Information extraction method of quantum classical interaction in quantum machine learning
Through the information extraction scheme of rotation and synchronous unbiased measurement basis, combined with quantum circuits and fully connected layers, the problem of low information interaction efficiency in quantum machine learning is solved, and the classification accuracy and data feature expression ability of the model are improved.
Patent Information
- Application Number
- CN202510862267.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-06-25
- Publication Date
- 2025-10-28
AI Technical Summary
In existing quantum machine learning algorithms, the efficiency of quantum-classical information interaction is low, and the complete information of the quantum state is not fully extracted, which limits the model's ability to express complex data features.
An information extraction scheme based on rotating unbiased measurement basis and synchronous unbiased measurement basis is adopted. Through steps such as zero-padding operation, sliding window, parameterized quantum circuit and fully connected layer, combined with multi-angle measurement of unbiased measurement basis, the characteristic information of quantum state is extracted.
It improves the classification accuracy of quantum neural networks, enhances the ability to express complex data features, and strengthens the representation ability of the model.
Smart Images
Figure CN120852847A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of quantum computing technology, specifically relating to a method for extracting information from quantum classical interactions in quantum machine learning. Background Technology
[0002] With the explosive growth of data and the increasingly urgent need to solve complex problems, traditional computers are gradually becoming unable to meet the demands due to their computational limitations. Quantum computing, based on principles such as quantum superposition and quantum entanglement, endows qubits with the ability to process multiple states simultaneously, enabling parallel computing. Theoretically, it possesses efficiency far exceeding that of classical computing and holds the promise of overcoming complex optimization, drug molecule simulation, and encryption cracking challenges that traditional computing struggles with. However, current quantum computing technology is still in the noisy medium-scale quantum (NISQ) era. At this stage, quantum computers have a limited number of qubits and are susceptible to environmental noise, resulting in a high error rate and insufficient stability. There is still a long way to go before building a universal, reliable, large-scale quantum computer.
[0003] Quantum machine learning, with its powerful parallel computing capabilities and high computational speed, has emerged as a new paradigm for solving robot learning tasks, perfectly aligning with the innovative solutions of the NISQ era. Given the limitations of NISQ devices, relying solely on quantum computing is insufficient for complex tasks. This algorithm organically combines the parallel processing and unique quantum properties of quantum computing with the mature algorithmic framework and data processing capabilities of classical machine learning. The quantum computer utilizes its quantum properties to process high-dimensional data and perform quantum state operations, uncovering latent features; the classical computer handles data preprocessing, algorithm flow control, and the interpretation and analysis of quantum computing results. This collaborative model effectively balances the advantages of quantum computing with hardware limitations, providing a practical solution for complex machine learning tasks such as pattern recognition and predictive analysis under NISQ conditions, thus propelling quantum computing technology towards practical applications. Current quantum machine learning algorithms employ a quantum-classical interaction model that projects the quantum state onto a single measurement basis, converting it into classical bits. This method significantly limits the advantages of superposition and entanglement in quantum computing. Summary of the Invention
[0004] To overcome the shortcomings of existing technologies, this invention provides a method for extracting information from quantum classical interactions in quantum machine learning. It presents two unbiased basis vector measurement schemes applicable to quantum machine learning algorithms, including an alternating unbiased measurement basis information extraction scheme and a synchronous unbiased measurement basis information extraction scheme. The aim is to mine more information from quantum superposition states in quantum machine learning for network training, unleash the superiority of quantum computing, and solve the efficiency problem of quantum classical information interaction in existing technologies.
[0005] The technical solution adopted by this invention to solve its technical problem is as follows:
[0006] Step 1: Expand the image matrix by padding with zeros;
[0007] Step 2: Use a sliding window to extract multiple sub-matrices;
[0008] Step 3: Flatten each sub-matrix into a vector and encode it into the qubit circuit as the amplitude to prepare the quantum state;
[0009] Step 4: Evolve the initial quantum state and extract features by parameterizing the quantum circuit, which includes alternating entanglement layers and rotation gate layers;
[0010] Step 5: Perform measurements in batches, using either a rotating unbiased measurement basis information extraction scheme or a synchronous unbiased measurement basis information extraction scheme for each batch;
[0011] Step 6: Combine the measurement results of each submatrix into an eigenvector;
[0012] Step 7: Reduce the dimension of the feature vector to the number of classes using a fully connected layer;
[0013] Step 8: Use the Softmax function to convert the real vector into a probability distribution to obtain the image classification result.
[0014] Preferably, step 1 specifically comprises:
[0015] The n×n input image matrix X is expanded into a (n+2p)×(n+2p) matrix X′ by zero-padding, where p is the number of padding pixels, used to preserve edge features and adapt to subsequent quantum convolution operations; the zero-padding operation satisfies:
[0016]
[0017] Where, X′ i，j X represents the value of a newly generated image pixel. i-p，j-p This represents the value of the original image pixel.
[0018] Preferably, step 2 specifically comprises:
[0019] Using a k×k sliding window, traverse X′ with a step size s to extract m submatrices {M1, M2, ..., M}. m},in:
[0020]
[0021] Preferably, step 3 specifically comprises:
[0022] Flatten each submatrix into k 2 dimensional vector vi Quantum states are prepared by amplitude encoding:
[0023]
[0024] Where ||v i || represents the vector magnitude, ensuring quantum state normalization; v i，j Represents the values in the submatrix, |ψ i > represents the encoded quantum state, and |j> represents the basis vector.
[0025] Preferably, step 4 specifically comprises:
[0026] Construct parameterized quantum circuits U(θ), including single-qubit gates RX, RY, RZ, H, SX, and two-qubit gates CRX, CNOT; RX, RY, RZ, and CRX are tunable, while H, SX, and CNOT construct inter-qubit correlations; the quantum state evolution process is represented as:
[0027] |φ i >=U(θ)|ψ i >
[0028] Where, |φ i > Represents the quantum state |ψ i The result after quantum circuit calculation.
[0029] Preferably, step 5 specifically comprises:
[0030] By cyclically switching the single-bit measurement basis {|0>,|1>}, Repeated preparation and evolution, or simultaneous use of single-bit measurement bases {|0>,|1>}, Measurement and evolution are performed to extract multi-angle features of quantum states.
[0031] Preferably, in step 6, the quantum circuit measurement results of each sub-matrix are recombined into a three-dimensional feature map, and then flattened into a one-dimensional vector.
[0032] Preferably, step 7 specifically comprises:
[0033] Applying a fully connected layer with trainable parameters, the output dimension is equal to the number of image categories, as shown below:
[0034]
[0035] Where z represents the result, and W represents the weight matrix. Let b represent the vector in step 6, and b represent the bias term.
[0036] Preferably, step 8 specifically comprises:
[0037] Applying the Softmax function to the output z yields the class probability distribution:
[0038]
[0039] Among them, z i Let z represent the i-th element. j K represents all elements, C represents a total of C categories, and K and C are of equal size.
[0040] Preferably, the scheme for extracting the unbiased measurement basis information in rotation is as follows:
[0041] During the training and testing of a quantum machine learning network, in each training cycle, when it is necessary to measure the quantum state of the quantum machine learning network, each measurement basis in a set of unbiased basis sets is used in turn. Specifically, if there are m measurement basis sets in the selected unbiased measurement basis set M, then in the first cycle of measuring the quantum state of the machine learning network, the first measurement basis M1 of the unbiased measurement basis set M is used for measurement, in the second cycle, the second measurement basis M2 of the unbiased measurement basis set M is used for measurement, and so on, with the m-th measurement basis M of the unbiased measurement basis set M being used in the m-th cycle. m The measurement is performed using the first measurement basis M1 of the unbiased measurement basis set M in the (m+1)th cycle, and the unbiased measurement basis is used cyclically.
[0042] Preferably, the synchronous unbiased measurement basis information extraction scheme is as follows:
[0043] During the training and testing of a quantum machine learning network, when the quantum state of the quantum machine learning network needs to be measured in each training cycle, each measurement basis in a set of unbiased basis sets is used simultaneously. Specifically, if there are m measurement basis sets in the selected set of unbiased measurement basis sets M, then in each cycle of quantum state measurement of the machine learning network, all measurement basis sets in the set of unbiased measurement basis sets M are used to measure the quantum state, and the average of the results obtained by all measurement basis sets is the result of that measurement.
[0044] Preferably, the unbiased measurement basis is defined as follows:
[0045] set up and It is a d-dimensional Hilbert space C d Two sets of orthonormal bases exist if and only if the following equation is satisfied. In space C d In this context, a set {M1, M2, ..., M} consisting of an orthonormal basis is given. m A set is called an unbiased basis if M1, M2, ..., M mIt is pairwise unbiased; if a physical system is in a characteristic state of M1, then when measuring this system with another set of bases M2, the probability of obtaining all measurement results is the same; specifically, for a qubit... |α| 2 +|β| 2 =1, and the corresponding set of unbiased measurement bases is the Pauli matrix {σ} x ,σ y ,σ z The three basis vectors formed by the eigenvectors of} are {|0>,|1>}, These three sets of bases are unbiased measurement bases.
[0046] The beneficial effects of the present invention are as follows:
[0047] Existing quantum neural networks, particularly quantum convolutional neural networks, typically only consider measurements using a single observation basis, failing to fully extract the complete information of the quantum state and limiting the model's ability to represent complex data features. This invention, based on joint observations of pauli-z, pauli-x, and pauli-y, can more comprehensively capture the global information of the quantum state. By jointly calculating the average value using multiple observation bases or dynamically switching measurement strategies, this invention effectively enhances the model's representational ability and improves classification accuracy. Attached Figure Description
[0048] Figure 1 This is a schematic diagram of the quantum neural network process for image classification according to the present invention.
[0049] Figure 2 A schematic diagram of the quantum circuit and its representation on the Bloch sphere for a scheme to extract information from a single-bit cyclic unbiased measurement basis;
[0050] Figure 3 A schematic diagram of the quantum circuit and its representation on the Bloch sphere for a scheme to extract synchronous unbiased measurement basis information on a single bit;
[0051] Figure 4 To compare the curves of MNIST data classification accuracy with training cycles using the method of this invention and computer measurement scheme;
[0052] Figure 5 To compare the classification accuracy curves of Fashioin MNIST data with training cycles using the method of this invention and computer measurement scheme;
[0053] Figure 6 To compare the curves of EMNIST data classification accuracy with training cycles using the method of this invention and computer measurement scheme;
[0054] Figure 7 To compare the CIFAR10 data classification accuracy curves with training cycles using the method of this invention and computer measurement scheme. Detailed Implementation
[0055] The present invention will be further described below with reference to the accompanying drawings and embodiments.
[0056] This invention proposes two unbiased basis vector measurement schemes suitable for quantum machine learning algorithms, aiming to extract more information from quantum superposition states in quantum machine learning for network training, unleash the superiority of quantum computing, and solve the efficiency problem of quantum classical information interaction in existing technologies.
[0057] The purpose of this invention is to provide an efficient quantum classical information interaction scheme applicable to quantum neural networks.
[0058] Quantum neural networks are composed of various single-qubit and multi-qubit quantum gates. The quantum classical information interaction scheme proposed in this invention is divided into two types: the first is a time-sequential unbiased measurement basis information extraction scheme, and the second is a synchronous unbiased measurement basis information extraction scheme.
[0059] The unbiased measurement basis is defined as follows: Let and It is a d-dimensional Hilbert space C d Two sets of orthonormal bases exist if and only if the following equation is satisfied. In space C d In this context, a set {M1, M2, ..., M} consisting of an orthonormal basis is given. m A set is called an unbiased basis if M1, M2, ..., M m It is pairwise unbiased. If a physical system is in a characteristic state of M1, then when measuring this system using another set of bases M2, we obtain all measurement results with the same probability. Specifically, for a qubit... |α| 2 +|β| 2 =1, and the corresponding set of unbiased measurement bases is the Pauli matrix {σ} x ,σ y ,σ z The three basis vectors formed by the eigenvectors of} are {|0>,|1>}, These three sets of bases are unbiased measurement bases.
[0060] The first quantum classical information extraction method in quantum machine learning of this invention—the time-sequential rotating unbiased measurement basis information extraction scheme—is as follows. During the training and testing of the quantum machine learning network, in each training cycle, when it is necessary to measure the quantum state of the quantum machine learning network, each measurement basis in a set of unbiased basis sets is used in turn. Specifically, if the selected unbiased measurement basis set M contains m measurement basis sets, then in the first cycle of quantum state measurement of the machine learning network, the first measurement basis M1 of the unbiased measurement basis set M is used for measurement; in the second cycle, the second measurement basis M2 of the unbiased measurement basis set M is used for measurement; and so on, with the m-th measurement basis M1 of the unbiased measurement basis set M being used in the m-th cycle. m The measurement is performed using the first measurement basis M1 of the unbiased measurement basis set M in the (m+1)th cycle, and the unbiased measurement basis is used cyclically.
[0061] The second method for extracting quantum classical information in quantum machine learning according to the present invention—the synchronous unbiased measurement basis information extraction scheme—is as follows. During the training and testing of the quantum machine learning network, in each training cycle, when it is necessary to measure the quantum state of the quantum machine learning network, each measurement basis in a set of unbiased bases is used simultaneously. Specifically, if the selected set of unbiased measurement bases M contains m measurement bases, then in each cycle of quantum state measurement in the machine learning network, all measurement bases in the set of unbiased measurement bases M are used to measure the quantum state, and the average of the results obtained from all measurement bases is the result of that measurement.
[0062] This invention further illustrates the application of quantum machine learning to image classification as an example. The specific inventive steps are as follows:
[0063] Step 1: Expand the image matrix by padding with zeros to avoid loss of edge information.
[0064] Step 2: Use a sliding window to extract multiple sub-matrices.
[0065] Step 3: Flatten each sub-matrix into a vector and encode it into the qubit circuit as the amplitude to prepare the quantum state.
[0066] Step 4: Evolve the initial quantum state and extract features by parameterizing the quantum circuit, which includes alternating entanglement layers and rotation gate layers.
[0067] Step 5: Perform measurements in batches, using either a rotating unbiased measurement basis information extraction scheme or a synchronous unbiased measurement basis information extraction scheme for each batch.
[0068] Step 6: Combine the measurement results of each submatrix into an eigenvector.
[0069] Step 7: Reduce the dimension of the feature vector to the number of categories using a fully connected layer.
[0070] Step 8: Use the Softmax function to convert the real vector into a probability distribution to obtain the image classification result.
[0071] In step one, the n×n input image matrix X is expanded into a (n+2p)×(n+2p) matrix X′ by zero-padding, where p is the number of padding pixels, used to preserve edge features and adapt to subsequent quantum convolution operations. The zero-padding operation satisfies:
[0072]
[0073] In step two, a k×k sliding window is used to traverse X′ with a step size s to extract m submatrices {M1,M2,...,M}. m},in
[0074]
[0075] In step three, each submatrix is flattened into k... 2 dimensional vector v i Quantum states are prepared by amplitude encoding.
[0076]
[0077] Where ||v i || represents the vector magnitude, ensuring quantum state normalization.
[0078] In step four, parameterized quantum circuits U(θ) are constructed, specifically including single-qubit gates RX, RY, RZ, H, SX, and two-qubit gates CRX, CNOT. RX, RY, RZ, and CRX are tunable, while H, SX, and CNOT establish inter-qubit correlations. The quantum state evolution process is represented as follows:
[0079] |φ i >=U(θ)|ψ i >
[0080] In step five, the single-bit measurement basis {|0>,|1>} is switched cyclically. Repeated preparation and evolution, or simultaneous use of single-bit measurement bases {|0>,|1>}, By performing measurements and evolution, we can extract multi-angle features of quantum states and capture quantum correlation features that are difficult to extract using classical methods.
[0081] In step six, the quantum circuit measurement results of each sub-matrix are recombined into a three-dimensional feature map, and then flattened into a one-dimensional vector.
[0082] In step seven, a classic fully connected layer with trainable parameters is applied, resulting in an output dimension equal to the number of image categories. This can be represented as...
[0083]
[0084] In step eight, the Softmax function is applied to the output z to obtain the class probability distribution:
[0085]
[0086] Example:
[0087] Part 1: An image classification model scheme using quantum convolutional networks, illustrating the complete steps of the method of this invention as follows:
[0088] Step 1: Classic image preprocessing, which involves padding the image matrix with zeros to prevent the loss of edge information. In this case, the 28×28×1 input image matrix is padded with zeros to expand it into a 30×30×1 matrix.
[0089] Step two: Use a sliding window to extract multiple submatrices. In this case, extract 3×3×1 submatrices, and traverse the entire input matrix with a step size of 3 to obtain 10×10 submatrices.
[0090] Step 3: Flatten each sub-matrix into a vector, encode it as amplitude into the qubit circuit, and prepare the quantum state. Flatten each sub-matrix, encode it into a 4-qubit circuit, prepare the quantum state, and perform the evolution of the quantum state.
[0091] Step four involves designing a parameterized quantum circuit, including alternating entanglement layers and rotation gate layers, to evolve the initial quantum state and extract features. The schematic diagram of the quantum neural network designed in this case is shown below. Figure 1 As shown.
[0092] Step 5: Perform measurements in batches, with each batch using a cyclic switching of the single-bit measurement base {|0>,|1>}. Repeated preparation and evolution, or simultaneous use of single-bit measurement bases {|0>,|1>}, Measurement and evolution are performed to extract multi-angle features of quantum states, capturing quantum correlation features that are difficult to extract using classical methods. The quantum circuit diagram and Bloch sphere representation of the scheme for extracting basis information from a single qubit using a rotating unbiased measurement method are shown below. Figure 2 As shown, the quantum circuit diagram and the representation on the Bloch sphere for a single-bit synchronous unbiased measurement basis information extraction scheme are as follows. Figure 3 As shown.
[0093] Step Six: Merge the measurement results of each sub-matrix into an eigenvector. Reassemble the quantum circuit measurement results of each sub-matrix into a three-dimensional feature map, and then flatten it into a one-dimensional vector.
[0094] Step 7: Reduce the dimension of the feature vector to the number of target categories using a fully connected layer.
[0095] Step 8: Use the Softmax function to convert the real vector into a probability distribution to obtain the image classification result.
[0096] This embodiment uses a quantum neural network to classify the MINST, Fashion MNIST, EMNIST, and CIFAR10 image datasets, and compares the results with those from direct computer projection measurement. The classification accuracy of the method of this invention is improved by an average of 24.5%, as shown in Table 1. Furthermore, the accuracy curves of the scheme of this invention and the computer measurement scheme as a function of the training period are compared for the aforementioned four types of data classification. Figures 4-7 As shown.
[0097] Table 1. Image classification accuracy of quantum neural networks under different quantum classical interaction information extraction schemes.
[0098] Computational base projection measurement Rotational unbiased basis measurement Synchronous unbiased basis measurement MNIST 65.3 96 95 Fashion MNIST 73.8 86.5 85.4 EMNIST 33.1 71 71 CIFAR10 34.4 52.1 53.5
[0099] The method for extracting information from quantum classical interactions in quantum neural networks proposed in this invention is not limited to the quantum neural network structure shown in this scheme, but can be applied to quantum machine learning of other types of quantum network structures.
Claims
1. A method for extracting information from quantum classical interactions in quantum machine learning, characterized in that, Includes the following steps: Step 1: Expand the image matrix by padding with zeros; Step 2: Use a sliding window to extract multiple sub-matrices; Step 3: Flatten each sub-matrix into a vector and encode it into the qubit circuit as the amplitude to prepare the quantum state; Step 4: Evolve the initial quantum state and extract features by parameterizing the quantum circuit, which includes alternating entanglement layers and rotation gate layers; Step 5: Perform measurements in batches, using either a rotating unbiased measurement basis information extraction scheme or a synchronous unbiased measurement basis information extraction scheme for each batch; Step 6: Combine the measurement results of each submatrix into an eigenvector; Step 7: Reduce the dimension of the feature vector to the number of classes using a fully connected layer; Step 8: Use the Softmax function to convert the real vector into a probability distribution to obtain the image classification result.
2. The method for extracting information from quantum classical interactions in quantum machine learning according to claim 1, characterized in that, Step 1 specifically involves: The n×n input image matrix X is expanded into a (n+2p)×(n+2p) matrix X′ by zero-padding, where p is the number of padding pixels, used to preserve edge features and adapt to subsequent quantum convolution operations; the zero-padding operation satisfies: Where, X′ i，j X represents the value of the newly generated image pixel. i-p，j-p This represents the value of the original image pixel.
3. The method for extracting information from quantum classical interactions in quantum machine learning according to claim 2, characterized in that, Step 2 specifically involves: Using a k×k sliding window, traverse X′ with a step size s to extract m submatrices {M1, M2, ..., M}. m },in:
4. The method for extracting information from quantum classical interactions in quantum machine learning according to claim 3, characterized in that, Step 3 specifically involves: Flatten each submatrix into k 2 dimensional vector v i Quantum states are prepared by amplitude encoding: Where ||v i || represents the vector magnitude, ensuring quantum state normalization; v i，j Represents the values in the submatrix, |ψ i > represents the encoded quantum state, and |j> represents the basis vector.
5. The method for extracting information from quantum classical interactions in quantum machine learning according to claim 4, characterized in that, Step 4 specifically involves: Construct parameterized quantum circuits U(θ), including single-qubit gates RX, RY, RZ, H, SX, and two-qubit gates CRX, CNOT; RX, RY, RZ, and CRX are tunable, while H, SX, and CNOT construct inter-qubit correlations; the quantum state evolution process is represented as: |f i 〉=U(θ)|ψ i 〉 Where, |φ i > Represents the quantum state |ψ i The result after quantum circuit calculation.
6. The method for extracting information from quantum classical interactions in quantum machine learning according to claim 5, characterized in that, Step 5 specifically involves: By cyclically switching single-bit measurement bases Repeated preparation and evolution, or simultaneous use of single-bit measurement bases Measurement and evolution are performed to extract multi-angle features of quantum states.
7. The method for extracting information from quantum classical interactions in quantum machine learning according to claim 6, characterized in that, In step 6, the quantum circuit measurement results of each sub-matrix are recombined into a three-dimensional feature map, and then flattened into a one-dimensional vector.
8. The method for extracting information from quantum classical interactions in quantum machine learning according to claim 7, characterized in that, Step 7 specifically involves: Applying a fully connected layer with trainable parameters, the output dimension is equal to the number of image categories, as shown below: Where Z represents the result, and W represents the weight matrix. Let b represent the vector in step 6, and b represent the bias term.
9. The method for extracting information from quantum classical interactions in quantum machine learning according to claim 8, characterized in that, Step 8 specifically involves: Applying the Softmax function to the output z yields the class probability distribution: Among them, z i Let z represent the i-th element. j K represents all elements, C represents a total of C categories, and K and C are of equal size.
10. The method for extracting information from quantum classical interactions in quantum machine learning according to claim 9, characterized in that, The specific details of the rotating unbiased measurement basis information extraction scheme and the synchronous unbiased measurement basis information extraction scheme are as follows: During the training and testing of a quantum machine learning network, in each training cycle, when it is necessary to measure the quantum state of the quantum machine learning network, each measurement basis in a set of unbiased basis sets is used in turn. Specifically, if there are m measurement basis sets in the selected unbiased measurement basis set M, then in the first cycle of measuring the quantum state of the machine learning network, the first measurement basis M1 of the unbiased measurement basis set M is used for measurement, in the second cycle, the second measurement basis M2 of the unbiased measurement basis set M is used for measurement, and so on, with the m-th measurement basis M of the unbiased measurement basis set M being used in the m-th cycle. m Measurements are taken using the first measurement basis M1 of the unbiased measurement basis set M in the (m+1)th cycle, and the unbiased measurement basis is used cyclically. During the training and testing of a quantum machine learning network, when it is necessary to measure the quantum state of the quantum machine learning network in each training cycle, each measurement basis in a set of unbiased basis sets is used simultaneously. Specifically, if there are m measurement basis sets in the selected set of unbiased measurement basis sets M, then in each cycle of quantum state measurement of the machine learning network, all measurement basis sets in the set of unbiased measurement basis sets M are used to measure the quantum state, and the mean of the results obtained by all measurement basis sets is the result of that measurement. The unbiased measurement basis is defined as follows: set up and It is a d-dimensional Hilbert space C d Two sets of orthonormal bases exist if and only if the following equation is satisfied. i,j=0,...,j,d-1; in space C d In this context, a set {M1, M2, ..., M} consisting of an orthonormal basis is given. m A set is called an unbiased basis if M1, M2, ..., M m It is pairwise unbiased; if a physical system is in a characteristic state of M1, then when measuring this system with another set of bases M2, the probability of obtaining all measurement results is the same; specifically, for a qubit... |α| 2 +|β| 2 =1, and the corresponding set of unbiased measurement bases is the Pauli matrix {σ} x ,σ y ,σ z The three sets of basis vectors formed by the eigenvectors of} are respectively These three sets of bases are unbiased measurement bases.