Quantum algorithm for Neighborhood Preserving Embedding

TL;DR

This paper presents a complete quantum algorithm for Neighborhood Preserving Embedding (NPE), offering polynomial speedup in data points and exponential speedup in data dimensionality over classical methods, with rigorous complexity analysis.

Contribution

The authors redesign the quantum sub-algorithms for NPE and provide a rigorous complexity analysis, improving upon previous incomplete quantum approaches.

Findings

Achieves polynomial speedup on data points m

Achieves exponential speedup on data dimensionality n

Demonstrates significant speedup over previous quantum NPE algorithms

Abstract

Neighborhood Preserving Embedding (NPE) is an important linear dimensionality reduction technique that aims at preserving the local manifold structure. NPE contains three steps, i.e., finding the nearest neighbors of each data point, constructing the weight matrix, and obtaining the transformation matrix. Liang et al. proposed a variational quantum algorithm (VQA) for NPE [Phys. Rev. A 101, 032323 (2020)]. The algorithm consists of three quantum sub-algorithms, corresponding to the three steps of NPE, and was expected to have an exponential speedup on the dimensionality $n$. However, the algorithm has two disadvantages: (1) It is incomplete in the sense that the input of the third sub-algorithm cannot be obtained by the second sub-algorithm. (2) Its complexity cannot be rigorously analyzed because the third sub-algorithm in it is a VQA. In this paper, we propose a complete quantum algorithm for NPE, in which we redesign the three sub-algorithms and give a rigorous complexity analysis. It is shown that our algorithm can achieve a polynomial speedup on the number of data points $m$ and an exponential speedup on the dimensionality $n$ under certain conditions over the classical NPE algorithm, and achieve significant speedup compared to Liang et al.'s algorithm even without considering the complexity of the VQA.