Execution of characteristic estimation using quantum gradient operation in a quantum computing system

Abstract

A quantum computing system and method for determining a state of a physical system is provided. In some examples, the method can include obtaining a defined function associated with the physical system, the defined function encoding an estimate of at least one characteristic to be simulated by the quantum computing system as a gradient of the defined function. The method can include implementing a quantum circuit on a plurality of qubits in the quantum computing system to perform a quantum operation (e.g., a Gilyen quantum gradient algorithm) on the plurality of qubits. The quantum operation is operable to determine a gradient of the defined function. The method can include determining an estimate of the at least one characteristic based at least on at least one of the plurality of qubits after implementing the quantum operation.

Description

Technical Field

[0001] Priority Claim This application claims the benefit of priority of U.S. Provisional Application No. 63 / 270,877, titled "Performing Property Estimation Using Quantum Gradient Operation on Quantum Computing System", filed on Oct. 22, 2021, which is incorporated herein by reference.

[0002] The present disclosure generally relates to quantum computing systems, and more particularly to quantum computing systems and methods operable to perform simulations of physical systems (e.g., quantum systems).

Background Art

[0003] Quantum computing is a computational method that utilizes quantum effects, such as superposition and entanglement of basis states, to perform certain calculations more efficiently than classical digital computers. In contrast to digital computers that store and manipulate information in the form of bits, e.g., "1" or "0", a quantum computing system can manipulate information using quantum bits ("qubits"). A qubit may refer to a quantum device that enables superposition of data in multiple states, e.g., both the "0" and "1" states, and / or the superposition of data in multiple states itself. According to conventional terminology, the superposition of the "0" and "1" states in a quantum system can be represented, for example, as |0〉 + b|1〉. The "0" and "1" states of a digital computer are respectively similar to the |0〉 and |1〉 basis states of a qubit.

Prior Art Documents

Non-Patent Documents

[0004]

Non-Patent Document 1

[0005] Aspects and advantages of embodiments of the present disclosure are described in part in the following description, or can be learned from the description, or can be learned through the practice of the embodiments.

[0006] One exemplary aspect of the present disclosure is directed to a method for determining the state of a physical system. The method includes obtaining, by one or more computing devices, a defined function associated with the physical system, the defined function encoding an estimated value of at least one property to be simulated by a quantum computing system as the gradient of the defined function. The method can include implementing, by one or more computing devices, a quantum circuit on a plurality of qubits within the quantum computing system to perform a quantum operation. The quantum operation is operable to determine the gradient of the defined function. The method can include determining, by one or more computing devices, an estimated value of at least one property based at least in part on at least one of the plurality of qubits after implementation of the quantum operation.

[0007] Other aspects of the present disclosure are directed to various systems, methods, apparatuses, non-transitory computer-readable media, computer-readable instructions, and computing devices.

[0008] These and other features, aspects, and advantages of the various embodiments of the present disclosure will be better understood with reference to the following description and the appended claims. The accompanying drawings, which are incorporated herein and constitute a part of this specification, illustrate exemplary embodiments of the present disclosure and, together with the description, explain the related principles.

[0009] A detailed description of embodiments directed to those skilled in the art is set forth herein, with reference to the accompanying figures.

Brief Description of the Drawings

[0010]

Figure 1

Figure 2

Figure 3

Figure 4

Embodiments for Carrying Out the Invention

[0011] Exemplary aspects of the present disclosure are directed to quantum computing systems and methods that can be used, for example, to simulate experiments in a physical system (e.g., a quantum system). The quantum computing systems and methods can be used to determine estimated properties associated with the system, such as, for example, the expected value of an observable or an unequal time correlation function related to the state of a quantum system. As an example, a quantum computing system can be used to determine properties in an electronic ground state problem, such as measuring the dipole moment and polarizability, electron density, forces experienced by classical nuclei, or other properties associated with the ground state. As another example, elements associated with a correlation function can be determined, for example, to understand aspects of many-body quantum phenomena in condensed matter physics and beyond.

[0012] The complexity associated with performing a simulation of a physical system using a quantum computing system can arise from the state preparation of qubits within the quantum computing system for simulating the state of the physical system. The state preparation of qubits can be computationally and operationally expensive and may, for example, require performing phase estimation and / or other state preparation operations on the qubits with high precision.

[0013] One approach for determining the expected values of multiple observables using a quantum computing system can include a sampling approach that also involves repeatedly preparing the state of qubits and performing measurements to determine an estimate after each state preparation. However, in the sampling approach, it is necessary to perform some state preparation steps that scale exponentially with respect to a defined error tolerance. Amplitude estimation techniques can provide good scaling with respect to the error tolerance. However, amplitude estimation techniques may be suitable for measuring only one estimate of a single observable at a time. As a result, using amplitude estimation to determine the expected values of multiple observables (e.g., M observables) may require scaling by a factor directly related to the number of observables.

[0014] According to an exemplary aspect of the present disclosure, the estimated values of multiple characteristics of a physical system can be determined by encoding, in a parameterized quantum circuit implemented in a quantum computing system, a defined function whose gradient yields the estimated values of the multiple characteristics. A quantum operation can then be performed via the implementation of the quantum circuit to determine the gradient. The estimated values can then be determined by measuring one or more qubits within the quantum computing system after the implementation of the quantum operation.

[0015] For example, a quantum computing system can be used to determine the expected value of an observable of a physical system, such as the expected value of the Pauli Z operator on each qubit (which can obtain the electron density, for example, in the context of a ground state problem) or the fermionic k-reduced density matrix, as the characteristic of interest. According to an exemplary aspect of the present disclosure, in a quantum system, a defined function whose gradient yields the expected values of multiple observables can be encoded. The quantum operation can be performed by implementing a quantum circuit on multiple qubits within the quantum computing system to determine the gradient. The expected value can then be determined from the multiple qubits after the implementation of the quantum operation.

[0016] More specifically, in some embodiments, the quantum operation can implement a quantum algorithm for determining the gradient of a defined function. In some embodiments, the quantum algorithm can be, for example, the gradient algorithm of Gilyen et al. described in Gilyen et al., "Optimizing quantum optimization algorithms via faster quantum gradient computation", In Proceedings of the 30th ACM - SIAM Symposium on Discrete Algorithms (SODA 2019), pages 1425 - 1444, which is incorporated herein by reference. The Gilyen quantum gradient algorithm is based on the quantum gradient algorithm disclosed in S. P. Jordan, Phys. Rev. Lett. 95, 050501 (2005), which demonstrates an exponential - speedup for computing the gradient of a function in a particular black - box access model, incorporated herein by reference. The Jordan quantum gradient algorithm queries a black - box oracle that yields the objective function in the quantum phase and uses phase kickback and the quantum Fourier transform to achieve a first - order approximation of the gradient of this objective function using several queries that do not depend on the dimension of the gradient vector. The Gilyen quantum gradient algorithm applies a modified version of the Jordan quantum gradient algorithm when the value of the function is encoded in the expectation value of a quantum observable, including optimizations arising from higher - order differential formulas.

[0017] Quantum calculus (e.g., quantum gradient algorithms) can be executed by implementing a quantum circuit containing a plurality of quantum gates on a plurality of qubits within a quantum computing system. In some embodiments, the plurality of qubits can include M registers of a first qubit, a system register of N second qubits, and ancillary qubits, where M is the number of observables to be determined and N is an integer (e.g., an integer power of 2). The N second qubits can be initialized to zero and can receive a state preparation operation to prepare the state of the qubits, for example, to simulate a physical system. The M registers of the first qubit can provide the expected value of each of the quantum observables. Each of the M registers can include b first qubits, where b is the number of digits used in the binary representation of the observable associated with each qubit register.

[0018] The quantum circuit can implement a state preparation unitary on the N second qubits as a state preparation operation to prepare the state of the qubits within the quantum computing system to simulate the state of a physical system. The quantum circuit can implement a probabilistic oracle of a defined function whose gradient encodes the expected value of an observable. The quantum circuit can encode the expected value of each of the M observables in the M registers of the first qubit. The quantum circuit can implement a Hadamard test (e.g., using one or more Hadamard gates and / or phase gates) on the ancillary qubits to encode the gradient of a defined function in the amplitude of the ancillary qubits. A double-controlled gate can implement the time evolution by an observable on the first qubit register. The double-controlled gate can be controlled based on the register of N qubits and the ancillary qubits.

[0019] The inventors have discovered that encoding the expected values of multiple observables as gradients with respect to a defined function and solving the gradients using quantum operations according to exemplary aspects of the present disclosure can lead to more efficient use of quantum computing resources for determining the expected values of observables. More specifically, quantum operations according to exemplary aspects of the present disclosure can include implementing a plurality of state preparation operations. The number of state preparation operations required to determine the expected value of an observable within a defined error tolerance can scale as a function of the square root of the number of observables determined using a quantum computing device. This provides improved scalability and more efficient use of quantum computing resources compared to, for example, sampling approaches or amplitude estimation approaches.

[0020] For purposes of illustration and discussion, aspects of the present disclosure are discussed with reference to determining the expected value of an observable in a physical system using a quantum computing system. Those of ordinary skill in the art using the disclosure provided herein will understand that aspects of the present disclosure can be used for other applications. For example, aspects of the present disclosure can be used, for example, for the evaluation of unequal time correlation functions.

[0021] More specifically, in some embodiments, to evaluate an unequal correlation function, a defined function whose gradient yields the element of interest (e.g., the matrix element of interest) can be constructed. Quantum operations (e.g., the Gilyen quantum gradient algorithm) can then be executed via the implementation of a quantum circuit to determine the gradient. The estimate can then be determined by measuring one or more qubits in the quantum computing system after the implementation of the quantum operation.

[0022] Aspects of the present disclosure provide several technical effects and benefits. For example, quantum computing systems and methods according to exemplary aspects of the present disclosure can be used to determine estimates of observables or other items of interest within a defined error tolerance using fewer quantum computing resources (e.g., fewer state preparation operations on multiple qubits) compared to other approaches such as sampling approaches or amplitude estimation approaches. As a result, quantum computing resources can be used to perform other functions of the quantum computing system, such as error correction, execution of other quantum algorithms, etc. Further, since fewer quantum computing operations are required to determine the expected value of an observable (e.g., fewer state preparation operations), there are fewer opportunities for errors that interfere with quantum computing, and the coherence of quantum computing operations is improved.

[0023] Exemplary aspects of the present disclosure are directed to accurately and efficiently estimating multiple properties from quantum computations. In some examples, the expected values of a set of M Hermitian operators {O j} with respect to a pure state ψ are evaluated. Each expected value can be evaluated within an additive error ε using as few calls as possible to a state preparation oracle for ψ (or its inverse). One approach is to repeatedly prepare ψ and projectively measure a commuting subset of {O j}. Alternatively, an amplitude estimation-based strategy achieves a quadratic speedup with respect to ε but involves measuring each observable separately.

[0024] Various “shadow tomography” techniques use joint measurements of multiple copies of ψ to achieve polylogarithmic scaling with respect to M at the expense of an undesirable 1 / ε 4 scaling. In certain situations, a randomized method based on the concept of a “classical shadow” of a state obtains 1 / ε 2 scaling while improving on sampling protocols with deterministic measurement settings.

[0025] The inventors have found that, in the exemplary systems and methods for estimating multiple properties disclosed herein, the same 1 / ε scaling as the amplitude-estimation-based method can be achieved, but

[0026]

Number

[0027] from

[0028]

Number

[0029] the scaling with respect to M from

[0030]

Number

[0031] to

[0032] Theorem 1: Let {O j} be a set of M Hermitian operators on N qubits such that for all j, the spectral norm ||O j || ≤ 1.

[0033]

Number

[0034] For various values of x such that

[0035]

Number

[0036] of

[0037]

Number

[0038] together with the gate, U ψ and

[0039]

Number

[0040] to

[0041]

Number

[0042] using the query of, for all j, with at least a 2 / 3 probability,

[0043]

Number

[0044] such that, for any N - qubit quantum state ψ prepared by the unitary U ψ the estimated value

[0045]

Number

[0046] there exists a quantum algorithm that outputs.

[0047] As shown in Inference 3 below, the complexity of this query is optimal in the worst case (

[0048]

Number

[0049] up to the logarithmic coefficient of the high-precision regime of

[0050] The lower bound of the subject matter referred to above can be obtained as a natural result. More specifically, J. van Apeldoorn, "Quantum probability oracles & multi-dimensional amplitude estimation", 16 th Conference on the Theory of Quantum 2021 ("Apeldoorn") established a lower bound in which results are expressed for a particular quantum access model of classical probability distributions.

[0051] Definition 1 (Sample oracle of a probability distribution): Let p be a probability distribution over M outcomes, i.e., p ∈ [0, 1] with ||p||1 = 1 M be the case. The sample oracle U for p p is a unitary operator that functions as follows.

[0052]

Number

[0053] where |φ j 〉 is an arbitrary normalized quantum state. Here, and throughout this disclosure, queries to the unitary oracle U and its inverse U † are counted with equal cost. Based on this, Theorem 2 can be established.

[0054] Theorem 2: Let M be a positive integer power of 2,

[0055]

Number

[0056] Let it be so. There exists a known matrix \(A\in{- 1,+1\}\) such that the following is true. M×M exists.

[0057]

Number

[0058] For all probability distributions \(p\) accessed via the sample oracle \(U\) p such that

[0059]

Number

[0060] Let it be an algorithm that outputs

[0061]

Number

[0062] (with at least a probability of \(2 / 3\)). Then, in the worst - case scenario,

[0063]

Number

[0064] needs to use queries of

[0065]

Number

[0066] to \(U\). p This theorem can be used to derive the following corollary, establishing the near - optimality of an algorithm in a specific regime.

[0067] Corollary 3: Let \(M\) be a positive power of 2,

[0068]

Number

[0069] Let it be.

[0070]

Number

[0071] Let A be an arbitrary algorithm that takes as input any set of M observables {O j}. For all quantum states |ψ〉 accessed via the state preparation oracle U ψ ,

[0072]

Number

[0073] suppose that outputs an estimated value of each 〈ψ|O j |ψ〉 within an additive error ε (with probability at least 2 / 3). Then, there exists a set of observables {O j} to which

[0074]

Number

[0075] needs to use queries to U ψ to

[0076]

Number

[0077] as follows. An exemplary mathematical proof is provided. j} exists.

[0078] To lead to a contradiction, for any {O j} and U ψRegarding the algorithm

[0079]

Number

[0080] is

[0081]

Number

[0082] used for the query of ψ in U j to estimate all 〈ψ|O p |ψ〉 within an error ε (with at least a 2 / 3 success probability). Assume an arbitrary sample oracle U

[0083]

Number

[0084] By rapid calculation, it is verified that the i-th entry of the vector Ap is equal to 〈ψ(U p )│Z i │ψ(U p )〉, where Z i denotes the Pauli Z operator acting on the i-th qubit. Since the matrix A is known, for a known unitary U A :

[0085]

Number

[0086] of

[0087]

Number

[0088] it is obvious that it is Therefore, the algorithm

[0089]

Number

[0090] is applicable for j ∈ {1, …, M} with O j = Z j and

[0091]

Number

[0092] It can be applied with. This means that using queries to U p to

[0093]

Number

[0094] to estimate each entry of Ap within error ε for all U p might contradict Theorem 2 and show that the proof is complete.

[0095] The framework for simultaneously estimating multiple expected values uses an improved Gilyen quantum algorithm for gradient estimation. The Gilyen algorithm is based on the Jordan algorithm, which demonstrated an exponential quantum speedup for calculating gradients in a specific black-box access model. Specifically, the Jordan algorithm uses one query to the binary oracle of the function f, along with phase kickback and the quantum Fourier transform, to obtain an approximation of the gradient ∇f.

[0096] The definition of the probabilistic oracle of the Gilyen algorithm is shown below. Definition 2 (Exemplary Probabilistic Oracle): Function

[0097]

Number

[0098] Consider the probabilistic oracle \(U_f\). f is a unitary operator that functions as follows,

[0099]

Number

[0100] where \(|x\rangle\) represents the discretization of the variable \(x\) encoded in a register of qubits, \(|0\rangle\) represents the all-zero state of a register of auxiliary qubits, and \(|\varphi_0(x)\rangle\) and \(|\varphi_1(x)\rangle\) are arbitrary quantum states.

[0101] The Gilyen algorithm uses such a probabilistic oracle to encode a finite difference approximation to the directional derivative of \(f\) in the phase of the auxiliary register. For example, the first-order approximation is implemented as follows.

[0102]

Number

[0103] Similar to the Jordan algorithm, the quantum Fourier transform can then be used to extract an approximate gradient from the phases accumulated in an appropriate superposition of the basis states. By using higher-order finite difference formulas, the Gilyen algorithm can estimate the gradient with optimal scaling (up to logarithmic factors) for a particular class of smooth functions. Exemplary properties of the Gilyen algorithm are provided by the following theorem.

[0104] Theorem 4: \(\varepsilon\),

[0105]

Number

[0106] Let \(\varepsilon\) be a fixed constant and \(\varepsilon\leq c\).

[0107]

Mathematics

[0108] and

[0109]

Mathematics

[0110] be set as

[0111]

Mathematics

[0112] all

[0113]

Mathematics

[0114] for all k-th partial derivatives of f with respect to x at x, the following limit (

[0115]

Mathematics

[0116] indicated by) holds. Assume that it is an analytic function. Then, with at least a probability of 1-δ,

[0117]

Mathematics

[0118] such an estimated value

[0119]

Mathematics

[0120] There is a quantum algorithm that outputs. This algorithm makes queries to a probabilistic oracle for f.

[0121]

Number

[0122] as follows.

[0123] Exemplary aspects of the present disclosure can be used to determine an expected value using a quantum gradient algorithm (e.g., the Gilyen algorithm). For example, a probabilistic oracle can be constructed for a function whose gradient encodes the expected value of interest, and a quantum gradient algorithm for the gradient can be applied. An exemplary proof of Theorem 1 is provided below.

[0124] The parameterized unitary is

[0125]

Number

[0126] in the case of

[0127]

Number

[0128] can be defined as

[0129]

Number

[0130] The derivative of this unitary with respect to is as follows.

[0131]

Number

[0132] O with respect to the state ψj To determine the expected value of [[ID=]], the following function f can be defined.

[0133]

Number

[0134] Using the above formula (7), it becomes as follows.

[0135]

Number

[0136] Therefore, the gradient ∇f(0) is exactly the set of the expected values of the target.

[0137] f can satisfy the conditions of Theorem 4. f is analytic, and the k-th partial derivative of f with respect to any set of exponents α ∈ {1,…,M} k Note that it takes the following form for some operators V(x,α) that depend on both a and x. ∂ α f(x) = (-2) k-1 Im(i k 〈ψ|V(x,α)|ψ〉), (10) Note that V is either unitary or a product of terms from {O j}. Since ||O j || ≤ 1 for all j, ||V|| ≤ 1, and thus |∂ α f(0)| ≤ 2 k-1 is true. By setting c = 2, the derivative condition of Theorem 4 is satisfied.

[0138] In some cases, to construct the probability oracle of f, the quantum circuit can encode f(x) into the amplitude of the ancilla. The quantum circuit can be constructed using the Hadamard test for the imaginary part of 〈ψ|U(x)|ψ〉.

[0139]

Number

[0140] wherein H represents a Hadamard gate, c-U(x) represents a U(x) gate controlled on the first qubit, and S := |0〉〈0| + i|1〉〈1| represents a phase gate.

[0141]

Number

[0142] When applied to, for some normalized states |φ0(x)〉 and |φ1(x)〉, the circuit can encode f(x) with the amplitudes regarding the computational basis states of the first qubit

[0143]

Number

[0144] of. Note that F(x) can be a single call to the oracle U ψ .

[0145] Quantum control can be added to the rotation of F(x) such that F(x) is controlled by the register encoding x. For example, for the unitary

[0146]

Number

[0147] consider, wherein

[0148]

Number

[0149] is

[0150]

Number

[0151] where it is distributed over an M - dimensional unit hypercube, the set of 2 nM points, and x max is a rescaling factor. The values of x max and n can be chosen to satisfy the requirements of the gradient algorithm. Here,

[0152]

Number

[0153] in the case of, |k〉 = |k1〉…|k M 〉 represents the basis state that stores the binary representation of k in an M - qubit index register of n qubits. For each O j the controlled time - evolution operator has n

[0154]

Number

[0155] exponentially - spaced values of x, each controlled on the appropriate qubit of the j - th index register, as shown in Fig. 3, and can be implemented as a product of gates.

[0156] U f is the probability oracle of the function f, and each call to U f is accompanied by a single call to the state - preparation oracle U ψ . Theorem 4 states that, with probability at least 2 / 3, all components of the gradient of f, and thus all expectation values 〈ψ|O j |ψ〉, are within f of the

[0157]

Number

[0158] means that it can be estimated within an error ε using the query of. The complexity regarding the controlled time evolution is U f the number of controlled time evolutions required for each query to U, i.e.,

[0159] [Number]

[0160] to the total number of queries, i.e.,

[0161] [Number]

[0162] is obtained by multiplying.

[0163] [Number]

[0164] is the result of the details of the proof of Theorem 4. This completes the exemplary demonstration of Theorem 1. Furthermore, the space complexity of the gradient algorithm may be the same as the space complexity of the probabilistic oracle up to an additive logarithmic factor. Therefore, the systems and methods according to the exemplary aspects of the present disclosure

[0165] [Number]

[0166] may use qubits of.

[0167] Aspects of the present disclosure are directed to simultaneously estimating the expected values of multiple observables with respect to the pure state ψ. The algorithm is U ψ and its reciprocal

[0168] [Number]

[0169] can be used for the application, where M represents the number of observables, ε represents the target error, and U ψ is unitary for preparing ψ. The lower bound of the closely related problem raised in Apeldoorn is U ψ For an algorithm given black-box access to U, the complexity of this query is

[0170] [Number]

[0171] means being optimal in the worst case up to the logarithmic factor when. In fact, aspects of the present disclosure positively solve the outstanding problem from Apeldoorn regarding the achievability of this limit for the simultaneous estimation of classical random variables. The result is that when requiring a scaling where ε -2 is replaced by ε -1 it suggests that the optimal cost for the estimation of the expected value may deteriorate exponentially with respect to M. Furthermore, the instances used in establishing the lower bound may involve a set of mutually commuting observables, which means that commutativity may not be useful when implementing the ε -1 scaling.

[0172] A comparison with other approaches for the estimation of the expected value is provided in Table I (Table 1) below.

[0173] [Table 1]

[0174] Table I (Table 1) provides a comparison of the (worst-case) complexity of different approaches for measuring multiple observables with respect to state preparation oracle queries. Three applications are provided: estimating the expected values of M commuting or non-commuting observables, and determining the fermionic k-RDM of an N-mode system. Here, ε represents the additive error with which each quantity is estimated. Strategies based on naive sampling, amplitude estimation, and shadow tomography are compared with the gradient-based approach according to examples of the present disclosure.

[0175] Aspects of the present disclosure

[0176] [Number]

[0177] Using the state preparation query of, each element of the k-body fermionic reduced density matrix (k-RDM) of an N-mode system can be estimated within an error ε. This provides an unconditional asymptotic speedup compared to existing methods when ε = o(N -k / 3 ), and can be particularly useful in practical applications where one wishes to achieve a large fixed error by measuring the 1- or 2-RDM and summing Ω(N) elements.

[0178] The gradient-based approach for estimating expected values according to examples of the present disclosure can be extended to other properties. For example, consider the task of evaluating a set of two-point dynamic correlation functions. These functions take the following form, C A,B (t) := 〈ψ|U(0,t)A † U(t,0)B|ψ〉, (14) where A and B are some simple operators, and U(t,t') is the time-evolution operator that maps the system from time t' to time t. These correlation functions are often directly accessible in experiments, such as in the case of angle-resolved photoemission spectroscopy, and may also be central to hybrid quantum-classical methods based on dynamical mean-field theory.

[0179] Examples of the present disclosure may include the time evolution by each of M observables. The total duration of the required time evolution is scaled as

[0180]

Number

[0181] and additional

[0182]

Number

[0183] qubits may be used, but this approach can be modified due to the trade-off between space and query complexity. When estimating O(N) expected values simultaneously, the asymptotic scaling of the space complexity can be only logarithmically larger than that which stores the system itself. This can hold in various situations, such as, for example, the evaluation of the momentum distribution. In other situations, the space overhead may be larger, but the ability of modern simulation algorithms that use so-called "dirty ancilla bits" (temporarily borrowing qubits in an arbitrary state) can offset this issue in some situations.

[0184] In some examples, the observables of interest may have different norms or the desired accuracy may vary. In these examples, aspects of the present disclosure may include measuring a particular observable using aspects of the present disclosure and measuring other observables using a sampling-based method. In some examples, the Gilyen gradient estimation algorithm can be generalized to adapt to functions where the gradient components are not necessarily uniformly bounded. This is

[0185]

Number

[0186] using queries of to any norm ||Oj ||(presumably greater than 1) having an observable {O j} that can enable simultaneous estimation of the expected value.

[0187] Extracting useful information from quantum computing, particularly quantum simulation, is a bottleneck for many applications. This is especially true in fields such as quantum chemistry and materials science, where it may be desirable to combine high-level quantum computing with coarser approximations at other length scales to describe macroscopic physical phenomena. The gradient-based approach to estimating expected values according to examples of the present disclosure can be a useful tool and starting point for related approaches to other problems.

[0188] One exemplary embodiment of the present disclosure is directed to a method for determining the state of a physical system. The method includes obtaining, by one or more computing devices, a defined function associated with the physical system, where the defined function encodes an estimated value of at least one property to be simulated by a quantum computing system as a gradient of the defined function. The method includes implementing, by one or more computing devices, a quantum circuit on a plurality of qubits within the quantum computing system to perform a quantum operation, where the quantum operation is operable to determine the gradient of the defined function. The method includes determining, by one or more computing devices, an estimated value of at least one property based at least in part on at least one of the plurality of qubits after implementation of the quantum operation.

[0189] In some embodiments, the property to be simulated includes an expected value of a quantum observable of the physical system. In some embodiments, the property to be simulated includes one or more elements associated with an unequal-time correlation function.

[0190] In some embodiments, the quantum operation is operable to determine the gradient of a defined function using the Gilyen quantum gradient algorithm.

[0191] In some embodiments, implementing a quantum circuit by one or more computing devices includes implementing a plurality of state preparation operations on at least a subset of a plurality of qubits using the quantum circuit by one or more computing devices.

[0192] In some embodiments, the quantum circuit implements a probability oracle of a defined function.

[0193] In some embodiments, the quantum circuit is implemented on M first qubit registers, an N-system register of second qubits, and at least one auxiliary qubit, where M is some quantum observables and N is an integer. The quantum circuit can encode a defined function in the amplitude of the auxiliary qubit. The quantum circuit can implement a Hadamard test on the auxiliary qubit using one or more Hadamard gates and one or more phase gates. The N second qubits can be initialized to zero. The quantum circuit may be operable to implement a state preparation unitary on the N second qubits.

[0194] In some embodiments, the quantum circuit can implement a controlled time evolution for a quantum observable using at least one double-controlled quantum gate on the M first qubit registers. The double-controlled gate can be at least partially based on one of the N second qubits and the auxiliary qubit.

[0195] In some embodiments, determining an estimated value of a characteristic by one or more computing devices includes performing a measurement on at least one of a plurality of qubits.

[0196] Another exemplary aspect of the present disclosure is directed to a quantum computing system. The quantum computing system includes a plurality of qubits, where the plurality of qubits includes M registers of a first qubit, N registers of a second qubit, and at least one auxiliary qubit. The quantum computing system includes one or more control devices configured to implement a quantum circuit on the plurality of qubits to determine a gradient of a function defined using a quantum operation, where the gradient of the defined function encodes an estimate of a property to be simulated by the quantum computing system.

[0197] Next, exemplary embodiments of the present disclosure will be described in further detail with reference to the figures. As used herein, the use of the term "about" in conjunction with a value refers to within 20% of that value.

[0198] FIG. 1 shows an exemplary quantum computing system 100. System 100 is an example of a system on one or more classical computers and / or quantum computing devices at one or more locations that can implement the systems, components, and techniques described below. Those skilled in the art using the disclosure provided herein will understand that other quantum computing devices or quantum computing systems may be used without departing from the scope of the present disclosure.

[0199] System 100 includes quantum hardware 102 that communicates with one or more classical processors 104. The classical processor 104 can be configured to execute computer-readable instructions stored in one or more memory devices to perform operations such as any of the operations described herein. The quantum hardware 102 includes components for performing quantum computing. For example, the quantum hardware 102 includes a quantum system 110, a control device 112, and a readout device 114 (e.g., a readout resonator). The quantum system 110 can include one or more multi-level quantum subsystems, such as a register of qubits (e.g., qubit 120). In some implementations, the multi-level quantum subsystem can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, etc.

[0200] The type of multi-level quantum subsystem utilized by system 100 can vary. For example, in some cases, it may be convenient to include one or more superconducting qubits, such as transmon qubits, flux qubits, gmon qubits, xmon qubits, or one or more readout devices 114 attached to other qubits. In other cases, ion traps, photonic devices, or superconducting cavities (e.g., that can prepare a state without requiring qubits) may be used. Further examples of multi-level quantum subsystems include fluxonium qubits, silicon quantum dots, or phosphorus impurity qubits.

[0201] A quantum circuit can be constructed and applied to a register of qubits contained within a quantum system 110 via a plurality of control lines coupled to one or more control devices 112. Exemplary control devices 112 operating on the register of qubits can be used to implement a quantum circuit having a quantum gate, or a plurality of quantum gates, such as Pauli gates, Hadamard gates, controlled NOT (CNOT) gates, controlled phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, and the like. One or more control devices 112 can be configured to operate on the quantum system 110 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystem can be a superconducting qubit, and the control device 112 can be configured to provide control pulses to the control lines to generate a magnetic field for adjusting the frequency of the qubit.

[0202] The quantum hardware 102 can further include a readout device 114 (e.g., a readout resonator). Measurement results 108 obtained via a measurement device can be provided to a classical processor 104 for processing and analysis. In some implementations, the quantum hardware 102 can include a quantum circuit and control devices 112, and the readout device 114 can implement one or more quantum logic gates that operate on the quantum hardware 102 through physical control parameters (e.g., microwave pulses) sent through wires contained within the quantum hardware 102. Further examples of control devices include any waveform generator in which a DAC (digital-to-analog converter) creates a signal.

[0203] The readout device 114 may be configured to perform a quantum measurement on the quantum system 110 and send the measurement result 108 to the classical processor 104. Additionally, the quantum hardware 102 may be configured to receive data specifying the physical control qubit parameter values 106 from the classical processor 104. The quantum hardware 102 may use the received physical control qubit parameter values 106 to update the actions of the control device 112 and the readout device 114 on the quantum system 110. For example, the quantum hardware 102 may receive data specifying new values representing the voltage strengths of one or more DACs included within the control device 112, and accordingly, may update the actions of the DACs on the quantum system 110. The classical processor 104 may be configured to initialize the quantum system 110 to an initial quantum state, for example, by sending data specifying an initial set of the parameters 106 to the quantum hardware 102.

[0204] In some implementations, the readout device 114 can measure the state of an element (e.g., a qubit) of the quantum system, such as a qubit, by utilizing the difference in impedance between the |0〉 state and the |1〉 state of the element. For example, when the qubit is in the state |0〉 or the state |1〉, the resonance frequency of the readout resonator may take on different values due to the non-linearity of the qubit. Thus, the microwave pulse reflected from the readout device 114 transmits an amplitude and phase shift that depends on the qubit state. In some implementations, a parcel filter may be used in conjunction with the readout device 114 to impede microwave propagation at the qubit frequency.

[0205] In some embodiments, the quantum system 110 can include, for example, a plurality of qubits 120 arranged within a two-dimensional grid 122. For clarity, the two-dimensional grid 122 shown in FIG. 1 includes 4×4 qubits, but in some implementations, the system 110 can include fewer or more qubits. In some embodiments, the plurality of qubits 120 can interact with each other via a plurality of qubit couplers, such as qubit coupler 124. The qubit coupler can define nearest-neighbor interactions between the plurality of qubits 120. In some implementations, the strength of the plurality of qubit couplers is an adjustable parameter. In some cases, the plurality of qubit couplers included in the quantum computing system 100 can be couplers having a fixed coupling strength.

[0206] In some implementations, the plurality of qubits 120 can include data qubits, such as qubit 126, and measurement qubits, such as qubit 128. Data qubits are qubits that participate in the calculations performed by the system 100. Measurement qubits are qubits that can be used to determine the results of the calculations performed by the data qubits. That is, during the calculation, the unknown state of the data qubit is transferred to the measurement qubit using appropriate physical operations and measured via appropriate measurement operations performed on the measurement qubit.

[0207] In some implementations, each qubit within the plurality of qubits 120 can operate using respective operating frequencies, such as an idling frequency and / or an interaction frequency and / or a readout frequency and / or a reset frequency. The operating frequencies can be different for each qubit. For example, each qubit can idle at a different operating frequency. The operating frequencies of the qubits 120 can be selected before the calculation is performed.

[0208] FIG. 1 shows one exemplary quantum computing system that can be used to implement the methods and operations according to exemplary aspects of the present disclosure. Without departing from the scope of the present disclosure, other quantum computing systems can be used.

[0209] FIG. 2 shows an overview of a system and method according to an exemplary embodiment of the present disclosure. As shown, aspects of the present disclosure can simulate an experiment on a physical system 202, for example, using a quantum computing system. The physical system can have M quantum observables such as quantum observables 204.a, 204.b, 204.c, ... 204.M. In some examples, the quantum observables can include, for example, dipole moment and polarizability, electron density, forces experienced by classical nuclei, or other properties associated with the ground state. A defined function 206 can be generated that encodes the expected value 208 of each of the M quantum observables 204.a, 204.b, 204c., ... 204d as the gradient 210 of a defined function.

[0210] The quantum operation 212 can be executed by implementing a quantum circuit 214 on one or more qubits 216 of the quantum computing system. The quantum operation 212 can be operable to determine the gradient 210 of the defined function 206. For example, the quantum operation 212 can be operable to determine the gradient 210 using the Gilyen quantum gradient algorithm 218. The quantum circuit 214 can implement a plurality of state preparation operations 220 on the qubits 216. The quantum circuit 214 can implement a probability oracle 222 of the defined function 206. An exemplary quantum circuit 214 will be described with reference to FIG. 3.

[0211] Referring to FIG. 2, the expected values 226 of the M quantum observables 204.a, 204.b, 204.c, ... 204.M can be determined after implementation of the quantum operation 212. For example, the expected values 226 of the M quantum observables 204.a, 204.b, 204.c, ... 204.M can be obtained based at least in part on the measurement values 224 of the qubits 216.

[0212] FIG. 3 shows an exemplary quantum circuit 214 according to an exemplary embodiment of the present disclosure. The quantum circuit 214 can be implemented on one or more qubits in a quantum computing system, such as the quantum computing system shown in FIG. 1. The quantum circuit 214 can implement a probabilistic oracle of a defined function for measuring an observable that acts on a prime qubit. The quantum circuit 214 includes M registers of first qubits 302, 304, 306, a register of N second qubits 308, and an auxiliary qubit 310. The M registers of the first qubits 302, 304, and 306 include b-bit (e.g., 4-bit) precision for each parameter. Ellipse 312 represents the input value of the defined function. Ellipse 314 represents the zero-initialized qubits on the register of the N second qubits 308. Block 316 indicates a state preparation unitary for preparing the state of the register of the N second qubits 308 to simulate a physical system. The auxiliary qubit 310 encodes the value of the defined function in its amplitude.

[0213] Gates 318 and 320 on the auxiliary qubit 310 represent a Hadamard and a phase gate for a Hadamard test on the auxiliary qubit. A series of ellipses represents a double-controlled gate 322 that implements time evolution by an observable acting on the M registers of the first qubits 302, 304, and 306. Each series of gates 322 acts on the M registers of the first qubits 302, 304, and 306 respectively and corresponds to a different observable. The rotation angle within a series of gates changes by a power of 2 for each gate.

[0214] FIG. 4 shows a flowchart of an exemplary method 400 for operating one or more qubits within a quantum computing system according to an exemplary embodiment of the present disclosure. Method 400 may be implemented using any suitable quantum computing system, such as the system shown in FIG. 1. As used herein, the term “computing device” may refer to a classical computing device, a quantum computing device, or a combination of a classical computing device and a quantum computing device. FIG. 4 shows operations performed in a particular order for purposes of illustration and explanation. Those skilled in the art using the disclosure provided herein will understand that any of the operations of the methods described herein may include steps that are extended, omitted, rearranged, and / or varied in various ways without departing from the scope of the present disclosure and without being illustrated.

[0215] At 402, method 400 includes accessing or obtaining a defined function associated with a physical system to be simulated by a quantum computing system. The defined function can encode an estimated value of at least one characteristic to be simulated by the quantum computing system as a gradient of the defined function.

[0216] At 404, method 400 can implement a quantum circuit on a plurality of qubits within the quantum computing system to perform a quantum operation. The quantum operation can be operable to determine the gradient of the function. For example, in some embodiments, the quantum operation can include the Gilyen quantum gradient algorithm.

[0217] At 406, method 400 can determine an estimated value of at least one characteristic based at least in part on a measurement of at least one of the plurality of qubits after implementation of the quantum operation. For example, in some embodiments, the estimated value can include an expected value of an observable within a physical system (e.g., a quantum system). In some embodiments, the estimated value can include elements associated with a time-dependent correlation function.

[0218] The digital, classical, and / or quantum mechanical subjects described herein, as well as the implementations of digital functional operations and quantum computations, can be implemented in digital electronic circuits, appropriate quantum circuits, or more generally, quantum computing systems, tangible digital and / or quantum computer software or firmware, digital and / or quantum computer hardware including the structures disclosed herein and their structural equivalents, or combinations of one or more of these. The term "quantum computing system" can include, without limitation, a quantum computer / computing system, a quantum information processing system, a quantum cryptographic system, or a quantum simulator.

[0219] The implementations of the digital and / or quantum subjects described herein can be implemented as one or more digital and / or quantum computer programs, i.e., as one or more modules of digital and / or quantum computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, a data processing apparatus. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubit / qubit structures, or combinations of one or more of these. Alternatively or in addition, the program instructions can be encoded on an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by a data processing apparatus.

[0220] The terms "quantum information" and "quantum data" refer to information or data that is conveyed by, held within, or stored in a quantum system, and the smallest non-trivial system is the qubit, i.e., the system that defines the unit of quantum information. It is understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as two-level systems in the corresponding context. Such quantum systems can include, for example, multi-level systems having two or more levels. As an example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational basis states are identified with the ground state and the first excited state, but it is understood that other settings are possible where the computational states are identified with higher levels of excited states (e.g., qubits).

[0221] The term "data processing device" refers to digital and / or quantum data processing hardware and encompasses all kinds of devices, apparatuses, and machines for processing digital and / or quantum data, including, by way of example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, or multiple digital and quantum processors or computers, and combinations thereof. The device can also be, or further include, special-purpose logic circuits, such as FPGAs (field-programmable gate arrays), or ASICs (application-specific integrated circuits), or quantum simulators, i.e., quantum data processing devices designed to simulate or generate information about a particular quantum system. In particular, a quantum simulator is a dedicated quantum computer that does not have the ability to perform general-purpose quantum computing. The device can, in some cases, in addition to hardware, include code that creates an execution environment for digital and / or quantum computer programs, such as code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

[0222] A digital or classical computer program, also called or describable as a program, software, software application, module, software module, script, or code, can be written in any form of programming language, including compiler-type or interpreter-type languages, or declarative or procedural languages, and can be deployed in any form, such as as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, also called or describable as a program, software, software application, module, software module, script, or code, can be written in any form of programming language, including compiler-type or interpreter-type languages, or declarative or procedural languages, can be converted into a suitable quantum programming language, or can be written in a quantum programming language, such as QCL, Quipper, Cirq, etc.

[0223] Digital and / or quantum computer programs, although not essential, may correspond to files in a file system. The program may be stored in a part of a file that holds other programs or data, such as one or more scripts stored in a markup language document, or in a single file dedicated to the program, or in a plurality of coordinated files, such as files that store one or more modules, subprograms, or portions of code. Digital and / or quantum computer programs may be deployed to execute on one digital or one quantum computer, or may be distributed across one site or multiple sites and interconnected by a digital and / or quantum data communication network on a plurality of digital and / or quantum computers. A quantum data communication network is understood to be a network that can transmit quantum data using a quantum system, such as qubits. In general, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0224] The processes and logical flows described herein can be executed by one or more programmable digital and / or quantum computers, operate on one or more digital and / or quantum processors, and, as necessary, execute one or more digital and / or quantum computer programs to function by operating on input digital and quantum data and generating an output. The processes and logical flows can also be executed by special-purpose logic circuits, such as FPGAs or ASICs, or quantum simulators, or by a combination of special-purpose logic circuits or quantum simulators and one or more programmed digital and / or quantum computers, and the apparatus can also be implemented as special-purpose logic circuits, such as FPGAs or ASICs, or quantum simulators.

[0225] For one or more digital and / or quantum computers or processor systems to be "configured to" or "operable to" perform a particular operation or action means that the system has installed thereon software, firmware, hardware, or combinations thereof that cause the system to perform the operation or action during operation. For one or more digital and / or quantum computer programs to be configured to perform a particular operation or action means that the one or more programs include instructions that, when executed by a digital and / or quantum data processing apparatus, cause the apparatus to perform the operation or action. A quantum computer may receive from a digital computer instructions that, when executed by a quantum computing apparatus, cause the apparatus to perform an operation or action.

[0226] Digital and / or quantum computers suitable for the execution of digital and / or quantum computer programs may be based on general-purpose or special-purpose digital and / or quantum microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, the central digital and / or quantum processing unit receives instructions and digital and / or quantum data from a read-only memory, a random access memory, or a quantum system suitable for transmitting quantum data, such as photons, or combinations thereof.

[0227] Some exemplary elements of a digital and / or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions as well as digital and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special-purpose logic circuitry or a quantum simulator. Generally, a digital and / or quantum computer includes, receives digital and / or quantum data from, transfers digital and / or quantum data to, or is operatively coupled to one or more mass storage devices for storing digital and / or quantum data, such as, for example, magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0228] Digital and / or quantum computer program instructions and digital and / or quantum computer-readable media suitable for storing digital and / or quantum data include, by way of example, all forms of non-volatile digital and / or quantum memories, media, and memory devices including semiconductor memory devices such as EPROM, EEPROM, and flash memory devices, magnetic disks such as internal hard disks or removable disks, magneto-optical disks, CD-ROM and DVD-ROM disks, and quantum systems such as trapped atoms or electrons. It is understood that a quantum memory is a device capable of storing quantum data for a long time with high fidelity and efficiency, such as an optical matter interface where light is used for transmission, and a substance for storing and preserving quantum characteristics of quantum data such as superposition or quantum coherence.

[0229] The various systems described herein, or control of portions thereof, can be implemented as a digital and / or quantum computer program product stored on one or more tangible, non-transitory machine-readable storage media and including instructions executable on one or more digital and / or quantum processing devices. The systems described herein, or portions thereof, can each be implemented as an apparatus, method, or electronic system including one or more digital and / or quantum processing devices and memory for storing executable instructions for performing the operations described herein.

[0230] This specification includes many details of particular implementations, but these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. The particular features described herein in the context of separate implementations can also be implemented in combination within a single implementation. Conversely, the various features described in the context of a single implementation can also be implemented separately, or in any suitable sub-combination, in multiple implementations. Further, features may be described above as acting in certain combinations and even initially claimed as such, but one or more features from a claimed combination may in some cases be excised from that combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0231] Similarly, the operations are presented in the drawings in a particular order, but this should not be construed as requiring that such operations be performed in that particular order or sequence, or that all of the illustrated operations be performed, to achieve a desirable result. In some situations, multitasking and parallel processing may be advantageous. Further, the separation of various system modules and components in the above-described implementations should not be understood as necessarily requiring such separation in all implementations, and it should be understood that the program components and systems described may generally be integrated together in a single software product or packaged into multiple software products.

[0232] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes illustrated in the accompanying figures do not necessarily require the particular order or sequence shown to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Explanation of Reference Numerals

[0233] 100 Quantum computing system 102 Quantum hardware 104 Classical processor 106 Physical control qubit parameter value 108 Measurement result 110 Quantum system 112 Control device 114 Readout device 120 Qubit 122 Two-dimensional grid 124 Qubit coupler 126 Qubit 128 Qubit 202 Physical system 204 Quantum observable 206 Defined function 208 Expected value 210 Gradient 212 Quantum operation 214 Quantum circuit 216 Qubit 218 Quantum gradient algorithm 220 State preparation operation 222 Probability oracle 224 Measurement value 226 Expected value 302 First qubit 304 First qubit 306 First qubit 308 Second qubit 310 Auxiliary qubit 312 Ellipse 318 Gate 320 Gate 322 Double-controlled gate

Claims

1. A method for determining the state of a physical system, comprising: obtaining, by one or more computing devices, a defined function associated with the physical system, the defined function encoding an estimated value of at least one characteristic to be simulated by a quantum computing system as a gradient of the defined function; implementing, by the one or more computing devices, a quantum circuit on a plurality of qubits within the quantum computing system to perform a quantum operation, the quantum operation being operable to determine the gradient of the defined function; determining, by the one or more computing devices, the estimated value of the at least one characteristic based at least in part on at least one of the plurality of qubits after implementation of the quantum operation. A method comprising the above steps.

2. The method according to claim 1, wherein the characteristic to be simulated comprises an expected value of a quantum observable of the physical system.

3. The method according to claim 1, wherein the characteristic to be simulated comprises one or more elements associated with an unequal-time correlation function.

4. The method according to claim 1, wherein the quantum operation is operable to determine the gradient of the defined function using the Gilyen quantum gradient algorithm.

5. The method according to claim 1, wherein the step of implementing the quantum circuit by the one or more computing devices comprises implementing, by the one or more computing devices, a plurality of state preparation operations on at least a subset of the plurality of qubits using the quantum circuit.

6. The method according to claim 1, wherein the quantum circuit implements a probability oracle of the defined function.

7. The method according to claim 1, wherein the quantum circuit is implemented on M first qubit registers, a system register of N second qubits, and at least one auxiliary qubit, where M is the number of quantum observables and N is an integer.

8. The method according to claim 7, wherein the quantum circuit encodes the defined function into the amplitude of the auxiliary qubit.

9. The method of claim 8, wherein the quantum circuit implements a Hadamard test on the auxiliary qubit using one or more Hadamard gates and one or more phase gates.

10. The method of claim 7, wherein the N second qubits are initialized to zero and the quantum circuit is operable to implement a state preparation unitary on the N second qubits.

11. The method of claim 7, wherein the quantum circuit implements a controlled time evolution for the quantum observable using at least one doubly controlled quantum gate on the M first qubit registers.

12. The method of claim 11, wherein the at least one doubly controlled quantum gate is based at least in part on one of the N second qubits and the auxiliary qubit.

13. The method of claim 6, wherein the step of determining the estimated value of the characteristic by the one or more computing devices includes performing a measurement on at least one of the plurality of qubits.

14. A quantum computing system, a plurality of qubits including M registers of first qubits, a register of N second qubits, and at least one auxiliary qubit, one or more control devices configured to implement a quantum circuit on the plurality of qubits to determine a gradient of a function defined using a quantum operation, wherein the gradient of the defined function encodes an estimated value of a characteristic to be simulated by the quantum computing system, a quantum computing system comprising.

15. The quantum computing system of claim 14, wherein the characteristic includes expected values of a plurality of observables of a physical system.

16. The quantum computing system of claim 14, wherein the characteristic includes one or more elements associated with an unequal time correlation function.

17. The quantum computing system of claim 14, wherein the quantum operation is operable to determine the gradient of the defined function using a Gilyen quantum gradient algorithm.

18. The quantum computing system of claim 14, wherein the quantum circuit encodes the defined function into the amplitude of the auxiliary qubit.

19. The quantum computing system according to claim 18, wherein the quantum circuit implements a Hadamard test on the auxiliary qubit using one or more Hadamard gates and one or more phase gates.

20. The quantum computing system according to claim 14, wherein the N second qubits are initialized to zero, the quantum circuit is operable to implement a state preparation unitary on the N second qubits, the quantum circuit implements a controlled time evolution for a quantum observable using at least one doubly controlled quantum gate on the M first qubit registers, the doubly controlled quantum gate is based at least in part on one of the N second qubits and the auxiliary qubit, and determining the estimated value of the characteristic by one or more computing devices includes performing at least one measurement of at least one of the M first qubit registers.

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