Variational quantum algorithms for dimensionality reduction and classification

Abstract

Dimensionality reduction and classification play an absolutely critical role in pattern recognition and machine learning. In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. These two algorithms have an exponential speedup over their respectively classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver (VQGE) that finds the generalized eigenvalues and eigenvectors of a matrix pencil (G, S) with coherence time O(1). We successfully conduct numerical experiment solving a problem size of 2 5 × 2 5 . Moreover, our results offer two optional outputs with quantum or classical form, which can be directly applied in another quantum or classical machine learning process.

Quantum algorithm for Neighborhood Preserving Embedding

“…(2) The complexity of our algorithm is less than the complexity of VQNPE, even without considering the complexity of the third sub-algorithm of VQNPE. Specifically, The complexity of the first sub-algorithm is O( m 2 ǫ 2 log 2 n), and the complexity of the second sub-algorithm is Ω(poly(n)) (we should mention that the complexity showed here are different with the original paper [26], see Appendix B for details), while the total complexity of our algorithm is O(m 1.5 polylog(mn)) (only consider the main parameters). The advantage of our first sub-algorithm is mainly coming from the parallel estimation of the distance of each pair of data points.…”

“…(1) Our algorithm is complete while VQNPE is not. In [26], the authors pointed out that it is not known how to obtain the input of the third sub-algorithm from the output of the second sub-algorithm. (2) The complexity of our algorithm is less than the complexity of VQNPE, even without considering the complexity of the third sub-algorithm of VQNPE.…”

“…For NPE, Liang et al proposed a Variational Quantum Algorithm (VQA) [26], called VQNPE. NPE contains three steps, i.e., finding the nearest neighbors of each data point, constructing the weight matrix, and obtaining the transformation matrix A. VQNPE includes three sub-algorithms, corresponding to the three steps of NPE.…”

On Depth, Robustness and Performance Using the Data Re-Uploading Single-Qubit Classifier

“…Quantum machine learning is a rapidly expanding domain, bringing promising performance enhancements through complex feature space representations [1][2][3][4][5] and lowering computational complexity of equivalent classical algorithms by exponential factors in cases [6][7][8][9][10][11]. Variational quantum circuits (VQCs) are currently an area of large interest in the field [12][13][14][15][16][17][18][19][20], and provide a natural progression point for developing quantum algorithms due to their optimization capability.…”

Error-mitigated photonic variational quantum eigensolver using a single-photon ququart

“…At the same time, quantum-classical hybrid algorithms-called variational quantum algorithms (VQAs)-are emerging as promising candidates for near-term practical use of quantum processors. VQAs have applications in a wide variety of fields ranging from chemistry to physics and machine learning [5][6][7][8][9][10][11][12][13][14]. VQAs operate by preparing a parameterized trial state on the quantum processor and evaluating a cost function of interest by measuring the state.…”