CN116310473A - Quantum neural network image classification method based on error mitigation - Google Patents

Quantum neural network image classification method based on error mitigation Download PDF

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CN116310473A
CN116310473A CN202211410414.2A CN202211410414A CN116310473A CN 116310473 A CN116310473 A CN 116310473A CN 202211410414 A CN202211410414 A CN 202211410414A CN 116310473 A CN116310473 A CN 116310473A
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王淑梅
慕晓毅
于翼铭
曲英杰
马鸿洋
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Abstract

The invention relates to the technical field of quantum neural networks, and discloses an image classification method of a quantum neural network based on error mitigation, which comprises the following steps: preprocessing a data set, loading classical feature vectors, relieving errors, judging a classical layer and outputting data. The invention compresses the input high-dimensional image into low-dimensional characteristic representation by using tensor network to match the input of variable component sub-circuits, maximally utilizes the existing quantum resources and opens up a new road for analyzing the image by quantum machine learning in NISQ era; the error bits are filtered by using the quantum automatic encoder based on detection, so that the influence of noise in a quantum circuit on an algorithm result is greatly relieved, the accuracy of algorithm classification is obviously improved, and the accuracy of the algorithm is ensured; the number of training parameters is reduced, fewer parameters are needed compared with a classical equivalent model, and a mixed classical quantum circuit with millions of parameters can be optimized relatively efficiently.

Description

Quantum neural network image classification method based on error mitigation
Technical Field
The invention relates to the technical field of quantum neural networks, in particular to an image classification method of a quantum neural network based on error mitigation.
Background
Recently, quantum computing has been used for machine learning, and it is expected that uncertainty in quantum computing can be a great advantage of probability-based modeling in machine learning, driving new research into noise mesoscale quantum (NISQ) devices. In recent years, variable component sub-circuits (VQC) have become one of the most effective methods of deep quantum learning when used with NISQ devices. With the advent of available quantum computing devices and quantum variation algorithms, quantum machine learning research has begun focusing on hybrid classical quantum algorithms that can be performed in short-term noise mesoscale quantum (NISQ) devices. Due to the limited number of qubits, these pave the way for machine learning applications that ultimately use the NISQ device in practice. VQC can be trained using classical optimization methods, which are considered to have certain expression advantages over classical neural network topologies, including quantum state-based Hopfield networks and their classical implementation using quantum acceleration matrix inversion, recursive quantum neural networks, quantum convolutional neural networks, and peak quantum neural networks. However, due to the large amount of noise in the quantum circuit, erroneous qubits are generated, affecting the final result of the algorithm. Therefore, reducing errors is important in quantum neural networks.
Among the many techniques developed for this purpose, the concept of subspaces is ubiquitous. In quantum error correction, some stabilizers define a subspace in which the quantum states are verified: the damaged condition detected by the syndrome measurement outside the subspace is corrected by the recovery operation. Alternatively, quantum computation may be performed in non-decoherence subspaces, which are chosen to be completely decoupled from certain ambient noise, thereby protecting the desired operation. It is expected that quantum algorithms can run completely successfully on approximately 50-100 qubits in the noisy medium-scale quantum (NISQ) era. However, quantum error correction on this scale of algorithms requires more controllable qubits, which presents a technical challenge. On the other hand, the incoherent subspace exists only in the selective error source, and the noise suppression capability is limited. This means that a more versatile approach is needed to reduce different types of errors within a limited number of controllable qubits. In view of this, we propose an image classification method based on the quantum neural network for error mitigation.
Disclosure of Invention
In order to overcome the problems in the prior art, the invention provides an image classification method of a quantum neural network based on error mitigation.
The technical scheme adopted for solving the technical problems is as follows: an image classification method of a quantum neural network based on error mitigation comprises the following steps:
s1, preprocessing a data set: compressing the input high-dimensional image into a low-dimensional feature representation using a tensor network to match the input of the variable component sub-line;
s2, loading classical eigenvectors into a quantum circuit: encoding the feature vector into a variable component sub-circuit using variable component encoding;
s3, error relief: filtering out erroneous bits using a quantum-based auto-encoder;
s4, outputting a classification result: the first four qubits are measured and the recovered data is subjected to classical processing using Softmax to calculate the probability of each potential class.
The specific method for preprocessing the data set in S1 includes the following steps:
s1.1, input image in data setFlattened into an N-dimensional vector x= (x) 1 ,x 2 ,...,x N ) And normalize each component to x i ∈[0,1]；
S1.2, mapping the vector subjected to normalization processing to a product state by using feature mapping, wherein a specific conversion formula is as follows:
Figure BDA0003938277810000021
where Φ (x) > represents a product state obtained by using the feature map.
S1.3, MPS is further processed to generate a compressed representation, MPS is taken as a feature extraction program, data is encoded into a product state, the product state is contracted with the MPS, class labels or outputs are generated, the product state is obtained in step S1.2, and then input data is loaded into the MPS through the following operations:
Figure BDA0003938277810000031
wherein f (l) (x) Representing the loaded input data
Figure BDA0003938277810000032
Representing feature extractor approximation, ++>
Figure BDA0003938277810000033
Representing the corresponding product state of each pixel point of the input image.
The specific method for loading the classical feature vector into the quantum circuit in the S2 comprises the following steps:
s2.1, a coding part, firstly creating an unbiased initial state by using a Hadamard gate, and then passing through a single quantum bit gate R y (arctan(x i ) Sum of (d)
Figure BDA0003938277810000034
Implementing phase encodingWherein the characteristic of the rotation angle is derived from an input of the tensor network, and subsequently processing the encoded states using a parametric optimized variable component sub-circuit;
s2.2, a variation part, which consists of a CNOT gate for entangling each qubit quantum state and a unitary gate R (alpha, beta, gamma) representing a general single qubit.
The above-mentioned image classification method based on the quantum neural network of error relief, in said S3, the concrete method of error relief includes the following steps:
s3.1, firstly, coding the unitary operator U e Applied to noise quantum state, the compression state is formed by
Figure BDA0003938277810000035
Becomes as follows
Figure BDA0003938277810000036
Wherein ε represents the probability of the quantum state being destroyed and ε [0,1 ]]Sigma represents the compressed quantum state, ρ err Representing error items->
Figure BDA0003938277810000037
Representing a decoding unitary operator, U e Representing encoded unitary operator->
Figure BDA0003938277810000038
Representing the noisy quantum state after compression, +.>
Figure BDA0003938277810000039
Represent noisy quantum states, U e Transferring the error-free term to the potential subspace, while most of the errors remain in the garbage subspace;
s3.2, error detection and measurement are carried out, error quantum bits are detected, corresponding quantum data are discarded, and the state is projected to a potential subspace by measurement, so that errors in a garbage subspace are detected and eliminated;
s3.3, unitary operator to be decoded
Figure BDA0003938277810000041
Applied to sigma' to obtain error states +.>
Figure BDA0003938277810000042
Can be written as:
Figure BDA0003938277810000043
wherein ρ' represents the error state, ε represents the probability of quantum state destruction, ε [0,1 ]]，
Figure BDA0003938277810000044
Trace representing projection of error term to supporting subspace, O (ε) 2 ) Representing the square order of the probability that the quantum state is destroyed, ρ represents the ideal quantum state,
Figure BDA0003938277810000045
representing the projection of the error term into the supporting subspace.
The above-mentioned method for classifying images of quantum neural network based on error mitigation, in the step S4, the specific method for outputting the classification result includes the following steps:
s4.1, measuring the first four quantum bits recovered by the quantum self-encoder, transmitting all measurement results to the next layer by using a full-connection layer, and outputting Pauli-Z expected values by a multi-time operation quantum circuit by using the measurement part;
s4.2, mapping all score values into a probability value by using a sigmoid function;
and S4.3, judging the category of the image according to the probability value.
The beneficial effects of the invention are as follows: 1. in the method for classifying the images based on the quantum neural network for error alleviation, the tensor network is utilized to compress the input high-dimensional images into low-dimensional characteristic representation so as to match the input of the variable component sub-circuit, a new road is opened up for analyzing the images by the quantum machine learning process in the NISQ era, a small number of quantum bits can be ensured to be used, the analysis and the processing of the images can be realized, the classical quantum boundary can be regulated according to the availability of the quantum resources by utilizing the corresponding relation between the tensor network and the quantum circuit, and the existing quantum resources can be utilized to the maximum extent;
2. in the method for classifying the images of the quantum neural network based on error alleviation, due to the error alleviation characteristic of the quantum self-encoder, the error bits are filtered by using the quantum self-encoder based on detection, so that the influence of noise in a quantum circuit on algorithm results is greatly relieved, the accuracy of algorithm classification is obviously improved compared with other quantum classification algorithms, and the accuracy of the algorithm is ensured;
3. the image classification method based on the error alleviation quantum neural network has a simpler overall scheme, ensures simple hardware operation through the algorithmic integrity, and is easy to realize in an actual quantum computer.
4. In a noise environment, the image classification method of the quantum neural network based on error alleviation has higher precision than that of the traditional neural network, the traditional neural network is difficult to modify the network weight of the traditional neural network in the noisy environment, quantum bits have inherent probability characteristics, and the original pixel value is effectively used for approximating noise in a modification quantum layer, so that the classification precision is improved;
5. because the parameterized VQC in the modified quantum layer reduces the number of training parameters, the image classification method of the quantum neural network based on error relief needs fewer parameters than a classical equivalent model thereof to be realized, and when hundreds of quantum bits are possessed in the future, the image classification method of the quantum neural network based on error relief can be easily realized to relatively efficiently optimize a mixed classical quantum circuit with millions of parameters.
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The invention will be further described with reference to the drawings and examples.
FIG. 1 is a flow chart of an overall scheme of an image classification method of a quantum neural network based on error mitigation in the present invention;
FIG. 2 is a schematic diagram of tensor network feature extraction in accordance with the present invention;
FIG. 3 is a schematic diagram of error mitigation in accordance with the present invention.
Detailed Description
The present invention will be described in detail below with reference to the drawings and detailed description to enable those skilled in the art to better understand the technical scheme of the present invention.
As shown in fig. 1, the present embodiment provides an image classification method of a quantum neural network based on error mitigation, and the overall transmission scheme of the present embodiment is mainly divided into four parts of a data set preprocessing stage, a classical feature vector loading stage, an error mitigation stage, and a classical layer judgment stage, and specifically includes the following steps:
s1, preprocessing a data set: compressing the input high-dimensional image into a low-dimensional feature representation using a tensor network to match the input of the variable component sub-line;
s2, loading classical eigenvectors into a quantum circuit: encoding the feature vector into a variable component sub-circuit using variable component encoding;
s3, error relief: filtering out erroneous bits using a quantum-based auto-encoder;
s4, outputting a classification result: the first four qubits are measured and the recovered data is subjected to classical processing using Softmax to calculate the probability of each potential class.
In this embodiment, in S1, a specific method for preprocessing a data set includes the following steps:
s1.1, flattening input images 0,1,2,3 with the size of 28 x 28 in the mnist data set into 784-dimensional vectors x= (x) respectively 1 ,x 2 ,...,x N ) And normalize each component to x i ∈[0,1]；
S1.2, mapping the vector subjected to normalization processing to a product state by using feature mapping;
s1.3, MPS is further processed to generate a compressed representation, MPS is the feature extraction procedure, as shown in fig. 2, the data is encoded into a product state, which contracts with MPS and generates class labels or outputs. The product state is obtained according to step S1.2, and then the input data is loaded into the MPS by the following operations.
As a further improvement of the present technical solution, in S1.2, a specific conversion formula is:
Figure BDA0003938277810000061
where Φ (x) > represents a product state obtained by using the feature map.
The product state is obtained according to step S1.2, and then the input data is loaded into MPS by:
Figure BDA0003938277810000062
wherein f (l) (x) Representing the MPS after loading of the input data,
Figure BDA0003938277810000063
representing feature extractor approximation, ++>
Figure BDA0003938277810000064
Representing the corresponding product state of each pixel point of the input image.
In this embodiment, in S2, the specific method for loading the classical feature vector into the quantum circuit includes the following steps:
s2.1, a coding part, firstly creating an unbiased initial state by using a Hadamard gate, and then passing through a single quantum bit gate R y (arctan(x i ) Sum of (d)
Figure BDA0003938277810000071
Implementing phase encoding, wherein the characteristic of the rotation angle is derived from the input of the tensor network, followed by processing the encoded states using a parametric optimized variable component sub-circuit;
s2.2, a variation part, which consists of a CNOT gate used for entanglement of each quantum bit quantum state and a unitary gate R (alpha, beta, gamma) representing a general single quantum bit, wherein three parameters alpha, beta and gamma need to be learned, the three parameters represent the rotation angle of the unitary gate, the three parameters are initialized randomly firstly, and then the three parameters are updated by adopting a counter-propagation algorithm based on an Adam optimizer.
In which S2.1 the outputs of the classical parts need to be encoded in order for the quantum circuits to use them. The general N-qubit quantum state can be expressed as:
Figure BDA0003938277810000072
wherein |ψ>Represents the N-qubit quantum state, |q N >Representing the Nth quantum state
Figure BDA0003938277810000073
Representing the amplitude of each quantum state.
We choose the variant encoding method to encode classical data into quantum states. Initial quantum state
Figure BDA0003938277810000074
Figure BDA0003938277810000075
First experience->
Figure BDA0003938277810000076
Operation of creating unbiased State->
Figure BDA0003938277810000077
Where H is Hadamard. Considering an n-qubit system, the corresponding unbiased initial state is:
Figure BDA0003938277810000078
wherein, the liquid crystal display device comprises a liquid crystal display device,
Figure BDA0003938277810000081
represents an unbiased initial state, h|0>Represents the quantum state in which the H-gate acts in the spin direction, |1>Representing a spin-down quantum state, |0>Represents the quantum state in the spin-up direction, |k>Representing unbiased quantum states, n representing n qubits.
In the present embodiment, S3In the specific method of error mitigation, as shown in fig. 3, the large ellipse in fig. 3 represents the complete hilbert space H, the small ellipse inside represents the potential subspace L, and the rest is the garbage subspace J, we use the points to represent the error-free term (1-epsilon) ρ, and the cross represents the error term epsilon ρ err ，U e Transferring the error-free term to L while most of the error remains in J, measuring the projection of the state to the potential subspace, thereby detecting and eliminating the error in J, and finally, applying
Figure BDA00039382778100000811
And the quantum data is recovered, and the error is reduced. The method comprises the following specific steps:
s3.1, firstly, coding the unitary operator U e Applied to the quantum state processed by the variable component sub-circuit, the quantum state is formed by
Figure BDA0003938277810000082
Become->
Figure BDA0003938277810000083
Wherein ε represents the probability of the quantum state being destroyed and ε [0,1 ]]Sigma represents the compressed quantum state, ρ err Representing error items->
Figure BDA0003938277810000084
Representing a decoding unitary operator, U e Representing encoded unitary operator->
Figure BDA0003938277810000085
Representing the noisy quantum state after compression, +.>
Figure BDA0003938277810000086
Represent noisy quantum states, U e Transferring the error-free term to the potential subspace, while most of the errors remain in the garbage subspace;
s3.2, error detection and measurement are carried out, error quantum bits are detected, corresponding quantum data are discarded, and the measurement projects states to a potential subspace, so that errors in a garbage subspace are detected and eliminated.
S3.3, finally, decoding the unitary operator
Figure BDA0003938277810000087
Applied to sigma' to obtain error states +.>
Figure BDA0003938277810000088
Specifically, in S3.3, the error state may be written as:
Figure BDA0003938277810000089
wherein ρ' represents the error state, ε represents the probability of quantum state destruction, ε [0,1 ]]，
Figure BDA00039382778100000810
Trace representing projection of error term to supporting subspace, O (ε) 2 ) Square order representing probability of destruction of quantum state, ρ representing ideal quantum state, +.>
Figure BDA0003938277810000091
Representing the projection of the error term into the supporting subspace.
In this embodiment, in S4, the specific method for outputting the classification result includes the following steps:
s4.1, measuring the first four quantum bits recovered by the quantum self-encoder, transmitting all measurement results to the next layer by using a full-connection layer, and outputting Pauli-Z expected values by a multi-time operation quantum circuit by using the measurement part;
s4.2, mapping all score values into a probability value by using a sigmoid function;
and S4.3, finally judging the category of the image according to the probability value.
Through the steps of the method, the embodiment combines the general quantum error mitigation protocol with the quantum neural network for image classification for the first time, does not need to use multiple bits as large equipment cost of a line gate, is applicable to NISQ equipment, in which the classification accuracy of the image classification method can reach 94.1%, the accuracy refers to the proportion of correctly classified samples to the total number of samples, the mnist data set is input into TN-VQC and RQNN models for image classification, and the classification accuracy of the two models is calculated, wherein the classification accuracy of TN-VQC is 75.5% and the classification accuracy of RQNN is 71.6%, and therefore, the image classification method of the embodiment is obviously higher than that of the RQNN and TN-VQC models in the prior art.
Wherein quantum data is compressed into a potential subspace while leaving errors outside, the latter then being eliminated by measurement and post-selection. In contrast to previously developed methods, our protocol does not require extra qubits on the one hand and on the other hand it has near optimal denoising capabilities, all errors detected outside the potential subspace can be removed, and errors detected in the subspace cannot be removed anyway, under reasonable requirements. Therefore, our method of image classification based on error mitigation quantum neural networks is particularly useful for recent quantum devices where controllable quantum bit limitation and noise reduction is important.
The above embodiments are only exemplary embodiments of the present invention and are not intended to limit the present invention, the scope of which is defined by the claims. Various modifications and equivalent arrangements of this invention will occur to those skilled in the art, and are intended to be within the spirit and scope of the invention.

Claims (5)

1. An image classification method of a quantum neural network based on error mitigation is characterized by comprising the following steps:
s1, preprocessing a data set: compressing the input high-dimensional image into a low-dimensional feature representation using a tensor network to match the input of the variable component sub-line;
s2, loading classical eigenvectors into a quantum circuit: encoding the feature vector into a variable component sub-circuit using variable component encoding;
s3, error relief: filtering out erroneous bits using a quantum-based auto-encoder;
s4, outputting a classification result: the first four qubits are measured and the recovered data is subjected to classical processing using Softmax to calculate the probability of each potential class.
2. The method for classifying images based on the quantum neural network for error mitigation according to claim 1, wherein in S1, the specific method for preprocessing the data set comprises the following steps:
s1.1, flattening the input image in the dataset into an N-dimensional vector x= (x) 1 ,x 2 ,...,x N ) And normalize each component to x i ∈[0,1]；
S1.2, mapping the vector subjected to normalization processing to a product state by using feature mapping, wherein a specific conversion formula is as follows:
Figure FDA0003938277800000011
where Φ (x) > represents a product state obtained by using the feature map.
S1.3, MPS is further processed to generate a compressed representation, MPS is taken as a feature extraction program, data is encoded into a product state, the product state is contracted with the MPS, class labels or outputs are generated, the product state is obtained in step S1.2, and then input data is loaded into the MPS through the following operations:
Figure FDA0003938277800000012
wherein f (l) (x) Representing the MPS after loading of the input data,
Figure FDA0003938277800000013
representing feature extractor approximation, ++>
Figure FDA0003938277800000021
Representing the corresponding product state of each pixel point of the input image.
3. The method for classifying images of a quantum neural network based on error mitigation according to claim 1, wherein in S2, the specific method for loading classical eigenvectors into a quantum circuit comprises the following steps:
s2.1, a coding part, firstly creating an unbiased initial state by using a Hadamard gate, and then passing through a single quantum bit gate R y (arctan(x i ) Sum of (d)
Figure FDA0003938277800000022
Implementing phase encoding, wherein the characteristic of the rotation angle is derived from the input of the tensor network, followed by processing the encoded states using a parametric optimized variable component sub-circuit;
s2.2, a variation part, which consists of a CNOT gate for entangling each qubit quantum state and a unitary gate R (alpha, beta, gamma) representing a general single qubit.
4. The method for classifying images based on the quantum neural network for error mitigation according to claim 1, wherein in S3, the specific method for error mitigation comprises the following steps:
s3.1, firstly, coding the unitary operator U e Applied to noise quantum state, the compression state is formed by
Figure FDA0003938277800000023
Becomes as follows
Figure FDA0003938277800000024
Wherein ε represents the probability of the quantum state being destroyed and ε [0,1 ]]Sigma represents the compressed quantum state, ρ err Representing error items->
Figure FDA0003938277800000025
Representing a decoding unitary operator, U e Representing encoded unitary operator->
Figure FDA0003938277800000026
Representing the noisy quantum state after compression, +.>
Figure FDA0003938277800000027
Represent noisy quantum states, U e Transferring the error-free term to the potential subspace, while most of the errors remain in the garbage subspace;
s3.2, error detection and measurement are carried out, error quantum bits are detected, corresponding quantum data are discarded, and the state is projected to a potential subspace by measurement, so that errors in a garbage subspace are detected and eliminated;
s3.3, unitary operator to be decoded
Figure FDA0003938277800000028
Applied to sigma' to obtain error states +.>
Figure FDA0003938277800000029
Can be written as:
Figure FDA00039382778000000210
wherein ρ' represents the error state, ε represents the probability of quantum state destruction, ε [0,1 ]]，
Figure FDA00039382778000000211
Trace representing projection of error term to supporting subspace, O (ε) 2 ) Square order representing probability of destruction of quantum state, ρ representing ideal quantum state, +.>
Figure FDA0003938277800000031
Representing the projection of the error term into the supporting subspace.
5. The method for classifying images based on the quantum neural network for error mitigation according to claim 1, wherein in S4, the specific method for outputting the classification result comprises the following steps:
s4.1, measuring the first four quantum bits recovered by the quantum self-encoder, transmitting all measurement results to the next layer by using a full-connection layer, and outputting Pauli-Z expected values by a multi-time operation quantum circuit by using the measurement part;
s4.2, mapping all score values into a probability value by using a sigmoid function;
and S4.3, judging the category of the image according to the probability value.
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CN117456561A (en) * 2023-12-22 2024-01-26 华侨大学 Pedestrian re-identification method based on component quantum learning
CN117456561B (en) * 2023-12-22 2024-03-12 华侨大学 Pedestrian re-identification method based on component quantum learning
CN117649563A (en) * 2024-01-29 2024-03-05 量子科技长三角产业创新中心 Quantum recognition method, system, electronic device and storage medium for image category
CN117649563B (en) * 2024-01-29 2024-05-10 量子科技长三角产业创新中心 Quantum recognition method, system, electronic device and storage medium for image category

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