A method and apparatus for quantum simulation based on quantum gates

Abstract

Embodiments of the present application provide a method and device for quantum simulation based on quantum gates, the method comprising: determining a Hamiltonian of a target system, an evolution time, dividing the evolution time into a plurality of time periods in sequence; preparing a first quantum state; determining a Hamiltonian evolution operation according to the Hamiltonian and the evolution time, the Hamiltonian evolution operation being composed of a plurality of sub-operation combinations corresponding to the plurality of time periods respectively; and the sub-operation combination being composed of a stage evolution operation and an error compensation operation.

Description

Technical Field

[0001] This invention relates to the field of quantum computing, and in particular to a method and apparatus for quantum simulation based on quantum gates. Background Technology

[0002] Quantum simulation is a crucial component of quantum computing and the most natural application of quantum computers. Most applications of quantum computing to solve problems are based on quantum simulation, or used as a subroutine. Quantum simulation has wide applications in numerous technological fields, such as solving for eigenstates and eigenvalues ​​of quantum systems, solving linear equations, solving the dynamical evolution of physical systems, and solving partial differential equations, among others.

[0003] The problem to be solved in quantum simulation is to realize the evolution of the Hamiltonian U = exp(iHt), where H is the Hamiltonian of the objective problem, and H has a dimension of 2. n ×2 n The complexity of solving the above problem using classical computation grows exponentially with respect to the system size (n). For quantum computing, how to implement U using programmable means is a very important issue. Summary of the Invention

[0004] This invention provides a method and apparatus for quantum simulation based on quantum gates. Using this method, the evolutionary operation of the Hamiltonian can be efficiently realized using quantum evolution circuits constructed from quantum gates.

[0005] The technical solution adopted by the present invention to solve the above-mentioned technical problems is, on the one hand, to provide a method for quantum simulation based on quantum gates, comprising:

[0006] The Hamiltonian and evolution time of the target system are determined, and the evolution time is divided into a series of time periods.

[0007] Preparation of the first quantum state;

[0008] Based on the Hamiltonian and the evolution time, a Hamiltonian evolution operation is determined. The Hamiltonian evolution operation is composed of a combination of multiple sub-operations corresponding to the multiple time periods. The combination of sub-operations consists of a stage evolution operation and an error compensation operation.

[0009] Preferably, the stage evolution operation is determined by a product formula.

[0010] Preferably, the product formula includes one of the first-order Trotter product formula and the higher-order Trotter product formula.

[0011] Preferably, the error compensation operation is constituted by a Pauli gate.

[0012] Preferably, the category of the Pauli gate is obtained through random sampling.

[0013] Preferably, determining the Hamiltonian and evolution time of the target system includes: determining the Hamiltonian of the target system, as well as the first evolution time and the second evolution time;

[0014] Determining Hamiltonian evolution operations based on the Hamiltonian and the evolution time includes: determining a first Hamiltonian evolution operation and a second Hamiltonian evolution operation based on the Hamiltonian, a first evolution time, and a second evolution time;

[0015] The method further includes:

[0016] The first Hamiltonian evolution operation and the second Hamiltonian evolution operation are applied to the first quantum state, and the observations of the target system are measured.

[0017] Preferably, the measurement of the observations of the target system includes: measuring the real and imaginary parts of the observations respectively.

[0018] Secondly, a device for quantum simulation based on quantum gates is provided, the device comprising:

[0019] The preparation unit is configured to determine the Hamiltonian and evolution time of the target system, and divide the evolution time into a series of time periods.

[0020] The initial state preparation unit is configured to prepare the first quantum state;

[0021] An evolution operation generation unit is configured to determine a Hamiltonian evolution operation based on the Hamiltonian quantity and the evolution time. The Hamiltonian evolution operation is composed of a combination of multiple sub-operations corresponding to the multiple time periods. The combination of sub-operations is composed of a stage evolution operation and an error compensation operation.

[0022] Thirdly, a quantum computer is provided, including one or more quantum circuits configured to implement the method described in the first aspect. Attached Figure Description

[0023] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0024] Figure 1 A flowchart illustrating a method for quantum simulation based on quantum gates, provided as an embodiment of the present invention;

[0025] Figure 2A schematic diagram of a Hamiltonian evolution circuit composed of quantum gates provided for an embodiment of the present invention;

[0026] Figure 3 A schematic diagram of the circuit for measuring and observing quantities provided in an embodiment of the present invention;

[0027] Figure 4 This is a structural diagram of a device for quantum simulation based on quantum gates, provided as an embodiment of the present invention. Detailed Implementation

[0028] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0029] To more clearly illustrate the advantages of the quantum gate-based quantum simulation schemes provided in the embodiments of this specification, a further summary of existing quantum simulation schemes and their disadvantages is given below.

[0030] Existing quantum simulation schemes include the product formula scheme, among which the representative scheme is the trotter method. The general process of the trotter method is as follows: obtain the initial state of the quantum system; decompose the Hamiltonian of the quantum system into L local sub-Hamiltonians H = ∑ l H l The dynamics of the system are fully described by the system's Hamiltonian. The evolution time T, time interval dt, and total evolution steps v = T / dt are set, starting from the initial moment. For each short time interval, the quantum system evolves according to the local Hamiltonian, where the evolution process is implemented through local quantum gates. L terms of Hamiltonian evolution are applied. This evolution update process is repeated until the target evolution time T is reached. The problem with the piecewise evolution scheme is that the algorithm itself introduces a large error, which is polynomially dependent on the target precision, making it difficult to achieve high-precision quantum simulation.

[0031] Currently known high-precision quantum simulation schemes include quantum signal processing (QS) and linear combination of unitaries. However, both schemes suffer from several drawbacks: they are based on complex quantum circuits and require a series of black-box operations, including phase encoding and state preparation operations. These operations necessitate multi-qubit controlled quantum gates, making them difficult to implement in quantum devices. While the linear combination of unitaries also requires time segmentation, the operations within each segment are performed through black-box operations (multi-qubit controlled operations). Furthermore, these schemes require complex coherent operations, consuming significant resources in terms of circuit depth and qubits. Additionally, post-processing of measurement results is necessary to determine the success of the quantum simulation, making both QS and linear combination of unitaries highly sensitive to noise.

[0032] To address the aforementioned technical problems, this specification provides a method for quantum simulation based on quantum gates. This method employs a series of random single-qubit or multi-qubit (two-qubit) quantum gate operations to programmatically implement quantum simulation. The core idea is to divide time into segments, with each segment first undergoing Hamiltonian evolution according to the product formula. Then, a unitary operation, randomly generated with a certain probability, compensates for the algorithmic error caused by the product formula evolution (error compensation operation). This method combines the advantages of random combinations of the product formula and unitary operations, while effectively avoiding the problems of both approaches.

[0033] Figure 1 This is a flowchart illustrating a method for quantum simulation based on quantum gates, provided as an embodiment of the present invention. Figure 1 As shown, the method includes at least the following steps:

[0034] Step 11: Determine the Hamiltonian and evolution time of the target system, and divide the evolution time into a series of time periods.

[0035] Step 12, prepare the first quantum state;

[0036] Step 13: Determine the Hamiltonian evolution operation based on the Hamiltonian and the evolution time. The Hamiltonian evolution operation is composed of a combination of multiple sub-operations corresponding to the multiple time periods. The combination of sub-operations is composed of a stage evolution operation and an error compensation operation.

[0037] First, in step 11, the Hamiltonian and evolution time of the target system are determined, and the evolution time is divided into a series of time periods.

[0038] In different embodiments, the target system can be a quantum system characterizing different specific problems. In one embodiment, the target system includes any one of a molecular system, a superconducting material system, a metallic crystal system, or a nanomaterial system.

[0039] In different embodiments, different specific methods can be used to obtain the evolution time. In one embodiment, the evolution time can be obtained based on a preset probability distribution, and the evolution time can be divided into a series of sequential time periods. In different embodiments, different specific methods can be used based on different probability distributions to obtain the evolution time.

[0040] In one embodiment, algorithmic errors introduced by the product formula in subsequent steps are compensated for through an error compensation operation. For example, for the product formula S(t), assuming the ideal evolution is U(t), the error compensation operation is as follows: Where S T (t) denotes the complex conjugate of the transpose of S(t). The specific expression for V(t) can be efficiently obtained from U(t) and S(t) (e.g., through classical computation), and can be expressed using Taylor expansion without loss of generality.

[0041]

[0042] Where c is the normalization coefficient, p r It is a probability distribution, K r It's a quantum operation. It can be performed using p... r t r The probability distribution of random sampling K r Finally, V1(t) can be obtained. It can be observed that U(t) = S(t)V(t). Therefore, the operation after error compensation can theoretically achieve U(t) without deviation. Because for a large r, the probability will decrease very rapidly as r increases. In one embodiment, only a portion of K can be selected. r This maintains sufficiently high precision. It is also noted here that in different embodiments, r can be sampled infinitely without excessively increasing the depth of the quantum circuit. This is because of the higher-order K... r These are all Pauli gates and can be merged, thus increasing the number of quantum gates to a maximum of N (where N is the number of qubits). In a specific embodiment, for each small segmented time interval, the following operations can be performed: implement the product formula S(t); by using K... r The corresponding probability distribution is randomly sampled K rV(t) can be realized statistically without bias, and then the idealized Hamiltonian evolution U(t) can be realized at each time interval.

[0043] In different embodiments, different product formulas can be selected, and different error compensation operations can be obtained accordingly. A series of quantum operations are generated through sampling, thereby realizing U(t) without deviation. In one embodiment, the above sampling can be implemented by a classical computing unit.

[0044] Next, in step 12, the first quantum state is prepared.

[0045] In different embodiments, the first quantum state can be prepared in different specific ways, and this specification does not limit this. For example, in one embodiment, the first quantum state |ψ0> can be prepared based on any one of the state preparation operation or the variable quantum eigenvalue solver. For example, in one embodiment, the first quantum state can be prepared on an initial state, which can be, for example, a simple product state. For example, in a chemical problem, the first quantum state can be obtained using a mean-field scheme, also known as the Hartree-Fock method. In different embodiments, the initial state can be any experimentally realizable state, such as a microproduct state. In one example, the initial product state can be... The initial state is prepared by applying a permitted operation on a simulated quantum device. In one example, a unitary operation can be used to prepare the initial state. In another example, Hermitian operations can also be used. In yet another example, a variational quantum eigenvalue solver can be used to take the variationally optimized state as the initial state.

[0046] Subsequently, in step 13, a Hamiltonian evolution operation is determined based on the Hamiltonian quantity and the evolution time. The Hamiltonian evolution operation is composed of a combination of multiple sub-operations corresponding to the multiple time periods. The combination of sub-operations is composed of a stage evolution operation and an error compensation operation.

[0047] In this step, the Hamiltonian evolution operation can be determined based on the evolution time obtained in steps 11 and 12, the time period divided according to the evolution time, and the Hamiltonian H corresponding to the target system. It can be expressed as U = e itH It is important to note that this Hamiltonian evolution operation is not implemented by a simulated quantum device, but rather consists of stage evolution operations and error compensation operations composed of multiple quantum gates (unlike simulated quantum devices that can independently simulate Hamiltonian evolution, individual quantum gates such as Pauli gates and controlled NOT gates cannot independently simulate Hamiltonian evolution).

[0048] In one embodiment, the quantum circuit constituting U can be determined through the following refinement process: First, based on several time segments obtained after dividing the evolution time of U, U = e iHt The process is divided into several time periods, each corresponding to a different evolutionary sub-operation. These evolutionary sub-operations can be represented as follows: Furthermore, the original Hamiltonian evolutionary operation can be expressed as like Figure 2 As shown in (a).

[0049] Secondly, to achieve U r In one embodiment, U r It can be done Figure 3 The circuit shown in (b) is implemented. That is, each U r This is achieved through staged evolution operations and error compensation operations. The staged evolution operations can be determined using a product formula. In a specific embodiment, the product formula may include one of a first-order Trotter product formula or a higher-order Trotter product formula, such as... Figure 3 As shown in (c), the error compensation operation can be constructed using a single-bit Pauli operation (Pauli gate). In a specific embodiment, the type of the single-bit Pauli operation can be obtained through random sampling using a classical computer. A Pauli gate can be represented as... It consists of only one layer of single-bit gates (P1, P2, ... P). n )constitute, This represents the direct product.

[0050] In another embodiment, U r It can be done Figure 3 The circuit shown in (d) is implemented. That is, each U r Through the Pauli revolving door V r And Pauli P r Implementation. Here, the Pauli revolving door can be represented as exp iVθ, where V is the Pauli door of a defined category. In one embodiment, the angle θ of the Pauli revolving door and the category of the Pauli door V can be obtained by sampling through classical computer post-processing. In one embodiment, as... Figure 3 As shown in (e), the Pauli rotation gate can be implemented by decomposing it into a basic Pauli gate, a CNOT gate (represented by a CNOT ladder), and a single-bit Pauli rotation gate.

[0051] It is important to note that U r It can be determined randomly. For example, in a specific case, for each time segment r, only one sampling can be used to obtain a final U. r sequence (i.e. To achieve U, in another example, sampling can be repeated several times to obtain several sets of U. r U is implemented using a sequence.

[0052] According to one implementation, the acquired Hamiltonian evolution operation can also be used to obtain the observations of the target system. Specifically, in one embodiment A, in steps 11 and 12, the Hamiltonian quantity of the target system, as well as the first evolution time and the second evolution time, can be determined; in step 13, the first Hamiltonian evolution operation and the second Hamiltonian evolution operation are determined based on the Hamiltonian quantity, the first evolution time, and the second evolution time. Thereafter, the first Hamiltonian evolution operation U is applied to the first quantum state. i Second Hamiltonian evolution operation U j And measure the observations of the target system. In one embodiment, it can utilize... Figure 3 The circuits shown in (a) and (b) respectively measure the real and imaginary parts of the observed quantity O. In another embodiment, the process of embodiment A can be executed multiple times to obtain multiple measurement results, and a result can be obtained based on the measurement results. The expected value, of which It is the transpose of the complex conjugate.

[0053] In summary, the advantages of the quantum gate-based quantum simulation method provided in this specification are as follows: First, it avoids the problems inherent in both product formula schemes and unitary operation linear combination schemes, enabling high-precision, algorithm-bias-free quantum simulations. Theoretically, it can achieve quantum simulations of arbitrary precision. Second, it is simple to operate. It is not based on existing quantum signal processing or unitary operation linear combination quantum black box operations. It does not require multi-qubit controlled quantum gates, only single-qubit or two-qubit quantum gate operations. Third, it has low requirements for system coherence time, thus exhibiting good robustness to noise. It can be applied to programmable quantum devices. It is compatible with general-purpose fault-tolerant quantum computing devices and with noisy quantum computing devices.

[0054] According to another embodiment, an apparatus for quantum simulation based on quantum gates is provided. Figure 4 A structural diagram of a device for quantum simulation based on quantum gates provided in an embodiment of the present invention is shown below. Figure 4 As shown, the device 400 includes:

[0055] The preparation unit 41 is configured to determine the Hamiltonian and evolution time of the target system, and divide the evolution time into a series of time periods.

[0056] Initial state preparation unit 42 is configured to prepare the first quantum state;

[0057] The evolution operation generation unit 43 is configured to determine a Hamiltonian evolution operation based on the Hamiltonian quantity and the evolution time. The Hamiltonian evolution operation is composed of a combination of multiple sub-operations corresponding to the multiple time periods. The combination of sub-operations is composed of a stage evolution operation and an error compensation operation.

[0058] According to yet another embodiment, a computer-readable medium is also provided, including a computer program stored thereon, which executes the methods described above when the computer is running.

[0059] According to yet another embodiment, a quantum computer is also provided, including one or more quantum circuits configured to implement the methods described above.

[0060] The foregoing has described specific embodiments of this specification. Other embodiments are within the scope of the appended claims. In some cases, the actions or steps recited in the claims may be performed in a different order than that shown in the embodiments and may still achieve the desired result. Furthermore, the processes depicted in the drawings do not necessarily require the specific or sequential order shown to achieve the desired result. In some embodiments, multitasking and parallel processing are possible or may be advantageous.

[0061] Those skilled in the art will further recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, computer software, or a combination of both. To clearly illustrate the interchangeability of hardware and software, the components and steps of the various examples have been generally described in terms of functionality in the foregoing description. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementations should not be considered beyond the scope of this invention.

[0062] The steps of the methods or algorithms described in conjunction with the embodiments disclosed herein can be implemented in hardware, a software module executed by a processor, or a combination of both. The software module can be located in random access memory (RAM), main memory, read-only memory (ROM), electrically programmable ROM, electrically erasable programmable ROM, registers, hard disk, removable disk, CD-ROM, or any other form of storage medium known in the art.

[0063] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for quantum simulation based on quantum gates, comprising: The Hamiltonian and evolution time of the target system are determined, and the evolution time is divided into a series of time periods. Preparation of the first quantum state; Based on the Hamiltonian and the evolution time, a Hamiltonian evolution operation is determined. This Hamiltonian evolution operation is composed of multiple sub-operations corresponding to the multiple time periods. The sub-operation combination consists of a stage evolution operation and an error compensation operation. The stage evolution operation is determined by the product formula S(t). The error compensation operation is used to compensate for algorithmic errors in the product formula S(t). The error compensation operation includes quantum operations, which are obtained through corresponding probability random sampling. These quantum operations are either single-qubit or double-qubit quantum gate operations. The sampling probability is based on the Taylor expansion of the error compensation expression V(t). In Determined, where c is the normalization coefficient. It is a probability distribution. It is a quantum operation; the error compensation expression V(t) is calculated using classical computation methods. Obtain; among them Representing the product formula The transpose of the complex conjugate, U(t), is the ideal evolution.

2. The method according to claim 1, wherein, The product formula includes one of the first-order Trotter product formula and the higher-order Trotter product formula.

3. The method according to claim 1, wherein, The error compensation operation is constituted by Pauli gates.

4. The method according to claim 3, wherein, The categories of the Pauli gates were obtained through random sampling.

5. The method according to claim 1, wherein, Determine the Hamiltonian and evolution time of the target system, including: determining the Hamiltonian of the target system, as well as the first evolution time and the second evolution time; Determining Hamiltonian evolution operations based on the Hamiltonian and the evolution time includes: determining a first Hamiltonian evolution operation and a second Hamiltonian evolution operation based on the Hamiltonian, a first evolution time, and a second evolution time; The method further includes: The first Hamiltonian evolution operation and the second Hamiltonian evolution operation are applied to the first quantum state, and the observations of the target system are measured.

6. The method according to claim 5, wherein, Measuring the observations of the target system includes measuring the real and imaginary parts of the observed quantities, respectively.

7. A device for quantum simulation based on quantum gates, the device comprising: The preparation unit is configured to determine the Hamiltonian and evolution time of the target system, and divide the evolution time into a series of time periods. The initial state preparation unit is configured to prepare the first quantum state; An evolution operation generation unit is configured to determine a Hamiltonian evolution operation based on the Hamiltonian and the evolution time. The Hamiltonian evolution operation is composed of multiple sub-operations corresponding to the multiple time periods. Each sub-operation combination consists of a stage evolution operation and an error compensation operation. The stage evolution operation is determined by a product formula S(t), and the error compensation operation is used to compensate for algorithmic errors in the product formula S(t). The error compensation operation includes a quantum operation, which is obtained through corresponding probability random sampling. The quantum operation is either a single-qubit quantum gate operation or a two-qubit quantum gate operation. The sampling probability is based on the Taylor expansion of the error compensation expression V(t). In Determined, where c is the normalization coefficient. It is a probability distribution. It is a quantum operation; the error compensation expression V(t) is calculated using classical computation methods. Obtain; among them Representing the product formula The transpose of the complex conjugate, U(t), is the ideal evolution.

8. A quantum computer comprising one or more quantum circuits configured to implement the method according to any one of claims 1 to 6.

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