Quantum linear solution method, device and equipment based on polynomial preprocessor

Abstract

The invention discloses a quantum linear solution method, device and equipment based on a polynomial preprocessor. The method comprises: obtaining information of a first matrix A and a first vector b in a linear system to be processed, calculating a polynomial p(A) of the first matrix A used for preprocessing the linear system, preprocessing the linear system according to the polynomial p(A), obtaining a second matrix A′ and a second vector b′, constructing a quantum circuit corresponding to an HHL algorithm, performing evolution and measurement operations of a quantum state according to the second matrix A′ and the second vector b′, and obtaining the final quantum state of the evolved quantum circuit, so as to solve the deficiencies in the prior art. The method can reduce the time complexity and the amount of calculation for solving linear problems, and at the same time reduce the occupation of hardware resources.

Description

Technical Field

[0001] The present invention belongs to the field of quantum computing technology, and in particular to a quantum linear solution method, device and equipment based on a polynomial preprocessor. Background Art

[0002] Quantum computers are physical devices that follow the laws of quantum mechanics to perform high-speed mathematical and logical operations, store and process quantum information. When a device processes and calculates quantum information and runs quantum algorithms, it is a quantum computer. Quantum computers have become a key technology under research because they have the ability to process mathematical problems more efficiently than ordinary computers. For example, they can speed up the time to crack RSA keys from hundreds of years to a few hours.

[0003] Quantum computing simulation is a simulation that uses numerical calculations and computer science to simulate calculations that follow the laws of quantum mechanics. As a simulation program, it uses the basic laws of quantum bits in quantum mechanics and the high-speed computing power of computers to characterize the spatiotemporal evolution of quantum states.

[0004] Solving systems of linear equations is at the core of many scientific and engineering problems. The classical algorithms for solving such problems are collectively referred to as linear system algorithms. In recent years, a very important achievement in the field of quantum computing is the quantum linear system algorithm, of which the most famous is the HHL algorithm proposed by Harrow, Hassidim, and Lloyd in 2009. However, as the condition number of the input matrix increases, the time complexity of solving linear problems will increase accordingly, and the complexity of the quantum linear solver is related to the polynomial of the condition number κ of the linear system, and the complexity is expressed as Furthermore, when the condition number of the linear system is too large, the exponential speedup of the quantum linear solver disappears.

[0005] Based on this, it is necessary to implement a quantum linear solution technology that can handle linear systems with too large condition numbers for use in quantum computing simulations. Summary of the invention

[0006] The purpose of the present invention is to provide a quantum linear solution method based on a polynomial preprocessor to address the deficiencies in the prior art, which can reduce the time complexity and computational complexity of solving linear problems while reducing the occupation of hardware resources.

[0007] An embodiment of the present application provides a quantum linear solution method based on a polynomial preprocessor, comprising:

[0008] Obtain information of the first matrix A and the first vector b in the linear system to be processed;

[0009] Calculating a polynomial p(A) of a first matrix A for preprocessing of the linear system;

[0010] Preprocessing the linear system according to the polynomial p(A) to obtain a second matrix A′ and a second vector b′;

[0011] A quantum circuit corresponding to the HHL algorithm is constructed, and according to the second matrix A′ and the second vector b′, the quantum state evolution and measurement operations are performed to obtain the final quantum state of the evolved quantum circuit.

[0012] Optionally, obtaining the second matrix A′ and the second vector b′ includes:

[0013] A second matrix A′=p(A)A and a second vector b′=p(A)b are obtained.

[0014] Optionally, performing the quantum state evolution and measurement operation according to the second matrix A′ and the second vector b′ to obtain the final quantum state of the evolved quantum circuit includes:

[0015] For the quantum circuit, input the initial quantum state |b′> and the operator , perform quantum state evolution and measurement operations to obtain the final quantum state of the quantum circuit after evolution, wherein the operator The t is a constant.

[0016] Optionally, the calculating the polynomial p(A) of the first matrix A for preprocessing the linear system includes:

[0017] Construct a function f that solves the polynomial m (x) and a preset interval S, wherein the function f m (x) satisfies f m (x) = p m (x)x, and f m (x i )+(-1) i E=1, m is the polynomial order, i=0, 1, 2, ..., m+1, S∈[1 / κ A , 1], E is the deviation amplitude, κ A is the condition number;

[0018] Select m+1 points in the preset interval and obtain the linear equation system f m (x i )+(-1) i The solution of E=1, wherein the m+1 points are x1, x2, ..., x m+1 And satisfy x1=1 / κ, x m+1 =1;

[0019] Substitute the obtained solution of the linear equations into f m (x), we get |1-f m (x)|The set N′ of points with local maxima;

[0020] Determine for any x∈N′, f m (x)-1 absolute value is the same and increases with x, f m (x)-1 Whether the positive and negative signs change alternately, if so, then the current f m (x) is the optimal polynomial.

[0021] Optionally, the method further includes:

[0022] If not, then replace the x1, x2, ..., x m+1 The elements in are replaced with elements in the set N′, and the replacement rule is: x1, x m+1 Keep unchanged and change x2,...,x m Replace with the elements in the set N', and then return to execute the method to obtain the linear equation system f m (x i )+(-1) i The steps of solving E=1 until the optimal f that meets the judgment conditions is obtained m (x).

[0023] Optionally, constructing a quantum circuit corresponding to the HHL algorithm includes:

[0024] A plurality of quantum bits including an auxiliary quantum bit, a first quantum bit, and a second quantum bit are obtained, wherein the initial states of the auxiliary quantum bit and the first quantum bit are set to |0>, and the initial state of the second quantum bit is set to The b′ i is the i-th element of the second vector b′, and N is the dimension of the second vector;

[0025] Determine the unitary matrix corresponding to the second matrix A′

[0026] Construct a first sub-quantum circuit module for phase estimation, which is used to decompose the |b′> in the eigenspace of the second matrix A′ into And transform the initial states of the first quantum bit and the second quantum bit |0>|b′> into Among them, the |μ j > is the eigenvector of the second matrix A′, the λ j is the eigenvalue of the second matrix A′, the β j is the amplitude of the eigenvector of the second matrix A′;

[0027] A second sub-quantum circuit module is constructed to perform a controlled rotation operation, which is used to convert |λ j >Using the auxiliary qubit as a control bit to rotate, we get Wherein, C is a normalization constant;

[0028] The third quantum circuit module for performing inverse phase estimation is constructed to convert |λ j >reset to |0>;

[0029] A measurement operation module for the auxiliary quantum bit is constructed so that when the quantum state of the auxiliary quantum bit is measured to be |1>, the following is obtained: The |x′〉∝|x〉;

[0030] The first sub-quantum circuit module, the second sub-quantum circuit module, the third sub-quantum circuit module and the quantum measurement operation module are sequentially combined to form a quantum circuit corresponding to the HHL algorithm.

[0031] Another embodiment of the present application provides a quantum linear solution device based on a polynomial preprocessor, the device comprising:

[0032] An acquisition module, used to acquire information of a first matrix A and a first vector b in a linear system to be processed;

[0033] A calculation module, used for calculating a polynomial p(A) of a first matrix A for preprocessing of the linear system;

[0034] An acquisition module, used for preprocessing the linear system according to the polynomial p(A) to obtain a second matrix A′ and a second vector b′;

[0035] The construction module is used to construct a quantum circuit corresponding to the HHL algorithm, perform quantum state evolution and measurement operations according to the second matrix A' and the second vector b', and obtain the final quantum state of the evolved quantum circuit.

[0036] Optionally, the obtaining module includes:

[0037] The first obtaining unit is used to obtain a second matrix A′=p(A)A and a second vector b′=p(A)b.

[0038] Optionally, the building block includes:

[0039] An input unit, used to input an initial quantum state |b′> and an operator for the quantum circuit Performing quantum state evolution and measurement operations to obtain the final quantum state of the quantum circuit after evolution, wherein the operator The t is a constant.

[0040] Optionally, the computing module includes:

[0041] The first construction unit is used to construct a function f for solving the polynomial m (x) and a preset interval S, wherein the function f m (x) satisfies f m (x) = p m (x)x, and f m (x i )+(-1) i E=1, m is the polynomial order, i=0, 1, 2, ..., m+1, S∈[1 / κ A , 1], E is the deviation amplitude, κ A is the condition number;

[0042] A selection unit is used to select m+1 points in the preset interval and obtain the linear equation system f m (x i )+(-1) i The solution of E=1, wherein the m+1 points are x1, x2, ..., x m+1 And satisfy x1=1 / κ, x m+1 =1;

[0043] A substitution unit is used to substitute the solution of the linear equation system into f m (x), we get |1-f m (x)|The set of points N′ of local maximum values;

[0044] The judgment unit is used to judge whether f m (x)-1 absolute value is the same and increases with x, f m (x)-1 Whether the positive and negative signs change alternately, if so, then the current f m (x) is the optimal polynomial.

[0045] Optionally, the calculation module further includes:

[0046] Returns the unit for converting x1, x2, ..., x m+1 The elements in are replaced with elements in the set N′, and the replacement rule is: x1, x m+1 Keep unchanged and change x2,...,x m Replace with the elements in the set N', and then return to execute the method to obtain the linear equation system f m (x i )+(-1) iThe steps of solving E=1 until the optimal f that meets the judgment conditions is obtained m (x).

[0047] Optionally, the building block includes:

[0048] The second obtaining unit is used to obtain a plurality of quantum bits including an auxiliary quantum bit, a first quantum bit, and a second quantum bit, wherein the initial states of the auxiliary quantum bit and the first quantum bit are set to |0>, and the initial state of the second quantum bit is set to The b′ i is the i-th element of the second vector b′, and N is the dimension of the second vector;

[0049] A determination unit, used to determine the unitary matrix corresponding to the second matrix A′

[0050] The second construction unit is used to construct a first sub-quantum circuit module for phase estimation, and is used to decompose the |b′> in the eigenspace of the second matrix A′ into And transform the initial states of the first quantum bit and the second quantum bit |0>|b′> into Among them, the |μ j > is the eigenvector of the second matrix A′, the λ j is the eigenvalue of the second matrix A′, the β j is the amplitude of the eigenvector of the second matrix A′;

[0051] The third construction unit is used to construct a second sub-quantum circuit module for performing a controlled rotation operation, and is used to convert |λ j >Using the auxiliary qubit as a control bit to rotate, we get Wherein, C is a normalization constant;

[0052] The fourth construction unit is used to construct a third sub-quantum circuit module for performing inverse phase estimation, and is used to convert |λ j >reset to |0>;

[0053] A fifth construction unit is used to construct a measurement operation module for the auxiliary quantum bit, so that when the quantum state of the auxiliary quantum bit is measured to be |1>, the following is obtained: The |x′〉∝|x〉;

[0054] The combining unit is used to sequentially combine the first sub-quantum circuit module, the second sub-quantum circuit module, the third sub-quantum circuit module and the quantum measurement operation module to form a quantum circuit corresponding to the HHL algorithm.

[0055] Yet another embodiment of the present application provides a storage medium, wherein the storage medium stores a computer program, wherein the computer program is configured to execute any of the above methods when running.

[0056] Yet another embodiment of the present application provides an electronic device, including a memory and a processor, wherein the memory stores a computer program, and the processor is configured to run the computer program to execute any of the methods described above.

[0057] Compared with the prior art, the present invention first obtains information of the first matrix A and the first vector b in the linear system to be processed, calculates the polynomial p(A) of the first matrix A used for preprocessing the linear system, preprocesses the linear system according to the polynomial p(A), obtains the second matrix A′ and the second vector b′, constructs a quantum circuit corresponding to the HHL algorithm, and executes quantum state evolution and measurement operations according to the second matrix A′ and the second vector b′ to obtain the final quantum state of the evolved quantum circuit. It can reduce the time complexity and calculation amount of solving linear problems, speed up the solution speed of the quantum linear algorithm, and reduce the occupation of hardware resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 A hardware structure block diagram of a computer terminal for a quantum linear solution method based on a polynomial preprocessor provided in an embodiment of the present invention;

[0059] Figure 2 A schematic flow chart of a quantum linear solution method based on a polynomial preprocessor provided in an embodiment of the present invention;

[0060] Figure 3 A construction operator provided in an embodiment of the present invention Quantum circuit diagram of

[0061] Figure 4 A schematic diagram of a first sub-quantum circuit module corresponding to phase estimation provided in an embodiment of the present invention;

[0062] Figure 5 A schematic diagram of the total quantum circuit corresponding to the HHL algorithm provided in an embodiment of the present invention;

[0063] Figure 6 A schematic structural diagram of a quantum linear solver based on a polynomial preprocessor provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0064] The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, but should not be construed as limiting the present invention.

[0065] The embodiment of the present invention first provides a quantum linear solution method based on a polynomial preprocessor, which can be applied to electronic devices such as computer terminals, specifically ordinary computers, quantum computers, etc.

[0066] The following describes it in detail by taking running on a computer terminal as an example. Figure 1 The hardware structure block diagram of a computer terminal for a quantum linear solution method based on a polynomial preprocessor provided in an embodiment of the present invention. Figure 1 As shown, the computer terminal may include one or more ( Figure 1 Only one is shown in the figure) a processor 102 (the processor 102 may include but is not limited to a processing device such as a microprocessor MCU or a programmable logic device FPGA) and a memory 104 for storing data. Optionally, the computer terminal may also include a transmission device 106 for communication functions and an input and output device 108. It can be understood by those skilled in the art that Figure 1 The structure shown is only for illustration and does not limit the structure of the above-mentioned computer terminal. Figure 1 More or fewer components as shown, or with Figure 1 Different configurations are shown.

[0067] The memory 104 can be used to store software programs and modules of application software, such as the program instructions / modules corresponding to the quantum linear solution method based on the polynomial preprocessor in the embodiment of the present application. The processor 102 executes various functional applications and data processing by running the software programs and modules stored in the memory 104, that is, implementing the above method. The memory 104 may include a high-speed random access memory, and may also include a non-volatile memory, such as one or more magnetic storage devices, flash memory, or other non-volatile solid-state memory. In some instances, the memory 104 may further include a memory remotely arranged relative to the processor 102, and these remote memories may be connected to the computer terminal via a network. Examples of the above-mentioned network include, but are not limited to, the Internet, an intranet, a local area network, a mobile communication network, and combinations thereof.

[0068] The transmission device 106 is used to receive or send data via a network. The specific example of the above network may include a wireless network provided by a communication provider of a computer terminal. In one example, the transmission device 106 includes a network adapter (Network Interface Controller, NIC), which can be connected to other network devices through a base station so as to communicate with the Internet. In one example, the transmission device 106 can be a radio frequency (RF) module, which is used to communicate with the Internet wirelessly.

[0069] It should be noted that a true quantum computer is a hybrid structure, which consists of two parts: one part is a classical computer, which is responsible for performing classical calculations and control; the other part is a quantum device, which is responsible for running quantum programs and thus realizing quantum computing. A quantum program is a sequence of instructions written in a quantum language such as QRunes that can be run on a quantum computer, which supports quantum logic gate operations and ultimately realizes quantum computing. Specifically, a quantum program is a sequence of instructions that operate quantum logic gates in a certain sequence.

[0070] In practical applications, due to the limitations of the development of quantum device hardware, quantum computing simulation is usually required to verify quantum algorithms, quantum applications, etc. Quantum computing simulation is the process of simulating the operation of quantum programs corresponding to specific problems using a virtual architecture (i.e., quantum virtual machine) built with the resources of ordinary computers. Usually, it is necessary to build a quantum program corresponding to a specific problem. The quantum program referred to in the embodiment of the present invention is a program written in a classical language to characterize quantum bits and their evolution, in which quantum bits, quantum logic gates, etc. related to quantum computing are represented by corresponding classical codes.

[0071] Quantum circuits, as a manifestation of quantum programs, are also called quantum logic circuits. They are the most commonly used general quantum computing model. They represent circuits that operate on quantum bits in an abstract concept. They are composed of quantum bits, circuits (timelines), and various quantum logic gates. Finally, the results often need to be read out through quantum measurement operations.

[0072] Unlike traditional circuits that are connected by metal wires to transmit voltage or current signals, in quantum circuits, the circuits can be seen as connected by time, that is, the state of the quantum bit evolves naturally over time, following the instructions of the Hamiltonian operator until it encounters a logic gate and is operated.

[0073] A quantum program as a whole corresponds to a total quantum circuit. The quantum program described in the present invention refers to the total quantum circuit, wherein the total number of quantum bits in the total quantum circuit is the same as the total number of quantum bits in the quantum program. It can be understood that a quantum program can be composed of a quantum circuit, measurement operations on quantum bits in the quantum circuit, registers for storing measurement results, and control flow nodes (jump instructions). A quantum circuit can contain dozens, hundreds, or even thousands of quantum logic gate operations. The execution process of a quantum program is the process of executing all quantum logic gates in a certain time sequence. It should be noted that the time sequence is the time order in which a single quantum logic gate is executed.

[0074] It should be noted that in classical computing, the most basic unit is the bit, and the most basic control mode is the logic gate, which can achieve the purpose of controlling the circuit through the combination of logic gates. Similarly, the way to process quantum bits is quantum logic gates. Using quantum logic gates, quantum states can evolve. Quantum logic gates are the basis of quantum circuits. Quantum logic gates include single-bit quantum logic gates, such as Hadamard gates (H gates, Hadamard gates), Pauli-X gates (X gates), Pauli-Y gates (Y gates), Pauli-Z gates (Z gates), RX gates, RY gates, RZ gates, etc.; multi-bit quantum logic gates, such as CNOT gates, CR gates, iSWAP gates, Toffoli gates, etc. Quantum logic gates are generally represented by unitary matrices, and unitary matrices are not only in matrix form, but also a kind of operation and transformation. The effect of general quantum logic gates on quantum states is calculated by multiplying the unitary matrix on the left by the matrix corresponding to the right vector of the quantum state.

[0075] The quantum state, that is, the logical state of the quantum bit, is represented by binary in the quantum algorithm (or quantum program). For example, a group of quantum bits is q0, q1, and q2, representing the 0th, 1st, and 2nd quantum bits, and is sorted from high to low as q2q1q0. The quantum state corresponding to this group of quantum bits is the superposition of the eigenstates corresponding to this group of quantum bits. The eigenstates corresponding to this group of quantum bits have 2 to the power of the total number of quantum bits, that is, 8 eigenstates (determined states): |000>, |001>, |010>, |011>, |100>, |101>, |110>, |111>, and the bit of each eigenstate corresponds to the quantum bit. For example, in the |000> state, 000 corresponds to q2q1q0 from high to low, and |> is the Dirac symbol.

[0076] Using a single qubit to illustrate, the logical state of a single qubit It may be in the |0> state, |1> state, or the superposition state of |0> state and |1> state (uncertain state), which can be specifically expressed as Among them, c and d are complex numbers representing the amplitude (probability amplitude) of the quantum state, and the square of the amplitude modulus |c| 2 and |d| 2 denote the probabilities of the |0> and |1> states, respectively, and |c| 2 +|d| 2 = 1. In short, the quantum state is a superposition state composed of various eigenstates. When the probability of other eigenstates is 0, it is in the only determined eigenstate.

[0077] In recent years, quantum algorithms for solving the linear equation system Ax=b have been proposed. For example, Childs et al. proposed an algorithm for constructing matrix inverse using series expansion and constructed a new quantum linear solver with a complexity of The exponential improvement on ∈ also improves the performance by VTAA, and they also optimize the complexity to Recently, work has been proposed to build a quantum linear solver using QSVT, which does not require VTAA optimization and has a complexity of From the above work, we can see that the complexity of the optimal quantum linear solver is linearly related to the system condition number κ, and in practical problems, we may encounter linear systems with large condition numbers, such as κ in the polynomial order of n. When solving such problems, the advantage of the quantum linear solver will be severely reduced, and there will no longer be a quantum advantage. When encountering an ill-conditioned linear system, the classical algorithm for solving a system of linear equations generally pre-processes the linear system to construct a new well-conditioned linear system, and the solution is the same as the original system, thereby effectively solving the ill-conditioned linear system. In this application, we add the classical polynomial preprocessing process to the HHL algorithm and construct an HHL algorithm with a polynomial preprocessor, which reduces the condition number of the linear system to be solved, thereby improving the performance of the HHL algorithm and extending the scope of application of the HHL algorithm.

[0078] See also Figure 2 , Figure 2 A schematic flow chart of a quantum linear solution method based on a polynomial preprocessor provided in an embodiment of the present invention may include the following steps:

[0079] S201: Obtain information of a first matrix A and a first vector b in a linear system to be processed.

[0080] A linear system is a mathematical model that refers to a system composed of linear operators and satisfies superposition and uniformity (also known as homogeneity). At present, linear systems are the core of many scientific and engineering fields. For the linear system Ax=b to be processed, the element information and dimension of the first matrix A and the first vector b can be obtained respectively. Among them, the first matrix A can be a reversible Hermitian matrix. The Hermitian matrix is ​​one of the many types of matrices. Simply put, the Hermitian matrix refers to a self-conjugate matrix. Each element in the i-th row and j-th column of the matrix is ​​conjugate and equal to the element in the j-th row and i-th column. The elements on the main diagonal of the Hermitian matrix are all real numbers, and its eigenvalues ​​are also real numbers.

[0081] S202: Calculate the polynomial p(A) of the first matrix A used for the linear system preprocessing.

[0082] Specifically, for the linear system to be processed Ax=b, where A is a reversible Hermitian matrix, its condition number is κ A =||A||||A -1||. The linear system preprocessing process is to find the polynomial p(A) of the first matrix A, and calculate the polynomial p(A) of the first matrix A for linear system preprocessing, which may include the following steps:

[0083] Step 1: Construct a function f that solves the polynomial m (x) and a preset interval S, wherein the function f m (x) satisfies f m (x) = p m (x)x, and f m (x i )+(-1) i E=1, m is the polynomial order, i=0, 1, 2, ..., m+1, S∈[1 / κ A , 1], where E is the deviation amplitude.

[0084] Step 2: Select m+1 points in the preset interval and obtain the linear equation system f m (x i )+(-1) i The solution of E=1, wherein the m+1 points are x1, x2, ..., x m+1 And satisfy x1=1 / κ, x m+1 =1.

[0085] Step 3: Substitute the obtained solution of the linear equations into f m (x), we get |1-f m The set N′ of points with local maxima of (x)|.

[0086] Step 4: Determine for any x∈N′, f m (x)-1 absolute value is the same and increases with x, f m (x)-1 Whether the positive and negative signs change alternately, if so, then the current f m (x) is the optimal polynomial.

[0087] Step 5: If not, replace x1, x2, ..., x m+1 The elements in are replaced with elements in the set N′, and the replacement rule is: x1, x m+1 Keep unchanged and change x2,...,x m Replace with the elements in the set N', and then return to execute the method to obtain the linear equation system f m (x i )+(-1) i The steps of solving E=1 until the optimal f that meets the judgment conditions is obtained m (x).

[0088] Specifically, the polynomial p(A) of the first matrix A used for linear system preprocessing is calculated so that κ A′ The minimum problem can be transformed into a minimax approximation problem, and then the minimax approximation problem can be solved by the Remez algorithm to obtain the optimal p(A).

[0089] First, we can define: ||l(x)-p(x)x|| S =max x∈S |l(x)-p(x)x|, where S = [-1, -1 / κ] ∪ [1 / κ, 1], l(x) is defined as: Considering that p(x) is an even function, l(x)-p(x)x is an odd function. Therefore, we only consider the interval S = [1 / κ, 1] where x > 0. The optimal p(x) satisfies: in, represents the set of even functions of order 2m.

[0090] The Remez algorithm is an iterative algorithm that approximates a function through the minimax norm. Assume that p(x) can be expressed as: definition: First, select m+1 points X={x1, x2, ..., x m+1}, where x1 = 1 / κ, x m+1 =1; Secondly, solve the linear equation system f m (x i )+(-1) i E=1,i=0,1,2,...,m+1,get the solution α1,α2,...,α m , E; then α1, α2, ..., α m Substitute f m (x), find g(x) = |1-f m (x)| is the set of points N′ of the local maximum value; finally, we determine if for any x∈N′, f m (x)-1 has the same absolute value and as x increases, f m (x)-1 changes its positive and negative signs alternately, so the current f m (x) is the optimal polynomial. If not, replace the elements in X with the elements in N′. The replacement rule is: x1, x m+1 Keep unchanged and change x2,...,x m Replace with the elements in N'. Then return to execute the method of obtaining the linear equation system p m (x i )+(-1) i E=1,i=1,2,...,m+1,until the optimal p that meets the judgment condition is obtainedm (x).

[0091] S203: Preprocess the linear system according to the polynomial p(A) to obtain a second matrix A′ and a second vector b′.

[0092] Specifically, based on the obtained optimal polynomial p(A), a new linear system A′x=b′ is constructed, including obtaining a second matrix A′=p(A)A and a second vector b′=p(A)b, so that the new linear system can be solved more efficiently. In general, when the condition number of a linear system of equations becomes smaller, the equation will be easier to solve. Therefore, p(A) needs to make the condition number of the preprocessed second matrix A′ κ A′ Satisfy κ A′ <<κ A .

[0093] It should be noted that there are many ways to select p(A), such as Neumann polynomial, Chebyshev polynomial, Least Square polynomial, etc., which are not limited here.

[0094] S204: construct a quantum circuit corresponding to the HHL algorithm, perform quantum state evolution and measurement operations according to the second matrix A′ and the second vector b′, and obtain the final quantum state of the evolved quantum circuit.

[0095] Solving linear equations is a fundamental problem in many fields. In recent years, algorithms for solving linear equations Ax = b have been proposed. In 2009, Harrow et al. proposed a quantum algorithm for solving linear equations, namely the HHL algorithm, whose query complexity is s represents the sparsity of the first matrix A, κ=||A||||A -1 || represents the condition number of the first matrix A.

[0096] Specifically, since the HHL algorithm has an exponential acceleration effect compared to the classical algorithm under certain conditions, it will be widely used in data processing, machine learning, numerical calculation, fluid mechanics problem processing and other scenarios in the future. In this embodiment, the HHL algorithm can be used to solve a problem of solving a linear equation, that is, for the quantum circuit, input the initial quantum state |b′> and the operator Performing quantum state evolution and measurement operations to obtain the final quantum state of the quantum circuit after evolution, wherein the operator The t is a constant.

[0097] Among them, the input initial quantum state |b′> and the operator Output the solution x that satisfies the linear equation A′x=b′, that is, x=A′ -1b′. Therefore, the second matrix A′ needs to satisfy the requirement of being a reversible matrix, and for the second vector b′, due to the need to load the data of the second vector b′ into the quantum circuit, the dimension of the second vector b′ needs to be expressible as a positive integer power of 2. If the dimension does not conform to the form of a positive integer power of 2, zeros are added to the elements of the second vector b′ until it satisfies the form of a positive integer power of 2.

[0098] Specifically, constructing the quantum circuit corresponding to the HHL algorithm includes the following steps:

[0099] Step 1: Obtain a plurality of quantum bits including an auxiliary quantum bit, a first quantum bit, and a second quantum bit, wherein the initial states of the auxiliary quantum bit and the first quantum bit are set to |0>, and the initial state of the second quantum bit is set to The b′ i is the i-th element of the second vector b′, and N is the dimension of the second vector.

[0100] Specifically, the number of quantum bits to be obtained can be determined by the user according to demand, or a more sufficient number of quantum bits can be set to meet computing needs if computing resources are sufficient.

[0101] A plurality of quantum bits including an auxiliary quantum bit, a first quantum bit, and a second quantum bit are obtained, which can be specifically represented by a quantum bit position. For example, if the initial quantum bit position is |0>, it means that the quantum state of the quantum bit is |0> state, and if it is |1>, it means that the initial quantum state is |1> state.

[0102] It should be noted that, for the convenience of subsequent distinction, the obtained quantum bits can be divided into auxiliary quantum bits, first quantum bits, and second quantum bits. The specific distinction names are not limited here, and the initial state of each quantum bit can be prepared by the existing amplitude coding method or quantum state coding method. Among them, the initial state of the auxiliary quantum bit and the first quantum bit is set to |0> state, and the initial state of the second quantum bit is set to For example, for a 4-dimensional second vector b′=[b′1, b′2, b′3, b′4], that is, N=4, the data of the second vector b′ is encoded into the quantum state amplitude, and the result is:

[0103]

[0104] The data of the second vector b′ is loaded onto the quantum state amplitude of two second quantum bits in the quantum circuit.

[0105] Step 2: Determine the unitary matrix corresponding to the second matrix A′

[0106] Specifically, since the second matrix A′ is a Hermitian matrix, the unitary matrix corresponding to the second matrix A′ can be directly determined The conversion from the Hermitian matrix to the unitary matrix can also be realized through Hamiltonian simulation to obtain the corresponding unitary matrix Where t is a constant, usually 2π. First, replace the operator According to the Jacobi-Anger expansion, the operator is determined The exponential expansion of The exponential expansion of is:

[0107]

[0108] Among them, J k is the kth-order Bessel function of the first kind, T k is a Chebyshev polynomial of the first kind of order k.

[0109] Operators that preserve order q precision The exponential expansion of , the operator with order q precision The exponential expansion of is:

[0110] e iγt =f1(β, t)+f2(β, t),

[0111] in,

[0112]

[0113]

[0114] β = P(A)A;

[0115] Secondly, the operator corresponding to f1(β, t) is prepared by quantum signal processing QSP The operator corresponding to f2(β, t) Finally, the LCU and the operator are linearly merged through the unitary operation Sum Operator Constructing Operators

[0116] Among them, when k is a positive integer, J k (-x) = (-1) k J k (x), T k (-x) = (-1) k T k (x), so f1(β, t) is an even function, and f2(β, t) is an odd function. Therefore, the operator It can also be prepared by QSP. When the operator After that, operators can be constructed by linear combination of unitary operators (LCU) like Figure 3 As shown, Figure 3 A construction operator provided in an embodiment of the present invention Quantum circuit diagram.

[0117] Build in this way The success rate is η. After amplitude estimation and amplitude amplification, the success rate can be amplified to η≈1. In the subsequent QPE process, O(-log∈) controlled The success rate of the whole process is Where δ = logη. Therefore, when η approaches 1, the entire algorithm process approaches 1.

[0118] Step 3: Construct a first sub-quantum circuit module for phase estimation, which is used to decompose the |b′> in the eigenspace of the second matrix A′ into And transform the initial states of the first quantum bit and the second quantum bit |0>|b′> into Among them, the |μ j > is the eigenvector of the second matrix A′, the λ j is the eigenvalue of the second matrix A′, the β j is the amplitude of the eigenvector of the second matrix A′.

[0119] Specifically, a first sub-quantum circuit module for phase estimation is constructed to decompose the |b′> in the eigenspace of the second matrix A′ into And transform the initial states of the first and second qubits |0>|b′> into Those skilled in the art can understand that phase estimation (Phaseestimation) is an important application of quantum Fourier transform QFT, which is reflected in that it is the basis of many quantum algorithms.

[0120] like Figure 4 The first sub-quantum circuit module schematic diagram corresponding to the phase estimation provided in this embodiment is shown, including: an H gate operation module, a controlled Operator operation module and quantum Fourier inverse transform module (Figure module), where The operator is the unitary matrix corresponding to the second matrix A′ After the first sub-quantum circuit module, the auxiliary quantum bit (corresponding to Figure 4 The quantum state of the top timeline) remains unchanged, and the first qubit (corresponding to Figure 4 The initial state of the timeline in the middle |0> is transformed into |λj >, the second quantum bit (corresponding to Figure 4 The initial state of the bottom timeline) |b′> decomposition

[0121] In fact, the output λ j is an estimated value, and the output accuracy of the phase estimation can be improved by increasing the number of the first quantum bits. In addition, in practical applications, an auxiliary quantum register, a first quantum register, and a second quantum register can be set to store the quantum states of the auxiliary quantum bit, the first quantum bit, and the second quantum bit, respectively.

[0122] Step 4: Construct a second sub-quantum circuit module for performing a controlled rotation operation to convert |λ j >Using the auxiliary qubit as a control bit to rotate, we get Wherein, C is a normalization constant.

[0123] Specifically, a second sub-quantum circuit module for performing a controlled rotation operation is constructed to convert |λ j >Using the auxiliary qubit as a control bit to rotate, we get The controlled rotation can also be called "extraction ratio" because after the phase estimation operation, the first quantum register will store a series of eigenvalues ​​λ j (Specifically stored in the base state |λ j >), and the input state stored in the second quantum register, namely the initial state |b′>, is decomposed in the eigenspace of the second matrix A′, and then the λ in the ground state is transformed by a controlled rotation operation. j The value is extracted from the amplitude, and the quantum state of the auxiliary quantum bit |0> is converted to obtain The quantum state of each quantum bit passes through the second quantum circuit route Convert to In order to reduce resource usage, the auxiliary quantum bit can be set to 1 bit, and C is a constant, generally 1.

[0124] It should be noted that the quantum state |x>=A′ can be obtained in the second quantum register -1 |b′> (more precisely, the close |x> quantum state), accompanied by a constant factor C. Later, amplitude amplification can be used to increase the probability of success, replacing the process of simply measuring and screening the results.

[0125] Step 5: Construct a third sub-quantum circuit module for performing inverse phase estimation to convert |λ j >Reset to |0>.

[0126] Specifically, those skilled in the art can understand that the inverse operation of phase estimation is the restoration process of the above phase estimation, or the transposed conjugate operation of phase estimation, the purpose of which is to convert |λ j >reset to |0>, specifically by resetting the quantum state Convert to the following:

[0127]

[0128] Step 6: Construct a measurement operation module for the auxiliary quantum bit so that when the quantum state of the auxiliary quantum bit is measured to be |1>, the following is obtained: The |x′>∝|x>.

[0129] Specifically, the quantum measurement operation is applied to the auxiliary qubit to measure the auxiliary qubit after the inverse operation of the phase estimation. After the measurement, the state of the auxiliary qubit collapses to a certain state, where the probability of collapsing to |0> is The probability of collapsing to |1> is When the quantum state of the auxiliary quantum bit is measured to be |1> and C=1, a definite quantum state can be obtained: It can be seen that The corresponding result of amplitude normalization is obtained. In practical applications, |x> can be obtained according to the application scenario required by the user, or |x′> can be directly used for subsequent scenario calculations.

[0130] Step 7: The first sub-quantum circuit module, the second sub-quantum circuit module, the third sub-quantum circuit module and the quantum measurement operation module are sequentially combined into a quantum circuit corresponding to the HHL algorithm.

[0131] For details, see Figure 5 , Figure 5 According to the execution sequence of the first sub-quantum circuit module, the second sub-quantum circuit module, the third sub-quantum circuit module and the quantum measurement operation module, a complete quantum circuit is sequentially formed, which is the total quantum circuit corresponding to the HHL algorithm.

[0132] In an optional embodiment, the quantum state |b> of the first vector b can be prepared, and the operator can be constructed based on the polynomial p(A) Then the operator Acting on the quantum state |b>, we get the quantum state |b′>, which is the quantum state corresponding to the second vector b′. Operators Acting on the quantum state |b>, we get the quantum state |b′>, the quantum state

[0133] Specifically, the operator corresponding to the polynomial p(A) is prepared by quantum signal processing QSP Among them, Quantum Signal Processing (QSP) ensures that quantum operators defined on polynomial mappings over the real number field can be prepared if and only if the polynomial is an odd function or an even function (i.e., the polynomial is of all odd order or all even order).

[0134] It can be seen that the HHL algorithm with polynomial preprocessing is constructed by combining the polynomial preprocessor with the quantum linear solver, which optimizes the complexity of the HHL algorithm. The optimization multiple increases linearly with the order of the polynomial preprocessing function, thereby improving the ability of the HHL algorithm to solve ill-conditioned linear systems and expanding the application range of the HHL algorithm. As the order m increases, the HHL algorithm transitions to the quantum linear solver, and the algorithm complexity increases from κ 2 Slowly becomes κ.

[0135] Compared with the prior art, the present invention first obtains information of the first matrix A and the first vector b in the linear system to be processed, calculates the polynomial p(A) of the first matrix A used for preprocessing the linear system, preprocesses the linear system according to the polynomial p(A), obtains the second matrix A′ and the second vector b′, constructs a quantum circuit corresponding to the HHL algorithm, and executes quantum state evolution and measurement operations according to the second matrix A′ and the second vector b′ to obtain the final quantum state of the evolved quantum circuit. It can reduce the time complexity and calculation amount of solving linear problems, speed up the solution speed of the quantum linear algorithm, and reduce the occupation of hardware resources.

[0136] See also Figure 6 , Figure 6 A schematic diagram of a quantum linear solution device based on a polynomial preprocessor provided by an embodiment of the present invention, and Figure 2 The process shown here corresponds to the following:

[0137] An acquisition module 601 is used to acquire information of a first matrix A and a first vector b in a linear system to be processed;

[0138] A calculation module 602, used for calculating a polynomial p(A) of a first matrix A for preprocessing of the linear system;

[0139] An acquisition module 603 is used to preprocess the linear system according to the polynomial p(A) to obtain a second matrix A′ and a second vector b′;

[0140] The construction module 604 is used to construct a quantum circuit corresponding to the HHL algorithm, perform quantum state evolution and measurement operations according to the second matrix A′ and the second vector b′, and obtain the final quantum state of the evolved quantum circuit.

[0141] Specifically, the acquisition module includes:

[0142] The first obtaining unit is used to obtain a second matrix A′=p(A)A and a second vector b′=p(A)b.

[0143] Specifically, the building blocks include:

[0144] An input unit, used to input an initial quantum state |b′> and an operator for the quantum circuit Performing quantum state evolution and measurement operations to obtain the final quantum state of the quantum circuit after evolution, wherein the operator The t is a constant.

[0145] Specifically, the calculation module includes:

[0146] The first construction unit is used to construct a function f for solving the polynomial m (x) and a preset interval S, wherein the function f m (x) satisfies f m (x) = p m (x)x, and f m (x i )+(-1) i E=1, m is the polynomial order, i=0, 1, 2, ..., m+1, S∈[1 / κ A , 1], E is the deviation amplitude, κ A is the condition number;

[0147] A selection unit is used to select m+1 points in the preset interval and obtain the linear equation system f m (x i )+(-1) i The solution of E=1, wherein the m+1 points are x1, x2, ..., x m+1 And satisfy x1=1 / κ, x m+1 =1;

[0148] A substitution unit is used to substitute the solution of the linear equation system into f m (x), we get |1-f m (x)|The set of points N′ of local maximum values;

[0149] The judgment unit is used to judge whether f m(x)-1 absolute value is the same and increases with x, f m (x)-1 Whether the positive and negative signs change alternately, if so, then the current f m (x) is the optimal polynomial.

[0150] Specifically, the calculation module also includes:

[0151] Returns the unit for converting x1, x2, ..., x m+1 The elements in are replaced with elements in the set N′, and the replacement rule is: x1, x m+1 Keep unchanged and change x2,...,x m Replace with the elements in the set N', and then return to execute the method to obtain the linear equation system f m (x i )+(-1) i The steps of solving E=1 until the optimal f that meets the judgment conditions is obtained m (x).

[0152] Specifically, the building blocks include:

[0153] The second obtaining unit is used to obtain a plurality of quantum bits including an auxiliary quantum bit, a first quantum bit, and a second quantum bit, wherein the initial states of the auxiliary quantum bit and the first quantum bit are set to |0>, and the initial state of the second quantum bit is set to The b′ i is the i-th element of the second vector b′, and N is the dimension of the second vector;

[0154] A determination unit, used to determine the unitary matrix corresponding to the second matrix A′

[0155] The second construction unit is used to construct a first sub-quantum circuit module for phase estimation, and is used to decompose the |b′> in the eigenspace of the second matrix A′ into And transform the initial states of the first quantum bit and the second quantum bit |0>|b′> into Among them, the |μ j > is the eigenvector of the second matrix A′, the λ j is the eigenvalue of the second matrix A′, and βj is the amplitude of the eigenvector of the second matrix A′;

[0156] The third construction unit is used to construct a second sub-quantum circuit module for performing a controlled rotation operation, and is used to convert |λ j >Using the auxiliary qubit as a control bit to rotate, we get Wherein, C is a normalization constant;

[0157] The fourth construction unit is used to construct a third sub-quantum circuit module for performing inverse phase estimation, and is used to convert |λ j >reset to |0>;

[0158] A fifth construction unit is used to construct a measurement operation module for the auxiliary quantum bit, so that when the quantum state of the auxiliary quantum bit is measured to be |1>, the following is obtained: The |x′〉∝|x〉;

[0159] The combining unit is used to sequentially combine the first sub-quantum circuit module, the second sub-quantum circuit module, the third sub-quantum circuit module and the quantum measurement operation module to form a quantum circuit corresponding to the HHL algorithm.

[0160] Compared with the prior art, the present invention first obtains information of the first matrix A and the first vector b in the linear system to be processed, calculates the polynomial p(A) of the first matrix A used for preprocessing the linear system, preprocesses the linear system according to the polynomial p(A), obtains the second matrix A′ and the second vector b′, constructs a quantum circuit corresponding to the HHL algorithm, and executes quantum state evolution and measurement operations according to the second matrix A′ and the second vector b′ to obtain the final quantum state of the evolved quantum circuit. It can reduce the time complexity and calculation amount of solving linear problems, speed up the solution speed of the quantum linear algorithm, and reduce the occupation of hardware resources.

[0161] An embodiment of the present invention further provides a storage medium, in which a computer program is stored, wherein the computer program is configured to execute the steps in any of the above method embodiments when running.

[0162] Specifically, in this embodiment, the above storage medium may be configured to store a computer program for performing the following steps:

[0163] S201: Obtain information of a first matrix A and a first vector b in a linear system to be processed;

[0164] S202: Calculate the polynomial p(A) of the first matrix A for preprocessing the linear system;

[0165] S203: Preprocessing the linear system according to the polynomial p(A) to obtain a second matrix A′ and a second vector b′;

[0166] S204: construct a quantum circuit corresponding to the HHL algorithm, perform quantum state evolution and measurement operations according to the second matrix A′ and the second vector b′, and obtain the final quantum state of the evolved quantum circuit.

[0167] Specifically, in this embodiment, the above-mentioned storage medium may include but is not limited to: a USB flash drive, a read-only memory (ROM), a random access memory (RAM), a mobile hard disk, a magnetic disk or an optical disk, and other media that can store computer programs.

[0168] An embodiment of the present invention further provides an electronic device, comprising a memory and a processor, wherein the memory stores a computer program, and the processor is configured to run the computer program to execute the steps in any of the above method embodiments.

[0169] Specifically, the electronic device may further include a transmission device and an input / output device, wherein the transmission device is connected to the processor, and the input / output device is connected to the processor.

[0170] Specifically, in this embodiment, the processor may be configured to perform the following steps through a computer program:

[0171] S201: Obtain information of a first matrix A and a first vector b in a linear system to be processed;

[0172] S202: Calculate the polynomial p(A) of the first matrix A for preprocessing the linear system;

[0173] S203: Preprocessing the linear system according to the polynomial p(A) to obtain a second matrix A′ and a second vector b′;

[0174] S204: construct a quantum circuit corresponding to the HHL algorithm, perform quantum state evolution and measurement operations according to the second matrix A′ and the second vector b′, and obtain the final quantum state of the evolved quantum circuit.

[0175] The above describes in detail the structure, features and effects of the present invention based on the embodiments shown in the drawings. The above is only a preferred embodiment of the present invention, but the present invention is not limited to the scope of implementation shown in the drawings. Any changes made according to the concept of the present invention, or modifications to equivalent embodiments with equivalent changes, which still do not exceed the spirit covered by the description and drawings, should be within the protection scope of the present invention.

Claims

1. A quantum linear solution method based on polynomial preprocessor, characterized in that: The method comprises: Get the first matrix of the linear system to be processed , the first vector information; Calculate the first matrix for preconditioning the linear system The polynomial ; According to the polynomial , preprocess the linear system to obtain the second matrix and the second vector ; A plurality of quantum bits including an auxiliary quantum bit, a first quantum bit, and a second quantum bit are obtained, wherein the initial states of the auxiliary quantum bit and the first quantum bit are set to , the initial state of the second quantum bit is set to , The second vector No. elements, the is the dimension of the second vector; Determine the second matrix The corresponding unitary matrix ; Constructing a first sub-quantum circuit module for phase estimation, In the second matrix The feature space is decomposed into , and the initial states of the first quantum bit and the second quantum bit Convert to , wherein the The second matrix The feature vector of The second matrix The characteristic value of The second matrix the amplitude of the eigenvector; A second sub-quantum circuit module is constructed to perform a controlled rotation operation. As the control bit, the auxiliary quantum bit is rotated to obtain , wherein the is the normalization constant; A third quantum circuit module is constructed to perform inverse phase estimation. Reset to ; Construct a measurement operation module for the auxiliary quantum bit so that the quantum state of the auxiliary quantum bit measured is When , we get: , ; The first sub-quantum circuit module, the second sub-quantum circuit module, the third sub-quantum circuit module and the measurement operation module are sequentially formed into a quantum circuit corresponding to the HHL algorithm; according to the second matrix and the second vector , performing quantum state evolution and measurement operations to obtain the final quantum state of the quantum circuit after evolution; The calculation is used for the first matrix of the linear system preprocessing. The polynomial , including: constructing a function to solve the polynomial and preset intervals , where the function satisfy ,and , is the polynomial order, , , is the deviation amplitude, is the condition number; Select in the preset interval points and obtain the linear equation system The solution of The points are And meet , ; Substitute the obtained solution of the linear equations into ,get The set of points with local maxima ; Judging any , Are the absolute values ​​the same and as Incremental, Whether the positive and negative signs change alternately, if so, the current is the polynomial p(A).

2. The method according to claim 1, characterized in that The second matrix is ​​obtained and the second vector ,include: Get the second matrix , and the second vector .

3. The method according to claim 2, characterized in that The second matrix and the second vector , performing quantum state evolution and measurement operations to obtain the final quantum state of the quantum circuit after evolution, including: For the quantum circuit, input the initial quantum state Sum Operator , perform quantum state evolution and measurement operations to obtain the final quantum state of the quantum circuit after evolution, wherein the operator , Is a constant.

4. The method according to claim 1, characterized in that: The method further comprises: If not, then The elements in the set are The elements in are replaced, and the replacement rules are: remain unchanged, Replace with a collection The elements in , and then return to execute the linear equation system The solution steps are repeated until the optimal solution that meets the judgment conditions is obtained. .

5. A quantum linear solver based on a polynomial preprocessor, characterized in that: The device comprises: The acquisition module is used to obtain the first matrix of the linear system to be processed , the first vector information; A calculation module, used to calculate a first matrix for preprocessing of the linear system The polynomial ; Obtaining module, for obtaining a polynomial according to the polynomial , preprocess the linear system to obtain the second matrix and the second vector ; Specifically including: constructing a function to solve the polynomial and preset intervals , where the function satisfy ,and , is the polynomial order, , , is the deviation amplitude, is the condition number; select points and obtain the linear equation system The solution of The points are And meet , ; Substitute the obtained solution of the linear equations into ,get The set of points with local maxima ; judge for any , Are the absolute values ​​the same and as Incremental, Whether the positive and negative signs change alternately, if so, the current is the polynomial p(A); A construction module is used to obtain a plurality of quantum bits including an auxiliary quantum bit, a first quantum bit, and a second quantum bit, wherein the initial states of the auxiliary quantum bit and the first quantum bit are set to , the initial state of the second quantum bit is set to , The second vector No. elements, the is the dimension of the second vector; Determine the second matrix The corresponding unitary matrix ; Constructing a first sub-quantum circuit module for phase estimation, for converting the In the second matrix The feature space is decomposed into , and the initial states of the first quantum bit and the second quantum bit Convert to , wherein the The second matrix The characteristic vector of The second matrix The characteristic value of The second matrix The amplitude of the eigenvector; constructing a second sub-quantum circuit module for performing a controlled rotation operation, As the control bit, the auxiliary quantum bit is rotated to obtain , wherein the is a normalization constant; a third sub-quantum circuit module for performing inverse phase estimation is constructed to convert Reset to ; construct a measurement operation module for the auxiliary quantum bit so that the quantum state of the auxiliary quantum bit measured is When , we get: , ; The first sub-quantum circuit module, the second sub-quantum circuit module, the third sub-quantum circuit module and the measurement operation module are sequentially formed into a quantum circuit corresponding to the HHL algorithm; according to the second matrix and the second vector , perform quantum state evolution and measurement operations to obtain the final quantum state of the quantum circuit after evolution.

6. The device according to claim 5, characterized in that The obtaining module comprises: Get unit, used to get the second matrix , and the second vector .

7. A storage medium, characterized in that: The storage medium stores a computer program, wherein the computer program is configured to execute the method according to any one of claims 1 to 4 when executed.

8. An electronic device, comprising a memory and a processor, characterized in that: A computer program is stored in the memory, and the processor is configured to run the computer program to perform the method according to any one of claims 1 to 4.

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