Data dimension reduction method based on quantum mechanical characteristics

Abstract

The invention provides a data dimension reduction method based on quantum mechanical characteristics. The method comprises the following steps: S1, performing nonlinear dimension reduction processing on high-dimensional data through quantum kernel principal component analysis; s2, performing linear dimensionality reduction on the output of the S1 based on a quantum neighborhood preserving embedding algorithm of mahalanobis distance; and S3, mapping the output of the S2 to a low-dimensional space by adopting a quantum variational manifold learning algorithm, and generating a final dimension reduction result. According to the method, efficient dimension reduction of high-dimensional data can be realized, the efficiency and precision of quantum machine learning preprocessing are improved, the calculation complexity is reduced, and the accuracy and reliability of a dimension reduction result are improved.

Description

Technical Field

[0001] This application relates to the field of quantum machine learning preprocessing, specifically to a data dimensionality reduction method based on quantum mechanical properties and its quantum kernel principal component analysis, quantum neighborhood-preserving embedding, and quantum variable manifold learning algorithm. Background Technology

[0002] In recent years, with the rapid development of internet technology, the amount of data has exploded, and the growth rate of data dimensions far exceeds the growth of the data itself. The coherence and redundancy problems commonly found in high-dimensional data pose a serious "curse of dimensionality" challenge to traditional machine learning methods. Data dimensionality reduction technology, as a key means to solve this problem, is becoming increasingly important. Quantum computing, with its computational characteristics that follow the laws of quantum mechanics, has demonstrated significant advantages over classical computing in certain computational problems. Quantum machine learning, as an intersection of quantum computing and artificial intelligence, has been listed as an important research direction for the development of artificial intelligence in my country.

[0003] Current research on quantum data dimensionality reduction still faces many technical bottlenecks: existing algorithms are mostly limited to the quantization of a single dimensionality reduction step, lacking a complete quantum dimensionality reduction process design; quantum state transitions and cooperative mechanisms between different dimensionality reduction steps have not yet been established; research on quantum implementation methods for nonlinear dimensionality reduction is severely insufficient; and efficient utilization schemes for quantum computing resources urgently need improvement. These problems prevent existing quantum dimensionality reduction algorithms from fully leveraging the advantages of quantum computing and failing to meet the needs of high-dimensional data processing in practical applications. Especially when processing high-dimensional data with complex nonlinear structures, existing quantum dimensionality reduction methods have significant shortcomings in preserving the data manifold structure and optimizing the dimensionality reduction effect.

[0004] To address the aforementioned issues, existing technologies urgently need improvement. Summary of the Invention

[0005] The purpose of this application is to provide a data dimensionality reduction method based on quantum mechanical properties and its quantum kernel principal component analysis, quantum neighborhood-preserving embedding and quantum variational manifold learning algorithms, which have the advantages of complete quantum dimensionality reduction process design, establishment of multi-step collaborative mechanism, improvement of nonlinear dimensionality reduction capability and optimization of quantum computing resource utilization efficiency.

[0006] This application provides a data dimensionality reduction method based on quantum mechanical properties, applied to the field of quantum machine learning preprocessing. The technical solution includes the following steps: 1. Perform nonlinear dimensionality reduction processing on high-dimensional data through quantum kernel principal component analysis; 2. Perform linear dimensionality reduction on the output of S1 using a quantum neighborhood-preserving embedding algorithm based on Mahalanobis distance; 3. Map the output of S2 to a low-dimensional space using a quantum variable manifold learning algorithm to generate the final dimensionality reduction result.

[0007] Furthermore, this application also proposes that the quantum nuclear principal component analysis of S1 includes:

[0008] Hamiltonian simulations are used to construct the high-dimensional feature space corresponding to the kernel function;

[0009] Unitary transform circuits are obtained through quantum linear equation solving algorithms;

[0010] To achieve quantum state transformations using Gaussian kernels or exponential kernel functions.

[0011] Furthermore, this application also proposes that the quantum neighborhood-preserving embedding algorithm for S2 includes:

[0012] Calculate the sample neighborhood matrix based on Mahalanobis distance;

[0013] Quantum state representation of the neighborhood matrix is ​​constructed using annihilation and generation operators;

[0014] The weight matrix is ​​reconstructed by solving a system of quantum linear equations.

[0015] Furthermore, this application also proposes that the S3 quantum variational manifold learning algorithm includes:

[0016] A nonlinear mapping matrix is ​​prepared using a parametric quantum gate circuit U(θ);

[0017] Design a loss function C(θ) to evaluate the mapping effect;

[0018] The parameter θ is updated iteratively using the classic gradient descent optimizer.

[0019] Furthermore, this application proposes that the implementation of the Gaussian kernel function is as follows:

[0020] Convert the Euclidean distance between sample vectors into a quantum Hamiltonian operator;

[0021] The eigenvalues ​​of the kernel matrix are calculated using quantum phase estimation.

[0022] Furthermore, this application proposes that the construction of the neighborhood matrix specifically includes:

[0023] Preparing the initial state of data using quantum random access memory;

[0024] Calculate Mahalanobis distance using quantum covariance matrix;

[0025] Solving for dimensionality reduction mapping vectors based on generalized eigenvectors.

[0026] Furthermore, this application proposes that the loss function C(θ) is designed to satisfy:

[0027] Measuring the consistency of popular structures between the original high-dimensional data and the low-dimensional mapped data;

[0028] The loss function value is obtained through quantum measurement.

[0029] Furthermore, this application also proposes that the quantum circuit be simulated and verified on the IBM Q quantum cloud computing platform, specifically including:

[0030] The unitary transform operation is implemented using quantum gate circuits;

[0031] The dimensionality reduction results are output using quantum state tomography.

[0032] Furthermore, this application proposes that the calculation of the Mahalanobis distance further includes:

[0033] Constructing a quantum version of the data covariance matrix;

[0034] The efficiency of nearest neighbor search is optimized using a quantum amplitude amplification algorithm.

[0035] Furthermore, this application proposes that the optimization of parameter θ further includes:

[0036] Compress the mapping matrix using variational quantum singular value decomposition;

[0037] The convergence of the loss function is accelerated by combining quantum convex optimization algorithms.

[0038] As can be seen from the above, the data dimensionality reduction method based on quantum mechanical properties provided in this application, along with its quantum kernel principal component analysis, quantum neighborhood-preserving embedding, and quantum variable manifold learning algorithms, constructs a complete quantum dimensionality reduction process through the three-stage collaborative processing of quantum kernel principal component analysis, quantum neighborhood-preserving embedding, and quantum variable manifold learning. This achieves an organic combination of nonlinear and linear dimensionality reduction within the quantum computing framework, while simultaneously optimizing the efficiency of quantum resource utilization. It has the advantages of a complete quantum dimensionality reduction process design, the establishment of a multi-step collaborative mechanism, improved nonlinear dimensionality reduction capabilities, and optimized efficiency of quantum computing resource utilization. Attached Figure Description

[0039] The above and other objects, features, and advantages of exemplary embodiments of the present invention will become readily apparent from the following detailed description taken in conjunction with the accompanying drawings. Several embodiments of the invention are illustrated in the drawings by way of example and not limitation, wherein:

[0040] Figure 1 This is a flowchart illustrating a data dimensionality reduction method based on quantum mechanical properties provided in an embodiment of the present invention. Detailed Implementation

[0041] The technical solutions of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of the embodiments. The components of this application described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of this application provided in the accompanying drawings is not intended to limit the scope of the claimed application, but merely represents selected embodiments of this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application without inventive effort are within the scope of protection of this application. It should be noted that similar reference numerals and letters in the following drawings indicate similar items; therefore, once an item is defined in one drawing, it does not need to be further defined and explained in subsequent drawings. Furthermore, in the description of this application, the terms "first," "second," etc., are used only to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0042] In existing technologies, quantum data dimensionality reduction algorithms mostly focus on accelerating and optimizing individual processing steps, lacking a systematic design for the overall process. Traditional methods often handle nonlinear and linear dimensionality reduction processes independently, resulting in low data transformation efficiency between steps and an inability to effectively preserve the topological structure of high-dimensional data. When processing high-dimensional financial transaction data, existing quantum algorithms often suffer from feature information loss due to improper connection between dimensionality reduction steps, affecting the prediction accuracy of subsequent machine learning models.

[0043] To address the aforementioned issues and the fragmentation problem of existing quantum dimensionality reduction processes, this paper first identifies the inefficiency of feature space construction in the quantum implementation of classical kernel principal component analysis. By analyzing the correlation between quantum phase estimation and Hamiltonian simulation, a proposal is made to map the kernel function to the quantum state space. Next, the shortcomings of the neighborhood-preserving algorithm in eliminating feature correlations are observed, and data relationships are reconstructed by combining Mahalanobis distance and the quantum covariance matrix. Finally, to solve the challenge of quantum implementation of manifold mapping, parameterized quantum gate circuits are designed to construct the nonlinear transformation matrix, forming a three-level processing framework.

[0044] Therefore, this application proposes a data dimensionality reduction method based on quantum mechanical properties, such as... Figure 1 As shown, this method is applied to the field of quantum machine learning preprocessing and includes the following steps: S1, performing nonlinear dimensionality reduction on high-dimensional data through quantum kernel principal component analysis; S2, performing linear dimensionality reduction on the output of the previous step using a quantum neighborhood-preserving embedding algorithm based on Mahalanobis distance; S3, using a quantum variable manifold learning algorithm to map the output of the previous step to a low-dimensional space to generate the final dimensionality reduction result.

[0045] Among them, quantum kernel principal component analysis refers to simulating the high-dimensional feature space corresponding to the kernel function using quantum computing. Specifically, it can use Hamiltonians to simulate and construct the kernel matrix, and obtain the unitary transform circuit by solving a system of quantum linear equations, which is used for feature extraction of nonlinear data structures. Quantum neighborhood-preserving embedding algorithm refers to calculating the Mahalanobis distance between samples based on the quantum covariance matrix. Specifically, it can use annihilation operators and generation operators to construct the quantum state representation of the neighborhood matrix, eliminating interference from correlations between features. Quantum variational manifold learning algorithm refers to realizing nonlinear mapping through parameterized quantum gate circuits. Specifically, it can design parameterized quantum gate circuits to prepare the mapping matrix, and combine classical optimization algorithms to adjust parameters to preserve the manifold structure.

[0046] Specifically, the quantum kernel principal component analysis stage converts the classical kernel function into a quantum-processable form through Hamiltonian simulation and accelerates the calculation of kernel matrix eigenvalues ​​using quantum phase estimation, providing optimized nonlinear features for subsequent processing. The quantum neighborhood-preserving embedding stage reconstructs the data neighborhood relationships based on the quantum covariance matrix and obtains the weight matrix by solving a system of quantum linear equations, effectively preserving the local linear structure. The quantum variational manifold learning stage uses parameterized quantum gates to construct nonlinear mappings and achieves globally optimal dimensionality reduction through iterative optimization of gate circuit parameters. These three processing stages form a complete quantum-classical cooperative dimensionality reduction path.

[0047] Compared to existing technologies, most quantum dimensionality reduction methods employ a single processing mode, such as only quantizing kernel principal component analysis or optimizing neighborhood-preserving algorithms separately. This scheme achieves an organic combination of nonlinear and linear dimensionality reduction through a three-level processing framework. Quantum kernel processing provides feature optimization input for subsequent steps, quantum neighborhood preservation ensures the integrity of the topological structure, and finally, variational mapping achieves global optimum. This progressive processing overcomes the shortcomings of traditional methods, such as fragmented steps and low information transfer efficiency.

[0048] Through the above technical solution, this application effectively solves the systemic connection problem of the quantum dimensionality reduction process. In the scenario of financial fraud detection, it can completely preserve the spatiotemporal correlation characteristics of high-dimensional transaction data, and the dimensionality-reduced data exhibits higher anomaly detection accuracy in machine learning models. At the same time, it takes into account both nonlinear feature extraction and linear structure preservation, avoiding the loss of key pathological features caused by the fragmented steps in traditional methods when processing medical image data.

[0049] This application further proposes quantum nuclear principal component analysis, which includes constructing a high-dimensional feature space corresponding to the kernel function using Hamiltonian simulation, obtaining a unitary transformation circuit through a quantum linear equation solving algorithm, and realizing quantum state transformation of Gaussian kernels or exponential kernel functions.

[0050] Hamiltonian simulation, specifically constructing the high-dimensional feature space corresponding to the kernel function, refers to mapping the classical kernel function to the energy evolution process of the quantum system. This can be achieved using quantum phase estimation and Hamiltonian time evolution operations, thereby encoding the high-dimensional feature space of the kernel matrix in the quantum state. Solving quantum linear equations to obtain the unitary transform circuit involves solving the linear equations corresponding to the kernel matrix using the HHL algorithm or its variants. This can be achieved by loading data into a quantum random access memory and constructing a coefficient matrix, thus generating the unitary transform circuit for dimensionality reduction. Realizing quantum state transformation using a Gaussian or exponential kernel function involves transforming the mathematical expression of the nonlinear kernel function into a superposition of quantum states. This can be achieved using quantum interference effects and controllable rotation gate operations, thereby characterizing the nonlinear properties of the kernel function in the quantum system.

[0051] Specifically, in quantum principal component analysis (PCA), the high-dimensional feature space corresponding to the classical kernel function is first mapped to the energy eigenstates of the quantum system through Hamiltonian simulation, and the feature information of the kernel matrix is ​​extracted using quantum phase estimation. Then, a quantum linear equation solving algorithm is used to handle the covariance structure of the kernel matrix, and an orthogonal transformation of the data in the feature space is achieved by constructing a quantum circuit. To address the nonlinear characteristics of Gaussian or exponential kernel functions, quantum state encoding of the kernel matrix is ​​achieved through quantum state superposition and phase rotation operations. For example, a controlled rotation gate is used to adjust the phase of the qubits to simulate the distance decay characteristics of the kernel function.

[0052] Compared to existing technologies, traditional quantum kernel methods are typically limited to linear kernel function processing and rely on classical covariance matrix calculations. Existing quantum principal component analysis requires explicit construction of the density matrix, and its computational complexity increases exponentially with the data dimensionality. Our proposed solution, however, directly constructs the nonlinear kernel space through Hamiltonian simulation, avoiding explicit matrix storage and computation. Furthermore, it employs a quantum linear algebra algorithm for eigenvalue decomposition, maintaining nonlinear feature extraction capabilities while reducing resource consumption.

[0053] Through the above technical solutions, this application solves the problem of efficient implementation of nonlinear kernel functions in a quantum environment, realizes quantum state encoding of high-dimensional feature space, improves the computational efficiency of kernel principal component analysis, and expands the ability of quantum dimensionality reduction algorithms to process complex data structures.

[0054] This application further proposes a technical solution that calculates the sample neighborhood matrix based on Mahalanobis distance, constructs the quantum state representation of the neighborhood matrix using annihilation and generation operators, and reconstructs the weight matrix by solving a system of quantum linear equations.

[0055] The Mahalanobis distance refers to the distance between samples adjusted by the data covariance matrix, which can be implemented using a quantum covariance matrix calculation module. This module obtains the data distribution characteristics through quantum density of states matrix operations. The annihilation operator and the generation operator are linear operators describing the change in the number of particles in a quantum system. They can be implemented using a combination of raising and lowering operators in the quantum harmonic oscillator model, used to transform classical neighborhood relations into quantum superposition states. Solving the quantum linear equations refers to a method that uses quantum parallelism to accelerate matrix inversion operations. This can be implemented using the HHL algorithm framework, obtaining the quantum state solution of the weight matrix through quantum phase estimation and controlled rotation operations.

[0056] Specifically, in the process of preserving embedding in the quantum neighborhood, a quantum covariance matrix is ​​first constructed to eliminate the interference of data dimensionality correlation on the distance metric, and parallel computation of Mahalanobis distance is achieved through quantum gate operations. Then, utilizing the algebraic properties of fermionic operators, the classical neighborhood matrix is ​​encoded into the particle-occupancy states of the quantum system, forming a quantum state representation with space compression properties. Finally, leveraging the exponential speedup capability of the quantum linear solver, the weight matrix containing neighborhood relationships is efficiently reconstructed, overcoming the time complexity limitations of classical algorithms in large matrix operations.

[0057] Compared to existing technologies, traditional quantum dimensionality reduction methods, which use Euclidean distance, suffer from a lack of covariance structure modeling. Storing the neighborhood matrix requires exponential qubit resources, and solving the weight matrix relies on classical computing interfaces. This scheme precisely characterizes the data distribution properties using the quantum covariance matrix, achieves compact encoding of neighborhood relationships using the particle number representation, and completes the fully quantumized calculation of the weight matrix based on a quantum solver, forming a complete quantum processing chain.

[0058] Through the above technical solution, this application effectively solves the problem of neighborhood relationship distortion caused by inaccurate modeling of data covariance structure, realizes quantum state compression storage of high-dimensional neighborhood matrix, reduces the computational complexity of reconstructing weight matrix from polynomial level of classical algorithm to logarithmic level, and avoids the data transmission bottleneck caused by quantum-classical hybrid computing.

[0059] This application further proposes a quantum variational manifold learning algorithm for S3, which includes preparing a nonlinear mapping matrix using a parameterized quantum gate circuit U(θ), ​​designing a loss function C(θ) to evaluate the mapping effect, and using a classical gradient descent optimizer to iteratively update the parameter θ.

[0060] The parametric quantum gate circuit U(θ) refers to a quantum circuit composed of a sequence of quantum gates controlled by an adjustable parameter θ. Specifically, it can be implemented using a combination of single-qubit rotation gates and two-qubit entangled gates. By adjusting the rotation angle parameter θ, the quantum state evolution path is altered, thereby generating a nonlinear mapping matrix adaptable to different data distributions. The loss function C(θ) is an evaluation function used to quantify the difference in manifold structure between high-dimensional data and low-dimensional mapped data. Specifically, it can obtain the difference in distance distribution between data points before and after dimensionality reduction through quantum measurements, for example, using quantum state fidelity or trace distance as evaluation metrics to establish a quantifiable optimization objective. The classical gradient descent optimizer is a parameter optimization algorithm based on classical computation. Specifically, it can use automatic differentiation techniques to calculate the gradient of the loss function with respect to parameter θ, and update parameter θ through backpropagation, achieving collaborative iteration between the quantum circuit parameters and the classical optimizer.

[0061] Specifically, in the quantum variational manifold learning process, the input data is first quantum-state encoded using a parametric quantum gate circuit U(θ), ​​generating a mapping matrix with nonlinear characteristics. Adjusting the parameters θ of this quantum circuit directly affects the generation of the mapping matrix, making the nonlinear mapping process adjustable. Subsequently, the dimensionality-reduced data distribution is obtained through quantum measurement operations, and the loss function C(θ) is calculated based on a pre-defined manifold structure preservation criterion. This loss function transforms the topological relationship between high-dimensional and low-dimensional data into a computable numerical index, such as by comparing the similarity between the original adjacency graph and the dimensionality-reduced adjacency graph for quantitative evaluation. Finally, the parameter θ is iteratively optimized using the classical gradient descent algorithm, adjusting the quantum gate parameters according to the gradient direction of the loss function to gradually reduce the manifold structure difference until convergence is achieved.

[0062] Compared to existing technologies, current quantum dimensionality reduction methods typically use fixed-structure quantum circuits to generate mapping matrices, which cannot dynamically adjust the mapping method according to data characteristics and lack a quantitative evaluation mechanism for the dimensionality reduction effect. In contrast, this scheme achieves flexible generation of nonlinear mappings through parameterized quantum circuits and constructs a closed-loop optimization system by combining quantum measurement with classical optimization algorithms. While maintaining the parallel advantages of quantum computing, it solves the problems of rigid mapping methods and unclear optimization objectives in traditional methods.

[0063] Through the above technical solutions, this application realizes the flexible preparation of nonlinear mapping matrices, establishes a quantitative evaluation system for manifold structure preservation, and improves parameter convergence efficiency through a quantum-classical hybrid optimization framework, effectively solving the technical obstacles of uncontrollable mapping process, unclear optimization objectives, and high computational resource consumption in quantum nonlinear dimensionality reduction.

[0064] This application further proposes to convert the Euclidean distance between sample vectors into a quantum Hamiltonian operator and to calculate the eigenvalues ​​of the kernel matrix through quantum phase estimation.

[0065] In this context, a quantum Hamiltonian operator refers to a linear operator used to describe the energy state of a quantum system. Specifically, it can be implemented using a Pauli matrix combination of qubits. By encoding the square of the Euclidean distance as the eigenvalues ​​of the operator, classical data relations are transformed into physical quantities that can be processed by quantum mechanics. Quantum phase estimation refers to an eigenvalue extraction method based on the quantum Fourier transform. Specifically, it can be implemented using controlled unitary operations and inverse Fourier transform circuits, obtaining the eigenspectral information of the kernel matrix through parallel quantum superposition states.

[0066] Specifically, the Euclidean distance between sample vectors is converted into the eigenvalues ​​of a quantum Hamiltonian operator, allowing distance relationships in the classical data space to be directly mapped to the energy eigenstates of the quantum system. The quantum phase estimation algorithm operates on the unitary transform corresponding to this operator, simultaneously calculating all eigenvalues ​​of the kernel matrix through quantum parallelism, avoiding the iterative process of sequentially calculating eigenvalue decompositions in classical algorithms. This process utilizes the superposition property of quantum states to complete the global analysis of the kernel matrix eigenvalues ​​in a single operation.

[0067] Compared to existing technologies, traditional quantum kernel methods are limited by the implementation of linear kernels or low-complexity kernel functions, making it difficult to handle the eigenvalue decomposition of nonlinear kernel functions such as Gaussian kernels. Existing techniques typically require explicit construction of the kernel matrix followed by classical eigenvalue calculation, leading to an exponential increase in computational complexity with data dimensionality. This scheme bypasses the explicit kernel matrix construction step by directly combining quantum mechanical operator mapping and phase estimation, achieving direct quantum state transformation and parallel eigenvalue calculation for nonlinear kernel functions.

[0068] Through the above technical solutions, this application solves the problem of the difficulty in efficiently implementing Gaussian kernel functions under the quantum computing framework, realizes the rapid calculation of kernel matrix eigenvalues ​​in nonlinear kernel principal component analysis, and provides a feasible quantum implementation path for nonlinear data dimensionality reduction in quantum machine learning.

[0069] This application further proposes the following specific methods for constructing the neighborhood matrix: preparing the initial data state using a quantum random access memory; calculating the Mahalanobis distance using a quantum covariance matrix; and solving for the dimension-reduced mapping vector based on the generalized eigenvector.

[0070] Quantum random access memory (QRAM) refers to a quantum storage device used for storing and quickly accessing high-dimensional data. Specifically, it can be implemented by encoding data points using a superposition of qubit states, and the storage efficiency of high-dimensional data is improved through parallel loading of quantum states. The quantum covariance matrix refers to a data correlation representation matrix constructed based on the inner product operation of quantum states. Specifically, it can be implemented by performing covariance operations on data points using a quantum phase estimation algorithm, and the calculation process of the covariance matrix is ​​accelerated through quantum parallelism. The generalized eigenvector solution is the core step in extracting the data manifold structure using quantum linear algebra algorithms. Specifically, it can be implemented by combining quantum principal component analysis with the HHL algorithm, and the dimensionality of the feature space is compressed through quantum state rotation operations.

[0071] Specifically, in quantum random access memory (QRAM), high-dimensional data is encoded as a superposition of quantum states, allowing all data points to be accessed and processed synchronously. The calculation of the quantum covariance matrix involves performing inner product operations on data points using quantum circuits, transforming the covariance calculation in classical statistics into a sequence of quantum gate operations. This utilizes quantum superposition to simultaneously handle the correlation analysis of all data pairs. The calculation of Mahalanobis distance is reconstructed as a phase difference measurement process between quantum states, obtaining the quantum state representation of the distance matrix through quantum interference effects. The solution for the generalized eigenvectors employs quantum phase estimation and conditional rotation operations, mapping the data manifold structure to a low-dimensional feature space while preserving the quantum state expression of neighborhood topological relationships.

[0072] Compared to existing technologies, traditional neighborhood matrix construction requires point-by-point computation of the covariance matrix and storage of intermediate results. This scheme, however, utilizes quantum superposition to achieve parallel data processing, reducing the computational complexity of covariance from polynomial to logarithmic levels. Existing quantum algorithms typically neglect the quantized reconstruction of the covariance matrix when processing Mahalanobis distance, while this scheme directly constructs the quantum expression of the covariance matrix through quantum phase estimation, avoiding the repeated loading of classical data. Existing technologies using classical eigenvalue decomposition algorithms face exponential complexity growth when processing high-dimensional data; this scheme achieves quantized compression of the feature space through quantum principal component analysis, reducing computational resource consumption while maintaining the manifold structure.

[0073] Through the above technical solutions, this application realizes the efficient quantum construction of high-dimensional data neighborhood matrices, solves the storage bottleneck problem of traditional quantum dimensionality reduction algorithms when dealing with covariance correlation operations, improves the parallel processing capability of Mahalanobis distance calculation, and maintains the integrity of the data manifold structure through quantum eigenvalue decomposition.

[0074] This application further proposes a loss function C(θ) designed to measure the manifold structure consistency between the original high-dimensional data and the low-dimensional mapped data, and obtains the loss function value through quantum measurement.

[0075] Among them, manifold structure consistency refers to the correspondence between high-dimensional and low-dimensional data in terms of topological structure. Specifically, it can be achieved by using quantum entanglement to associate the geometric relationships between data points of different dimensions. This design can capture the inherent topological constraints of the data. Quantum measurement to obtain the loss function value refers to extracting a quantitative index of the mapping effect through quantum state projection operations. Specifically, it can be achieved by using a quantum phase estimation circuit to realize parallel measurement of multiple qubits. This method utilizes the characteristics of quantum superposition to simultaneously obtain structural difference information of multiple data points.

[0076] Specifically, in the quantum variational manifold learning process, after preparing a nonlinear mapping matrix using parametric quantum gate circuits, the difference between the original high-dimensional data and the manifold structure of the dimensionality reduction result is encoded as entanglement relationships between quantum states. The quantum measurement module performs joint measurements on the entangled states, generating a probability amplitude distribution reflecting structural consistency. This probability amplitude distribution, after classical transformation, forms a loss function value, driving the parameter optimization module to adjust the quantum gate parameter θ, ensuring that the low-dimensional space mapping maintains the topological characteristics of the original data.

[0077] Compared to existing technologies, traditional quantum dimensionality reduction methods typically only calculate point-to-point distance differences without establishing a global structural constraint mechanism. Existing technologies use classical covariance matrices to analyze manifold structures, and their computational complexity increases exponentially with data dimensionality. This scheme characterizes the topological relationships of manifolds through quantum entangled states, directly constructs structural consistency constraints in Hilbert space, and simultaneously acquires global structural information using a quantum parallel measurement mechanism.

[0078] Through the above technical solutions, this application effectively solves the problem of manifold structure distortion during quantum nonlinear dimensionality reduction, and achieves preservation of high-dimensional data topological features in quantum machine learning preprocessing scenarios. Simultaneously, the loss function calculation method based on quantum measurement breaks through the dimensionality limitations of classical algorithms in evaluating efficiency, providing feasibility for processing large-scale datasets.

[0079] This application further proposes to perform simulation verification on a quantum cloud computing platform, specifically including using quantum gate circuits to implement unitary transformation operations and using quantum state tomography to output dimensionality reduction results.

[0080] Among these, the implementation of unitary transformation operations using quantum gate circuits refers to constructing unitary matrix transformations through combinations of logic gates supported by quantum hardware. Specifically, this can be achieved using the standardized quantum gate instruction set provided by the IBM Q platform, such as combining Hadamard gates, CNOT gates, and rotation gates to form specific unitary transformation circuits. This method directly maps to the physical implementation of quantum hardware and can reflect the noise interference and decoherence effects during the operation of real quantum devices. The output dimensionality reduction result of quantum state tomography refers to reconstructing the density matrix by measuring the projection distribution of quantum states under multiple basis vectors. Specifically, maximum likelihood estimation or Bayesian inference methods can be used to process the measurement data. This technique can completely acquire quantum state information, avoiding the verification bias caused by traditional simulations that only obtain partial measurement basis vector data.

[0081] Specifically, the core steps of the quantum dimensionality reduction algorithm are converted into quantum gate circuits running on the IBM Q platform. When unitary transformation operations are performed through quantum hardware, actual parameters such as coupling errors between qubits and gate operation fidelity are automatically incorporated into the simulation process, thus realistically reflecting the algorithm's operating state in the physical device. Quantum state tomography reconstructs the full information of the dimensionality-reduced quantum state. By repeatedly preparing the same quantum state and performing projection measurements under different measurement bases, complete statistical distribution data of the quantum state are obtained, thereby verifying the accuracy of the dimensionality reduction results.

[0082] Compared to existing technologies, traditional simulation verification typically involves numerically simulating quantum circuits on classical computers. This approach fails to accurately reflect the noise models and decoherence effects unique to quantum hardware, leading to discrepancies between the verification results and the actual operation of quantum devices. In contrast, this solution directly constructs the verification environment on a quantum cloud computing platform. It physically implements the core algorithm steps using quantum gate circuits and combines quantum state tomography to obtain complete quantum state information, ensuring complete compatibility between the verification process and the real quantum computing environment.

[0083] Through the above technical solutions, this application can accurately evaluate the performance of quantum dimensionality reduction algorithms in real quantum hardware environments, effectively identify the robustness of the algorithms to quantum noise, and ensure the verification integrity of the dimensionality reduction results through full-information quantum state reconstruction, thus providing a reliable experimental verification basis for the actual deployment of quantum machine learning algorithms.

[0084] This application further proposes a quantum version of constructing the data covariance matrix during quantum data dimensionality reduction and optimizes the nearest neighbor search efficiency through a quantum amplitude amplification algorithm.

[0085] The quantum version of the data covariance matrix refers to transforming the covariance matrix, which describes the correlation between data dimensions in classical statistics, into a quantum-state operable expression. Specifically, this can be achieved by using the quantum superposition principle to perform parallel computation on multidimensional data features. This feature allows the characterization of covariance relationships to be completed directly at the quantum state level. The quantum amplitude amplification algorithm refers to a probability amplitude control mechanism based on an improved Grover search algorithm. Specifically, it can be achieved by using quantum phase rotation operations to directionally amplify the probability amplitude corresponding to the target's nearest neighbors. This feature transforms the nearest neighbor search process, which requires traversal computation in traditional algorithms, into a quantum state amplitude control process.

[0086] Specifically, in a quantum computing device, the input high-dimensional dataset is first quantum-state encoded. Utilizing quantum parallelism, the covariance relationships between each dimension are simultaneously calculated, generating a quantum version of the covariance matrix. This matrix is ​​embedded into the quantum circuitry through quantum gate operations, providing the mathematical foundation for subsequent distance calculations. In the nearest neighbor search phase, the quantum amplitude amplification algorithm iteratively executes reflection and phase-flip operations to gradually increase the probability amplitude of the target nearest neighbor quantum states, enabling the measurement to obtain the desired nearest neighbor samples with a higher probability. The synergistic effect of these two technical features allows the Mahalanobis distance calculation process to retain the mathematical rigor of classical statistics while fully leveraging the parallel advantages of quantum computing.

[0087] In some specific implementations, the construction of the quantum covariance matrix can be achieved by quickly recalling data features through a quantum random access memory, and the phase rotation angle of the quantum amplitude amplification algorithm can be dynamically adjusted according to the characteristics of the dataset.

[0088] Compared to existing technologies, traditional methods, which require correlation analysis dimension-by-dimensional calculation of the classical covariance matrix, utilize quantum parallel computing to simultaneously acquire covariance relationships. Existing nearest neighbor searches typically require traversing the entire dataset, while this approach reduces the search complexity from linear to square root level through a probability amplitude-oriented amplification mechanism.

[0089] Through the above technical solution, this application effectively solves the problems of low efficiency in covariance calculation and time-consuming nearest neighbor search in the process of quantum data dimensionality reduction, enabling the distance calculation of high-dimensional data to be accelerated under the quantum computing framework, and providing more efficient preprocessing support for subsequent dimensionality reduction operations.

[0090] This application further proposes an optimization method for quantum data dimensionality reduction, which compresses the mapping matrix through variational quantum singular value decomposition and accelerates the convergence of the loss function by combining quantum convex optimization algorithm.

[0091] Variational quantum singular value decomposition (VSD) refers to a quantum computing method that approximates the singular value decomposition of a matrix using parameterized quantum circuits. Specifically, it can be implemented by constructing a variational quantum circuit with parameterized quantum gate sequences, and by optimizing the circuit parameters, the VSD of the target matrix is ​​approximated. This feature leverages the linear algebraic acceleration capability of quantum computing to reduce the dimension of the mapping matrix, thereby reducing the computational resources required for parameter optimization.

[0092] Quantum convex optimization algorithms are optimization methods based on parallel search for optimal solutions using quantum superposition states. Specifically, they can be implemented using a combination of quantum amplitude amplification and phase estimation algorithms, simultaneously evaluating the loss function values ​​of multiple candidate solutions through the superposition property of quantum states. This feature leverages quantum parallel computing mechanisms to rapidly traverse the parameter space, overcoming the convergence speed limitations of classical optimization algorithms.

[0093] Specifically, in the quantum variational manifold learning process, the high-dimensional mapping matrix is ​​first subjected to singular value decomposition using a parameterized quantum circuit, and minor singular values ​​are truncated according to a preset threshold to achieve matrix compression. The compressed low-dimensional matrix is ​​then converted into classical data using quantum state tomography, which serves as the input for subsequent optimization processes. In the loss function optimization stage, the quantum convex optimization algorithm encodes the parameter space as a quantum superposition state, accurately calculates the gradient of the loss function corresponding to each parameter combination through quantum phase estimation, and utilizes quantum interference effects to screen out parameter update directions that rapidly reduce the loss function.

[0094] Compared to existing technologies, traditional parameter optimization methods rely on the classical gradient descent algorithm, whose computational complexity increases exponentially with matrix dimension and is prone to getting trapped in local optima in non-convex optimization scenarios. This scheme achieves matrix dimension compression through quantum singular value decomposition, reducing the size of the optimization parameters to the logarithmic level. Simultaneously, the quantum convex optimization algorithm leverages quantum parallelism to achieve superlinear convergence speed, effectively avoiding local optimum traps.

[0095] Through the above technical solutions, this application significantly reduces the computational resource consumption of the parameter optimization process while maintaining the integrity of the data manifold structure. At the same time, it achieves rapid convergence of the loss function through a quantum parallel search mechanism, thus solving the core technical bottlenecks of low parameter optimization efficiency and slow convergence speed in quantum dimensionality reduction algorithms.

[0096] The above description is merely an embodiment of this application and is not intended to limit the scope of protection of this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the scope of protection of this application.

Claims

1. A data dimensionality reduction method based on quantum mechanical properties, applied to the field of quantum machine learning preprocessing, characterized in that, Includes the following steps: S1. Perform nonlinear dimensionality reduction on high-dimensional data using quantum kernel principal component analysis; S2. The quantum neighborhood-preserving embedding algorithm based on Mahalanobis distance performs linear dimensionality reduction on the output of S1; S3. The quantum variable manifold learning algorithm is used to map the output of S2 to a low-dimensional space to generate the final dimensionality reduction result.

2. The method according to claim 1, characterized in that: The quantum nuclear principal component analysis of S1 includes: Hamiltonian simulations are used to construct the high-dimensional feature space corresponding to the kernel function; Unitary transform circuits are obtained through quantum linear equation solving algorithms; To achieve quantum state transformations using Gaussian kernels or exponential kernel functions.

3. The method according to claim 1, characterized in that, The quantum neighborhood-preserving embedding algorithm of S2 includes: Calculate the sample neighborhood matrix based on Mahalanobis distance; Quantum state representation of the neighborhood matrix is ​​constructed using annihilation and generation operators; The weight matrix is ​​reconstructed by solving a system of quantum linear equations.

4. The method according to claim 1, characterized in that, The quantum variational manifold learning algorithm of S3 includes: A nonlinear mapping matrix is ​​prepared using a parametric quantum gate circuit U(θ); Design a loss function C(θ) to evaluate the mapping effect; The parameter θ is updated iteratively using the classic gradient descent optimizer.

5. The method according to claim 2, characterized in that, The Gaussian kernel function is implemented as follows: Convert the Euclidean distance between sample vectors into a quantum Hamiltonian operator; The eigenvalues ​​of the kernel matrix are calculated using quantum phase estimation.

6. The method according to claim 3, characterized in that, The construction of the neighborhood matrix specifically includes: Preparing the initial state of data using quantum random access memory; Calculate Mahalanobis distance using quantum covariance matrix; Solving for dimensionality reduction mapping vectors based on generalized eigenvectors.

7. The method according to claim 4, characterized in that, The design of the loss function C(θ) satisfies: Measuring the consistency of popular structures between the original high-dimensional data and the low-dimensional mapped data; The loss function value is obtained through quantum measurement.

8. The method according to claim 1, characterized in that, The method also includes simulation verification of the quantum circuit on the IBMQ quantum cloud computing platform, specifically including: The unitary transform operation is implemented using quantum gate circuits; The dimensionality reduction results are output using quantum state tomography.

9. The method according to claim 3 or 6, characterized in that, The calculation of the Mahalanobis distance further includes: Constructing a quantum version of the data covariance matrix; The efficiency of nearest neighbor search is optimized using a quantum amplitude amplification algorithm.

10. The method according to claim 4 or 7, characterized in that, The optimization of the parameter θ further includes: Compress the mapping matrix using variational quantum singular value decomposition; The convergence of the loss function is accelerated by combining quantum convex optimization algorithms.

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