Quantum phase estimation of multiple eigenvalues

Abstract

Methods, systems, and apparatus, including computer programs encoded on a computer storage medium, for quantum phase estimation. In one aspect, a method includes: performing a plurality of phase estimation experiments on an ancilla qubit, wherein each experiment includes: preparing the ancilla qubit in an initial quantum state; applying a unitary operator to the ancilla qubit prepared in the initial quantum state and a quantum register prepared in an arbitrary quantum state one or more times; and measuring the state of the ancilla qubit; determining an expectation value of the state of the ancilla qubit based on results of the plurality of phase estimation experiments; and inferring a phase of an eigenvalue of the unitary operator by optimizing a mixture model representing probabilities of the state of the ancilla qubit based on the determined expectation value of the state of the ancilla qubit and the measured state of the ancilla qubit.

Description

[0001] This application is a divisional application of the invention patent application filed on December 30, 2015, with application number 201580085813.8 and title "Quantum Phase Estimation of Multiple Eigenvalues". Technical Field

[0002] This manual relates to quantum computing. Background Technology

[0003] Quantum phase estimation is an important subroutine in many quantum computing applications. The eigenspectrum of all unitary operators is a complex number with an identity norm. The purpose of quantum phase estimation can be to estimate the eigenphase of the eigenvectors of a unitary operator. For example, for a unitary operator U (i.e., U|ψ>=e2), the eigenphase can be estimated. -iφ The quantum state of the eigenstate of |ψ> is given by |ψ>. The purpose of quantum phase estimation is to estimate the value of φ. Summary of the Invention

[0004] This specification describes a technique for inducing and inferring multiple phases of the individual eigenvalues ​​of a unitary operator on a physical quantum state by performing systematic quantum experiments, which are then interpreted using a classical hybrid model. The individual phases of the eigenvalues ​​of a unitary operator applied to a quantum state can be known without the quantum state being an eigenstate of the unitary operator.

[0005] Generally, an innovative aspect of the subject matter described in this specification can be implemented as a method comprising: accessing a first quantum register including at least one ancilla qubit and a second quantum register including one or more qubits, wherein the second quantum register is prepared in a quantum state of an eigenstate of a unitary operator that is not operated on the first or second quantum registers; performing a phase estimation experiment on the first quantum register to determine the expected value of the state of the ancilla qubit; and obtaining the phase of the eigenvalue of the unitary operator based on the determined expected value of the state of the ancilla qubit.

[0006] Other implementations of this include corresponding computer systems, devices, and computer programs stored on one or more computer storage devices, each configured to perform the actions of the method. A system of one or more computers can be configured to perform specific operations or actions by means of software, firmware, hardware, or combinations thereof installed on the system that cause the system to perform actions during operation. One or more computer programs can be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform actions.

[0007] The foregoing and other implementations may optionally include one or more of the following features, individually or in combination. In some implementations, performing a phase estimation experiment on the first quantum register includes repeatedly measuring the state of the auxiliary qubit for each phase estimation experiment.

[0008] In other implementations, the phase of knowing the eigenvalues ​​of the unitary operator includes: representing the result of a measurement of the state of the auxiliary qubit as a binary random variable; defining a model representing the probability of the value of the binary random variable based on the binary random variable and the determined expected value; and optimizing the model to know the phase of the eigenvalues ​​of the unitary operator.

[0009] In some implementations, the model is a hybrid model.

[0010] In some implementations, the method further includes providing the phase of the known eigenvalues ​​of the unitary operator for use in quantum computing.

[0011] In other implementations, measuring the state of the auxiliary qubit includes randomly sampling pairs (M, θ), where M represents the number of applications of the unitary operator, and θ represents the angle of rotation applied by a phase gate operating on the first quantum register.

[0012] In some cases, from [0, M] max Sample M uniformly, where M max The predetermined maximum number of times the unitary operator is applied, and θ is sampled uniformly from [0, 2π].

[0013] In some implementations, the binary random variable is constrained by the expected value of the state of the determined auxiliary qubit.

[0014] In other implementations, the mixture model representing the probability of the value of a binary random variable is given by the following:

[0015]

[0016] Among them, Z anc a represents the value of the state of the auxiliary qubit. k =|<k|ψ> | 2 , where |ψ> is a quantum state and |k> is the k-th eigenstate of the unitary operator, φ k Let θ represent the phase of the eigenvalue corresponding to the k-th eigenstate |k> of the unitary operator, M represent the number of applications of the unitary operator, θ represent the angle of rotation applied by the phase gate, N represent the number of qubits in the second register, and δ represent the Kronecker delta.

[0017] In a further implementation, optimizing the model to determine the phase of the eigenvalues ​​of the unitary operator includes optimizing the hybrid model with respect to a unique {a, φ} to determine the phase of the eigenvalues ​​of the unitary operator to the required precision, wherein the lowest φ is within a predetermined distance to the ground state of the quantum state.

[0018] In some cases, the hybrid model uses Bayesian inference for estimation.

[0019] In some implementations, the mixture model representing the probability of the value of a binary random variable includes an additional term representing a noise source.

[0020] In some cases, the noise is depolarization noise.

[0021] In some implementations, optimizing the model to determine the phase of the eigenvalues ​​of the unitary operator includes performing an expectation-maximization (EM) algorithm on the model.

[0022] The subject matter described in this specification can be implemented in a particular manner to achieve one or more of the following advantages.

[0023] Determining the quantum phase of multiple eigenvalues ​​of a unitary operator is a computationally challenging task, and in some cases, a tricky one. Typically, determining the eigenphase φ of multiple eigenvalues ​​of a unitary operator is a computationally difficult task. k This is equivalent to classical inference of multiple Fourier modes, given the ability to sample noisy channels at arbitrary times. However, this is computationally inefficient due to the exponential number of eigenphases. Even in some simplified cases, such as when the expectation of the measurement of the auxiliary qubits from which the phase is to be extracted is sparse, determining the quantum phase is simply a matter of finding the eigenphase φ. k The problem involves estimating polynomial numbers to n-bit precision. This can be achieved by measuring auxiliary qubits at various time points and performing Fourier transforms, but this requires multiples of M = 1, 2, 4, 8, ..., 2. n-1 Measuring auxiliary qubits is computationally challenging and requires coherent quantum evolution over an exponential timescale of n.

[0024] Unlike methods that require inference of Fourier moduli, systems that enable quantum phase estimation of multiple eigenvalues ​​of a unitary operator allow for the computation of individual eigenphases without sampling to large values ​​of M, the number of applications of the unitary operator. Therefore, the computational resources required for quantum phase estimation can be reduced compared to other systems that require large M-cycles.

[0025] Furthermore, systems that use hybrid or nonparametric Bayesian models to achieve quantum phase estimation of multiple eigenvalues ​​of unitary operators enable the model to recognize and remove non-trivial noise sources that may contaminate the system.

[0026] Compared to systems that require preparing eigenstates of unitary operators, systems that enable quantum phase estimation of multiple eigenvalues ​​of unitary operators can be applied to a wider range of computational tasks because they do not require preparing eigenstates of unitary operators, but rather any quantum state.

[0027] Systems that enable quantum phase estimation of multiple eigenvalues ​​of a unitary operator allow for the inference of multiple eigenphases of a physical quantum state, providing them for a wide range of computationally valuable industrial applications. For example, the inferred eigenphases can be used to perform quantum simulations, such as quantum algorithms for simulating chemical and molecular reactions, quantum metrology, spectroscopy, factorization algorithms, sequential search algorithms, discrete logarithm calculations, database search algorithms, or well-state sparse systems for solving linear equations.

[0028] Furthermore, systems that enable phase estimation of multiple eigenvalues ​​of a unitary operator can facilitate quantum acceleration of computational tasks. This results in an exponential reduction in the physical resources required to solve computational tasks compared to the most well-known classical algorithms. For example, quantum acceleration can be achieved for computational tasks including Shor's factorization algorithm or quantum simulation algorithms.

[0029] In one aspect, this disclosure provides a method comprising: performing a plurality of phase estimation experiments on an auxiliary qubit, wherein each experiment comprises: preparing the auxiliary qubit in an initial quantum state; applying a unitary operator to the auxiliary qubit prepared in the initial quantum state and to a quantum register prepared in an arbitrary quantum state once or multiple times; and measuring the state of the auxiliary qubit; determining an expected value of the state of the auxiliary qubit based on the results of the plurality of phase estimation experiments; and determining the phase of the eigenvalues ​​of the unitary operator by optimizing a mixture model representing the probability of the state of the auxiliary qubit based on the determined expected value of the state of the auxiliary qubit and the measured state of the auxiliary qubit.

[0030] In another aspect, this disclosure provides an apparatus comprising: a quantum circuit including: at least one auxiliary qubit; a quantum register including one or more qubits, wherein the second quantum register is prepared in an arbitrary quantum state; a phase knowing system configured to know the phase of the eigenvalue of a unitary operator, wherein the apparatus is configured to: perform a plurality of phase estimation experiments on the auxiliary qubit, wherein each experiment includes: preparing the auxiliary qubit in an initial quantum state; applying the unitary operator to the auxiliary qubit prepared in the initial quantum state and the quantum register prepared in an arbitrary quantum state once or multiple times; and measuring the state of the auxiliary qubit; determining an expected value of the state of the auxiliary qubit based on the results of the plurality of phase estimation experiments; and, based on the determined expected value of the state of the auxiliary qubit and the measured state of the auxiliary qubit, the phase of the eigenvalue of the unitary operator is known by the phase knowing system by optimizing a mixture model representing the probability of the state of the auxiliary qubit.

[0031] In another aspect, this disclosure provides a method comprising: determining an expected value of a state of an auxiliary qubit based on the results of multiple phase estimation experiments, wherein each result corresponds to a corresponding number of applications to the auxiliary qubit and a unitary operator of a quantum register prepared in an arbitrary quantum state; and, based on the determined expected value of the state of the auxiliary qubit, obtaining the phase of an eigenvalue of the unitary operator by optimizing a mixture model representing the probability of the state of the auxiliary qubit; wherein the arbitrary quantum state is not an eigenstate of the unitary operator.

[0032] In another aspect, this disclosure provides an apparatus comprising: a quantum circuit including: at least one auxiliary qubit; a quantum register including one or more qubits, wherein the quantum register is prepared in an arbitrary quantum state; a phase knowing system configured to know the phase of an eigenvalue of a unitary operator, wherein the apparatus is configured to: determine an expected value of a state of the auxiliary qubit based on the results of a plurality of phase estimation experiments, wherein each result corresponds to a corresponding number of applications to the auxiliary qubit and the unitary operator of the quantum register prepared in an arbitrary quantum state; and, based on the determined expected value of the state of the auxiliary qubit, know the phase of the eigenvalue of the unitary operator by optimizing a mixture model representing the probability of the state of the auxiliary qubit; wherein the arbitrary quantum state is not an eigenstate of the unitary operator.

[0033] Details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, drawings, and claims. Attached Figure Description

[0034] Figure 1 An example phase estimation system is described.

[0035] Figure 2 This is a flowchart of an example procedure for determining the phase of multiple eigenvalues ​​of a unitary operator.

[0036] Figure 3 This is a flowchart of an example process for determining the phase of the eigenvalues ​​of a unitary operator using data generated from the expected values ​​of the states of the determined auxiliary qubits.

[0037] The same reference numerals and labels in each of the accompanying figures indicate the same elements. Detailed Implementation

[0038] This specification describes apparatus and methods for inducing and inferring multiple phases of individual eigenvalues ​​of a unitary operator on a physical quantum state by performing systematic quantum experiments subsequently interpreted using a classical mixture model. The system employs methods for obtaining probabilistic models that combine maximum a posteriori estimation, mixture models, and nonparametric Bayesian inference on a classical computer. The system enables the determination of individual phases of the eigenvalues ​​of a unitary operator applied to a quantum state, without requiring the quantum state to be an eigenstate of the unitary operator. The system also enables the modeling and mitigation of noise sources that could contaminate the signal from which the model derives multiple phases.

[0039] Example operating environment

[0040] Figure 1 An example phase estimation system 100 is depicted. The example system 100 is an example of a system implemented as a classical or quantum computer program on one or more classical computers or quantum computing devices at one or more locations, in which the systems, components and techniques described below can be implemented.

[0041] Phase estimation system 100 can receive input data (e.g., input data 108) including data specifying quantum circuit parameters, and can generate output data, e.g., output data 110, of multiple known phases as eigenvalues ​​of a specified unitary operator. Quantum circuit parameters may include parameters related to the number of times quantum gates are applied in the quantum circuit, or quantum gate parameters. The eigenvalues ​​of the unitary operator specified by input data 106 are complex numbers with a unit norm, e.g., e^(-1 / 2). -iφ , where φ represents the phase of the corresponding eigenvalue. Since the unitary operator has more than one eigenvector and a corresponding eigenvalue, the known phase 110 includes multiple known phases.

[0042] Phase estimation system 100 may include quantum circuit 102 and phase acquisition subsystem 104. Quantum circuit 102 may include two quantum registers, such as quantum register 112 and quantum register 114. Quantum register 112 may include at least one auxiliary qubit and is initially prepared in a quantum state (e.g., state |0>). In some implementations, quantum circuit 102 may include multiple quantum registers containing a single auxiliary qubit. Quantum register 114 may include one or more qubits and may be initially prepared in an arbitrary quantum state |ψ>. In some implementations, the arbitrary quantum state |ψ> is not an eigenstate of the unitary operator specified by input data 106.

[0043] The quantum circuit 102 may also include a set of quantum gates. This set of quantum gates may include at least two Hadamard gates (e.g., Hadamard gates 116a and 116b), a phase gate (e.g., phase gate 118), a measurement operator (e.g., measurement operator 120), and a unitary operator (e.g., unitary operator 122) specified by the input data 106. The unitary operator 122 operates on the first quantum register 112 and the second quantum register 114, and can be applied multiple times, e.g., M times. The application of the unitary operator 122 on auxiliary qubits (e.g., in the |1> state of the auxiliary qubits) can be experimentally controlled.

[0044] exist Figure 1 In the example shown, quantum circuit 102 uses a hadamard gate 116a to map an auxiliary qubit to a quantum superposition, for example, quantum circuit 102 maps |0>→(|0>+|1>) / 2, and then uses a phase gate 118 to rotate the auxiliary qubit by an angle θ around the Z-axis. A unitary operator 122 is applied to both quantum registers, followed by a second hadamard gate 116b. Finally, a measurement is performed on the auxiliary qubit via a measurement operator 120 to determine the expected value of the auxiliary qubit's state. For clarity, Figure 1 An auxiliary qubit is described, but in some implementations, the quantum circuit 102 may include more than one auxiliary qubit.

[0045] Phase acquisition subsystem 104 is configured to perform a phase estimation experiment on quantum circuit 102. To perform the phase estimation experiment on quantum circuit 102, phase acquisition subsystem 104 can repeatedly measure the state of an auxiliary qubit using measurement operator 120. Phase acquisition subsystem 104 can optionally store the results from the measurements of the auxiliary qubit's state in a data storage (e.g., in measurement result data storage 106) for post-processing. For example, phase acquisition subsystem 104 can use the results of the measurements of the auxiliary qubit's state to determine the expected value of the auxiliary qubit's state.

[0046] The phase-knowing subsystem 104 can be configured to represent the result of measuring the state of the auxiliary qubit as a binary random variable, and based on this binary random variable and the expected value of the determined state of the auxiliary qubit, to define a model representing the probability of the value of the binary random variable. (See below for reference.) Figure 3 A more detailed description of the model that defines the probability of the value of a binary random variable.

[0047] The phase-finding subsystem 104 can be configured to optimize a model representing the probability of the value of a binary random variable to determine the phase of the eigenvalues ​​of a unitary operator. (See below for reference.) Figure 3 A more detailed description is given of the model that optimizes the probabilities of the values ​​of binary random variables to determine the phase of the eigenvalues ​​of a unitary operator specified by the input data.

[0048] The phase-aware subsystem 104 can provide output data of the known phase generated by specifying the eigenvalues ​​of the unitary operator as the output of the phase estimation system 100, for example, as output data 110. Output data 110 can be provided to a classical or quantum processor for further processing. For example, the phase estimation system 100 can be a subsystem of a main system that receives input data 106 and generates output data 110 from the input data 106 as part of a subroutine of a computation (e.g., a quantum simulation computation) performed by the main system.

[0049] Hardware programming

[0050] Figure 2 This is a flowchart of an example process 200 for obtaining the phase of multiple eigenvalues ​​of a unitary operator. For convenience, process 200 will be described as being performed by a system of one or more classical or quantum computing devices located at one or more locations. For example, a phase-obtaining system appropriately programmed according to this specification (e.g., Figure 1 The phase acquisition subsystem 104) can execute process 200.

[0051] System access to quantum circuits (e.g., Figure 1 (Step 202). The quantum circuit may include a first quantum register that may contain at least one auxiliary qubit and a set of quantum gates, said set of quantum gates may include at least two Hadamard gates, a phase gate that rotates the auxiliary qubit by an angle θ about the Z-axis, a unitary operator U, and a measurement operator. in Quantum circuits may also include a second quantum register with one or more qubits. The second quantum register can be prepared in any quantum state. The unitary operator U acts on both the first and second quantum circuits and can be applied M times.

[0052] In some implementations, the arbitrary quantum state may not be an eigenstate of the unitary operator, and the action of the unitary operator U on the arbitrary quantum state |ψ> can be given by the following equation (1).

[0053]

[0054] In equation (1), N represents the number of qubits in the second quantum register, and the eigenvalues ​​of U are... The eigenvectors {|k>} satisfy Where {φ k The phase of multiple eigenvalues ​​of U is defined.

[0055] The system performs a phase estimation experiment on the quantum circuit (step 204). The system can repeatedly measure the state of the auxiliary qubit for each phase estimation experiment to determine the expected value of the state of the auxiliary qubit. In the case of multiple eigenstates, the expected value of the state of the auxiliary qubit can be given by the following equation (2).

[0056]

[0057] In equation (2), <Z anc > represents the expected value of the state of the auxiliary qubit, a k =|<k|ψ> | 2 Where |ψ> represents an arbitrary quantum state, M represents the number of times the unitary operator U is applied, and φ k Let θ represent the phase of the eigenvalue corresponding to the k-th eigenstate |k> of the unitary operator, and let θ represent the angle of rotation applied by the phase gate.

[0058] In some implementations, the system can randomly sample (M, θ) the state of the auxiliary qubit before measuring its state. For example, the system can sample from [0, M... max Uniformly sample M, where M max A predetermined maximum number of times the unitary operator is applied, for example, 10 times, and θ is sampled uniformly from [0, 2π]. In other examples, the system can be based on a... k and φ k The values ​​of M and θ are assumed to be extracted from a special distribution designed to provide the most information. For example, it might be interesting to know the value of φ0 with high precision, for example, for applications in quantum chemistry. Therefore, the system can extract the value of θ from a normal distribution chosen such that the sample maximizes the information known about the lowest phase. A similar strategy can be implemented to inform the favorable choice of M.

[0059] The system obtains the phase of the eigenvalues ​​of the unitary operator (step 206). For example, the system can use the results from the phase estimation experiment performed with reference to step 204 to obtain the phase of the eigenvalues ​​of the unitary operator. See below for reference... Figure 3 The phase of the unitary operator's eigenvalues ​​is described in more detail using the expected value of the state of the determined auxiliary qubit.

[0060] The system provides the phase of the known eigenvalues ​​of the unitary operator for use in quantum computing (step 208). There may be numerous computational applications that require the phase of the known eigenvalues ​​of the unitary operator. For example, the phase of the known eigenvalues ​​of the unitary operator can be used in quantum algorithms for simulating chemistry and molecular reactions, quantum simulation, quantum metrology, factoring, searching databases, well-state sparse systems solving linear equations, order finding, or computational discrete algorithms. In some implementations, the phase of the known eigenvalues ​​of the unitary operator can be provided for quantum algorithms that offer quantum speedups compared to the most well-known classical algorithms (e.g., factoring algorithms or quantum simulation algorithms).

[0061] Classical inference using a mixture model

[0062] Figure 3 This is a flowchart of an example process 300 for determining the phase of a unitary operator's eigenvalues ​​using generated data including the expected values ​​of the states of the determined auxiliary qubits. For convenience, process 300 will be described as being performed by a system of one or more classical computing devices located in one or more locations. For example, a phase-knowing system appropriately programmed according to this specification (e.g., Figure 1 The phase acquisition subsystem 104) can execute process 300.

[0063] The system represents the measurement result of the auxiliary qubit's state as a binary random variable (step 302). After the measurement, the auxiliary qubit's state can be aligned with either the positive or negative Z-axis. The auxiliary qubit's state Z... anc Therefore, it can be written as a binary random variable of {0, 1}, which is given below by equation (3).

[0064]

[0065] In equation (3), Z anc The value of the state of the auxiliary qubit is represented, and <Z anc > represents the expected value of the state of the auxiliary qubit, which, for example, is shown above. Figure 2 The determination is made at step 204.

[0066] Based on the binary random variable defined in equation (3) and the determined expected value <Z anc The system defines a model representing the probability of the value of a binary random variable (step 304). In some implementations, the model representing the probability of the value of a binary random variable is p(Z). anc ) can be a mixture model p(Z) anc |a k , φ k ; M, θ). This mixture model can represent the probability of the value of the binary random variable and can be given by the following equation (4).

[0067]

[0068] In equation (4), Z anc a represents the value of the state of the auxiliary qubit. k =|<k|ψ> | 2 , where |ψ> is an arbitrary quantum state and |k> is the k-th eigenstate of the unitary operator, φ k Let M represent the phase of the eigenvalue corresponding to the k-th eigenstate |k> of the unitary operator, M represent the number of times the unitary operator is applied, θ represent the angle of rotation applied by the phase gate, N represent the number of qubits in the second register, and δ represent Kronecker δ.

[0069] In some implementations, the model p(Z) represents the probability of the value of a binary random variable. anc ) is a hybrid model p(Z) for modeling noise sources. anc |a k , φ k ; M, θ), and can be given by the following equation (5).

[0070]

[0071] In equation (5), Z anc a represents the value of the state of the auxiliary qubit. k =|<k|ψ> | 2 Where |ψ> is an arbitrary quantum state and |k> is the k-th eigenstate of the unitary operator, φ k Let represent the phase of the eigenvalue corresponding to the k-th eigenstate |k> of the unitary operator, M represent the number of unitary operator applications, θ represent the rotation angle applied by the phase gate, N represent the number of qubits in the second register, and δ represent the Kronecker δ. In some implementations, This can represent depolarization noise. In other implementations, It can represent other types of noise.

[0072] The system optimizes the model to determine the phase of the eigenvalues ​​of the unitary operator (step 306). The system can optimize the model to maximize the probability p(Z). anc |a k , φ k ; M, θ) of a k and φ k The probability p(Z) anc |a k , φ k M, θ) is a parameter a k and φ k Let M and θ be the probability of observing the corresponding measurement result given the condition (where M and θ have known values).

[0073] In some implementations, the system can optimize the model to determine the phase of the eigenvalues ​​of the unitary operator by optimizing a hybrid model with respect to a unique {a, φ} (where the lowest φ is within a predetermined distance (e.g., an acceptable distance) to the ground state of any quantum state), so as to determine the phase of the eigenvalues ​​of the unitary operator to the required precision.

[0074] In some implementations, the system performs an expectation-maximization (EM) algorithm on the mixture model to optimize it and determine the phase of the unitary operator's eigenvalues. For example, the system may introduce an additional auxiliary variable q. k The additional auxiliary variable q k satisfy This allows the hybrid model p(Z) to be approximated by the following equation (6). anc |a k , φ k ;M,θ).

[0075]

[0076] The system can apply the expectation-maximization algorithm by alternatively performing the E-step and the M-step. The E-step produces a function of the expectation of the log-likelihood evaluated with respect to the current estimate of the parameters, and may include optimizing q by ensuring that the inequalities given in equation (6) above are as tight as possible. k The M-step calculates the parameters that maximize the expected log-likelihood found in the E-step, and includes the parameters with a fixed q. k a k and φ k For q k and a k and φ k The update can be given by equations (7) and (8) below.

[0077]

[0078]

[0079] The expectation-maximization algorithm can generate parameters 'a' in a model representing the probability of the value of a binary random variable. k and φ k The maximum likelihood estimate, i.e., the phase φ of the known eigenvalues ​​of the unitary operator. k .

[0080] In some implementations, nonparametric Bayesian inference can be used to estimate the mixture model p(Z). anc |a k , φ k ; M, θ). When the magnitude of the mixed components is large, the system can introduce for a k Prior knowledge. Assume a k From k=0 to 2 N If the sum of -1 equals 1, then in some examples, we can introduce a condition for a = {a1, a2, ..., a...}. K The Dirichlet process D, for example, p(a) = DP(a, α). In some implementations, other normalized stochastic measures can also be applied.

[0081] In some implementations, it is possible to add parameters for φ. k The priors, and the mixture model p(Z) can be estimated by the following equation (9). anc |a k , φ k ;M,θ).

[0082]

[0083] In equation (9), Z anc a represents the value of the state of the auxiliary qubit. k =|<k|ψ> | 2 , where |ψ> is a quantum state and |k> is the k-th eigenstate of the unitary operator, φ k Let represent the phase of the eigenvalue corresponding to the k-th eigenstate |k> of the unitary operator, M represent the number of applications of the unitary operator, θ represent the angle of rotation applied by the phase gate, and a is the phase of the eigenvalue for {a k The prior of}.

[0084] The digital and / or quantum themes described in this specification, as well as the implementations of digital functional operations and quantum operations, can be implemented in digital electronic circuits, suitable quantum circuits (or more generally, quantum computing systems), tangibly implemented digital and / or quantum computer software or firmware, digital and / or quantum computer hardware (including the structures disclosed in this specification and their structural equivalents), or combinations thereof. The term "quantum computing system" can include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0085] The implementation of the digital and / or quantum themes described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., one or more modules of digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution by a data processing device or for controlling the operation of a data processing device. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or sequential access memory device, one or more qubits, or a combination of one or more of these. Alternatively or additionally, the program instructions can be encoded on artificially generated propagating signals capable of encoding digital and / or quantum information (e.g., machine-generated electrical, optical, or electromagnetic signals that are generated to encode digital and / or quantum information for transmission to a suitable receiver device for execution by a data processing device).

[0086] The terms “quantum information” and “quantum data” refer to information or data carried, held, or stored in quantum systems, the smallest nontrivial system being a qubit, i.e., a system that defines the unit of quantum information. It is important to understand that the term “qubit” encompasses all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems can include multi-level systems, for example, systems with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the fundamental state of computation is identified using the ground state or the first excited state; however, it is to understand that other settings, such as identifying the computational state using higher-level excited states, are also possible. 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, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multi-digital and quantum processors or computers, and combinations thereof. The device may also be, or further include, dedicated logic circuitry (e.g., FPGA (Field-Programmable Gate Array), ASIC (Application-Specific Integrated Circuit)) or a quantum simulator (i.e., a quantum data processing device designed to simulate or generate information about a specific quantum system). In particular, a quantum simulator is a dedicated quantum computer that does not have the capability to perform general-purpose quantum computing. In addition to hardware, the device may optionally include code that creates an execution environment for digital and / or quantum computer programs, such as code constituting processor firmware, protocol stack, database management system, operating system, or one or more combinations thereof.

[0087] Digital computer programs (also referred to or described as programs, software, software applications, modules, software modules, scripts, or code) can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and can be deployed in any form, including as standalone programs or as modules, components, subroutines, or other units suitable for use in digital computing environments. Quantum computer programs (also referred to or described as programs, software, software applications, modules, software modules, scripts, or code) can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and can be translated into a suitable quantum programming language, or can be written in a quantum programming language (such as QCL or Quipper).

[0088] Digital and / or quantum computer programs may, but do not necessarily, correspond to files in a file system. Programs may be stored as a portion of a file holding other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the target program, or in multiple collaborating files (e.g., a file storing portions of one or more modules, subroutines, or code). Digital and / or quantum computer programs can be deployed to execute on a single digital or quantum computer, or on multiple digital and / or quantum computers located at a single site or distributed across multiple sites and interconnected via digital and / or quantum communication networks. A quantum data communication network should be understood as a network that can use quantum systems to transmit quantum data (e.g., qubits). Typically, digital data communication networks cannot transmit quantum data, but quantum data communication networks can transmit both quantum and digital data.

[0089] The processes and logic flows described in this specification can be executed by one or more programmable digital and / or quantum computers, which, where appropriate, operate with the aid of one or more digital and / or quantum processors to execute one or more digital and / or quantum computer programs, performing functions by processing input digital and quantum data and producing outputs. The processes and logic flows can also be executed by dedicated logic circuits, and the device can be implemented as dedicated logic circuits, such as FPGAs or ASICs, or quantum simulators, or through a combination of dedicated logic circuits or quantum simulators and one or more programmable digital and / or quantum computers.

[0090] For a system of one or more digital and / or quantum computers to be "configured" to perform a specific operation or action, it means that the system has software, firmware, hardware, or a combination thereof installed thereon that causes the system to perform the operation or action in operation. For one or more digital and / or quantum computer programs to be configured to perform a specific operation or action, it means that one or more programs include instructions that, when executed by a digital and / or quantum data processing device, cause the device to perform the operation or action. A quantum computer can receive instructions from a digital computer that, when executed by a quantum computing device, cause the device to perform an operation or action.

[0091] Digital and / or quantum computers suitable for executing digital and / or quantum computer programs can be based on general-purpose or special-purpose digital and / or quantum processors or both, or any other kind of central digital and / or quantum processing unit. Typically, the central digital and / or quantum processing unit receives instructions and digital and / or quantum data from read-only memory, random access memory, or a quantum system suitable for transmitting quantum data (e.g., photons), or combinations thereof.

[0092] Essential components of a digital and / or quantum computer are a central processing unit (CPU) for executing or running instructions and one or more memory devices for storing instructions and digital and / or quantum data. The CPU and memory may be supplemented by dedicated logic circuitry or a quantum simulator, or may be integrated therein. Typically, a digital and / or quantum computer will also include one or more mass storage devices (e.g., magnetic disks, magneto-optical disks, optical disks) or quantum systems suitable for storing quantum information, or may be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to or to the mass storage device or quantum system, or both. However, a digital and / or quantum computer need not have such devices.

[0093] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memories, media, and storage devices, including, for example, semiconductor storage 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. To understand this, a quantum memory is a device capable of storing quantum data for extended periods with high fidelity and efficiency, using a light-matter interface where light is used for transmission and matter is used to store and preserve quantum characteristics (such as superposition or quantum coherence) of the quantum data.

[0094] Control of the various systems or parts thereof described in this specification may be implemented as a digital and / or quantum computer program product, comprising instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or parts thereof described in this specification may be implemented as devices, methods, or systems that may include one or more digital and / or quantum processing devices and a memory storing executable instructions for performing the operations described in this specification.

[0095] Although this specification contains many specific implementation details, these details should not be construed as limiting the scope of protection that can be claimed, but rather as descriptions of features that may be specific to a particular implementation. Certain features described in the context of different implementations in this specification may also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation may also be implemented individually in multiple implementations or in any suitable sub-combination. Furthermore, although features may be described as functioning in certain combinations as stated above, and are initially claimed in this way, in some cases one or more features from the claimed combination may be excluded from the combination, and the claimed combination may be for a sub-combination or a variation of the sub-combination.

[0096] Similarly, although operations are depicted in a specific order in the accompanying drawings, this should not be construed as requiring the operations to be performed in the specific order shown or in a sequential order, or as requiring the execution of all illustrated operations to achieve the desired result. In some implementations, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system modules and components in the above implementations should not be interpreted as requiring such separation in all implementations; rather, it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0097] Specific implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions described in the claims may be performed in different orders but still achieve the desired result. As an example, the processes depicted in the figures do not necessarily require the specific or sequential order shown to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Claims

1. A method for quantum phase estimation, comprising: Multiple phase estimation experiments are performed on auxiliary qubits, each of which includes: Prepare auxiliary qubits from the initial quantum state; The unitary operator is applied once or multiple times to auxiliary qubits prepared in the initial quantum state and to quantum registers prepared in arbitrary quantum states; and Measuring the state of auxiliary qubits; The expected value of the state of the auxiliary qubit is determined based on the results of the multiple phase estimation experiments; and Based on the expected value of the determined state of the auxiliary qubit and the measured state of the auxiliary qubit, the phase of the eigenvalues ​​of the unitary operator is obtained by optimizing a mixture model representing the probability of the state of the auxiliary qubit.

2. The method according to claim 1, wherein, The hybrid model comprises a sum of terms, each term in the sum of terms comprising the probability of the state of the auxiliary qubit for the corresponding eigenstate of the unitary operator.

3. The method according to claim 1, wherein, The mixture model representing the probability of the state of the auxiliary qubit is given by the following: p( | , ;M, ) = in, This represents the value of the state of the auxiliary qubit. ,in It is a quantum state and Let k be the k-th eigenstate of the unitary operator. This represents the k-th eigenstate of the unitary operator. The phase of the corresponding eigenvalue, M represents the number of times the unitary operator is applied. The angle of rotation applied by the phase gate is represented, and N represents the number of qubits in the quantum register. Kronek .

4. The method according to claim 3, wherein, The mixture model representing the probability of the state of the auxiliary qubit includes additional terms representing sources of noise.

5. The method according to claim 4, wherein, The noise mentioned is depolarization noise.

6. The method according to claim 3, wherein, Optimizing the hybrid model to obtain the phase of the eigenvalues ​​of the unitary operator includes: regarding the unique The hybrid model is optimized to determine the phase of the eigenvalues ​​of the unitary operator to the required accuracy, where the lowest Within a predetermined distance to the ground state of the quantum state.

7. The method according to claim 1, wherein, The phase of knowing the eigenvalues ​​of the unitary operator includes: The results of measuring the state of the auxiliary qubit are represented as binary random variables; and Based on the binary random variable and the determined expected value, a mixture model is defined to represent the probability of the value of the binary random variable.

8. The method according to claim 7, wherein, The binary random variable is constrained by the expected value of the determined state of the auxiliary qubit.

9. The method according to claim 1, wherein, Performing each phase experiment includes: Before performing operations on the auxiliary qubits and quantum registers with unitary operators, the first Hadamard gate is applied to the auxiliary qubits; Applying phase gates to auxiliary qubits; and After performing operations on the auxiliary qubit and the quantum register with unitary operators, a second Hadamard gate is applied to the auxiliary qubit.

10. The method of claim 1, further comprising providing the phase of the known eigenvalues ​​of the unitary operator for use in quantum computing.

11. The method according to claim 1, wherein, Performing the multiple phase estimation experiments includes pairing... Perform random sampling, where M represents the number of applications of the unitary operator, and This represents the angle of rotation applied by the phase gate that operates on the auxiliary qubit.

12. The method according to claim 11, wherein, from M is sampled uniformly, where The predetermined maximum number of applications of the unitary operator, and from right Sampling should be done uniformly.

13. The method according to claim 1, wherein, Optimizing the hybrid model to determine the phase of the eigenvalues ​​of the unitary operator includes performing the expectation-maximization (EM) algorithm on the hybrid model.

14. The method according to claim 1, wherein, The arbitrary quantum state is not an eigenstate of the unitary operator.

15. An apparatus for quantum phase estimation, comprising: Quantum circuits, including: At least one auxiliary qubit; A quantum register comprising one or more qubits, wherein the quantum register is prepared in an arbitrary quantum state; A phase-aware system is configured to know the phase of the eigenvalues ​​of a unitary operator. The device is configured as follows: Multiple phase estimation experiments are performed on auxiliary qubits, each of which includes: Prepare auxiliary qubits from the initial quantum state; The unitary operator is applied once or multiple times to auxiliary qubits prepared in the initial quantum state and to quantum registers prepared in arbitrary quantum states; and Measuring the state of auxiliary qubits; The expected value of the state of the auxiliary qubit is determined based on the results of the multiple phase estimation experiments; and Based on the expected value of the state of the determined auxiliary qubit and the measured state of the auxiliary qubit, the phase-knowing system obtains the phase of the eigenvalues ​​of the unitary operator by optimizing a hybrid model representing the probability of the state of the auxiliary qubit.

16. The device according to claim 15, wherein, The quantum circuit also includes at least (i) two Hadamard gates, (ii) a phase gate and (iii) a measurement operator.

17. The device according to claim 15, wherein, The arbitrary quantum state is not an eigenstate of the unitary operator.

18. The device according to claim 15, wherein, The hybrid model comprises a sum of terms, each term in the sum of terms comprising the probability of the state of the auxiliary qubit for the corresponding eigenstate of the unitary operator.

19. The device according to claim 15, wherein, The mixture model representing the probability of the state of the auxiliary qubit is given by the following: p( | , ;M, ) = in, This represents the value of the state of the auxiliary qubit. ,in It is a quantum state and Let k be the k-th eigenstate of the unitary operator. This represents the k-th eigenstate of the unitary operator. The phase of the corresponding eigenvalue, M represents the number of times the unitary operator is applied. The angle of rotation applied by the phase gate is represented, and N represents the number of qubits in the quantum register. Kronek .

20. The device according to claim 19, wherein, The mixture model representing the probability of the state of the auxiliary qubit includes additional terms representing sources of noise.

21. A method for quantum phase estimation, comprising: The expected value of the state of the auxiliary qubit is determined based on the results of multiple phase estimation experiments, wherein each result corresponds to a corresponding number of unitary operators applied to the auxiliary qubit and the quantum register prepared with arbitrary quantum states; and Based on the expected value of the state of the determined auxiliary qubit, the phase of the eigenvalue of the unitary operator is obtained by optimizing the mixture model representing the probability of the state of the auxiliary qubit. Wherein, the arbitrary quantum state is not an eigenstate of the unitary operator.

22. The method according to claim 21, wherein, The hybrid model comprises a sum of terms, each term in the sum of terms comprising the probability of the state of the auxiliary qubit for the corresponding eigenstate of the unitary operator.

23. The method according to claim 21, wherein, The mixture model representing the probability of the state of the auxiliary qubit is given by the following: p( | , ;M, ) = in, This represents the value of the state of the auxiliary qubit. ,in To assist the quantum state of the qubit and Let k be the k-th eigenstate of the unitary operator. This represents the k-th eigenstate of the unitary operator. The phase of the corresponding eigenvalue, M represents the number of times the unitary operator is applied. The angle of rotation applied by the phase gate is represented, and N represents the number of qubits in the quantum register. Kronek .

24. The method according to claim 23, wherein, The mixture model representing the probability of the state of the auxiliary qubit includes additional terms representing sources of noise.

25. The method according to claim 24, wherein, The noise mentioned is depolarization noise.

26. The method according to claim 22, wherein, Optimizing the hybrid model to obtain the phase of the eigenvalues ​​of the unitary operator includes: regarding the unique The hybrid model is optimized to determine the phase of the eigenvalues ​​of the unitary operator to the required accuracy, where the lowest Within a predetermined distance to the ground state of the quantum state.

27. The method according to claim 21, wherein, The phase of knowing the eigenvalues ​​of the unitary operator includes: The results of measuring the state of the auxiliary qubit are represented as binary random variables; and Based on the binary random variable and the determined expected value, a mixture model is defined to represent the probability of the value of the binary random variable.

28. The method according to claim 27, wherein, The binary random variable is constrained by the expected value of the determined state of the auxiliary qubit.

29. The method of claim 21, further comprising providing the phase of the known eigenvalues ​​of the unitary operator for use in quantum computing.

30. A device for quantum phase estimation, comprising: Quantum circuits, including: At least one auxiliary qubit; A quantum register comprising one or more qubits, wherein the quantum register is prepared in an arbitrary quantum state; A phase-aware system is configured to know the phase of the eigenvalues ​​of a unitary operator. The device is configured as follows: The expected value of the state of the auxiliary qubit is determined based on the results of multiple phase estimation experiments, wherein each result corresponds to a corresponding number of unitary operators applied to the auxiliary qubit and the quantum register prepared with arbitrary quantum states; and Based on the expected value of the state of the determined auxiliary qubit, the phase of the eigenvalue of the unitary operator is obtained by optimizing the mixture model representing the probability of the state of the auxiliary qubit. Wherein, the arbitrary quantum state is not an eigenstate of the unitary operator.

31. The device according to claim 30, wherein, The hybrid model comprises a sum of terms, each term in the sum of terms comprising the probability of the state of the auxiliary qubit for the corresponding eigenstate of the unitary operator.

32. The device according to claim 30, wherein, The phase of knowing the eigenvalues ​​of the unitary operator includes: The results of measuring the state of the auxiliary qubit are represented as binary random variables; and Based on the binary random variable and the determined expected value, a mixture model is defined to represent the probability of the value of the binary random variable.

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