Parametrized Complexity of Quantum Inspired Algorithms

TL;DR

This paper reviews recent quantum-inspired algorithms for low rank matrix approximation, explores their complexity using parametrized analysis, and conjectures their fixed parameter tractability.

Contribution

It introduces a parametrized complexity approach to quantum-inspired algorithms and hypothesizes their fixed parameter tractability based on low rank and sampling techniques.

Findings

Reviewed progress in quantum-inspired low rank approximation algorithms

Proposed using parametrized complexity for practical analysis

Conjectured fixed parameter tractability of these algorithms

Abstract

Motivated by recent progress in quantum technologies and in particular quantum software, research and industrial communities have been trying to discover new applications of quantum algorithms such as quantum optimization and machine learning. Regardless of which hardware platform these novel algorithms operate on, whether it is adiabatic or gate based, from theoretical point of view, they are performing drastically better than their classical counterparts. Two promising areas of quantum algorithms quantum machine learning and quantum optimization. These are based on performing matrix operations using quantum states and operation, in order to speed up data analysis where quantum computing can efficiently work with high dimensional vectors. Motivated by that, quantum inspired algorithms (e.g. for recommendation systems and principal component analysis) are developed to cope with high dimensionality using probabilistic techniques that are inspire from quantum computing. In this paper we review recent progress in the area of quantum inspired algorithms for low rank matrix approximation. We further explore the possibility of using parametrized complexity for such algorithms to refine practical complexity analysis. Finally, we conjecture that quantum inspired algorithms that use low rank approximation and also sample and query technique for input representations are Fixed Parameter Tractable (FPT).