instance, and the results may be averaged to obtain a more accurate estimate of tire expectation value.

[0050] Figure 3 illustrates in the quantum phase estimation (QPE) subroutine 308 (e.g,, the QPE operation 201 as described above) that the outer phase register qubits may be divided into m subsets of register qubits, each of which is prepared in the null state before having a Hadamard gate applied. In addition, the order of the rotation operators330 and328 is reversed relative to Figure 2. It is within the scope of the described embodiments to make various modifications to the order of operations within the unitary operator U, without adversely affecting the estimation of the expectation value. For example, flipping the order of and simply introduces a minus sign to the phase of the measured quantity. For example, in some embodiments iQPE[H]† 324 is performed first, followed by BE[F] 318,Attorney Docket No. 7246-02601PsiQ-600W01iQPE[H] 320, and REFLECT 322 for each instance of U. Various further aspects of Figure 3 will be explained in greater detail below in reference to the flowchart of Figure 4.Figure 4 - Flowchart for Calculating Expectation Value of an Observable

[0051] Figure 4 is a flowchart diagram illustrating a method for calculating an expectation value of an observable in a physical quantum system. Tire method shown in Figure 4 may be used in conjunction with any of the computer systems or devices shown in the above Figures, among other devices. For example, the method shown in Figure 4 may be performed by a quantum or classical / quantum hybrid computing device or the quantum computing system 101 as illustrated in Figure 1. In some embodiments, the described quantum circuit may be implemented in any of a variety of types of quantum computing systems, including but not limited to photonic, semiconductor, superconducting and / or topological quantum computing systems. The quantum computing system may be configured to direct the described method steps, and may include (or be coupled to) tire classical computing system 103 for processing classic information and directing operations of the quantum computing device.

[0052] In some embodiments, the methods described in reference to Figure 4 may be used in combination with the quantum circuit diagrams shown in Figures 2A-C, 3, and / or 10. It is to be understood this method may be used by any of a variety of types of quantum computing architectures, and these other types of systems should be considered within the scope of the embodiments described herein. As illustrated, the method shown in Figure 4 may proceed as follows.

[0053] At 402, multiple quantum registers are received that have been prepared in a plurality’ of respective initial states (e.g., a null (|0)) state or another initial state). The quantum registers may include an inner quantum register and an outer quantum phase register 312. The inner quantum register may include an inner phase register 334, a Hermitian operator register (enc[H], 336), and an observable operator register (enc[F], 340). The inner phase register 334 (sometimes referred to as a first subset of the inner quantum register) may be utilized to store phase information within the unitary operation that extracts information related to the expectation value of tire observable, and the outer phase register 312 may be utilized to store phase information that will be measured to obtain classical measurement results. The observable operator register (sometimes referred to as a second subset of the inner quantum register) may be utilized as a workspace for temporary storage of quantum informationAttorney Docket No. 7246-02601PsiQ-600W01related to the first operator, where the first operator F corresponds to the observable whose expectation value is to be calculated, Fmay be any type of operator that corresponds to a physical observable quantity. Similarly, the Hermitian operator register (sometimes referred to as a third subset of the inner quantum register) may be utilized as a quantum workspace for performing quantum phase estimation on the Hermitian operator.

[0054] The plurality of received quantum registers may also include a state simulation register (sim, 338) prepared in a reference eigenstate of the Hermitian operator, where the reference eigenstate has a reference eigenvalue. In some embodiments, the Hermitian operator is the Hamiltonian, and the reference eigenstate may be a ground state or an excited state of the Hamiltonian. More generally, the reference eigenstate may be an eigenstate of an arbitrary’ Hermitian operator. For specificity, explicit examples are discussed herein in reference to the energy eigenstate of the Hamiltonian, however, it is within the scope of some embodiments to calculate the expectation value of an observable for a quantum system in an eigenstate of an arbitrary Hermitian operator.

[0055] In some embodiments, the combination of the inner quantum register and the state simulation register is collectively referred to as the simulation qubits |Ψ〉sim.

[0056] In some embodiments, a Hadamard gate is applied to the outer phase register. In some embodiments, the Hadamard gate transforms the outer phase register from its initial state (e.g, a null 0) state) to a superposition of states (e.g., a superposition of the |0) and |1) states such as the | +) state). When the outer phase register is prepared in a superposition of |0 and |1) states, the conditional execution of the unitary circuits I / " on a state (such as the 1) state) of the outer phase register results in phase kickback, to transfer phase information from tire inner quantum register to the outer phase register. For example, the unitary circuit U extracts phase information related to the expectation value of the observable, and the conditional execution of U transfers this information to the outer phase register via phase kickback. Phase kickback may be understood as a result of the quantum entanglement between the inner quantum register and the outer phase register. Wren the unitary’ circuit U operates controlled on the 11) state of the outer phase register, and the outer phase register is in a superposition of the |0 and 1) states, the |1) subspace of the state of the outer phase register will accumulate a phase eiφextracted by the unitary U, whereas the 1) subspace of the state of the outer phase register will not accumulate this phase. Note that, in some embodiments, the quantum circuit may be modified such that the outer phase register is prepared in another initial state (e.g., the 11) state or the | +) state). More generally, the outer phase register may be prepared in a- S3 -Attomey Docket No. 7246-02601PsiQ-600W01superposition of at least one state that controls the conditional execution of tire unitary circuits Unand at least one other state.

[0057] In some embodiments, double phase kickback is performed wherein the first unitary circuit U is applied conditioned on the |1〉 state of the outer phase register and the inverse of the first unitary circuit U† is applied conditioned on the |0〉 state of the outer phase register.

[0058] At 404, a first unitary circuit U 316 (or an inverse of the first unitary circuit U†) is applied to the simulation qubits |Ψ〉sim. Tire first unitary circuit may receive as input the inner quantum register and the state simulation register, and outputs phase information to the outer quantum phase register (e.g., via the entanglement introduced through phase kickback) based at least in part on the input. The output phase information may be related to the reference eigenstate and a plurality of eigenvalues of the first operator F corresponding to an observable.

[0059] In more detail, applying the first unitary’ circuit may include performing a block encoding of the first operator on the state simulation register and the observable operator register 328. Note that the “first unitary circuit” refers to the circuit or operator U 316 in Figure 3, whereas the “first operator” refers to the operator F whose expectation value is being calculated. Performing the block encoding of the first operator on the state simulation register and the observable operator register outputs the result of applying the first operator to the state simulation register into the observable operator register, in a subspace of the observable operator register that remain in the first initial state (e.g., the null state or another initial state). The block encoding of the first operator may be performed conditionally when the Hermitian operator register is in the first initial state (i.e., it may be performed controlled on the state of the Hermitian operator, as indicated at 318 and 1038). Performing a block encoding of the first operator may extract phase information related to the plurality of eigenvalues of the first operator into the inner quantum register. The block encoding of the first operator may be further controlled on a state of the inner phase register, as shown at 1038 of Figure 10. Advantageously, this may facilitate the effectiveness of applying a window function to the inner phase register.

[0060] Tire first unitary operation may further include applying an inner quantum phase estimation (iQPE) gate 320 of the Hermitian operator on the state simulation register and the Hermitian operator register. Applying the iQPE gate inputs first information related to the reference eigenvalue into the inner phase register, and applying the iQPE gate entangles the first information with the state simulation register. Said another way, applying the iQPE gateAttorney Docket No. 7246-02601PsiQ-600W01entangles the first information related to the reference eigenvalue with respective eigenstates of the Hermitian operator encoded in the state simulation register. In some embodiments, the first unitary' circuit implements qubitization for the Hermitian operator when applying the iQPE gate.

[0061] In some embodiments, the iQPE operation utilizes a single value expectation estimation (SEVE) procedure to input the first information into the inner phase register. The SEVE procedure selects a total number of qubits in the inner phase register to be larger than a minimum number of qubits for encoding eigenvalues of the first Hermitian operator. In some embodiments, a SEVE+ procedure is used to implement the iQPE operation. An example of SEVE+ is illustrated in Figure 7. SEVE+ involves repeatedly inputting the first information into the inner phase register to perform a rounding procedure for the reference eigenvalue.

[0062] In some embodiments, the iQPE operation proceeds as described in the quantum circuit diagram shown in Figure 10, For example, applying the iQPE gate may include applying a window function to the inner phase register, applying a plurality of qubitization operators to the state simulation register and the Hermitian operator register controlled on qubits of the inner phase register, and performing an inverse quantum Fourier transform on the inner phase register.

[0063] In some embodiments, a reflect gate is applied to reflect the reference eigenvalue in the inner phase register 322. Applying the reflect gate to the inner phase register introduces a minus sign to the reference eigenstate in the state simulation register, where the minus sign is introduced via the entanglement of the first information with the state simulation register. Reflecting the reference eigenvalue may be performed conditionally when qubits of the observable operator register are in the null state.

[0064] In some embodiments, an inverse QPE operation is performed 324, where the inverse QPE operation reverts the inner phase register toward their initial state (e.g., the null state). Note that finite precision effects may cause the inverse QPE operation to revert the inner phase register qubits closer to, but not exactly to the initial state. Some embodiments apply a window function W to the inner phase register to reduce the impact of these finite precision effects on the expectation value calculation.

[0065] Note that while Figures 2 and 3 illustrate an ordered sequence of iQPE, followed by Refl, followed by iQPE†. in some embodiments the order of these operations may be rearranged without adversely affecting the outcome of the computation (e.g., an overall phaseAttorney Docket No. 7246-02601PsiQ-600W01may be introduced into the output of the computation, which may be corrected for while determining the expectation value of the observable).

[0066] Advantageously, tire combination of the QPE operation, the conditional reflection, and the inverse QPE extracts phase information related to the eigenvalue of the reference state into the inner phase register. The phase information may be based on a summation over the plurality of eigenvalues of the first operator of a product of the reference eigenstate, respective ones of the plurality of eigenvalues of the first operator, and respective ones of a plurality of eigenstates of the first operator. As described below, conditionally performing the first unitary’ circuit based on a state of the outer phase register results in phase kickback to extract this phase information (along with phase information related to the eigenvalues of the first operator) into the outer phase register.

[0067] In some embodiments, the first unitary operation is performed with finite numerical precision, which may cause the inverse QPE operation to not revert the Hamiltonian register qubits entirely to the initial state. For example, the finite numerical precision may introduce artifacts whereby the QPE operation causes the Hermitian operator register to contain small components that are not in the initial state. To address this, in these embodiments performing the inverse QPE reverts the Hermitian operator register closer to the null state. Again, finite precision effects may cause the inverse QPE operation to revert the Hermitian operator register qubits closer to, but not exactly to the initial state.

[0068] In some embodiments, the reference eigenvalue is precomputed, and the reference eigenvalue is reflected based at least in part on the precomputation of the reference eigenvalue.

[0069] In some embodiments, the first unitary operation is applied conditionally based on a particular state (e.g., the |1) state) of a subset of qubits within the outer phase register (e.g., via the control 216), Applying the first unitary operation conditionally in this manner transfers the phase information from the inner quantum register and the state simulation register to the subset of qubits of the outer phase register.

[0070] At 406, an inverse quantum Fourier transform (QFT†) circuit is applied to the outer phase register 312. Tire QFT†operation may transform the outer phase register into a form that is more readily measured to produce decodable classical measurement results. Note that, when applied to a unitary' matrix, the adjoint operation QFT†is the same as the inverse operation QFT-1, An example of a QFT†circuit is shown in Figure 9.Attorney Docket No. 7246-02601PsiQ-600W01

[0071] At 408, a projective measurement is performed on the outer phase register (314) after performing QFT† to obtain first classical measurement results. Each subset of the outer phase register qubits (e.g., the m sub-registers illustrated at 312 of Figure 3) may be measured to produce classical measurement results that contain information related to subsequent significant digits (in binary') of the expectation value estimation.

[0072] At 410, an expectation value of the first operator for the reference eigenstate is determined based at least in part on the first classical measurement results. For example, the quantity shown in Equation 2 may be determined from the first classical measurement results, from which the expectation value of the operator F may be determined. Advantageously, determining the expectation value of the observable for the reference eigenstate based at least in part on the first classical measurement results may be able to distinguish the sign of the expectation value.

[0073] In some embodiments, the expectation value may be determined with a first level of numerical precision. The level of numerical precision may be received as user input prior to running the computation, or it may be preconfigured for the computation. In these embodiments, the inner quantum register may be implemented at a second level of numerical precision independent of the first level of precision.

[0074] In some embodiments, the method further includes successive application of the first unitary operation 2ntimes to obtain subsequent significant digits of the determined expectation value of the observable. For example, for n={l,...,m-l }, where m is a positive integer greater than 1, the first unitary operation may be applied 2” times. This is illustrated in Figure 3 as the sequence of gates U, U2, U4,...,The first unitary operation may be applied 2ntimes conditionally on a state (e.g., the 11) state) of a respective second subset of the outer phase register qubits. Conditionally applying the first unitary operation transfers respective second phase information from the state simulation register to the respective second subset of the outer phase register qubits. The respective second phase information transferred to the respective second subset of the outer phase register encodes a respective significant digit of the expectation value of the first operator.Additional Technical Detail

[0075] The following numbered paragraphs provide additional technical detail and description regarding embodiments herein, discuss the inner workings of the described methods, and develop an expression for the computational complexity of the computation. InAttorney Docket No. 7246-02601PsiQ-600WOlsome embodiments, the expectation value estimation is based on the quantum phase estimation computation. Quantum phase estimation utilizes the phase kickback of a unitary operator U. It may be desirable for If to project an input state into eigenstates of the operator U. while also outputting the phase angle 6 of the corresponding eigenvalue exp( / 2zr on a separate register. To estimate expectation values, an instance of the phase estimation routine may be utilized, called quantum phase estimation (QPE). QPE features a first unitary’ circuit U, depicted in 2C. In some embodiments, U is the product of two self-in verse operators.

[0076] The expectation value estimation procedure defines the unitary operator U, such that at least one of its eigenphases θ is a function of 〈ψE|F̂|ψE〉, namely Eq. (2), Since the projection into eigenstates happens probabilistically according to the overlap of output and input states, it may be desirable to prepare an input state with substantial overlap with the eigenstates used to estimate tire expectation value. The unitary operator shown in Figure 2(c) may be divided into two subcircuits, ℛπandThe subcircuitis a block encoding of F, while the subcircuit ℛπuses block encodings of H and information about the eigenenergy E, as well as the spectral gap A of H. As one example, the 1-norms of H and F after factorization are identified as λHandrespectively. Since the subcircuit ℛπfeatures quantum phase estimation routines (inside this quantum phase estimation), it makes at least O(λH / Δ) queries to the block encoding of H. However, due to discretization errors, it may be desirable to improve the accuracy of these phase estimation routines to match the demands of the entire expectation value estimation computation, To achieve a target error e, the complexity of the iQPE routines may be increased by a factor χ(ε̃). To estimate the expectation value up to an error ε, the operator U may be repeated a total of O(λF / ε) times. Matching the target error of iQPE, with the resolution of QPE, ε̃ = ε / λF, we find that the entire expectation value estimation computation has a query complexity of

[0077]

[0078] The upper phase register qubits 204a-d in Figure 2A may contain O(log(> Q) qubits. The |Ψ〉simqubits 205 are split into four subsets as follows:

[0079] 1) phase: a register of at least O(λH / Δ) qubits to store expressions for qubitization eigenphases of H.

[0080] 2) enc[H]: the auxiliary qubits for the block encoding of H.

[0081] 3) sim: a collection of qubits representing the simulated quantum system.

[0082] 4) enc[F]: the auxiliary qubits in the block encoding of the observable F̂.Attorney Docket No. 7246-02601PsiQ-600W01

[0083] The operators H and F may be defined to have the following spectral decomposition:

[0084] Ĥ = ΣEE|ψE〉〈ψE|, (4)

[085] (5)

[0086] where |ψE〉 and |φη〉 are eigenstates corresponding to the energies E and eigenvalues η, respectively, which are normalized to have values in the range from -1 to +1. Knowing the complete spectrum of H and F (e.g., the exact values of E and in the sums of Eq. (4) and Eq. (5)) is not required. Rather, the method may be deployed with knowledge of a single eigenenergy energy E = Ejthat corresponds to the eigenstate in which we are determining the expectation value of the observable.

[0087] The unitary operator U may be defined as the product of two reflection operators Rπand Rτ, which each are defined through their respective projectors π̂ and τ̂:

[0088] Rπ= 1 - 2π̂,

[0089] Rf= l ” 2t. (6)

[0090] The projectors π̂ and τ̂ are not necessarily known. Only their respective reflections need to be constructible. Also, the projectors are of arbitrary rank. The singular values of their product may be described as:

[0091] τ̂ · π̂ = Σkwk|tk〉〈pk|, (7)

[0092] Left and right singular vectors \pk), \tk) of the same singular value wk> 0 may now be used to construct two eigenstates |vk+〉 and |vk-〉 of the iterate RτRπ. The result is

[0093] |vk±〉 = (1 / √2)(|pk〉 ± i|pk⊥〉) = (exp(±i·arccos wk) / √2)(|tk〉 ∓ i|tk⊥〉)

[0094] where \p ) is the component of \tk) orthogonal to \pkand vice-versa for |tk):

[0095] |pk⊥〉 = (|tk〉 - wk|pk〉) / √(1-wk2), |tk⊥〉 = (|pk〉 - wk|tk〉) / √(1-wk2). (9)Nk k

[0096] The corresponding eigenvalues of |vk±〉 are expression of wkas

[097] U|vk±〉 = exp(±i2arccos wk)|vk±〉. (10)

[0098] Using the operator 풰 =within phase estimation would thus estimate the eigenphases ±i2arccos wk. Tire strategy for expectation value estimation consists of making sure at least one of the singular values wkis an expression of 〈ψG|F̂|ψG〉 such that by measuring the eigenvalues wk, the expectation value may be computed.

[0099] The subcircuit ℛπcontains phase estimation routines to be performed on the phase register, enc[H], enc[F], and / or the state simulation qubits. Within the iQPE subroutines, qubitization may be performed, as a specific example of the more general method. InAttorney Docket No. 7246-02601PsiQ-600W01qubitization, ℛτis a reflection on the all-zero state |0〉 in the enc[H] register, whileℛπ= 풷[Ĥ] is a block encoding of Ĥ on the sim and enc[H] registers. Block encodings are defined to be self-inverse, and therefore they are reflections. With the singular values √(1 - E) / 2, the qubitization iterate has eigenphases TarccosE associated with eigenstates QE,±)'.

[0100] |QE,±〉 = (1 / √2)(1 ± ⊗ |0〉enc[H]■ (11)

[0101] With iQPE operating on the phase register, it would ideally output computational basis states |ΘE,±〉, such that

[0102] iQPE|QE,±〉 ⊗ |0〉phase= |QE,±〉 ⊗ |ΘE,±〉phase, (12)

[0103] where |ΘE,±〉 is the finite-qubit binary representation of an integer ΘE,±. related to the eigenphase TarccosE. Reflections may then be indirectly implemented on certain qubitization eigenstates |QE,σ〉 via operations on the combined sim and enc[H] registers controlled by the phase register. To that end, the Re fl (Reflect) operation is introduced, which is an arithmetic reflection that tags a set M of integers m in the computational basis of the phase register by means of Toffoli gates and data-loaders, such that

[0104] Refl = 1 - 2 Σm∈ℳ|m〉〈m|phase, (13)

[0105] where |m〉 is a binary representation for the integer m just like |ΘE,±〉 is for ΘE,±. Actually knowing the integers ΘE,σfor energies E and signs σ = ± allows implementation of a reflection on the corresponding qubitization eigenstates,

[0106] iQPE†Refl iQPE = 1 - 2 ΣE,σ|QE,σ〉〈QE,σ| ⊗ ϱ(E,σ)phase, (14)

[0107] where ϱ(E, σ) is a projector in the phase register, that is equal to the projection onto the all-zero state |0〉〈0| only for tuples (E', σ') where ΘE',σ'∈ ℳ. This reflection may be utilized for the expectation value estimation computation.Expectation value estimation

[0108] The expectation value estimation routine is another example of the singular value estimation. The latter is a block encoding ℬ[F̂] of the observable F̂ on the sim and enc[F] registers. The block encoding ℬ[F̂] is a reflection on the state |ωη〉 associated with the eigenvalues η, such that

[0109] ℬ[F̂] = 1 - 2 Ση|ωη〉〈ωη| (15)

[0110] The states |ωη〉 have a relationship with the eigenstates of F̂.

[0111] Using the shorthandAttorney Docket No. 7246-02601PsiQ-600W01

[0112] |000) = |0,,)sira® |0)ent[F], (16)

[0113] we can write

[0114] (tV|0,; O) = <5^ (17)

[0115] Indeed, setting

[0116] (18)

[0117] where |0^; O1) is the orthogonal component of m^) with respect to 0^; 0), gives B [F] its characteristic matrix form, verifying Eq. (16). Th controlled block encoding of F is related to the reflectionas follows:

[0118] 3?T= |0><0|enc[w]® B[F] + 1 - |0)(0|enc[Hj(19)

[0119] such that

[0120] t = |0)<0|enc[ff]® hM L (20)

[0121] whileis identified with a version of Eq. (14) where Refl is controlled on the all¬ zero state in the enc[F] register yielding

[0122] 7T — 10)(01enc[j=-] 0 Jfyu I QE TKQE. IJ I 0?(fy °’)phase (21)

[0123] = X. \K 0| ■ I QE,. XQE,. I 0 ^phase- (22)

[0124] Using Eq. (16), the product of the two reflections is

[0125] f • 7T = ET; (|0>enCpq ® ■ (<(M ® (OlencfF]) 0p(E, <7)phase (2-)

[0126] The square of the singular values may be obtained by solving the eigenvalue problem of

[0127] t • ft • f = Xfc Wk tk){tk(24)

[0128] — Y, E, E,(T,a |0)(0|enc[F] ® | QE,< J){QE,8 I 0 i?(F, <?) phase ' Q (E> ^Ophase

[0129] X S,;^>E|0fy(0,|0g), (25)

[0130] where the spectral decomposition of the observable in Eq. (5) has been utilized. When in Eq. (14) only contains 0G sfor a fixed o — s (determined by the state preparation preceding the expectation value estimation routine), then only p(G,s) = 10)(01 holds, and the other p(F, cr) are orthogonal to it, such that Q(G, S) • p(F, cr) —8E G 1;|0)(0. One solution to Eq. (24) is thereforeAttorney Docket No. 7246-02601PsiQ-600W01

[0131] |QG, S) ® l°)enc[Fj ® l°)phase (26)

[0132] and its eigenvalue is (ipGFPhase estimation with the iterateTtntherefore allows estimation of the phase angles

[0133] ±2arccosJ^^l£>= ±7r+arccos22^£i^2. (27)

[0134] when the input state is in Eq. (26). Note that there are variations of the unitary operator U. In some embodiments, Eq. (14) may be used as a stand-in for the reflection 1 - 2|\| / G)(1| / G| on the sim register. Such a reflection may be implemented with QSVT techniques instead, but this would involve finding highly-optimized phase factors. An example of these embodiments is shown in Figure 5A. In some embodiments, another version of the circuit may be used where QPE is used within Rn, but with a Hamiltonian simulation method different from qubitization. While qubitization has an asymptotically optimal scaling for normalized Hamiltonians, other methods may be more efficient in some cases. The corresponding circuit diagram for these embodiments is shown in Figure 5B. Another variation of the first unitary’ circuit is similar to what is shown in Figure 2C, but the set M in Refl includes 0C,+and 0G_ at the same time. All three variations of the iterate may allow estimation of the angles ±arccos(i / iG|F|i ic, which is twice as much signal as in Eq. (27).

[0135] Figure 6 is a quantum circuit diagram illustrating an example implementation of a reflect operation, according to some embodiments. The illustrated quantum circuit includes a collection of multi-qubit Toffoli gates to implement the reflect operation, which apply phase flips to a bit string "s’ such that Refl |s) = — |s). In the illustrated example, the bit strings are m = { 0111, 1000}.

[0136] Figure 7 is a quantum circuit diagram illustrating a single expectation value estimation (SEVE) implementation (referred to as SEVE+) of an inner quantum phase estimation (iQPE) operation, according to some embodiments. Figure 7 is one example of how iQPE 320 in Figure 3 may be implemented. X(a) and Z([3) are X and Z rotations about the angles a and p, respectively, following the convention X(α) ≔ cos α + i sin αX. The shaded areas are repeated d times with individual angles φk. SEVE+ differs from SEVE in that SEVE+ repeats the Vncircuit d times, which performs a rounding procedure for the reference eigenvalue (e.g., the ground state energy). SEVE+ inputs a more accurate estimate of the reference eigenvalue into the inner phase register qubits by repeating the V circuit. In contrast, an SEVE implementation of iQPE utilizes a larger number of inner phase register qubits to reduce the impact of finite precision effects on the input reference eigenvalue. ForAttorney Docket No. 7246-02601PsiQ-600W01example, in SEVE, a total number of qubits in the inner phase register qubits is selected to be larger than a minimum number of qubits for encoding eigenvalues of the first Hermitian operator.

[0137] Figures 8A and 8B are quantum circuit diagrams illustrating example circuits to implement a Fx and a Qubitization (Q) operation, respectively, according to some embodiments. Figure 8A illustrates a quantum phase estimation building block that rotates the qubit phase [x] according to the phase kickback of the oracles Q on the last register. Here, 0 is the qubitization iterate and the last register is the combination of the qubits sim and enc[H], A circuit Vxwith an n-qubit phase register makes an equivalent of 2x-l queries to Q for all x = (1,..., n}. Figure 8B illustrates a qubitization iterate Q, featuring the block encoding of the Hamiltonian H and a reflection on the all-zero state of the enc|H] register.

[0138] Figure 9 is a quantum circuit diagram illustrating an example circuit to implement an inverse quantum Fourier transform (QFT†), according to some embodiments. The illustrated example is designed to operate on an input of 4 qubits, although the QFT1' may be generalized to operate on any number of input qubits. In both Figures 8A and 9, Hadamard gates are denoted as H, and Rk are phase rotations Rk = |0)(0| + exp(-iπ / 2k-1)\ 1)(1| controlled on the less significant phase qubits for phase feedback.Figure 10 - Method for Windowed Calculation of Expectation Value of an Observable

[0139] Figure 10 is a quantum circuit diagram illustrating subroutines for estimating an expectation value of an observable using a window function on the inner phase qubits, according to some embodiments. The quantum circuit diagram shown in Figure 10 is similar in some aspects to the quantum circuit diagrams shown in Figures 2A-C and 3. For example, given an observable related to an operator F, as well as a Hamiltonian H, Figure 10 illustrates a quantum circuit for determining the expectation value of F with respect to an eigenstate of the Hamiltonian \(I E\F\I / / E). where / yd = \ E), and only the eigenenergy E corresponding to the eigenstate | ps) is known.

[0140] While in some embodiments the operator H is a Hamiltonian, more generally H may be any type of Hermitian operator. While some descriptions herein describe methods determining an expectation value of an operator F for an eigenstate of a Hamiltonian, it may be understood by one of skill in the art how the described embodiments may also be applied to determine expectation values of eigenstates of another type of Hermitian operator with a corresponding eigenspectrum and eigenvalues.Attorney Docket No. 7246-02601PsiQ-600W01

[0141] Blocks 1002, 1004 and 1006 illustrate nested subroutines of the quantum circuit at different levels of detail. At a high level of abstraction of the method, the quantum circuit shown at 1002 may be used as an alternative implementation of the QPE operation 201 shown in Figure 2A, in some embodiments. As illustrated, simulation qubits |l)Sim 1001 are received and prepared in an ansatz state using ansatz state preparation (ASP) at gate 1018. Note that as used herein, a “gate” refers to a quantum circuit configured to perform a particular operation or subroutine on one or more logical qubits. Gates may exist hierarchically, e.g., agate may itself contain one or more sub-gates, as illustrated in Figure 10 for the gate U.

[0142] In some embodiments, the simulation qubits may include an inner quantum register that serves as a “workspace” to store working information related to the encoded Hamiltonian BH, 1024), the block encoded operator F (BF, 1028), and phase information (inner phase, 1022) during the computation, and state simulation qubits (sys, 1026) to simulate the state of the quantum system. The simulation qubits may be operated on by the unitary operators Um, where m — {2°, 21,and N0= 2n-1. The quantum phase estimation (QPE) operation 201 may be performed on the qubits, and the outer quantum phase register is then measured at 1016 to produce classical measurement results to extract the desired expectation value 0. Measuring the qubits at 1016 produces, with high probability, the measurement outcome as the binary representation of a fixed-point number:

[0143] M = ½(1 ± 1 / π arccos p (1+⟨ψE|F̂|ψE⟩) / 2), (2)

[0144] where ± indicates that two solutions are possible, p is a tunable parameter between 0 and 1, and F is normalized to< 1 by the block encoding of F. The expectation value may- then be extracted from Eq. (2), for example, (IPE\F\I / / E) = 2cos ±n(2M — 1)) — 1.Advantageously, described embodiments may be able to distinguish which of the two solutions are obtained, to accurately extract the desired expectation value. Said another way, unlike some other estimation computations, the expectation value estimation described according to embodiments herein is unambiguous in the sign of the expectation value, meaning that the sign ± in Eq. (2) does not obfuscate the sign of {' E\F I / E). Tire first unitary-circuit U 1004 shown in Figure 10 may therefore be implemented without additional controls that are characteristic of some overlap estimation routines. Also, in some embodiments the computation may allow for a phase estimation with a larger response to (^E\F\t / E) - enabling estimation of the angles ±arccos(y / EiF| I / / E) and reducing the complexity by a factor of two.Attorney Docket No. 7246-02601PsiQ-600W01

[0145] As illustrated in Figure 10, Hadamard gates are depicted with a non-italicized capital H (to distinguish from the Hamiltonian or other Hermitian operator, which is depicted as an italicized capital II) and are applied to the outer phase register 1008 to place these qubits into the state. A sequence of ( / "unitary operators for m — {2°, 21,..., 2n°-1} are then applied to the |VQSimqubits 1001, controlled on the state of distinct subsets of the outer phase register (as illustrated in Figure 10, the control is based on the |1) state, illustrated as ablack dot 1012, although other controls are also possible). In various embodiments, larger or smaller values for no may be utilized to increase or decrease the numerical precision of the estimated expectation value with a longer or shorter computation, respectively. Note that U2refers to a sequential application of two instances of the ( / operation 1004, U4performs four such applications, etc.

[0146] Figure 10 illustrates the control of the unitary operations as black dot (e.g., 1012), indicating a control over the 11 state of a subset of the outer phase register qubits, but it is within the scope of the described embodiments to control the unitary operators based on the |0) state, or more generally based on any state of the qubits in the outer phase register.Controlling the operators Umon the state of the outer quantum phase register in this manner performs phase kickback to transfer phase information from the inner quantum register to the outer quantum phase register. Accordingly, the desired information used to determine the expectation value of the observable may be obtained by measuring the outer quantum phase register at gate 1016 after performing an inverse quantum Fourier transform QFT1at gate 1014. Subsequent terms in the sequence of Umare utilized to transmit subsequent significant digits (in binary representation) of the information related to the expectation value of the observable, which are stored in respective subsets of the outer quantum phase register.

[0147] The quantum circuit elements of the unitary' operator U 1004 are illustrated in greater detail in the center block of Figure 10, according to some embodiments. As illustrated, the unitary operator U is divided into two rotation operators,and 5T. The ℛπoperation consists of an inner quantum phase estimation (iQPE, 1006), a reflect operator (Refl, 1030), and an inverse iQPE (iQPE†, 1034). Note that “inner” QPE is simply an identifier to distinguish the QPE subroutine that occurs within the unitary operator ( / from the QPE operation 201 that performs the overall estimation of the quantum phase. A similar distinction is used between the inner phase register qubits that are operated on within the unitary' operator ( / and the outer quantum phase register which are measured at step 1016. The operator 52^ acts upon the state simulation register, the Hermitian operator register, theAttorney Docket No. 7246-02601PsiQ-600W01observable operator register, and the inner phase register. The REFLECT operator (labelled Refl 1030) operates on the inner phase register and is controlled on a state of the observable operator register (controlled on the null |0) state as illustrated by an open circle 1032 in Figure 10, although the control could be on another state of tire F operator register).

[0148] Tireoperation consists of a block encoding of the F operator, which acts on the state simulation qubits and the observable operator register, controlled on a state of both the inner phase register and Hermitian operator register (controlled on the null |0) state as illustrated by the two open circles 1038 in Figure 10, although the control could be on another state of the inner phase register and the Hermitian operator register). The R- operation shown in Figure 10 differs from the RToperation illustrated in Figure 2C in that, in Figure 10, Ti-ris controlled on a null state of both the inner phase register and the Hermitian operator register, whereas in Figure 2C,is controlled on a null state of only the Hermitian operator register. Advantageously, controllingon a null state of both the inner phase register and the Hermitian operator register enable the finite precision mitagation effects of the window function W 1040 in the iQPE subroutine 1006. For example, if ℛτis controlled on the Hermitian operator register only, performing W (in iQPE) and JK' (in iQPE*) will have a trivial effect on the inner phase register, by applying and then un-applying the window function. Controlling ℛτon a state of the inner phase register entangles the inner phase register with the block encoded observable operator, so that the combination of applying W and W†will have a nontrivial effect. Said another way, without controlling ℛτon a state of the inner phase register, the effect of W and W' on the inner phase register would be equal and opposite, therefore cancelling out and having a trivial effect.

[0149] The iQPE subroutine 1006 is illustrated in greater detail in the bottom block in Figure 10. As illustrated, the iQPE applies a window function V 1040 to the inner phase register 1022. The window function filters the inner phase qubits to remove finite precision artifacts and edge effects that result from the finite number n of logical qubits in the inner phase register. In exemplary embodiments, the window function is a Kaiser window, which may take the form

[0150] w0(x) ≜ {(1 / L) I0[β√(1-(2x / L)2)] / I0[β], |x| ≤ L / 2( o, |x| > L / 2;

[0151] where / o is the zeroth-order modified Bessel function of the first kind, x is the value encoded in binary' in tire inner phase register, L=2nis the window duration, and is a non-Attorney Docket No. 7246-02601PsiQ-600W01negative real number that determines the shape of the window. Note that, in some embodiments, the phase may be encoded in the inner phase register as <p = 2 / rx / L. In the frequency domain, [3 determines the trade-off between main-lobe width and side lobe level, which is a central decision in window design. Advantageously, applying a window function to the inner phase qubits enables utilization of desirable signal processing properties of the window function, e.g., by reducing spectral leakage. For example, tapering off signals at the ends of the time domain may eliminate aliasing effects in the frequency domain. In some embodiments, utilizing a Kaiser window function may enable the expectation value determination to proceed 10 times faster than methods that do not utilize a Kaiser window.

[0152] Figure 11 is a plot of the Kaiser w indow function for various values of the parameter P, according to some embodiments. Larger values of (3 correspond to a narrower distribution, and therefore a Fourier transform with a wider bandwidth and a lower amplitude in the sidebands. Setting p equal to zero corresponds to a rectangular window, and increasing p from zero increases the suppression of the side bands of the Fourier transform. In the context of phase estimation, this reduces the spectral leakage up to a point, at which further increase of P increases the spectral leakage again. Accordingly, in some embodiments the parameter [3 may be set to a “sweet spot” that results in a maximum suppression of spectral leakage.

[0153] Figure 10 further illustrates n applications of one or more powers of a qubitization operator Q[H] (1042, 1046). An example subroutine for a single power of the qubitization operator is illustrated in Figure 8B. Note that n, the number of qubits in the inner phase register, is independent of and may be different from no, the number of qubits in the outer phase register. Higher powers of the qubitization operator Q[H]Nmay be obtained by repeating the subroutine shown in Figure 8B N times, where N=2n-1. Finally, the iQPE subroutine performs an inverse quantum Fourier transform QFT11048. Note that while Figure 10 illustrates the window' function IF being applied prior to the qubitization operators Q[H]Nand the inverse quantum Fourier transform QFT’ 1048, the window function W may- more generally be applied in the middle, or at the end of the iQPE subroutine.

[0154] It should be understood that all numerical values used herein are for purposes of illustration and may be varied. In some instances, ranges are specified to provide a sense of scale, but numerical values outside a disclosed range are not precluded.

[0155] It should also be understood that all diagrams herein are intended as schematic.Unless specifically indicated otherwise, the drawings are not intended to imply any particular physical arrangement of the elements shown therein, or that all elements shown areAttorney Docket No. 7246-02601PsiQ-600WO1necessary. Those skilled in the art with access to this disclosure will understand that elements shown in drawings or otherwise described in this disclosure may be modified or omitted and that other elements not shown or described may be added.

[0156] This disclosure provides a description of the claimed invention with reference to specific embodiments. Those skilled in the art with access to this disclosure will appreciate that the embodiments are not exhaustive of the scope of the claimed invention, which extends to all variations, modifications, and equivalents.

[0157] The terminology used in the description of the various described embodiments herein is for the purpose of describing particular embodiments only and is not intended to be limiting. As used in the description of the various described embodiments and the appended claims, the singular forms "‘a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will also be understood that the term “and / or” as used herein refers to and encompasses any and all possible combinations of one or more of the associated listed items. It will be further understood that the terms “includes,” “including,” “comprises,” and / or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof.

[0158] It will also be understood that, although the terms first, second, etc., are, in some instances, used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first switch could be termed a second switch, and, similarly, a second switch could be termed a first switch, without departing from the scope of the various described embodiments. The first switch and the second switch are both switches, but they are not the same switch unless explicitly stated as such.

[0159] As used herein, the term “if” is, optionally, construed to mean “when” or “upon” or “in response to determining” or “in response to detecting” or “in accordance with a determination that,” depending on the context.

[0160] The foregoing description, for purpose of explanation, has been described with reference to specific embodiments. However, the illustrative discussions above are not intended to be exhaustive or to limit the scope of the claims to the precise forms disclosed. Many modifications and variations are possible in view of the above teachings. The embodiments were chosen in order to best explain the principles underlying the claims andAttorney Docket No. 7246-02601PsiQ-600W01their practical applications, to thereby enable others skilled in the art to best use the embodiments with various modifications as are suited to the particular uses contemplated.

Claims

Attorney Docket No. 7246-02601PsiQ-600W01ClaimsWhat is claimed is:

1. A quantum circuit, comprising:an outer quantum phase register;an inner quantum register prepared in a first initial state, wherein the inner quantum register comprises an inner phase register, a Hermitian operator register, and an observable operator register;a state simulation register prepared in a reference eigenstate of a Hermitian operator, wherein the reference eigenstate is associated with a reference eigenvalue;a first unitary circuit configured to:receive as input the inner quantum register and the state simulation register; apply a block encoding gate for a first operator of an observable, wherein applying the block encoding gate for the first operator inputs a plurali ty of eigenvalues of the first operator into the state simulation register;apply an inner quantum phase estimation (iQPE) gate, wherein, in applying the iQPE gate, the first unitary' circuit is configured to:apply a window function to the inner phase register;apply a plurality of qubitization operators to the state simulation register and the Hermitian operator register controlled on qubits of the inner phase register; andperform an inverse quantum Fourier transform on the inner phase register; andmeasurement circuitry configured to measure the outer quantum phase register to obtain classical measurement results; anda classical processor configured to determine an expectation value of the observable for the reference eigenstate based at least in part on the classical measurement results.

2. Tire quantum circuit of claim 1,wherein the block encoding gate is applied to the state simulation register and the observable operator register conditionally when the inner phase register and the Hermitian operator register are in the first initial state.Attorney Docket No. 7246-02601PsiQ-600W013. The quantum circuit of claim 1,wherein the window function comprises a Kaiser window function.

4. The quantum circuit of claim 1,wherein the first unitary circuit is configured to operate conditionally based on at least one state of a subset of qubits of the outer quantum phase register, wherein the conditional operation of the first unitary circuit transfers phase information from the inner quantum register and the state simulation register to the subset of qubits of the outer quantum phase register,wherein the phase information is based on a summation over the plurality of eigenvalues of the first operator of a product of the reference eigenstate, respective ones of the plurality of eigenvalues of the first operator, and respective ones of a plurality of eigenstates of the first operator.

5. The quantum circuit of claim 4,wherein the expectation value of the observable comprises an analytic function of the summation over the plurality of eigenvalues of the first operator of the product.

6. The quantum circuit of claim 4,wherein determining the expectation value of the observable for the reference eigenstate based at least in part on the classical measurement results distinguishes into which of two eigenstates of the first unitary circuit the subset of qubits of the outer quantum phase register was collapsed from measuring the outer quantum phase register.

7. The quantum circuit of claim 1,wherein applying the iQPE gate inputs first information related to the reference eigenvalue into the inner phase register, andwherein applying the iQPE gate entangles the first information with the state simulation register.

8. The quantum circuit of claim 7,wherein the first unitary circuit is further configured to:Attorney Docket No. 7246-02601PsiQ-600W01apply a reflect gate to the inner phase register controlled on the observable operator register to introduce a minus sign to the reference eigenstate in the state simulation register wherein the minus sign is introduced via the entanglement of the first information with the state simulation register; andapply an inverse iQPE gate.

9. The quantum circuit of claim 7,wherein applying the iQPE gate comprises repeatedly inputting the first information into distinct qubits of the inner phase register to perform a rounding procedure for the reference eigenvalue.

10. The quantum circuit of claim 1,wherein inputting the plurality of eigenvalues of the first operator into the state simulation register comprises outputting a result of applying the first operator to the state simulation register into a subspace of the observable operator register that remains in the first initial state.

11. The quantum circuit of claim 1,wherein the Hermitian operator comprises a Hamiltonian, andwherein the reference eigenstate comprises a ground state or an excited state of the Hamiltonian.

12. The quantum circuit of claim 1,wherein the quantum circuit is configured to, before applying the first unitary circuit, prepare the outer quantum phase register in a null state and subsequently apply a Hadamard gate to the outer quantum phase register.

13. The quantum circuit of claim 1,wherein quantum circuit is configured to receive user input specifying a first level of numerical precision for determining the expectation value, andwherein the quantum circuit is configured to implement the inner quantum register at a second level of numerical precision.Attorney Docket No. 7246-02601PsiQ-600W0114. The quantum circuit of claim 1, further comprising:for n={2,...,m-l}, where m is a positive integer greater than 1:2” instances of the first unitary circuit, wherein the 2" instances of the first unitary circuit are configured to be applied conditionally based on a state of a respective subset of qubits of the outer quantum phase register, wherein the conditional application of the 2ninstances of the unitary' circuit transfers respective second phase information from the state simulation register to the respective subset of qubits of the outer quantum phase register, and wherein the respective second phase information transferred to the respective subset of qubits of the outer quantum phase register encodes an nthbinary significant digit of a function of the expectation value of the observable.

15. The quantum circuit of claim 1,wherein the reference eigenvalue is precomputed before implementing the first unitary circuit.

16. The quantum circuit of claim 1,wherein the quantum circuit is configured to apply an inverse quantum Fourier transform (QFT) to the outer quantum phase register prior to measuring the outer quantum phase register.

17. Tire quantum circuit of claim 1,wherein a total number of qubits in the inner phase register is selected to be larger than a minimum number of qubits for encoding eigenvalues of the Hermitian operator.

18. A non-transitory computer-readable memory medium storing program instructions which, when executed by a processor, cause a quantum computing system to operate the quantum circuit of any of claims 1-17.

19. A method for operating the quantum circuit of any of claims 1-17.

20. A method, comprising:receiving an outer quantum phase register;Attorney Docket No. 7246-02601PsiQ-600W01receiving an inner quantum register prepared in a first initial state, wherein tire inner quantum register comprises an inner phase register, a Hermitian operator register, and an observable operator register;receiving a state simulation register prepared in a reference state of a Hermitian operator;applying a block encoding gate for a first operator of an observable, wherein applying the block encoding gate for the first operator inputs a plurality of eigenvalues of the first operator into tire state simulation register;applying an inner quantum phase estimation (iQPE) gate, wherein applying the iQPE gate comprises:applying a window function to the inner phase register;applying a plurality of qubitization operators to the state simulation register and the Hermitian operator register controlled on qubits of the inner phase register; and performing an inverse quantum Fourier transform on the inner phase register; andmeasuring, with measurement circuitry’, the outer quantum phase register to obtain classical measurement results; anddetermining, by a classical processor, an expectation value of the observable for the reference state based at least in part on the classical measurement results.

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