Repeated preparation of a stationary quantum state using a quantum computer

Abstract

A method, system and apparatus for preparing a target quantum state of a quantum system, the target quantum state being stationary with respect to a parameterized many-body qubit operator. In one aspect, the method includes preparing an initial quantum state as an input state for a first iteration, and iteratively evolving the initial quantum state and subsequent input quantum states as inputs for subsequent iterations until an approximation of the target stationary quantum state is obtained, the steps including, for each iteration, calculating by quantum computation parameter values ​​of the many-body qubit operator for the iteration, calculating by quantum computation an evolution time for the iteration, the evolution time including evaluating changes in elements of the 2-RDM for the iteration, and evolving the initial quantum state or the subsequent input quantum state for the iteration with the calculated parameter values ​​and evolution time to generate the subsequent input quantum state for the subsequent iteration.

Description

Technical Field

[0001] This specification relates to quantum computing.

Background Art

[0002] Classical computers have memory consisting of bits, each of which can represent either zero or one. Quantum computers maintain a sequence of quantum bits called qubits, each of which can represent zero, one, or any quantum superposition of zero and one. Quantum computers operate by setting qubits to an initial state and then controlling the qubits, for example, according to a sequence of quantum logic gates.

Summary of the Invention

Means for Solving the Problems

[0003] This specification describes an iterative process for constructing a stationary quantum state on a quantum computer.

[0004] Generally, one inventive aspect of the subject matter described in this specification can be implemented as a method for preparing a target quantum state of a quantum system, the target quantum state being stationary with respect to a parameterized many-body qubit operator, the method comprising preparing an initial quantum state as an input state for a first iteration and repeatedly evolving the initial quantum state and subsequent input quantum states as inputs for subsequent iterations until an approximation of the target stationary quantum state is obtained, the step comprising, for each iteration, calculating, by quantum computing, a parameter value of a many-body qubit operator for the iteration and calculating, by quantum computing, a development time for the iteration, the calculating including evaluating a change in an element of a two-electron reduced density matrix for the iteration, and generating a subsequent input quantum state for a subsequent iteration by evolving the initial quantum state or a subsequent input quantum state for the iteration using the calculated parameter value and development time.

[0005] Other implementations of these aspects include corresponding computer systems, each configured to perform the actions of the method, an apparatus, and a computer program recorded on one or more computer storage devices. A system consisting of one or more classical and / or quantum computers can be configured to perform specific operations or actions during operation, thanks to software, firmware, hardware, or combinations thereof installed on the system. One or more computer programs can be configured to perform specific operations or actions, as they include instructions that, when executed by a data processing device, cause the device to perform actions.

[0006] The above and other implementations can each optionally include one or more of the following features, either alone or in combination. In some implementations, evaluating the change in the elements of the two-electron reduced density matrix for iteration involves evaluating the first and second derivatives of the elements of the two-electron reduced density matrix for iteration.

[0007] In some implementations, calculating the evolution time for iteration by quantum computing involves

[0008]

Number

[0009] calculating, where in the above formula, λ i represents the evolution time for iteration i, H represents the Hamiltonian characterizing the quantum system,

[0010]

Number

[0011] represents the first derivative of the elements of the two - electron reduced density matrix for iteration,

[0012] [Number]

[0013] represents the second derivative of the elements of the two - electron reduced density matrix for iteration.

[0014] In some implementations, evaluating the first and second derivatives of the elements of the two - electron reduced density matrix for iteration involves using cumulant expansion.

[0015] In some implementations, calculating the parameter values of the many - body qubit operator for iteration by quantum computing involves performing measurements of the 3 - RDM.

[0016] In some implementations, using the calculated parameter values and evolution time to evolve an initial quantum state or a subsequent input quantum state for iteration to generate an input quantum state for subsequent iteration involves approximating the time evolution of the many - body qubit operator at the calculated parameter values using low - rank double factorization.

[0017] In some implementations, the low - rank factorization is implemented on a linear lattice of qubits with linear depth.

[0018] In some implementations, using the calculated parameter values and evolution time to evolve an initial quantum state or a subsequent input quantum state for iteration to generate an input quantum state for subsequent iteration involves approximating the time evolution of the many - body qubit operator at the calculated parameter values using unitary compression of the many - body qubit operator.

[0019] In some implementations, the unitary compression of the many - body qubit operator represents the many - body qubit operator at the calculated parameter values in the form of a sum of squares.

[0020] In some implementations, the target stationary quantum state includes a ground or excited quantum state.

[0021] In some implementations, the initial quantum state is non - orthogonal to the target quantum state.

[0022] In some implementations, the target quantum state includes the state of a quantum system characterized by a Hamiltonian, and the target quantum state 〈ψ|[G,H]|ψ〉 = 0 satisfies the stationary condition given by the above formula, where |ψ〉 represents the target quantum state, H represents the Hamiltonian characterizing the quantum system, and G represents a parameterized multi - body qubit operator.

[0023] In some implementations, the stationary condition includes a first - order stationary condition of energy with respect to changes in the parameters of the multi - body qubit operator.

[0024] In some implementations, the parameterized multi - body qubit operator includes a fermionic two - body qubit operator.

[0025] In some implementations, the fermionic two - body qubit operator

[0026]

Number

[0027] is given by, where i, j, k, l are indices representing quantum system orbits, θ represents a real - valued coefficient,

[0028]

Number

[0029] , a krepresents the creation and annihilation operators, and the associated energy is E(Θ)=〈ψ|e -Θ He Θ |ψ〉, where |ψ〉 represents the target quantum state and H represents the Hamiltonian characterizing the quantum system.

[0030] In some implementations, the target quantum state is

[0031]

Number

[0032] represented by a sequence of wave function transformations given by, where λ i represents the evolution time for iteration i, A i represents a many-body qubit operator with a parameter value for iteration i, and |ψ0〉 represents the initial quantum state.

[0033] The subject matter described herein can be implemented in a particular manner to achieve one or more of the following advantages.

[0034] The techniques described in this application can be used to construct a quantum circuit that produces a target stationary quantum state without the need for variational minimization. The target stationary quantum state can be a ground state or an excited state.

[0035] Furthermore, the quantum circuit is compiled using low-rank decomposition or unitary compression and can thus have a lower circuit depth compared to prior art that implements exact rotations, for example, to minimize energy. Thus, preparing the target quantum state using the techniques described herein requires fewer computational resources and improves the computational runtime. Additionally, due to the reduction in circuit depth, the techniques described herein are particularly suitable for implementations using short-term quantum computing devices, such as noisy intermediate-scale quantum (NISQ) devices, where long circuits can have noise that overwhelms any signal in the circuit due to gate errors and T1 decay rates.

[0036] Furthermore, a system implementing the techniques described herein can prepare the target quantum state with improved accuracy compared to conventional methods that are based on, for example, the propagation of the anti-Hermitian contracted Schrödinger equation (ACSE) and suffer from inexact two-RDM evolution and error accumulation.

[0037] Details of one or more implementations of the subject matter described in 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, the drawings, and the claims.

Brief Description of the Drawings

[0038]

Figure 1

Figure 2

Figure 3

Modes for Carrying Out the Invention

[0039] Like reference numerals and designations in the various drawings indicate like elements.

[0040] This specification describes techniques for constructing a quantum circuit that produces a target quantum state that is stationary with respect to a parameterized multi-body qubit operator. The initial quantum state is repeatedly evolved until an approximation of the target stationary quantum state is obtained. In each iteration, quantum computations are performed to calculate the parameter values of the multi-body qubit operator for the iteration and the evolution time for the iteration. The evolution time is calculated by evaluating the change in the elements of the two-electron reduced density matrix for the iteration.

[0041] FIG. 1 is a block diagram of an exemplary quantum computing system 100 for implementing the quantum state preparation techniques described in this application. The exemplary quantum computing system 100 is an example of a system implemented as classical and quantum computer programs on one or more classical computers and quantum computing devices in one or more locations where the systems, components, and techniques described in this specification can be implemented.

[0042] The exemplary quantum computing system 100 includes a control system 102 configured to operate a quantum system included in the quantum computing system 100, such as qubits, and to perform classical subroutines or computations. For example, the control system 102 can be programmed to send control signals to the qubits included in the quantum computing system, for example, to implement a quantum circuit, and to receive readout signals from the qubits, for example, as part of performing a measurement operation. In FIG. 1, the control system 102 performs operations to prepare the quantum system 104 in the target quantum state |ψ i 〉. The target quantum state is a quantum state that is stationary with respect to a parameterized multi-body qubit operator and can be a ground or excited quantum state. Exemplary target quantum states, stationarity conditions, and parameterized multi-body qubit operators are described below with reference to FIG. 2.

[0043] To prepare the quantum system 104 in the target quantum state, in step (A), the control system 102 performs an operation to set the quantum system 104 to the initial quantum state |ψ0〉, for example, by resetting the qubits included in the quantum system 104. The control system 102 then repeatedly evolves the initial quantum state |ψ0〉 until the target stationary quantum state |ψ i 〉 is obtained.

[0044] In each iteration, the control system 102 performs steps (B) to (D). In step (B), the control system performs quantum computation to calculate the parameter values of the many-body qubit operator for iteration. As will be described below with respect to equations (5) and (6), the parameter value A calculated in step (B) ijkl defines the operator A, which, when applied to the input quantum state with respect to time λ, evolves the input quantum state to minimize energy and approximately satisfy stationarity.

[0045] To calculate the parameter values, the control system 102 performs quantum and classical computations. For example, for each parameter A defined in equation (6) below, the control system 102 can prepare a copy of the input quantum state for iteration. The control system 102 then performs qubit encoding of the observable quantity to be measured, for example, the observable quantity defined in equation (6) below ijkl For each parameter A, the control system 102 can prepare a copy of the input quantum state for iteration. The control system 102 then performs qubit encoding of the observable quantity to be measured, for example, the observable quantity defined in equation (6) below

[0046]

Number

[0047] The quantum circuit 108 that affects the qubit encoding of the terms in [the relevant context] may be applied to the input quantum state. The control system 102 may then perform measurements to obtain each measurement result. The control system 102 can repeat the state preparation, quantum circuit application, and measurement steps multiple times to obtain multiple measurement results. The control system 102 may then have the classical processor 110 post-process the measurement results and calculate the expected value of the parameter value. The control system 102 can repeat this process for each parameter value until the many-body qubit operator A can be sufficiently specified.

[0048] In step (C), the control system performs quantum calculations to calculate the evolution time for iteration. For example, the control system 102 can calculate the evolution time λ by calculating the first and second derivatives of the elements of the two-electron reduced density matrix for iteration, as described below with respect to equations (9)-(13). To calculate the derivatives, the control system 102 performs classical and quantum operations similar to those described above for step (B). For example, to calculate the first derivative of the 2-RDM element, the control system 102 may just prepare a copy of the input quantum state for iteration. The control system 102 may then apply a quantum circuit that affects the qubit encoding of the observable quantity to be measured, for example, the qubit encoding of the terms in the observable quantity. The control system 102 may then perform measurements to obtain each measurement result. The control system 102 can repeat the state preparation, quantum circuit application, and measurement steps multiple times to obtain multiple measurement results. The control system 102 may then have the classical processor 110 post-process the measurement results and calculate the expected value of the parameter value. The control system 102 can repeat this process for the terms in equation (13) until the evolution time λ can be sufficiently specified.

[0049] In stage (D), the control system 102 uses the calculated parameter values and development time to develop an input quantum state for repetition. For example, the control system 102 can apply the unitary operator U = e iAλ to the input quantum state for repetition to develop the input quantum state and generate a corresponding output quantum state. In some implementations, the control system 102 can store data specifying the unitary operator for repetition in a cache to assist, for example, in preparing a copy of the input quantum state as part of stages (B) and (C) in the next repetition.

[0050] If the repetition is the first repetition, the input quantum state is the initial quantum state |ψ0〉, and the output quantum state is the developed quantum state

[0051]

Number

[0052] For the j-th subsequent repetition, the input quantum state is the output quantum state |ψ j-1 〉 for the immediately preceding (j - 1)-th repetition, and the output quantum state is the developed quantum state

[0053]

Number

[0054] The control system 102 may repeatedly perform stages (B) to (D) until a predetermined number of repetitions are completed or the output quantum state converges within a predetermined convergence threshold. The predetermined convergence threshold may vary based on the target accuracy of the final output state. For example, a smaller convergence threshold may result in a more accurate final output state.

[0055] When the iterative process ends, the quantum computing system 100 can provide the target quantum state for use in subsequent quantum computations. In some implementations, the quantum computing system 100 can store data representing a unitary operator used to evolve an initial quantum state to the target quantum state, e.g., the unitary operator defined in Equation (8) below. Then, when the quantum computing system 100 is requested, for example, to perform a quantum computation, to copy the target quantum state, the quantum computing system can retrieve the data representing the unitary operator and apply the unitary operator to the initial quantum state.

[0056] Exemplary Process for Preparing a Target Stationary Quantum State FIG. 2 is a flowchart of an exemplary process 200 for preparing a target quantum state. For convenience, process 200 is described as being implemented by a system that includes a classical computer and a quantum computer. For example, system 100 of FIG. 1, appropriately programmed in accordance with this specification, can implement process 200.

[0057] The target quantum state can be the ground state or an excited state of a quantum system characterized by a Hamiltonian H. The target quantum state is a stationary quantum state. For example, the target quantum state is stationary with respect to a parameterized multi-body qubit operator (e.g., a generator of rotations), 〈ψ|[G,H]|ψ〉=0 (1) and can satisfy the stationary condition given by, where |ψ〉 represents the target quantum state, H represents the Hamiltonian characterizing the quantum system, and G represents a parameterized multi-body qubit operator, i.e., the commutator of the parameterized multi-body qubit operator and the expectation value of the Hamiltonian with respect to the target quantum state is zero. These stationary conditions are the first-order stationary conditions for the energy of the quantum system with respect to changes in the parameters of the multi-body qubit operator.

[0058] In some implementations, the parameterized multi-body qubit operator can be a fermionic two-body qubit operator. For example, the fermionic two-body qubit operator can be

[0059]

Number

[0060] given by, where i, j, k, l are indices representing quantum system orbits, θ represents a real-valued coefficient,

[0061]

Number

[0062] , a k represents creation and annihilation operators. The associated energy can be E(Θ)=〈ψ|e -Θ He Θ |ψ〉 (3) given by, where |ψ〉 represents the target quantum state and H represents the Hamiltonian characterizing the quantum system. By applying the Baker-Campbell-Hausdorff (BCH) expansion and differentiation at Θ = 0,

[0063]

Number

[0064] is given, which indicates that the commutation relation is a first-order stationary condition.

[0065] In an implementation of the present disclosure, the parameterized multi-body qubit operator can be represented by an operator A whose coefficients are defined by the stationary condition when given some trial state |ψ T 〉.

[0066]

Number

[0067] As shown by the BCH expansion, for a certain amount of time λ, this operator applied to the trial state develops the state to minimize energy and approximately satisfy stationarity. This is because the BCH expansion can be regarded as a Taylor expansion of the expectation value with respect to the rotation parameters. By appropriately selecting λ, the low-order Taylor expansion is minimized. Determining the amount of time λ for developing the state is an important step in the exemplary process 200.

[0068] For convenience, the remaining description of the exemplary process 200 is described for the case of particle-conserving fermionic generators (many-body qubit operators), where the stationarity equation given in Equation (1) is the Brillouin condition. In this case,

[0069]

Number

[0070] There are individual stationarity conditions for two-body fermionic generators without constraints on the coefficients, where n represents the number of spin-orbit basis functions. The exemplary process 200 is used to construct the target quantum state |ψ f 〉 using a quantum computer. This starts from the initial state |ψ i 〉, calculates the gradient using Equation (4), and then develops for a certain amount of time λ such that the next iteration |ψ i+1 〉 has a smaller residual value with respect to the gradient, that is, performs gradient descent on the wave function. Mathematically, this sequence is i achieved by developing for a certain amount of time λ. Mathematically, this sequence is

[0071]

Number

[0072] represented as, where Ai represents a many-body qubit operator evaluated using the parameters for the current iteration i.

[0073] Since the objective function depends non-linearly on the rotation angles, multiple evolution steps are required. That is, the i-th state |ψ i 〉 is generated by transforming the (i-1)-th state |ψ i-1 〉, and the (i-1)-th state is generated by transforming the (i-2)-th state |ψ i-2 〉, and so on. That is, the target quantum state is

[0074]

Number

[0075] represented by a sequence of wave function transformations given by, where λ i represents the evolution time for iteration i, A i represents a many-body qubit operator with the parameter value for iteration i, and |ψ0〉 represents the initial quantum state.

[0076] At each iteration, the gradient operator based on equation (4) is obtained from the current iteration, and λ i for the current iteration is determined. The conventional procedure performs a line search optimization solution for λ such that the energy is minimized. However, the variational search or line search dependence can be eliminated by approximating the Hessian by the short-time propagation of the 2-RDM through cumulant reconstruction and solving the second-order BCH expansion.

[0077] The above equations (1)-(8) track how the energy 〈ψ i |H|ψ i 〉 evolved with respect to the operator given by equation (4). However, n is the number of spin-orbital basis functions, and all n of the 2-RDM 4Through the inspection of how the elements develop, the similar quadratic approximation to those values in the $i$-th wave function can be obtained through truncating the BCH expansion to the second order. For example, if all $n$ 4 elements of the 2-RDM are denoted as $D$ i , the BCH expansion up to the second order gives

[0078]

Math

[0079] , where primes and double primes represent derivatives with respect to $\lambda$ i-1 . Each term in the 2-RDM terms at $i$ can be approximated by the BCH expansion

[0080]

Math

[0081] , which defines that the first derivatives of all 2-RDM elements have the following form.

[0082]

Math

[0083] Therefore, the task becomes minimizing a quadratic function and only depends on the gradient and approximate Hessian of the 2-RDM at step $i - 1$.

[0084]

Math

[0085] In Eq. (13), $\lambda$ i-1 represents the development time for iteration $i - 1$, $H$ represents the Hamiltonian characterizing the quantum system,

[0086]

Math

[0087] represents the first derivative of the elements of the two-electron reduced density matrix for iteration,

[0088]

Number

[0089] represents the second derivative of the elements of the two-electron reduced density matrix for iteration. The inner product, i.e.,

[0090]

Number

[0091] and

[0092]

Number

[0093] are the Hilbert-Schmidt inner products between the row representation of the Hamiltonian coefficient tensor and the first and second derivatives of the 2-RDM, respectively. The development time step for iteration i - 1 is thus given by the negative of the Hilbert-Schmidt inner product between the Hamiltonian and the first derivative of the 2-RDM, divided by the Hilbert-Schmidt inner product between the Hamiltonian and the second derivative of the 2-RDM.

[0094] In other words, using the shorthand notation for D i for representing all the expected values of the 2-RDM (i.e., Equation (10)) formed by matrix variables, the first and second derivatives of all the 2-RDM elements are, respectively,

[0095]

Number

[0096] and

[0097] [Number]

[0098] are defined using equation (11) which should hold. These are matrices of all coefficients corresponding to the first and second derivatives of the corresponding 2-RDM elements. The optimal time-evolution step is then determined from a second-order approximation to a similarity transformation of a two-body operator that can be expressed as equation (13), where

[0099] [Number]

[0100] and

[0101] [Number]

[0102] is the Hilbert-Schmidt inner product between the matrix representation of the Hamiltonian coefficient tensor and the 2-RDM derivative.

[0103] The determination of the optimal time step uses the Hilbert-Schmidt inner product between matrices of size O(N^4) which are not exponentially large quantum states. They can thus be determined efficiently.

[0104] The cumulant expansion can be used to evaluate the derivatives to obtain an approximation of the parameter λ i-1 that defines how much to evolve along the gradient term A i-1 The classical version of this theory propagates the anti-Hermitian contracted Schrödinger equation (ACSE), but this classical version suffers from non-exact 2-RDM evolution and error accumulation. The quantum version of this algorithm described in the present application mitigates this problem.

[0105] In view of the above equations (1) to (13), exemplary process 200 may proceed as follows. The system prepares an initial quantum state |ψ0〉 as described in equation (8) above (step 202). The initial quantum state can be chosen to be non - orthogonal to the target quantum state. The system then repeatedly evolves the initial quantum state and subsequent quantum states until an approximation of the target stationary quantum state is obtained (step 204). The accuracy of the approximation may depend on a predetermined acceptable accuracy, for example, based on a predetermined convergence threshold.

[0106] In each iteration, the system performs a quantum calculation to calculate the parameter values of the many - body qubit operator for the iteration (step 204a). That is, the system calculates the value of parameter A given by equation (6) to define the many - body qubit operator for iteration A i of the many - body qubit operator for iteration A ijkl of equation (6).

[0107] Furthermore, in each iteration, the system performs a quantum calculation to calculate the evolution time for the iteration (step 204b). Calculating the evolution time for the iteration includes, for example, using cumulant expansion to evaluate the first and second derivatives of the elements of the two - electron reduced density matrix for the iteration. That is, the system performs a measurement operation to determine λ i in accordance with equation (13) above.

[0108] iteration A i Once the parameter values of the many - body qubit operator for iteration A and the evolution time for iteration λ i are calculated, the system uses the calculated parameter values and evolution time to evolve the initial quantum state or the subsequent input quantum state for the iteration to generate the input quantum state for the next iteration (step 204c). That is, the system evolves the initial quantum state or the subsequent quantum state through equation (7) above.

[0109] For a sufficiently general A iIn this case, this is prohibitively expensive. As described above,

[0110]

Number

[0111] terms are in A i . Generally, the gradient does not reach full rank, and thus, in some implementations, the time evolution under A i can be approximated through a low-rank factorization of A i , where a significant number of terms are discarded. This can drastically reduce the circuit depth required to implement the evolution under A i compared to the exact two-body rotation that should minimize the energy. Each low-rank factorization can be implemented on a linear lattice of qubits with linear depth. In some implementations, A becomes low-rank, and thus, a very small O(n)-depth circuit is required to implement the Trotterized form of A. This also follows from spin blocking, since A is constructed by examining the rotations within each two-electron spin sector {αα, ββ, αβ}. The highest l-rank of each component can then be evolved through a low-rank factorization scheme. Due to the low-rank nature of A, very few factors need to be implemented.

[0112] In other implementations, the system can approximate the time evolution under A i by performing a unitary compression of the many-body qubit operator, where the unitary compression represents the many-body qubit operator at the calculated parameter values in the form of a sum of squares. Techniques for compressing many-body operators are described below.

[0113] Compression of Many-Body Operators Using Sequential Orbital Optimization General tensors associated with non-positive semi-definite operators decompose into a sum of squares of normal operators and fermionic Gaussian unitaries and n 2can be developed using individual Ising interaction operators, where the coefficient matrix for the Ising operator has rank 1. The cost of implementing such an operator scales with the number of square-normal operators required to represent the operator. Prior art has attempted to reduce this number by increasing the rank of the coefficient matrix for the Ising operator. In some cases, the two-body operator to be compressed is the Coulomb interaction term by the least squares method. The drawback of this approach is that the number of free parameters is large and the number of optimization steps required to fit the tensor is large.

[0114] This disclosure generalizes to the case of general two-body operators and provides a greedy approach that necessarily leads to fewer variables. This is achieved by sequentially finding a single-particle basis such that A has large coefficients for n i n j terms. This component is then removed, leaving the remaining tensor. The procedure is repeated until the remainder is numerically zero or the norm falls below a pre-set threshold. This greedy approach is always guaranteed to converge. A by-product of this procedure is a J ij (l) matrix higher than rank 1.

[0115] Consider the one-body transformation of the generator. Given a two-body generator

[0116]

Number

[0117] the orbital rotation generator is given by the following,

[0118]

Number

[0119] and by doing so,

[0120]

Number

[0121] of n i n j an objective function that maximizes the coefficients of the components

[0122]

Number

[0123] can be expressed as. To optimize,

[0124]

Number

[0125] the gradient that results in is taken. For the parameters of the generator κ

[0126]

Number

[0127] to calculate the partial derivative of the coefficients of the

[0128]

Number

[0129] the form of is first derived as follows,

[0130]

Number

[0131] In the above equation, G ab is an anti-Hermitian operator obtained from the Duhamel's (Wilcox) formula for differentiation with respect to the parameter κ a,b and also,

[0132]

Number

[0133] and this is

[0134] [Number]

[0135] is used. Thus,

[0136] [Number]

[0137] and this is a reduction that can be evaluated in O(n 5 ), where n represents the number of spin orbits. When considering the real and imaginary components, there are a total of n 2 κ ab terms, and thus the total derivative is simply obtained in O(n 7 ) operations.

[0138] This method of obtaining the gradient is general and

[0139] [Number]

[0140] is sufficient for any continuous cost function. General scaling can be reduced by considering functions that are in the process of optimization and applying the chain rule. To see how this works, consider the differential equation (15) for κ ab , and

[0141] [Number]

[0142] and in the above equation,

[0143]

Number

[0144] and this is obtained for all {c, d} once per total derivative call in O(n 5 ). The n 2 intermediates are reused for each κ ab derivative. Thus, the overall scaling is O(n 5 ). Using the Duamel or Wilcox identities, the partial derivatives of the unitary or its Hermitian conjugate are

[0145]

Number

[0146] and. The check that this all agrees with the O(n 7 ) scaling method is to multiply Equation (24) by

[0147]

Number

[0148] and sum over {c, d}, which returns the appropriate terms in Equation (21).

[0149] κ = 0 Gradient Calculation An exemplary method for calculating the derivative is to

[0150]

Number

[0151] update the local basis of, thereby, for κ ab with respect to

[0152]

Number

[0153] The gradient is always calculated around κ = 0. To look at this first Taylor,

[0154]

Number

[0155] is expanded around κ = 0 using BCH expansion

[0156]

Number

[0157] When extended using, the derivative with respect to κ gives a Kronecker delta function that reduces the complexity of the sum from fourth order to third order, ab which amounts to copying the appropriate t coefficients into a new tensor to represent the gradient. After performing the BCH expansion to second order and evaluating the derivative that gives the Kronecker delta function, the Hessian can be calculated in a similar manner.

[0158]

Number

[0159] This amounts to copying the appropriate t coefficients into a new tensor to represent the gradient. After performing the BCH expansion to second order and evaluating the derivative that gives the Kronecker delta function, the Hessian can be calculated in a similar manner.

[0160] Recursive fitting of two-body generators For the objective function and gradient to maximize the n i n j components of the two-body operator, we described above. Using this infrastructure, a recursive technique for generating compressed two-body operators is introduced. Starting from the target operator T, equation (16) gives, for n i n jIt is maximized to obtain a basis such that the coefficients are the largest in terms of scale. These coefficients are selected and stored with κ rotations. The operator represented by the diagonal coefficients is rotated and returned to the original basis with κ obtained from the optimization, and then the tensor is subtracted from the original to generate a remainder. This procedure is repeated until the norm of the subtracted remainder becomes small and falls below a predetermined threshold. Using this method, the cost of optimization never exceeds that of sequential trajectory optimization. When the optimization is seeded with rotations generated from the Takagi decomposition, the procedure is guaranteed to converge.

[0161] Exemplary Operating Environment FIG. 3 shows an exemplary classical / quantum computer 300 for performing some or all of the classical and quantum operations described herein. The exemplary classical / quantum computer 300 includes an exemplary quantum computing device 302. The quantum computing device 302 is intended to represent various forms of quantum computing devices. The components shown herein, their connections and relationships, and their functions are illustrative only and do not limit the implementations of the invention described and / or claimed herein.

[0162] The exemplary quantum computing device 302 includes a qubit assembly 352 and a control and measurement system 304. The qubit assembly includes a plurality of qubits, such as qubit 306, used to perform algorithm operations or quantum computations. The qubits shown in FIG. 3 are arranged in a rectangular array, but this is a schematic description and is not intended to be limiting. The qubit assembly 352 also includes adjustable coupling elements, such as coupler 308, that enable interactions between the coupled qubits. In the schematic description of FIG. 3, each qubit is adjustably coupled to each of its four adjacent qubits using respective coupling elements. However, this is an exemplary arrangement of qubits and couplers, and other arrangements are possible, including non-rectangular arrangements, arrangements that enable coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits.

[0163] Each qubit may be a physical two-level quantum system or device having levels representing logical values of 0 and 1. The specific physical realizations of the plurality of qubits and how they interact with each other depend on various factors including the type of the quantum computing device 302 included in the exemplary computer 300 or the type of quantum computing being performed by the quantum computing device. For example, in an atomic quantum computer, the qubits may be realized by atoms, molecules or solid-state quantum systems, such as ultrafine atomic states. As another example, in a superconducting quantum computer, the qubits may be realized by superconducting qubits or semiconductor qubits, such as superconducting transmon states. As another example, in an NMR quantum computer, the qubits may be realized by nuclear spin states.

[0164] In some implementations, quantum computing may proceed, for example, by loading qubits from a quantum memory and applying a sequence of unitary operators to the qubits. Applying a unitary operator to a qubit may include, for example, applying a corresponding sequence of quantum logic gates to the qubit to implement a quantum algorithm such as a quantum principal component algorithm. Exemplary quantum logic gates include single-qubit gates, such as Pauli X, Pauli Y, Pauli Z (also called X, Y, Z), Hadamard gate, S gate, rotation, two-qubit gates, such as controlled-X, controlled-Y, controlled-Z (also called CX, CY, CZ), controlled-NOT gate (also called CNOT), controlled-swap gate (also called CSWAP), and gates involving three or more qubits, such as a Toffoli gate. The quantum logic gates can be implemented by applying control signals 310 generated by the control and measurement system 304 to the qubits and to the couplers.

[0165] For example, in some implementations, qubits in the qubit assembly 352 can be frequency-variable. In these examples, each qubit can have an associated operating frequency that can be adjusted by applying a voltage pulse via one or more drive lines coupled to the qubit. Exemplary operating frequencies include qubit idle frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to the corresponding idle frequency may put the qubit in a state where it does not strongly interact with other qubits and can be used to perform single qubit gates. As another example, in cases where qubits interact via a coupler with a fixed coupling, the qubits may be configured to interact with each other by setting their respective operating frequencies to some gate-dependent frequency detuning from a common interaction frequency. In other cases, for example, when qubits interact via a tunable coupler, the qubits can be configured to interact with each other by first setting the parameters of their respective couplers to enable interaction between the qubits, and then setting the respective operating frequencies of the qubits to some gate-dependent frequency detuning from their common interaction frequency. Such interactions may be performed to implement multi-qubit gates.

[0166] The type of control signal 310 used depends on the physical implementation of the qubit. For example, the control signal can include RF or microwave pulses in an NMR or superconducting quantum computer system, or optical pulses in an atomic quantum computer system.

[0167] By using each control signal 310 to measure the state of the qubit, quantum computing can be completed, for example, using a quantum observable such as X or Z. Through measurement, a readout signal 312 representing the measurement result is communicated back to the measurement and control system 304. The readout signal 312 can include RF, microwave, or optical signals depending on the physical scheme for the quantum computing device and / or qubit. For convenience, the control signal 310 and the readout signal 312 shown in FIG. 3 are shown as addressing only selected elements (i.e., the upper and lower rows) of the qubit assembly, but during operation, the control signal 310 and the readout signal 312 can address each element within the qubit assembly 352.

[0168] The control and measurement system 304 is an example of a classical computer system that can be used to perform various operations on the qubit assembly 352 as described above, as well as other classical subroutines or calculations. The control and measurement system 304 includes one or more classical processors, such as the classical processor 314, one or more memories, such as the memory 316, and one or more I / O units, such as the I / O unit 318, connected by one or more data buses. The control and measurement system 304 may be programmed to send a sequence of control signals 310 to the qubit assembly, for example, to perform a selected series of quantum gate operations, and to receive a sequence of readout signals 312 from the qubit assembly, for example, as part of performing a measurement operation.

[0169] The processor 314 is configured to process instructions for execution within the control and measurement system 304. In some implementations, the processor 314 is a single-threaded processor. In other implementations, the processor 314 is a multi-threaded processor. The processor 314 is capable of processing instructions stored in the memory 316.

[0170] Memory 316 stores information within control and measurement system 304. In some implementations, memory 316 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, memory 316 can include a storage device capable of providing mass storage to system 304, such as a hard disk device, an optical disk device, a storage device shared by multiple computing devices via a network (e.g., a cloud storage device), and / or any other mass storage device.

[0171] Input / output device 318 provides input / output operations for control and measurement system 304. Input / output device 318 can include a D / A converter, an A / D converter, and / or an RF / microwave / optical signal generator, transmitter, and receiver for sending control signal 310 to qubit assembly and receiving readout signal 312 from qubit assembly as required by the physical system for a quantum computer. In some implementations, input / output device 318 can also include one or more network interface devices, such as an Ethernet card, a serial communication device (e.g., an RS-232 port), and / or a wireless interface device (e.g., an 802.11 card). In some implementations, input / output device 318 can include a driver device configured to receive input data and send output data to other external devices, such as a keyboard, a printer, and a display device.

[0172] In FIG. 3, an exemplary control and measurement system 304 is shown, but the implementations of the subject matter and functional operations described herein can be implemented in other types of digital electronic circuitry, or in computer software, firmware, or hardware including the structures disclosed herein and their structural equivalents, or in combinations of one or more of them.

[0173] The exemplary system 300 also includes an exemplary classical processor 350. The classical processor 350 can be used to perform classical computing operations described herein according to some implementations.

[0174] Implementations of the subject matter and the operations described in this specification can be realized in digital electronic circuitry, analog electronic circuitry, appropriate quantum circuitry, or, more generally, in a quantum computing system, in software or firmware tangibly embodied in a computer hardware including the structures disclosed in this specification and their structural equivalents, or in a combination of one or more of them. The term "quantum computing system" can include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0175] Implementations of the subject matter described in this specification can be implemented as one or more computer programs, i.e., as one or more modules of 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 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 qubits, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially generated propagated signal capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal generated for encoding digital and / or quantum information for transmission to a suitable receiver device for execution by a data processing apparatus.

[0176] The terms "quantum information" and "quantum data" refer to information or data that is carried by and held or stored in a quantum system, where the smallest non-trivial system defines the qubit, i.e., the unit of quantum information. It will be 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. By way of example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational basis states are identified with the ground and first excited states, but it will be understood that other setups are possible where the computational states are identified with higher-level excited states.

[0177] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including, by way of example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The apparatus may further include, or alternatively be, special-purpose logic circuit configurations, such as FPGAs (field-programmable gate arrays), ASICs (application-specific integrated circuits), or quantum simulators, i.e., quantum data processing apparatuses designed to simulate or give rise to information about a specific quantum system, or may further include them. In particular, a quantum simulator is a special-purpose quantum computer that does not have the ability to perform universal quantum computing. The apparatus may optionally include, in addition to the hardware, code for creating an execution environment for digital and / or quantum computer programs, such as processor firmware, protocol stacks, database management systems, operating systems, or code constituting one or more combinations thereof.

[0178] A digital computer program, which may be called or known as a program, software, software application, module, software module, script, 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 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, which may be called or known as a program, software, software application, module, software module, script, or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, may be translated into a suitable quantum programming language, or may be written in a quantum programming language, such as QCL or Quipper.

[0179] A computer program may or may not correspond to a file in a file system. The program may be held in other programs or data, such as a file portion that holds one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple cooperating files, such as files that store one or more modules, subprograms, or portions of code. The computer program may be deployed to be executed on one computer located in one place or on multiple computers, or may be distributed across multiple places and interconnected by digital and / or quantum data communication networks. A quantum data communication network is understood to be a network that can transmit quantum data using a quantum system, such as qubits. Generally, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0180] The processes and logical flows described herein can, where appropriate, be implemented by one or more programmable computers that operate with one or more processors to execute one or more computer programs to perform functions by operating on input data and generating output. The processes and logical flows can also be implemented by, or the apparatus can be implemented as, special purpose logic circuitry, such as an FPGA or ASIC, or a quantum simulator, or a combination of special purpose logic circuitry or a quantum simulator and one or more programmed digital and / or quantum computers.

[0181] A system comprising one or more computers being "configured" to perform a particular operation or action means that the system has installed therein software, firmware, hardware, or a combination thereof that, when operating, causes the system to perform the operation or action. One or more computer programs being configured to perform a particular operation or action means that the one or more programs include instructions that, when executed by a data processing apparatus, cause the apparatus to perform the operation or action. For example, 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.

[0182] A computer suitable for the execution of a computer program can be based on a general-purpose microprocessor or a dedicated processor, or other types of central processing units. Generally, the central processing unit receives instructions or data from a read-only memory, a random access memory, or a quantum system suitable for transmitting quantum data, such as photons, or a combination thereof.

[0183] The elements of a computer include a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital, analog, and / or quantum data. The central processing unit and the memory can be supplemented by or incorporated into dedicated logic circuits or quantum simulators. Generally, a computer also includes one or more mass storage devices for storing data, such as magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information, or is operatively coupled to a mass storage device to receive data from, transfer data to, or both perform these operations on the mass storage device. However, a computer need not have such a device.

[0184] A quantum circuit element (also referred to as a quantum computing circuit element) includes circuit elements for performing quantum processing operations. That is, a quantum circuit element is configured to utilize quantum mechanical phenomena such as superposition and entanglement to perform operations on data in a non-deterministic manner. Some quantum circuit elements, such as qubits, can represent information that is in multiple states simultaneously and can be configured to operate on the information. Examples of superconducting quantum circuit elements include, in particular, circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUIDs or DC-SQUIDs).

[0185] In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively implement the instructions of a computer program by performing basic arithmetic, logical, and / or input / output operations on data, where the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to send data to and / or receive data from quantum circuit elements through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuit configurations, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices, and ERSFQ devices, which are energy-efficient versions of RSFQ that do not use bias resistors.

[0186] In some cases, some or all of the quantum and / or classical circuit elements may be implemented using, for example, superconducting quantum and / or classical circuit elements. The assembly of superconducting circuit elements may involve the deposition of one or more materials, such as superconductors, dielectrics, and / or metals. Depending on the materials selected, these materials may be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among other deposition processes. The processes for manufacturing the circuit elements described herein may involve the removal of one or more materials from the device during assembly. Depending on the materials to be removed, the removal process may include, for example, wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein can be patterned using known lithography techniques (e.g., photolithography or electron beam lithography).

[0187] During operation of a quantum computing system using superconducting quantum circuit elements and / or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled within a cryostat to a temperature at which the superconducting material can exhibit superconducting properties. A superconducting (alternatively superconducting) material can be understood as a material that exhibits superconducting properties below the superconducting critical temperature. Examples of superconducting materials include aluminum (superconducting critical temperature of 1.2 Kelvin) and niobium (superconducting critical temperature of 9.3 Kelvin). Thus, superconducting structures such as superconducting traces and superconducting base planes are formed from materials that exhibit superconducting properties below the superconducting critical temperature.

[0188] In some implementations, control signals for quantum circuit elements (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form.

[0189] Computer-readable media suitable for storing computer program instructions and 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; and CD-ROM and DVD-ROM disks; and quantum systems such as trapped atoms or electrons. Quantum memory is understood to be a device, such as an optical matter interface where light is used for transmission and matter is used to store and preserve quantum features of quantum data such as superposition or quantum coherence, capable of storing quantum data for long periods of time with high fidelity and efficiency.

[0190] The control of the various systems described herein, or portions thereof, can be implemented as a computer program product including instructions stored on one or more non-transitory machine-readable storage media and executable on one or more processing devices. The systems described herein, or portions thereof, can each be implemented as a system that can include an apparatus, a method, or one or more processing devices and a memory for storing executable instructions for performing the operations described herein.

[0191] This specification includes many specific implementation details, but these should not be construed as limitations on the scope of what can be claimed, but rather as descriptions of features that may be specific to a particular implementation form. Also, some of the features described in this specification in the context of separate implementation forms can be implemented in combination in a single implementation form. Conversely, various features described in the context of a single implementation form can also be implemented separately in multiple implementation forms, or in any suitable partial combination. Furthermore, features are described above as acting in some combinations and may initially be claimed as such, but one or more features from the claimed combination can, in some cases, be deleted from that combination, and the claimed combination may be directed to a partial combination or a variant of a partial combination.

[0192] Similarly, operations are shown in the drawings in a particular order, but this should not be understood as requiring that such operations be performed in the particular order shown or in a sequential order to achieve the desired result, or that all of the illustrated operations be performed. In some situations, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementation forms described above should not be understood as requiring such separation in all implementation forms, and it should be understood that the described program components and systems can generally be integrated together into a single software product or packaged into multiple software products.

[0193] Specific implementation forms of the subject matter have been described. Other implementation forms 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 the desired result. As an example, the processes shown in the accompanying drawings 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.

Description of the Reference Numerals

[0194] 100 Quantum computing system, system 102 Control system 104 Quantum system 108 Quantum circuit 110 Classical processor 300 Classical / quantum computer 302 Quantum computing device 304 Control and measurement system, system 306 Qubit 308 Coupler 310 Control signal 312 Readout signal 314 Classical processor, processor 316 Memory 318 I / O unit, input / output device 350 Classical processor 352 Qubit assembly

Claims

Claim 1 A computer-implemented method for preparing a target quantum state of a quantum system, wherein the target quantum state is stationary with respect to a parameterized multi-body qubit operator, the method comprising: preparing an initial quantum state as an input state for a first iteration; iteratively evolving the initial quantum state and subsequent input quantum states as inputs for subsequent iterations until an approximation of the target stationary quantum state is obtained, the method for each iteration comprising: computing, by quantum computation, a parameter value of the multi-body qubit operator for the iteration; computing, by quantum computation, a development time for the iteration, including evaluating a change in an element of the two-electron reduced density matrix for the iteration; using the computed parameter value and development time to evolve the initial quantum state or the subsequent input quantum state for the iteration to generate a subsequent input quantum state for the subsequent iteration; computing, by quantum computation, the development time for the iteration comprises: 【Number 1】 including computing, where λi represents the development time for iteration i, H represents the Hamiltonian characterizing the quantum system, 【Number 2】 represents the first derivative of an element of the two-electron reduced density matrix for the iteration, 【Number 3】 represents the second derivative of an element of the two-electron reduced density matrix for the iteration, a method. Claim 2 The method of claim 1, wherein evaluating a change in an element of the two-electron reduced density matrix for the iteration includes evaluating a first derivative and a second derivative of an element of the two-electron reduced density matrix for the iteration. Claim 3 The method of claim 2, wherein evaluating the first derivative and the second derivative of an element of the two-electron reduced density matrix for the iteration includes using a cumulant expansion. Claim 4 The method of claim 1, wherein computing, by quantum computation, a parameter value of the multi-body qubit operator for the iteration includes performing a measurement of a 3-RDM. Claim 5 Using the calculated parameter values and the evolution time to evolve the initial quantum state or the subsequent input quantum state for the iteration to generate the input quantum state for the subsequent iteration involves approximating the time evolution of the many-body qubit operator at the calculated parameter values using low-rank double factorization, the method according to claim 1.

6. The method according to claim 5, wherein the low-rank double factorization is implemented on a linear lattice of qubits with linear depth.

7. Using the calculated parameter values and the evolution time to evolve the initial quantum state or the subsequent input quantum state for the iteration to generate the input quantum state for the subsequent iteration involves approximating the time evolution of the many-body qubit operator at the calculated parameter values using unitary compression of the many-body qubit operator, the method according to claim 1.

8. The method according to claim 7, wherein the unitary compression of the many-body qubit operator represents the many-body qubit operator at the calculated parameter values in the form of a sum of squares.

9. The method according to claim 1, wherein the target stationary quantum state includes a ground or excited quantum state.

10. The method according to claim 1, wherein the initial quantum state is non-orthogonal to the target quantum state.

11. The target quantum state includes the state of a quantum system characterized by a Hamiltonian, and the target quantum state 〈ψ|[G,H]|ψ〉=0 satisfies the stationary condition given by the above formula, where |ψ〉 represents the target quantum state, H represents the Hamiltonian characterizing the quantum system, and G represents the parameterized many-body qubit operator, the method according to claim 1.

12. The method according to claim 11, wherein the stationary condition includes a first-order stationary condition of energy with respect to changes in the parameters of the many-body qubit operator.

13. The method according to claim 1, wherein the parameterized many-body qubit operator includes a fermionic two-body qubit operator.

14. The fermionic two-body qubit operator is [Number 4] given by, where i, j, k, l are indices representing quantum system orbits, and θ represents a real-valued coefficient. 【Number 5】 , a k represents a creation operator and an annihilation operator, and the associated energy is E(Θ) = 〈ψ|e -Θ He Θ |ψ〉, where |ψ〉 represents the target quantum state and H represents the Hamiltonian characterizing the quantum system, the method according to claim 13.

15. The target quantum state is 【Number 6】 which is represented by a sequence of wave function transformations given by, in the above equation, λ i represents the evolution time for iteration i, and A i represents the many-body qubit operator with the parameter value for iteration i, and |ψ 0 〉 represents the initial quantum state, the method according to claim 1.

16. A system for operating a quantum computer, comprising: one or more processors; one or more I / O devices coupled to the one or more processors, the one or more I / O devices being configured to send control signals and receive readout signals to and from the quantum computer; one or more memories storing computer-readable instructions, the instructions being configured to cause the one or more processors and the one or more I / O devices to practice the preparation step and the iterative development step in the method according to any one of claims 1 to 15 using the quantum computer. A system comprising one or more memories.

17. The system according to claim 16, further comprising the quantum computer.

18. A non-transitory computer-readable storage medium storing instructions for execution by one or more computers configured to send control signals and receive readout signals to and from a quantum computer, the instructions causing the one or more computers to practice the preparation step and the iterative development step in the method according to any one of claims 1 to 15 using the quantum computer. A non-transitory computer-readable storage medium.

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