Quantum systems and methods for estimating 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. A plurality of qubits is received, including inner and outer register qubits and state simulation qubits prepared in a reference eigenstate of a Hermitian operator having a reference eigenvalue. A first unitary operation is performed that receives the inner register and state simulation qubits as input and outputs phase information. The phase information is related to the reference eigenvalue and a plurality of eigenvalues of a first operator corresponding to the observable. The outer phase register qubits are measured to obtain classical measurement results that are used to determine an expectation value of the observable for the reference state.

Description

Priority Information

[001] This application claims priority to U.S. Provisional Patent Application No. 63 / 446,276, titled “Quantum Systems and Methods for Estimating Expectation Values of Arbitrary Observables on Quantum States”, and filed on February 16, 2023, which is hereby incorporated by reference in its entirety as though fully and completely set forth herein. Technical Field

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

[003] 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 O(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 systems and methods used to calculate expectation values of observables are desirable.

[004] Any discussion of the prior art throughout the specification should in no way be considered as an admission that such prior art is widely known or forms part of common general knowledge in the field. Summary

[005] In accordance with a first aspect of the present invention, there is provided a quantum circuit, comprising: a plurality of outer phase register qubits; a plurality of inner register qubits prepared in a first initial state; a plurality of state simulation qubits 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 plurality of inner register qubits and the plurality of state simulation qubits; apply 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 the plurality of state simulation qubits; apply an 2024223246  29 Aug 2025 inner quantum phase estimation (iQPE) gate to input first information related to the reference eigenvalue into a first subset of the inner register qubits, wherein applying the iQPE gate entangles the first information with the plurality of state simulation qubits; and output phase information, 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, wherein the first unitary circuit is configured to operate conditionally based on at least one state of a first subset of the outer phase register qubits, wherein the conditional operation of the first unitary circuit transfers the phase information from the inner register qubits and the state simulation qubits to the subset of the outer phase register qubits; measurement circuitry configured to measure the outer phase register qubits to obtain classical measurement results; and a classical processor configured to determine an expectation value of the observable for the reference state based at least in part on the first classical measurement results.

[006] In accordance with another aspect of the present invention, there is provided a method, comprising: receiving a plurality of outer phase register qubits; receiving a plurality of inner register qubits prepared in a first initial state; receiving a plurality of state simulation qubits prepared in a reference eigenstate of a Hermitian operator, wherein the reference eigenstate is associated with a reference eigenvalue; applying a first unitary circuit or an inverse of the first unitary circuit, comprising: receiving as input the plurality of inner register qubits and the plurality of state simulation qubits; 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 the plurality of state simulation qubits; applying an inner quantum phase estimation (iQPE) gate to input first information related to the reference eigenvalue into a first subset of the inner register qubits, wherein applying the iQPE gate entangles the first information with the plurality of state simulation qubits; and outputting phase information, 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, wherein the first unitary circuit is configured to operate conditionally based on at least one state of a first subset of the outer phase register qubits, wherein the conditional operation of the first unitary circuit transfers the phase information from the inner register qubits and the state simulation qubits to the subset of the outer phase register qubits; measuring the outer phase register qubits to obtain classical measurement results; and determining an expectation value of the observable for the reference state based at least in part on the first classical measurement results. 2024223246  29 Aug 2025

[007]

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

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

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

[011] Performing the first unitary operation may include performing a block encoding of the first operator on the state simulation qubits and the first operator register qubits, performing a quantum phase estimation (QPE) operation of the Hermitian operator on the state simulation qubits and the Hermitian operator register qubits, reflecting the reference eigenenergy in the inner phase register qubits and performing an inverse QPE operation that reverts the inner phase register qubits toward the first initial state. In some embodiments, performing the QPE operation outputs eigenvalues of the Hermitian operator into the inner phase register qubits, where the output eigenvalues are entangled with respective Hermitian operator eigenstates of the state simulation qubits.

[012] 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 the 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.

[013] 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 state based at least in part on the first classical measurement results. 2024223246  29 Aug 2025

[014] 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.

[015] 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.

[016] Unless the context clearly requires otherwise, throughout the description and the claims, the words “comprise”, “comprising”, and the like are to be construed in an inclusive sense as opposed to an exclusive or exhaustive sense; that is to say, in the sense of “including, but not limited to”. Brief Description of the Drawings

[017] 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.

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

[019] 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;

[020] 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;

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

[022] Figures 5A-5B are quantum circuit diagrams illustrating examples implementations of the ^! and ^" subroutines, according to some embodiments;

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

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

[025] Figures 8A-B are quantum circuit diagrams illustrating example circuits to implement a Vx and a Qubitization (Q) operation, respectively, according to some embodiments; and 2024223246  29 Aug 2025

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

[027] 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. DETAILED DESCRIPTION

[028] 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.

[029] 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. Qubits

[030] 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. 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 to encode 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 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. 2024223246  29 Aug 2025

[031] 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).

[032] Some embodiments described below relate to physical implementations of unitary operations that couple modes of a quantum system, which may be understood as transforming the quantum state of the system. For instance, if the initial state of the quantum system (prior to mode coupling) is one in which one mode is occupied with probability 1 and another mode is unoccupied with probability 1 (e.g., a state |10) in the Fock notation), mode coupling may result in a state in which both modes have a nonzero probability of being occupied, e.g., a state a#|10) + d$|01), where |a#l$ + |a$ |2 = 1. In some embodiments, operations of this kind may be implemented by coupling modes together and applying phase shifts to one or more modes. Figure 1 - Quantum Computing System

[033] Figure 1 is a system diagram of a quantum computing system, according to some embodiments. As illustrated, the system includes a classical computing system 103 coupled to a quantum computing system 105 over a classical channel 112. The classical channel may relay classical information between the classical and quantum computing systems.

[034] In some embodiments, classical computing system 103 includes one or more non-transitory computer-readable memory media 104, one or more central processing units (CPUs) or processor(s) 102, a power supply, an input / output (I / O) subsystem, and a communication bus or 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 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).

[035] 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 2024223246  29 Aug 2025 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 computer 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 quantum computing system, and generate output from those instructions. The processor 102 may store that output back into the memory 104.

[036] The quantum computing system 105 may include a plurality of qubits and a controller 106 configured to interface with a plurality of qubits 110. 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 quantum controller 106 during a quantum computation, and they may additionally include classical results of the quantum computation. 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 quantum computing system to perform a quantum computation. For example, the classical computing system 103 may provide classical data signals used for quantum state preparation within the quantum computing system 105, in response to which the controller may prepare the states of the qubits 110 into a desired initial state for a particular quantum computation. Calculating an Expectation Value of an Observable of a Quantum System

[037] 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.

[038] 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 2024223246  29 Aug 2025 reconstructed from the 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” state preparation or increase the 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.

[039] 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.

[040] 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

[041] 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 + , as well as a Hamiltonian - , it may be desirable to determine the expectation value of + with respect to an eigenstate of the Hamiltonian | ^e ): 2024223246  29 Aug 2025

[042] ( we |+1 we ), (1)

[043] where H\we= = E\we), and only the eigenenergy E corresponding to the eigenstate |we) is known. In some embodiments, the expectation value represented by Equation 1 may be computed by repeatedly querying block encodings of + and -. 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 qubits | )%&' 205 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[H]), the block encoded operator F (enc[F]), and phase information (inner phase) during the computation, and state simulation qubits |    ) to simulate the state of the quantum system. The 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:

[044] M = # >1 ± # arccos HF*!^*^ ,                    (2)

[045] where ± indicates that two solutions are possible, and F is normalized to ||r|| < 1 by the block encoding of + . The expectation value may then be extracted from Eq. (2), for example, (wE|+|wE) = 2   F± (2  - 1)L - 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 (wE|+|wE). The first unitary circuit 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 ( we |+1 We ) — enabling estimation of the angles ±arccos(wE|+|wE) and reducing the complexity by a factor of two.

[046] 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 2024223246  29 Aug 2025 phase register qubits 204a-d to place them into the |+) state. A sequence of U2n unitary operators for n = {1,2,3,4} are then applied to the IW)%&‘ 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 |1) 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., nmax=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 U2 to a sequential application of two instances of the U operation 206, U4 performs four such applications, etc.

[047] Figure 2B illustrates the control as a black dot 216, 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 206 based on the |0) state, or more generally based on any state of the outer phase register qubits. Controlling the operators Un 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 222 after performing an inverse quantum Fourier transform QFT^ at gate 218. Subsequent terms in the sequence of U2n are 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.

[048] 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, ^! 216 and ^r 218. The ^! operation 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 procedure 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 .'ft! acts upon the state simulation qubits, the 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 ^r operation 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 2024223246  29 Aug 2025 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).

[049] 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 the expectation value.

[050] Figure 3 illustrates in the quantum phase estimation (QPE) subroutine 308 (e.g., QPE 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 have a Hadamard gate applied. In addition, the order of the rotation operators K! 330 and ^r 328 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 R! and K" 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, iQPE[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

[051] Figure 4 is a flowchart diagram illustrating a method for calculating an expectation value of an observable in a physical quantum system. The 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 system 101 as illustrated in Figure 1. In some 2024223246  29 Aug 2025 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) a classical computing system 103 for processing classic information and directing operations of the quantum computing device. 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.

[052] At 402, a plurality of qubits is received that has been prepared in a plurality of respective initial states. The plurality of qubits may include a plurality of register qubits prepared in a first initial state (e.g., a null (|0)) state or another initial state). The register qubits may include inner register qubits (334, 336, and 340) and outer phase register qubits 312. The inner register qubits may include inner phase register qubits 334, Hermitian operator register qubits (enc[H], 336), and first operator register qubits (enc[F], 340). The inner phase register qubits 334 (sometimes referred to as a first subset of the inner register qubits) may be utilized to store phase information within the unitary operation that extracts information related to the expectation value of the observable, and the outer phase register qubits 312 may be utilized to store phase information that will be measured to obtain classical measurement results. The first operator register qubits (sometimes referred to as a second subset of the inner register qubits) may be utilized as a workspace for temporary storage of quantum information related to the first operator, where the first operator F corresponds to the observable whose expectation value is to be calculated. F may be any type of operator that corresponds to a physical observable quantity. Similarly, the Hermitian operator register qubits (sometimes referred to as a third subset of the inner register qubits) may be utilized as a quantum workspace for performing quantum phase estimation on the Hermitian operator.

[053] The plurality of received qubits may also include a plurality of state simulation qubits (   , 308) 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. 2024223246  29 Aug 2025

[054] In some embodiments, a Hadamard gate is applied to the outer phase register qubits. In some embodiments, the Hadamard gate transforms the outer phase register qubits from their initial state (e.g, a null |0) state) to a superposition of states (e.g., a superposition of the |0) and 11) states such as the | +) state). When the outer phase register qubits are prepared in a superposition of |0) and 11) states, the conditional execution of the unitary circuits Un on a state (such as the |1) state) of the outer phase register qubits results in phase kickback, to transfer phase information from the inner register qubits to the outer phase register qubits. 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 qubits via phase kickback. Phase kickback may be understood as a result of the quantum entanglement between the inner register qubits and the outer phase register qubits. When the unitary circuit U operates controlled on the |1) state of the outer phase register qubits, and the outer phase register qubits are in a superposition of the |0) and |1) states, the |1) subspace of the state of the outer phase register qubits will accumulate a phase e'i^ extracted by the unitary U, whereas the |1) subspace of the state of the outer phase register qubits will not accumulate this phase. Note that, in some embodiments, the quantum circuit may be modified such that the outer phase register qubits are prepared in another initial state (e.g., the |1) state). More generally, the outer phase register qubits may be prepared in a superposition of at least one state that controls the conditional execution of the unitary circuits Un and at least one other state.

[055] 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 qubits and the inverse of the first unitary circuit U is applied conditioned on the |0) state of the outer phase register qubits.

[056] At 404, a first unitary circuit U 316 (or an inverse of the first unitary circuit U) is applied to the qubits. The first unitary circuit may receive as input the plurality of inner register qubits and the plurality of state simulation qubits, and outputs phase information based at least in part on the input. The output phase information is related to the reference eigenstate and a plurality of eigenvalues of the first operator F corresponding to an observable.

[057] In more detail, applying the first unitary circuit may include performing a block encoding of the first operator on the state simulation qubits and the first operator register qubits 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 qubits and the first operator register qubits outputs the first operator applied to the state simulation qubits in a subspace of the first operator register qubits that remain in the first initial state (e.g., the null state or another initial 2024223246  29 Aug 2025 state). The block encoding may be performed conditionally when the Hamiltonian register qubits are in the first initial state 318. 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 register qubits.

[058] The first unitary operation may further include applying an inner quantum phase estimation (iQPE) gate 320 of the Hermitian operator on the state simulation qubits and the Hermitian operator register qubits. Applying the iQPE gate inputs first information related to the reference eigenvalue into a first subset of the inner register qubits (i.e., the inner phase register qubits), and applying the iQPE gate entangles the first information with the plurality of state simulation qubits. For example, applying the iQPE gate entangles the first information related to the reference eigenvalue with respective eigenstates of the Hermitian operator encoded in the state simulation qubits. In some embodiments, the first unitary circuit implements qubitization for the Hermitian operator when applying the iQPE gate.

[059] In some embodiments, the iQPE operation utilizes a single value expectation estimation (SEVE) procedure to input the first information into the inner phase register qubits. The SEVE procedure selects a total number of qubits in the inner phase register qubits 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 qubits to perform a rounding procedure for the reference eigenvalue.

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

[061] In some embodiments, an inverse QPE operation is performed 324, where the inverse QPE operation reverts the inner phase register qubits 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. 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 phase may be introduced into the output of the computation, which may be corrected for while determining the expectation value of the observable). 2024223246  29 Aug 2025

[062] Advantageously, the 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 qubits 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.

[063] 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 qubits 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.

[064] 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.

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

[066] At 406, an inverse quantum Fourier transform (QFTl) circuit is applied to the outer phase register qubits 312. The QFT^ operation may transform the outer phase register qubits into a form that is more readily measured to produce decodable classical measurement results. An example of a QFT^ circuit is shown in Figure 9.

[067] At 408, a projective measurement is performed on the outer phase register qubits (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.

[068] At 410, an expectation value of the first operator for the reference state is determined based at least in part on the first classical measurement results. For example, the quantity shown 2024223246  29 Aug 2025 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 state based at least in part on the first classical measurement results may be able to distinguish the sign of the expectation value.

[069] 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 plurality of inner register qubits may be implemented at a second level of numerical precision independent of the first level of precision.

[070] In some embodiments, the method further includes successive application of the first unitary operation 2n times to obtain subsequent significant digits of the determined expectation value of the observable. For example, for n={1,.,m-1}, where m is a positive integer greater than 1, the first unitary operation may be applied 2n times. This is illustrated in Figure 3 as the sequence of gates U, U2, U4, ..., U^m-d. The first unitary operation may be applied 2ntimes conditionally on a state (e.g., the |1) 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 qubits 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

[071] 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. In some 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 U to project an input state into eigenstates of the operator U, while also outputting the phase angle 9 of the corresponding eigenvalue exp( i2 nd) 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-inverse operators.

[072] The expectation value estimation procedure defines the unitary operator U, such that at least one of its eigenphases 9 is a function of (^e|+|^e), namely Eq. (2). Since the projection into eigenstates happens probabilistically according to the overlap of output and input states, it may 2024223246  29 Aug 2025 be desirable to prepare an input state with substantial overlap with the eigenstates used to estimate the expectation value. The unitary operator shown in Figure 2(c) may be divided into two subcircuits, ^! and ^r. The subcircuit ^r is a block encoding of +, while the subcircuit ^! uses block encodings of - and information about the eigenenergy E, as well as the spectral gap A of -. As one example, the 1-norms of - and + after factorization are identified as Xh and Xf, respectively. Since the subcircuit ^! features quantum phase estimation routines (inside this quantum phase estimation), it makes at least O(XH / A) queries to the block encoding of - . 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 f, the complexity of the iQPE routines may be increased by a factor X(f). To estimate the expectation value up to an error e, the operator U may be repeated a total of O ( Xf / s ) times. Matching the target error of iQPE, with the resolution of QPE, / = / / Ff, we find that the entire expectation value estimation computation has a query complexity of

[073] 0 S / 02#^ >.                                        (3)

[074] The upper phase register qubits 204a-d in Figure 2A may contain O(log(XF / s)) qubits. The \ ¥')%&‘ qubits 205 aresplit into four subsets as follows:

[075] 1) phase: a register of at least O(XH / A) qubits to store expressions for qubitization eigenphases of - .

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

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

[078] 4) enc[F]: the auxiliary qubits in the block encoding of the observable + .

[079] The operators - and + may be defined to have the following spectral decomposition:

[080] - = £3 E\^3 )^ 31,                                                 (4)

[081] + = ^4 n\04 [\ 04\,                                                       (5)

[082] where | ^e) and |¢,) are eigenstates corresponding to the energies E and eigenvalues n, respectively, which are normalized to have values in the range from -1 to +1. Knowing the complete spectrum of - and + (e.g., the exact values of E and n 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 = Ej that corresponds to the eigenstate in which we are determining the expectation value of the observable.

[083] The unitary operator U may be defined as the product of two reflection operators R n and Rt, which each are defined through their respective projectors ] and t:

[084] Rn = 1 — 2] , 2024223246  29 Aug 2025

[085] Rt = 1 - 2f .                                                                  (6)

[086] The projectors ] and f 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:

[087] f •rt = ^5W5|kk}{p5|, (7)

[088] 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 R tR n. The result is

[089] |Vk±> =   (|Pk>±>|p8>) = e:;(±&="?@sB$) (|tk) + t|tk8»          (8)

[090] where | k8> is the component of | k> orthogonal to | k> and vice-versa for | k8>:

[091] | k8 > = |d$)eb$If$) G# B$ | k8 > = If$)eb$|d$) G# B$ (9)

[092] The corresponding eigenvalues of | k±> are expression of k as

[093] K ^! | Vk± > = exp(±i2arccos wk )| Vk±>.                        (10)

[094] Using the operator Tl = KTK! within phase estimation would thus estimate the eigenphases ± 2arccos k. The strategy for expectation value estimation consists of making sure at least one of the singular values wk is an expression of (ipH |P|xpH > such that by measuring the eigenvalues k, the expectation value may be computed.

[095] The subcircuit K! contains phase estimation routines on the inner registers phase and enc[ ] and the state simulation qubits. Within the iQPE subroutines, qubitization may be performed, as a specific example of the more general method. In qubitization, KT is a reflection on the all-zero state |0> in the enc[H] register, while K! = B[H] is a block encoding of - on the sim and enc[ ] registers. Block encodings are defined to be self-inverse, and therefore they are reflections. With the singular values t (1 — F) / 2, the qubitization iterate has eigenphases ±arccos associated with eigenstates | 3,±>:

[096] |Q3,±> =   (1 ± i^^B |¢3>sop ® ReQ?^ .                (11)

[097] With iQPE operating on the phase register, it may output computational basis states |03,±>, such that

[098] iQPE|Q3,±> ® |0>;R=se = |Q3± ® |03,±>;R=se ,                 (12)

[099] where |03,±> is the finite-qubit binary representation of an integer 03,± related to the eigenphase ±arccos . Reflections may then be indirectly implemented on certain qubitization eigenstates | 3,S > via operations on the combined sim and enc[ ] registers controlled by the phase register. To that end, the Refl (Reflect) operation is introduced, which is an arithmetic 2024223246  29 Aug 2025 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

[0100] Refl = 1 - 2 S‘eM |m>(m|;haS9,                               (13)

[0101] where |m) is a binary representation for the integer m just like |03,±) is for 03,±. Actually knowing the integers 03 S for energies E and signs a = ± allows implementation of a reflection on the corresponding qubitization eigenstates,

[0102] iQPEV Refl iQPE = 1 -2^ | QE,a )( QE, | ® q (E, a)^ ,    (14)

[0103] where ( , ) is a projector in the phase register, that is equal to the projection onto the all-zero state 10)(01 only for tuples (EW, crW) where 03& S& G M. This reflection may be utilized for the expectation value estimation computation. Expectation value estimation

[0104] The expectation value estimation routine is another example of the singular value estimation. The latter is a block encoding B[F] of the observable + on the sim and enc[F] registers. The block encoding B[F] is a reflection on the state | a>4), associated with the eigenvalues , such that

[0105] B[F] = 1-2^ |^4)(^41.                                (15)

[0106] The states | 4 ) have a relationship with the eigenstates of + .

[0107] Using the shorthand

[0108] |¢4; 0) = |¢4)aop ®|0)9q?[3],                                      (16)

[0109] we can write

[0110] <o> 4| ¢4 ;0> = «44'J#$4.   (17)

[0111] Indeed, setting

[0112] | <04 ) = J#$41 ¢4:0)- / (41 <l>4; 08),                           (18)

[0113] where | 4; 8) is the orthogonal component of | 4) with respect to | 4; ), gives B [F] its characteristic matrix form, verifying Eq. (16). The controlled block encoding of + is related to the reflection 'Fr as follows:

[0114] ^ = |0)(0|9q?[l] ® B[F] + 1 - |0)(0|9q?[l]                     (19)

[0115] such that

[0116] t = |0)(0|9q?[l] ^4 |^4)(W4|,                                 (20)

[0117] while ^! is identified with a version of Eq. (14) where Refl is controlled on the all-zero state in the enc[ ] register, yielding: 2024223246  29 Aug 2025

[0118] ] = |O)(OU[F] ® ^3,s |Q3,sXQ3,s | ® £?(^,^;R=A9            (21)

[0119] X4 | ¢4 ;OX 04 ;O| -^ | Q3,s )( Q3,s | ® ((E, (t)^.         (22)

[0120] Using Eq. (16), the product of the two reflections is

[0121] f - ] = X4 1.3S (04103F|O»9Q?[L] ® 1)) - (( Q3,s | ® (O^F]) ® ( , );R=A9                                                                             (23)

[0122] The square of the singular values may be obtained by solving the eigenvalue problem of

[0123] f-^-f = S5W$ | tk )( tk |                                            (24)

[0124] = ^3,Yss |OXO|9nc[F] ® | Q3,sXQ3,S | ® e(£>);R=A9 ’ Q(E, 0%r=a9

[0125] *%4#r W3104 X04103),                      (25) ) <*!| — \*!]

[0126] where the spectral decomposition of the observable in Eq. (5) has been utilized. When M in Eq. (14) only contains 0H,% for a fixed cr = s (determined by the state preparation preceding the expectation value estimation routine), then only q(G, s) = 10)(01 holds, and the other q(E, o’) are orthogonal to it, such that q(G, s) - q(E, ct) = SE G 8S,% 10)(01. One solution to Eq. (24) is therefore

[0127] |Qg,%) ® |0)9nc[F] ® |0);R=A9                                       (26)

[0128] and its eigenvalue is (0G |+10G ) / 4. Phase estimation with the iterate ^r^! therefore allows estimation of the phase angles

[0129] ±2arccos ^^,,^, = ±77- + arccos 1(*G^]*G1_            (27)

[0130] 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 - | G)( 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 0G,( and 0G,_ at the same time. All three variations of the iterate may allow estimation of the angles ±arccos( G|+| G), which is twice as much signal as in Eq. (27). 2024223246  29 Aug 2025

[0131] 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}.

[0132] Figure 7 is a quantum circuit diagram illustrating a modified 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(0) are X and Z rotations about the angles a and 0, respectively, following the convention X(a) = cos a + i sin aX. The shaded areas are repeated d times with individual angles ^k. SEVE+ differs from SEVE in that SEVE+ repeats the Vn circuit 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, a SEVE implementation of iQPE utilizes a larger number of inner phase register qubits than SEVE+ to reduce the impact of finite precision effects on the input reference eigenvalue. For example, 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.

[0133] Using the general expression for overall complexity in Eq. (3), the computational complexity for SEVE and SEVE+ becomes, respectively:

[0134] 0 > —: and 0 ' — log >£)) .                         (28)

[0135] The maximum success probability of SEVE is fixed to roughly — ~ 81%, while the maximum success probability approaches 1 in SEVE+.

[0136] Figures 8A and 8B are quantum circuit diagrams illustrating example circuits to implement a Vx 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 sim and enc[H] registers. Here, Q is the qubitization iterate. A circuit Vx with an n-qubit phase register makes an equivalent of 2x-1 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.

[0137] 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 QFT^ may be generalized to operate on any number of input qubits. In both Figures 8A and 9, Hadamard gates are denoted as 2024223246  29 Aug 2025 H, and Rk are phase rotations Rk = |0)(0| + exp(~in / 2k-i)\ 1X 1| controlled on the less significant phase qubits for phase feedback.

[0138] 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.

[0139] 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 are necessary. 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.

[0140] 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.

[0141] 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.

[0142] 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. 2024223246  29 Aug 2025

[0143] 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.

[0144] 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 and their 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.

[0145] Hence, the foregoing describes a number of specific embodiments. Modifications, obvious to those skilled in the art, can be made thereto without departing from the scope of the invention.

Claims

1. A quantum circuit, comprising:a plurality of outer phase register qubits;a plurality of inner register qubits prepared in a first initial state;a plurality of state simulation qubits 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 plurality of inner register qubits and the plurality of state simulation qubits;apply 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 the plurality of state simulation qubits;apply an inner quantum phase estimation (iQPE) gate to input first information related to the reference eigenvalue into a first subset of the inner register qubits, wherein applying the iQPE gate entangles the first information with the plurality of state simulation qubits; andoutput phase information, 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, wherein the first unitary circuit is configured to operate conditionally based on at least one state of a first subset of the outer phase register qubits, wherein the conditional operation of the first unitary circuit transfers the phase information from the inner register qubits and the state simulation qubits to the subset of the outer phase register qubits;measurement circuitry configured to measure the outer phase register qubits to obtain classical measurement results; anda classical processor configured to determine an expectation value of the observable for the reference state based at least in part on the first classical measurement results.

2. The quantum circuit of claim 1,wherein the first unitary circuit is further configured to:apply a reflect gate to the first subset of the inner register qubits to introduce a minus sign to the reference eigenstate in the plurality of state simulation qubits, wherein the2024223246  29 Aug 2025minus sign is introduced via the entanglement of the first information with the plurality of state simulation qubits; andapply an inverse iQPE gate.

3. The quantum circuit of claim 2,wherein the block encoding is applied conditionally when a second subset of the inner register qubits are in the first initial state, andwherein the reflect gate is applied conditionally when a third subset of the inner register qubits are in the first initial state.

4. The quantum circuit of claim 3,wherein inputting the plurality of eigenvalues of the first operator into the plurality of state simulation qubits comprises applying the first operator to the plurality of state simulation qubits in a subspace of the third subset of inner register qubits that remain in the first initial state.

5. The quantum circuit of claim 1,wherein the expectation value of the first operator comprises an analytic function of the summation over the plurality of eigenvalues of the first operator of a product of the reference eigenstate and respective ones of the plurality of eigenvalues of the first operator, and respective ones of a plurality of eigenstates of the first operator.

6. 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.

7. The quantum circuit of claim 1,wherein the quantum circuit is configured to prepare the plurality of outer phase register qubits in a superposition of the at least one state and at least one other state.

8. The quantum circuit of claim 7,wherein the quantum circuit is configured to prepare the outer phase register qubits in a null state and subsequently apply a Hadamard gate to the plurality of outer phase register qubits before applying the first unitary circuit.2024223246  29 Aug 20259. The quantum circuit of claim 1,wherein determining the expectation value of the observable for the reference state based at least in part on the first classical measurement results distinguishes into which of two eigenstates of the unitary operator the first subset of the outer phase register qubits was collapsed from measuring the outer phase register qubits.

10. 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;wherein the quantum circuit is configured to implement the plurality of inner register qubits at a second level of numerical precision.

11. The quantum circuit of claim 1, further comprising:for n={1,...,m-1}, where m is a positive integer greater than 1:applying a combination of a plurality of instances of the unitary circuit, wherein the combination of the plurality of instances of the unitary circuit is applied conditionally based on the at least one state of a respective second subset of the outer phase register qubits, wherein the conditional application of the combination of the plurality of instances of the unitary circuit transfers respective second phase information from the state simulation qubits to the respective second subset of the outer phase register qubits, and wherein the respective second phase information transferred to the respective second subset of the outer phase register encodes a respective nth binary significant digit of a function of the expectation value of the observable.

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

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

14. The quantum circuit of claim 1,2024223246  29 Aug 2025wherein a total number of qubits in the first subset of the inner register qubits is selected to be larger than a minimum number of qubits for encoding eigenvalues of the first Hermitian operator.

15. The quantum circuit of claim 1,wherein applying the iQPE gate comprises repeatedly inputting the first information into the first subset of the inner register qubits to perform a rounding procedure for the reference eigenvalue.

16. The quantum circuit of claim 1,wherein the first unitary circuit implements qubitization for the Hermitian operator when applying the iQPE gate.

17. A method, comprising:receiving a plurality of outer phase register qubits;receiving a plurality of inner register qubits prepared in a first initial state;receiving a plurality of state simulation qubits prepared in a reference eigenstate of a Hermitian operator, wherein the reference eigenstate is associated with a reference eigenvalue;applying a first unitary circuit or an inverse of the first unitary circuit, comprising:receiving as input the plurality of inner register qubits and the plurality of state simulation qubits;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 the plurality of state simulation qubits;applying an inner quantum phase estimation (iQPE) gate to input first information related to the reference eigenvalue into a first subset of the inner register qubits, wherein applying the iQPE gate entangles the first information with the plurality of state simulation qubits; andoutputting phase information, 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, wherein the first unitary circuit is configured to operate conditionally based on at least one state of a first subset of the outer phase register qubits, wherein the conditional operation of the first unitary circuit transfers the phase2024223246  29 Aug 2025information from the inner register qubits and the state simulation qubits to the subset of the outer phase register qubits;measuring the outer phase register qubits to obtain classical measurement results; and determining an expectation value of the observable for the reference state based at least in part on the first classical measurement results.

18. A method for operating the quantum circuit of any of claims 1-16.

19. A non-transitory computer-readable memory medium storing program instructions which, when executed by a processor, cause a quantum computing system to:receive a plurality of outer phase register qubits;receive a plurality of inner register qubits prepared in a first initial state;receive a plurality of state simulation qubits prepared in a reference eigenstate of a Hermitian operator, wherein the reference eigenstate is associated with a reference eigenvalue;apply a first unitary circuit or an inverse of the first unitary circuit, wherein in applying the first unitary circuit or the inverse of the first unitary circuit, the program instructions are executable to cause the quantum computing system to:receive as input the plurality of inner register qubits and the plurality of state simulation qubits;apply 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 the plurality of state simulation qubits;apply an inner quantum phase estimation (iQPE) gate to input first information related to the reference eigenvalue into a first subset of the inner register qubits, wherein applying the iQPE gate entangles the first information with the plurality of state simulation qubits; andoutput phase information, 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, wherein the first unitary circuit is configured to operate conditionally based on at least one state of a first subset of the outer phase register qubits, wherein the conditional operation of the first unitary circuit transfers the phase information from the inner register qubits and the state simulation qubits to the subset of the outer phase register qubits;measure the outer phase register qubits to obtain classical measurement results; and2024223246  29 Aug 2025determine an expectation value of the observable for the reference state based at least in part on the first classical measurement results.

20. 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 1-16.

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