CN114819168A - Quantum comparison method and device for matrix eigenvalues - Google Patents
Quantum comparison method and device for matrix eigenvalues Download PDFInfo
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Abstract
The invention discloses a quantum comparison method and a device of matrix eigenvalues, wherein the method comprises the following steps: acquiring an eigenvalue quantum state carrying eigenvalue information of a target data matrix and a preset value of a specific parameter, wherein the eigenvalue quantum state corresponds to n comparison quantum bits; acquiring at least n auxiliary qubits and 1 qubit for outputting the result of the comparison; and determining a quantum logic gate of the quantum comparison circuit to be constructed according to a preset value, constructing and operating the quantum comparison circuit for comparing the characteristic value quantum state with the preset value by combining the n-bit comparison quantum bit, the n-bit auxiliary quantum bit and the 1-bit result quantum bit, and outputting a result quantum state representing the comparison result. By utilizing the embodiment of the invention, the application of the quantum algorithm to the comparative estimation of the matrix eigenvalue can be realized, the advantages of quantum computation are fully exerted, the important blank of the quantum computation direction of a linear system is filled, and the invention has important pioneering significance and practical application value.
Description
Technical Field
The invention belongs to the technical field of quantum computation, and particularly relates to a quantum comparison method and device for matrix eigenvalues.
Background
Quantum computers are physical devices that perform high-speed mathematical and logical operations, store and process quantum information in compliance with the laws of quantum mechanics. When a device processes and calculates quantum information and runs quantum algorithms, the device is a quantum computer. Quantum computers are a key technology under study because they have the ability to handle mathematical problems more efficiently than ordinary computers, for example, they can speed up the time to break RSA keys from hundreds of years to hours.
The characteristic value of the matrix deeply reveals the inherent property of the matrix, has an important function on solving the problem of a linear system related to the matrix, and is widely applied to the field of numerical solution of differential equations and corresponding practical problems. The condition number of the matrix depends on the eigenvalue of the matrix, which is an important derivative of the eigenvalue problem and has important application in the related background. For example, fast solution of the extreme values of some eigenvalues of the matrix can result in condition number estimation of the matrix, thereby giving important reference information to the above problem. Compared with the classical method, the method has the advantages that the characteristic value of the matrix is estimated through numerical solution, the calculation power is extremely consumed when the matrix scale is large, and the corresponding quantum algorithm is lacked at present so as to fully exert the advantages of quantum calculation, which is a problem to be solved urgently.
Disclosure of Invention
The invention aims to provide a matrix eigenvalue quantum comparison method and a matrix eigenvalue quantum comparison device, which are used for solving the defects in the prior art, can realize the application of a quantum algorithm to the comparison and estimation of the matrix eigenvalue, give full play to the advantages of quantum computation, fill up the important blank of the quantum computation direction of a linear system, and have important pioneering significance and practical application value.
One embodiment of the present application provides a method for quantum comparison of matrix eigenvalues, the method comprising:
acquiring an eigenvalue quantum state carrying eigenvalue information of a target data matrix and a preset value of a specific parameter, wherein the eigenvalue quantum state corresponds to n comparison quantum bits used for comparing with the preset value;
acquiring at least n auxiliary qubits and 1 qubit for outputting the result of the comparison;
and determining a quantum logic gate of a quantum comparison line to be constructed according to the preset value, constructing and operating a quantum comparison line for comparing the characteristic value quantum state with the preset value by combining n comparison quantum bits, n auxiliary quantum bits and 1 result quantum bit, and outputting a result quantum state representing a comparison result.
Optionally, the target data matrix is a hermitian matrix, and before the obtaining of the eigenvalue quantum state carrying the eigenvalue information of the target data matrix, the method further includes:
and obtaining quantum states containing Hermite matrixes, constructing and operating corresponding quantum phase estimation QPE circuits, and evolving the quantum states of the Hermite matrixes into characteristic value quantum states carrying characteristic value information of the Hermite matrixes.
Optionally, the hermitian matrix is a hermitian matrix corresponding to the transaction data matrix, and the obtaining a quantum state including the hermitian matrix includes:
and carrying out normalization processing on the maximum singular value of the transaction data matrix, and preparing a quantum state comprising a hermite matrix corresponding to the transaction data matrix after the normalization processing, wherein the maximum eigenvalue of the hermite matrix is equal to the maximum singular value 1 of the transaction data matrix after the normalization processing.
Optionally, the specific parameter is a reference condition number k i The preset value is
The initial value of i is 0, the method further comprising:
if the result quantum state indicates that the characteristic value exists in the characteristic value quantum state and is smaller than the preset value, adding 1 to the i, and returning to the step of executing the quantum logic gate for determining the quantum comparison circuit to be constructed according to the preset value until the output result quantum state indicates that the characteristic value does not exist in the characteristic value quantum state and is smaller than the preset value;
and determining approximate estimation of the minimum eigenvalue in the eigenvalue quantum state according to the current preset value.
Optionally, the method further includes:
calculating the condition number of the Hermite matrix according to the approximate estimation of the maximum eigenvalue and the minimum eigenvalue of the Hermite matrix;
and searching for a coordinated pair in the transaction data matrix according to the size of the condition number.
Optionally, the determining, according to the preset value, a quantum logic gate of a quantum comparison line to be constructed includes:
and calculating a binary complement corresponding to the preset value, and determining a quantum logic gate of a corresponding bit according to each bit of the binary complement, wherein when the bit is 0, the corresponding quantum logic gate is a Toffoli gate, and when the bit is 1, the corresponding quantum logic gate is a logic OR gate.
Optionally, the constructing and operating a quantum comparison circuit for comparing the characteristic value quantum state with the preset value, and outputting a result quantum state representing a comparison result includes:
acquiring a preset quantum logic gate, and adding the quantum logic gate and the preset quantum logic gate which are correspondingly determined according to the preset value to corresponding quantum bits to obtain a quantum comparison circuit for comparing the characteristic value quantum state with the preset value;
and operating the quantum comparison circuit and outputting a result quantum state representing a comparison result.
Yet another embodiment of the present application provides a quantum comparison apparatus for matrix eigenvalues, the apparatus comprising:
the data acquisition module is used for acquiring a characteristic value quantum state carrying characteristic value information of a target data matrix and a preset value of a specific parameter, wherein the characteristic value quantum state corresponds to n comparison quantum bits used for comparing with the preset value;
a bit obtaining module for obtaining at least n auxiliary qubits and 1 result qubit for outputting a comparison result;
and the construction output module is used for determining a quantum logic gate of the quantum comparison circuit to be constructed according to the preset value, constructing and operating the quantum comparison circuit for comparing the characteristic value quantum state with the preset value by combining the n comparison quantum bits, the n auxiliary quantum bits and the 1 result quantum bit, and outputting a result quantum state representing the comparison result.
A further embodiment of the application provides a storage medium having a computer program stored thereon, wherein the computer program is arranged to perform any of the methods described above when executed.
Yet another embodiment of the present application provides an electronic device comprising a memory having a computer program stored therein and a processor configured to execute the computer program to perform the method of any one of the above.
Compared with the prior art, the quantum comparison method of the matrix eigenvalue, provided by the invention, comprises the steps of firstly obtaining an eigenvalue quantum state carrying eigenvalue information of a target data matrix and a preset value of a specific parameter, wherein the eigenvalue quantum state corresponds to n comparison quantum bits used for comparing with the preset value; acquiring at least n auxiliary qubits and 1 qubit for outputting the result of the comparison; and determining a quantum logic gate of the quantum comparison circuit to be constructed according to a preset value, constructing and operating the quantum comparison circuit for comparing the characteristic value quantum state with the preset value by combining the n-bit comparison quantum bit, the n-bit auxiliary quantum bit and the 1-bit result quantum bit, and outputting a result quantum state representing the comparison result. By comparing the characteristic value quantum state with the preset value, the application of the quantum algorithm to the matrix characteristic value estimation is further realized, so that the advantages of quantum calculation are fully exerted, the important blank of the linear system quantum calculation direction is filled, and the method has important pioneering significance and practical application value.
Drawings
Fig. 1 is a block diagram of a hardware structure of a computer terminal of a quantum comparison method for matrix eigenvalues according to an embodiment of the present invention;
fig. 2 is a schematic flowchart of a method for quantum comparison of matrix eigenvalues according to an embodiment of the present invention;
fig. 3 is a schematic structural diagram of a quantum comparison circuit according to an embodiment of the present invention;
FIG. 4 is a schematic diagram of a logic OR gate according to an embodiment of the present invention;
fig. 5 is a schematic structural diagram of a quantum comparison device for matrix eigenvalue according to an embodiment of the present invention.
Detailed Description
The embodiments described below with reference to the drawings are illustrative only and should not be construed as limiting the invention.
The embodiment of the invention firstly provides a quantum comparison method of matrix eigenvalues, which can be applied to electronic equipment, such as computer terminals, specifically common computers, quantum computers and the like.
This will be described in detail below by way of example as it would run on a computer terminal. Fig. 1 is a block diagram of a hardware structure of a computer terminal of a quantum comparison method for matrix eigenvalues according to an embodiment of the present invention. As shown in fig. 1, the computer terminal may include one or more (only one shown in fig. 1) processors 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, and optionally, a transmission device 106 for communication functions and an input-output device 108. It will be understood by those skilled in the art that the structure shown in fig. 1 is only an illustration and is not intended to limit the structure of the computer terminal. For example, the computer terminal may also include more or fewer components than shown in FIG. 1, or have a different configuration than shown in FIG. 1.
The memory 104 may be used to store software programs and modules of application software, such as program instructions/modules corresponding to the quantum comparison method for matrix eigenvalues in the embodiment of the present application, and the processor 102 executes various functional applications and data processing by running the software programs and modules stored in the memory 104, so as to implement the above-mentioned method. The memory 104 may include high speed random access memory, and may also include non-volatile memory, such as one or more magnetic storage devices, flash memory, or other non-volatile solid-state memory. In some examples, the memory 104 may further include memory located remotely from the processor 102, which may be connected to a computer terminal over a network. Examples of such networks include, but are not limited to, the internet, intranets, local area networks, mobile communication networks, and combinations thereof.
The transmission device 106 is used for receiving or transmitting data via a network. Specific examples of the network described above may include a wireless network provided by a communication provider of the computer terminal. In one example, the transmission device 106 includes a Network adapter (NIC) that can be connected to other Network devices through a base station 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 in a wireless manner.
It should be noted that a true quantum computer is a hybrid structure, which includes two major components: one part is a classic computer which is responsible for executing classic calculation and control; the other part is quantum equipment which is responsible for running a quantum program to further realize quantum computation. The quantum program is a string of instruction sequences which can run on a quantum computer and are written by a quantum language such as a Qrun language, so that the support of the operation of the quantum logic gate is realized, and the quantum computation is finally realized. In particular, a quantum program is a sequence of instructions that operate quantum logic gates in a time sequence.
In practical applications, due to the development of hardware limited to quantum devices, quantum computation simulation is usually required to verify quantum algorithms, quantum applications, and the like. The quantum computing simulation is a process of realizing the simulation operation of a quantum program corresponding to a specific problem by means of a virtual architecture (namely a quantum virtual machine) built by resources of a common computer. In general, it is necessary to build quantum programs for a particular problem. The quantum program referred in the embodiment of the invention is a program written in a classical language for representing quantum bits and evolution thereof, wherein the quantum bits, quantum logic gates and the like related to quantum computation are all represented by corresponding classical codes.
A quantum circuit, which is an embodiment of a quantum program and also a weighing sub-logic circuit, is the most common general quantum computation model, and represents a circuit that operates on a quantum bit under an abstract concept, and the circuit includes the quantum bit, a circuit (timeline), and various quantum logic gates, and finally, a result is often read through a quantum measurement operation.
Unlike conventional circuits that are connected by metal lines to pass either voltage or current signals, in quantum circuits, the lines can be viewed as being connected by time, i.e., the state of a qubit evolves naturally over time, in the process being operated on as indicated by the hamiltonian until a logic gate is encountered.
The quantum program refers to the total quantum circuit, wherein the total number of the quantum bits in the total quantum circuit is the same as the total number of the quantum bits of the quantum program. It can be understood that: a quantum program may consist of quantum wires, measurement operations for quantum bits in the quantum wires, registers to hold measurement results, and control flow nodes (jump instructions), and a quantum wire may contain tens to hundreds or even thousands of quantum logic gate operations. The execution process of the quantum program is a process executed for all the quantum logic gates according to a certain time sequence. It should be noted that timing is the time sequence in which the single quantum logic gate is executed.
It should be noted that in the classical calculation, the most basic unit is a bit, and the most basic control mode is a logic gate, and the purpose of the control circuit can be achieved through the combination of the logic gates. Similarly, the way qubits are handled is quantum logic gates. The quantum state can be evolved by using quantum logic gates, which are the basis for forming quantum circuits, including 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, and the like; two-bit or multi-bit quantum logic gates such as CNOT gates, CR gates, CZ gates, iSWAP gates, Toffoli gates, and the like. Quantum logic gates are typically represented using unitary matrices, which are not only matrix-form but also an operation and transformation. The function of a general quantum logic gate on a quantum state is calculated by multiplying a unitary matrix by a matrix corresponding to a quantum state right vector.
Referring to fig. 2, fig. 2 is a schematic flowchart of a quantum comparison method for matrix eigenvalues according to an embodiment of the present invention, where the method may include the following steps:
s201, obtaining an eigenvalue quantum state carrying eigenvalue information of a target data matrix and a preset value of a specific parameter, wherein the eigenvalue quantum state corresponds to n comparison quantum bits used for comparing with the preset value;
the specific parameter is a preset parameter used for comparing with the characteristic value quantum state, and the specific preset value of the parameter can be determined according to practical application and requirements. The n comparison qubits corresponding to the eigenvalue quantum states and used for comparing with the preset value mean that the eigenvalue quantum states are prepared on the n quantum bits, namely the quantum states of the n quantum bits, and the n quantum bits can be called comparison qubits for the convenience of distinguishing. It should be noted that not only the characteristic value quantum state but also the quantum state carrying the value information of other data may be compared with the preset value.
For example, in one financial transaction application scenario, the target data matrix may be a hermite matrix, such as a hermite matrix corresponding to the transaction data matrix, and the transaction data matrix may contain a plurality of financial transaction data.
The financial transaction data may be transaction data of High Frequency Transaction (HFT), for example: discrete time point price vectors for multiple stocks, etc. There may be a co-integration between partial price vectors whose linear combination has some properties that do not change over time. For example, the linear combined total price of some stock price vectors may be fixed to be a constant (generally subject to some distribution).
In practical application, for example, before obtaining the eigenvalue quantum state carrying the eigenvalue information of the hermitian matrix, the quantum state containing the hermitian matrix may also be obtained, a corresponding quantum phase estimation QPE line is constructed and operated, and the quantum state of the hermitian matrix is evolved into the eigenvalue quantum state carrying the eigenvalue information of the hermitian matrix.
Specifically, the transaction data matrix may be subjected to normalization processing of the maximum singular value, and the quantum state including the hermitian matrix corresponding to the transaction data matrix after the normalization processing is prepared, so that the following is achieved: quantum states comprising hermitian matrices are obtained and the maximum eigenvalue of the hermitian matrix is made equal to the maximum singular value of the normalized transaction data matrix of 1. The maximum eigenvalue referred to in this application is the maximum of the absolute values of the eigenvalues and the minimum eigenvalue is the minimum of the absolute values of the eigenvalues.
For the transaction data matrix X, and also for facilitating subsequent practical applications such as condition number calculation, the transaction data matrix X may be normalized based on the maximum singular value as:
where max λ (X) is the largest singular value of the transaction data matrix X. The matrix can then be constructed by means of existing quantum wires
Corresponding Hermitian matrix (Hermitian) matrix
I.e. to prepare the quantum state comprising the hermite a. At this time, the maximum eigenvalue of the Hermitian matrix A is equal to the matrix
Maximum singular value of
In practical applications, max λ (X) can be estimated based on the following conclusions of matrix two-norm and F (Frobenius) norm, specifically: the Frobenius norm of the transaction data matrix X is denoted as | X | F 2 norm is | X | 2 P, the dimension is as follows:
thus, the Frobenius norm | X | can be utilized F Estimating by dividing the size of the channel F As the maximum singular value upper bound max λ (X) of available X, the 2-norm solution may be difficult to solve without consideration.
Because the absolute value of the eigenvalue of the Hermite matrix A and the matrix after normalization processing
The singular values are the same and can reflect the properties of transaction data and condition numbers, and the matrix can be determined by one-to-one mapping of the coordinated pair of the Hermite matrix A
To determine a coordinated pair of the transaction data matrix X. And, in order to adapt to the requirements of the quantum transformation on the matrix form (requiring the form of the hermitian matrix), the subsequent processing of the transaction data matrix X may be replaced with hermitian matrix a.
Specifically, the preparation of the eigenvalue quantum states carrying eigenvalue information of the hermitian matrix can be realized by constructing and operating a corresponding quantum phase estimation QPE line.
Among them, QPE (Quantum Phase Estimation) is an important application of Quantum fourier transform QFT, and its importance is that it is the basis of many Quantum algorithms, such as HHL algorithm and so on. QPE quantum wires mainly include: h door operation module, C-U j The operation (controlled U operator operation) module and the quantum Fourier inverse transformation module solve the essential problem of matrix eigenvalue estimation, namely solving the eigenvalue of the given matrix. For the quantum state of the hermitian matrix a, it can be transformed into an eigenvalue quantum state containing each eigenvalue of the hermitian matrix a through QPE quantum line (in the quantum domain, the quantum state is a superposition state, and thus can carry all eigenvalue information). In practical application, the transformation of the characteristic value quantum state is reasonably feasible by constructing other existing or improved quantum linesPlease do not limit this.
S202, acquiring at least n auxiliary quantum bits and 1 result quantum bit for outputting a comparison result;
specifically, since the characteristic value quantum state corresponds to n comparison qubits, n auxiliary qubits input by the user can be obtained, and 1 result qubit used for outputting the comparison result.
The n auxiliary qubits can be used as carry qubits. The size comparison is carried out by adopting a binary subtraction idea similar to the classic, and the carry qubits are used for storing the carry of each bit corresponding to the characteristic value quantum state and each bit of the complement of the preset value. And finally, outputting a comparison result through the result quantum bit.
S203, determining a quantum logic gate of a quantum comparison line to be constructed according to the preset value, constructing and operating a quantum comparison line for comparing the characteristic value quantum state with the preset value by combining n comparison quantum bits, n auxiliary quantum bits and 1 result quantum bit, and outputting a result quantum state representing a comparison result.
Specifically, a binary complement corresponding to the preset value may be calculated, and a quantum logic gate corresponding to a bit is determined according to each bit of the binary complement, where when the bit is 0, the corresponding quantum logic gate is a toffee gate, and when the bit is 1, the corresponding quantum logic gate is a logic or gate.
It should be noted that, only when the first bit of the two's complement is 0, there is no corresponding quantum logic gate operation; when the first bit of the two's complement is 1, the corresponding CNOT gate operates. In a specific implementation, the OR gate (OR gate) can be constructed by toffsol gate and X gate, and other quantum logic gates equivalent to toffsol gate and logic OR gate are reasonably feasible.
Specifically, a preset quantum logic gate may be obtained, and a quantum logic gate and a preset quantum logic gate determined in correspondence to the preset value are added to a corresponding quantum bit, so as to obtain a quantum comparison line (the quantum comparison line is also a quantum line) for comparing a characteristic value quantum state with the preset value; and operating a quantum comparison circuit and outputting a result quantum state representing the comparison result.
For example, fig. 3 is a schematic structural diagram of a quantum comparison circuit. Wherein i 1 、i 2 …i n Representing a comparison qubit, a 1 、a 2 …a n Representing an auxiliary qubit, c representing a result qubit, t [1 ]]、t[2]……t[n]The 1 st bit (least significant bit) and the 2 nd bit … … nth bit (most significant bit) of the two's complement representing the predetermined value. If t [1 ]]1, the quantum logic gate correspondingly determined is represented by i 1 Bit control a 1 Bit CNOT gate if t [1 ]]When the value is equal to 0, correspondingly determining a quantum-free logic gate; if t 2]When the value is 0, the correspondence is determined by i 2 Bit, a 1 Bit control a 2 Bit Toffoli gate, if t 2]Corresponding to 1 is determined by i 2 Bit, a 1 Bit control a 2 Logical OR gate of bits, the rest being the same as t [2 ]]… … up to a time t n]When the value is 0, the correspondence is determined by i n Bit, a n-1 Bits (a in the figure) n-1 Not shown) control a n Bit Toffoli gate, if t [ n ]]Corresponding to 1 is determined by i n Bit, a n-1 Bit control a n A logic OR gate of bit, the last quantum logic gate is a n The bit controls the c-bit CNOT gate (i.e., the predetermined quantum logic gate) to output the result quantum state of the comparison result. The number n of the required quantum bits is related to the number of bits of the characteristic value quantum state and the size of the preset value required to be compared, and can be set according to specific requirements. Alternatively, one configuration of the or logic gate may be as shown in fig. 4, where the left line includes 3X gates, 1 toffil gate, and 2X gates in that order.
It will be appreciated by those skilled in the art that in a classical computer, the basic unit of information is a bit, one bit has two states, 0 and 1, and the most common physical implementation is to represent these two states by the high and low of the levels. In quantum computing, the basic unit of information is a qubit, one qubit also having two states, 0 and 1, denoted as |0>And |1>However, it can be in a superimposed state of two states of 0 and 1, and can be expressed as
Wherein a and b represent |0>State, |1>Complex number of state amplitudes (probability amplitudes), which classical bits do not possess. After measurement, the state of the qubit collapses to a certain state (eigenstate, here | 0)>State, |1>State) in which the probability of collapsing to |0> is | a 2 The probability of collapsing to |1> is | b 2 ，|a| 2 +|b| 2 1, | > is a dirac symbol.
Taking the foregoing example as an example, the output c-bit result quantum state is a superposition state, which can be obtained by measurement, when the |0> state is measured, the default value is larger, and when the |1> state is measured, the opposite is indicated. But the measurement can only measure the |0> state or the |1> state which is collapsed with a certain probability, so that a plurality of times of measurement can be carried out, and the number of times of measurement can be determined according to actual requirements. Theoretically, after multiple measurements, if the |0> state is measured in all the multiple measurements, the preset value is considered to be larger than all value information in the characteristic value quantum state; if the |1> state is measured in multiple measurements, the preset value is considered to be less than or equal to all value information in the characteristic value quantum state; if multiple measurements are made of both the |0> state and the |1> state, the predetermined value is deemed to be greater than some of the eigenvalue quantum states and less than others of the eigenvalue quantum states.
Assuming a characteristic value quantum state of
The default value is 4, the complement of binary is 100, and we can obtain: n is 3, t 1]＝0、t[2]＝0、t[3]1, the corresponding quantum comparison circuit comprises: 3 bit comparison qubits, 3 bit auxiliary qubits, 1 bit result qubit, still include in proper order: a is composed of 2 Bit sum a 1 Bit control a 2 Toffoli gate of bits, one from i 3 Bit sum a 2 Bit control a 3 Logic or gate of bits, one from 3 The bit controls the c-bit CNOT gate. The corresponding quantum comparison circuit is operated to obtain the result quantum state of c bit
Assuming that 100 measurements are performed, the times of measuring |0> and |1> are approximately half each, which indicates that the preset value 4 is both greater than and less than the characteristic quantum state, i.e., the comparison result is in a superposition state of greater than and less than.
Specifically, taking the application scenario as an example, the specific parameter is the reference condition number k i Reciprocal of (1), preset value
i is a non-negative integer. If the characteristic value of the result quantum state representation characteristic value quantum state is smaller than the preset value, adding 1 to the value i, returning to the step of executing the quantum logic gate for determining the quantum comparison circuit to be constructed according to the preset value until the output result quantum state representation characteristic value quantum state does not have the characteristic value smaller than the preset value; and determining approximate estimation of the minimum eigenvalue in the eigenvalue quantum state according to the current preset value.
Exemplary, construct reference condition number k i Inverse of, i.e. preset value
Initializing i to 0; constructing and operating a quantum comparison circuit corresponding to the current preset value, and if the quantum state of the measurement result is |0>States, representing the presence of a certain eigenvalue in an eigenvalue quantum state
Otherwise, ending; (since the maximum eigenvalue of the Hermite matrix is normalized to 1, k is the first comparison 0 The resulting quantum state is affirmatively |0>State)
The resulting quantum state is |0>Adding 1 to i in the state, updating the preset value of the reference condition number, returning to the step of determining the quantum logic gate of the to-be-constructed quantum comparison line according to the preset value until the measured result quantum state is |1>A state, indicating the absence of a characteristic value in the characteristic value quantum state that is less than the reciprocal of the reference condition number. Assume that the reference condition number is increased to 2 at this time m The following can be obtained: 2 -m ≤minλ A <2 -m+1 I.e. Hermite matrixMinimum eigenvalue min λ of A A The estimation interval is [2 ] -m ,2 -m+1 ) Wherein m is a positive integer.
Specifically, in practical application, the condition number of the hermitian can be calculated according to the approximate estimation of the maximum eigenvalue and the minimum eigenvalue of the hermitian; and searching for a coordinated pair in the transaction data matrix according to the size of the condition number.
For example, the quotient of the maximum eigenvalue of the hermi matrix and the approximately estimated minimum eigenvalue may be determined as the condition number of the hermi matrix and used as the condition number of the transaction data matrix.
In statistics, multicollinearity refers to the case where some of the explanatory variables in a multiple regression model have a highly linear relationship. To detect and measure the degree of multiple collinearity, a condition number κ was introduced in the field of numerical analysis. For matrix X:
i.e. the ratio of the maximum singular value to the minimum singular value of X, is in this application the quotient of the maximum eigenvalue absolute value and the minimum eigenvalue absolute value of the corresponding hermitian matrix a. The larger the condition number of the matrix, the more severe the degree of multicollinearity. Multilinear can be detected by searching for a large condition number system, where the system is more likely to have co-aligned pairs. Based on this, the problem of the existence of the collaborative pair corresponding to the statistical arbitrage (pairing transaction) problem can be weakened into the problem of the estimation of the number of conditions.
Taking the above as an example, the maximum eigenvalue of the Hermite matrix A is normalized to 1 and the minimum eigenvalue is estimated to be [2 ] -m ,2 -m+1 ) Obtaining a condition number estimation interval of (2) m-1 ,2 m ]It is simply understood that any condition number within the interval can be used as a specific value of the estimated condition number.
It should be noted that, in practical application, the minimum eigenvalue of the hermitian matrix a may be processed into a preset fixed value, and the maximum eigenvalue upper bound of the hermitian matrix a is iteratively approximated through eigenvalue estimation, so as to obtain an estimation interval of the maximum eigenvalue, and further estimate the corresponding condition number.
Specifically, a synergistic pair refers to financial transaction data having a synergistic relationship. If the condition number is larger than the preset condition number, performing the co-integration check to determine whether a co-integration pair exists; if the fact that the covariance pair exists is determined, searching the covariance pair in the Hermite matrix; and obtaining the co-integration pair in the target data matrix through mapping according to the co-integration pair in the Hermite matrix.
Specifically, in one implementation, if the estimated condition number is greater than or equal to the preset condition number, performing a co-integration check to determine whether a co-integration pair exists; if the fact that the covariance pair exists is determined, searching the covariance pair in the Hermite matrix; and obtaining the co-integration pair in the transaction data matrix through mapping according to the co-integration pair in the Hermite matrix. And if the condition number is smaller than the preset condition number or the coordination check fails, the transaction data matrix does not find the coordination pair.
The preset condition number may be set based on a specific problem context and requirements. In fact, if the condition number of the matrix is too small, then it is assumed that there are no co-ordination pairs, otherwise it is assumed that there is a high probability of co-ordination pairs.
In the case that the condition number is greater than or equal to the preset condition number, that is, it is considered that there is a large possibility of a co-integration pair, at this time, a co-integration check may be performed, for example, using a quantum residual sequence generation algorithm or the like, to determine whether there is a co-integration pair really. And (4) the method comprises the steps of passing the harmony check, showing that the harmony pair exists, finding out a specific harmony pair for the hermitian matrix by adopting methods of quantum linear regression to judge the stationarity of a residual sequence and the like, and further obtaining financial transaction data, such as stock price vectors, with the harmony relation in the original transaction data matrix through mapping. The method for determining the stationarity of the residual sequence by using the quantum residual sequence generation algorithm and the quantum linear regression is the prior art, and the method is not repeated herein.
In the process, the problem can be simplified, namely, assuming that the stock price vectors are not only coordinated but also the linear combination of the stock price vectors is a constant, the stock price vectors forming the matrix A can be linearly regressed to obtain a residual sequence, so that the stationarity of the residual sequence is judged, and the linear regression corresponding to the stationary residual sequence is the stock price vector with the coordinated relation.
In a financial transaction application scene, the existing problem of the collaborative integration pair corresponding to the statistical arbitrage (paired transaction) problem is weakened into the problem of estimating the number of conditions, the parallel computing advantage of a quantum algorithm is exerted, the computing complexity is reduced, the condition number estimation which is an important pre-preselection problem of the collaborative integration pair is quickly solved, and an important data support advantage is provided for the statistical arbitrage; by applying the quantum algorithm in the field of financial transactions, a coordination pair with coordination relation in financial transaction data can be searched, so that the requirement of high-frequency transaction is met, the blank of related technologies is filled, and the method has important pioneering significance and practical application value. It should be noted that the financial transaction application scenario is only an example and is not to be construed as a limitation of the present invention.
Therefore, the method realizes the application of the quantum algorithm to the matrix eigenvalue estimation by comparing the eigenvalue quantum state with the preset value, thereby fully playing the advantages of quantum computation, filling the important blank of the linear system quantum computation direction, and having important pioneering significance and practical application value.
Referring to fig. 5, fig. 5 is a schematic structural diagram of a quantum comparison apparatus for matrix eigenvalue according to an embodiment of the present invention, corresponding to the flow shown in fig. 2, the apparatus includes:
a data obtaining module 501, configured to obtain a characteristic value quantum state carrying characteristic value information of a target data matrix and a preset value of a specific parameter, where the characteristic value quantum state corresponds to n comparison quantum bits used for comparing with the preset value;
a bit obtaining module 502, configured to obtain at least n auxiliary qubits and 1 result qubit for outputting a comparison result;
and a construction output module 503, configured to determine a quantum logic gate of the quantum comparison circuit to be constructed according to the preset value, construct and operate a quantum comparison circuit for comparing the characteristic value quantum state with the preset value by combining the n comparison qubits, the n auxiliary qubits, and the 1 result qubit, and output a result quantum state representing a comparison result.
Specifically, the target data matrix is a hermitian matrix, and before the data obtaining module, the method further includes:
and the construction evolution module is used for obtaining the quantum state containing the Hermite matrix, constructing and operating a corresponding quantum phase estimation QPE circuit, and evolving the quantum state of the Hermite matrix into the characteristic value quantum state carrying the characteristic value information of the Hermite matrix.
Specifically, the hermitian matrix is a hermitian matrix corresponding to the transaction data matrix, and the structure evolution module is specifically configured to:
and carrying out normalization processing on the maximum singular value of the transaction data matrix, and preparing a quantum state comprising a hermite matrix corresponding to the transaction data matrix after the normalization processing, wherein the maximum eigenvalue of the hermite matrix is equal to the maximum singular value 1 of the transaction data matrix after the normalization processing.
Specifically, the specific parameter is a reference condition number k i The preset value is
The initial value of i is 0, the apparatus further comprising:
an execution module, configured to, when the result quantum state indicates that a characteristic value exists in the characteristic value quantum state and is smaller than the preset value, add 1 to i, and return to the step of executing the quantum logic gate that determines the quantum comparison line to be constructed according to the preset value until the output result quantum state indicates that no characteristic value exists in the characteristic value quantum state and is smaller than the preset value;
and the determining module is used for determining approximate estimation of the minimum eigenvalue in the eigenvalue quantum state according to the current preset value.
Specifically, the apparatus further comprises:
the calculation module is used for calculating the condition number of the Hermite matrix according to the approximate estimation of the maximum eigenvalue and the minimum eigenvalue of the Hermite matrix;
and the searching module is used for searching the coordinated pairs in the transaction data matrix according to the size of the condition number.
Specifically, the construction output module is specifically configured to:
and calculating a binary complement corresponding to the preset value, and determining a quantum logic gate of a corresponding bit according to each bit of the binary complement, wherein when the bit is 0, the corresponding quantum logic gate is a Toffoli gate, and when the bit is 1, the corresponding quantum logic gate is a logic OR gate.
Specifically, the construction output module is specifically configured to:
acquiring a preset quantum logic gate, and adding the quantum logic gate and the preset quantum logic gate which are correspondingly determined according to the preset value to corresponding quantum bits to obtain a quantum comparison circuit for comparing the characteristic value quantum state with the preset value;
and operating the quantum comparison circuit and outputting a result quantum state representing the comparison result.
Therefore, the method realizes the application of the quantum algorithm to the matrix eigenvalue estimation by comparing the eigenvalue quantum state with the preset value, thereby fully playing the advantages of quantum computation, filling the important blank of the linear system quantum computation direction, and having important pioneering significance and practical application value.
An embodiment of the present invention further provides a storage medium, in which a computer program is stored, where the computer program is configured to execute the steps in any of the above method embodiments when running.
Specifically, in the present embodiment, the storage medium may be configured to store a computer program for executing the steps of:
s1, obtaining an eigenvalue quantum state carrying eigenvalue information of a target data matrix and a preset value of a specific parameter, wherein the eigenvalue quantum state corresponds to n bits and is used for comparing with the preset value;
s2, obtaining at least n auxiliary quantum bits and 1 result quantum bit for outputting comparison result;
s3, determining a quantum logic gate of the quantum comparison circuit to be constructed according to the preset value, constructing and operating the quantum comparison circuit for comparing the characteristic value quantum state with the preset value by combining n comparison quantum bits, n auxiliary quantum bits and 1 result quantum bit, and outputting a result quantum state representing the comparison result.
Specifically, in this embodiment, the storage medium may include, but is not limited to: various media capable of storing computer programs, such as a usb disk, a Read-Only Memory (ROM), a Random Access Memory (RAM), a removable hard disk, a magnetic disk, or an optical disk.
An embodiment of the present invention further provides an electronic apparatus, which includes a memory and a processor, and is characterized in that the memory stores a computer program, and the processor is configured to execute the computer program to perform the steps in any of the above method embodiments.
Specifically, the electronic apparatus 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.
Specifically, in this embodiment, the processor may be configured to execute the following steps by a computer program:
s1, obtaining an eigenvalue quantum state carrying eigenvalue information of a target data matrix and a preset value of a specific parameter, wherein the eigenvalue quantum state corresponds to n bits and is used for comparing with the preset value;
s2, obtaining at least n auxiliary quantum bits and 1 result quantum bit for outputting comparison result;
s3, determining a quantum logic gate of the quantum comparison circuit to be constructed according to the preset value, constructing and operating the quantum comparison circuit for comparing the characteristic value quantum state with the preset value by combining n comparison quantum bits, n auxiliary quantum bits and 1 result quantum bit, and outputting a result quantum state representing the comparison result.
The construction, features and functions of the present invention are described in detail in the embodiments illustrated in the drawings, which are only preferred embodiments of the present invention, but the present invention is not limited by the drawings, and all equivalent embodiments modified or changed according to the idea of the present invention should fall within the protection scope of the present invention without departing from the spirit of the present invention covered by the description and the drawings.
Claims (10)
1. A method for quantum comparison of matrix eigenvalues, the method comprising:
acquiring an eigenvalue quantum state carrying eigenvalue information of a target data matrix and a preset value of a specific parameter, wherein the eigenvalue quantum state corresponds to n comparison quantum bits used for comparing with the preset value;
acquiring at least n auxiliary qubits and 1 qubit for outputting the result of the comparison;
and determining a quantum logic gate of a quantum comparison circuit to be constructed according to the preset value, constructing and operating a quantum comparison circuit for comparing the characteristic value quantum state with the preset value by combining n comparison quantum bits, n auxiliary quantum bits and 1 result quantum bit, and outputting a result quantum state representing a comparison result.
2. The method of claim 1, wherein the target data matrix is a hermitian matrix, and before the obtaining eigenvalue quantum states carrying eigenvalue information of the target data matrix, the method further comprises:
and obtaining quantum states containing Hermite matrixes, constructing and operating corresponding quantum phase estimation QPE circuits, and evolving the quantum states of the Hermite matrixes into characteristic value quantum states carrying characteristic value information of the Hermite matrixes.
3. The method of claim 2, wherein the hermitian matrix is a hermitian matrix corresponding to a transaction data matrix, and wherein obtaining quantum states comprising the hermitian matrix comprises:
and carrying out normalization processing on the maximum singular value of the transaction data matrix, and preparing a quantum state comprising a hermite matrix corresponding to the transaction data matrix after the normalization processing, wherein the maximum eigenvalue of the hermite matrix is equal to the maximum singular value 1 of the transaction data matrix after the normalization processing.
4. The method according to claim 3, wherein the specific parameter is a reference condition number k i The preset value is
The initial value of i is 0, the method further comprising:
if the result quantum state indicates that the characteristic value exists in the characteristic value quantum state and is smaller than the preset value, adding 1 to the i, and returning to the step of executing the quantum logic gate for determining the quantum comparison circuit to be constructed according to the preset value until the output result quantum state indicates that the characteristic value does not exist in the characteristic value quantum state and is smaller than the preset value;
and determining approximate estimation of the minimum eigenvalue in the eigenvalue quantum state according to the current preset value.
5. The method of claim 4, further comprising:
calculating the condition number of the Hermite matrix according to the approximate estimation of the maximum eigenvalue and the minimum eigenvalue of the Hermite matrix;
and searching for a coordinated pair in the transaction data matrix according to the size of the condition number.
6. The method according to any one of claims 1 to 5, wherein the determining the quantum logic gate of the quantum comparison line to be constructed according to the preset value comprises:
and calculating a binary complement corresponding to the preset value, and determining a quantum logic gate of a corresponding bit according to each bit of the binary complement, wherein when the bit is 0, the corresponding quantum logic gate is a Toffoli gate, and when the bit is 1, the corresponding quantum logic gate is a logic OR gate.
7. The method of claim 6, wherein constructing and operating a quantum comparison circuit for comparing the eigenvalue quantum states with the preset values and outputting a result quantum state representing a comparison result comprises:
acquiring a preset quantum logic gate, and adding the quantum logic gate and the preset quantum logic gate which are correspondingly determined according to the preset value to corresponding quantum bits to obtain a quantum comparison circuit for comparing the characteristic value quantum state with the preset value;
and operating the quantum comparison circuit and outputting a result quantum state representing the comparison result.
8. A quantum comparison apparatus for matrix eigenvalues, the apparatus comprising:
the data acquisition module is used for acquiring a characteristic value quantum state carrying characteristic value information of a target data matrix and a preset value of a specific parameter, wherein the characteristic value quantum state corresponds to n comparison quantum bits used for comparing with the preset value;
a bit obtaining module for obtaining at least n auxiliary qubits and 1 result qubit for outputting a comparison result;
and the construction output module is used for determining a quantum logic gate of the quantum comparison circuit to be constructed according to the preset value, constructing and operating the quantum comparison circuit for comparing the characteristic value quantum state with the preset value by combining the n comparison quantum bits, the n auxiliary quantum bits and the 1 result quantum bit, and outputting a result quantum state representing the comparison result.
9. A storage medium, in which a computer program is stored, wherein the computer program is arranged to perform the method of any of claims 1 to 7 when executed.
10. An electronic device comprising a memory and a processor, wherein the memory has stored therein a computer program, and wherein the processor is arranged to execute the computer program to perform the method of any of claims 1 to 7.
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