A Low Complexity Quantum Principal Component Analysis Algorithm

In the era of big data,many machine learning algorithms have higher requirements for data processing methods such as data dimensionality reduction.Principal component analysis,as an important data dimensionality reduction algorithm,occupies a pivotal position in classical machine learning,and its quantization algorithm was first proposed by Lloyd in2014,giving it a theoretical basis for its implementation on a quantum computer.Quantum principal component analysis(q PCA),as an important algorithm in quantum machine learning,has been favored by many scientific researchers in recent years.The currently developed q PCA algorithm can directly extract larger feature components and reduce the amount of sampling,but there are some shortcomings: one is not estimated accurate,due to the unitary operator with parameters in the quantum gate setting,the accuracy is low; second is that quantum circuits are more redundant.In the design of quantum circuits,there are repeated similar quantum operations in the horizontal direction.When redundant quantum operations are applied to quantum computers,the influence of noise is likely to be enlarged.In order to solve the above problems,we proposed a low complexity quantum principal component analysis algorithm,and gave a quantum circuit diagram.The design of the algorithm circuit mainly relies on unitary operations such as phase estimation,threshold operation,controlled operation,quantum measurement,etc.The threshold is set for the eigenvalues of the data matrix,and the principal components that are larger than the threshold are selected.The threshold operation is mainly performed by Newton iterative algorithm implementation.This algorithm has two advantages: one is that the complexity of the quantum circuit is greatly reduced,and the required time complexity is about 3/5 of that of the current advanced algorithm,and it also reduces the influence of noisei for algorithm implementation;The parameterized quantum gate reduces the influence of parameter estimation and improves the accuracy of the algorithm.Finally,we conducted experiments with different dimensions and thresholds on the improved algorithm on the quantum computing cloud platform: IBM Quantum Experience.The experimental results show that the real-time quantum circuit calculation results of the algorithm are completely consistent with the classical calculation results,and the results on the QASM quantum simulation computer also closely match the classical vector results.The experiments with different thresholds and different dimensions are in line with the expectations of the algorithm.In short,this algorithm provides a candidate solution for reducing the circuit complexity of quantum principal component analysis.