Tensor Decompositions and Adiabatic Quantum Computing for Discovering Practical Matrix Multiplication Algorithms

TL;DR

This paper explores quantum algorithms for tensor decompositions to discover efficient matrix multiplication algorithms, leveraging quantum annealers and optimization techniques to improve computational efficiency.

Contribution

It introduces two quantum algorithms for tensor decompositions, including a decompositional and a holistic approach, to find practical and potentially faster matrix multiplication algorithms.

Findings

Discovered a tensor decomposition corresponding to Strassen's algorithm using quantum annealers.

Developed a holistic algorithm capable of finding fixed-length tensor decompositions.

Theoretical evidence that shorter fixed-length decompositions can lead to faster matrix multiplication.

Abstract

Quantum computing and modern tensor-based computing have a strong connection, which is especially demonstrated by simulating quantum computations with tensor networks. The other direction is less studied: quantum computing is not often applied to tensor-based problems. Considering tensor decompositions, we focus on discovering practical matrix multiplication algorithms and develop two algorithms to compute decompositions on quantum computers. The algorithms are expressed as higher-order unconstrained binary optimization (HUBO) problems, which are translated into quadratic unconstrained binary optimization (QUBO) problems. Our first algorithm is decompositional to keep the optimization problem feasible for the current quantum devices. Starting from a suitable initial point, the algorithm discovers tensor decomposition corresponding to the famous Strassen matrix multiplication algorithm, utilizing the current quantum annealers. Since the decompositional algorithm does not guarantee minimal length for found tensor decompositions, we develop a holistic algorithm that can find fixed-length decompositions. Theoretically, by fixing a shorter length than the length for the best-known decomposition, we can ensure that the solution to the holistic optimization problem would yield faster matrix multiplication algorithms.