Quantum algorithms for matrix operations and linear systems of equations

TL;DR

This paper introduces quantum algorithms for fundamental matrix operations and solving linear systems, utilizing a Sender-Receiver model to encode data into quantum states and extract results via unitary transformations.

Contribution

It presents novel quantum algorithms for matrix operations and linear system solutions, expanding the toolkit for quantum numerical methods.

Findings

Quantum algorithms for matrix-vector and matrix-matrix products.

Quantum protocols for calculating determinants and inverses.

An alternative quantum algorithm for solving linear systems.

Abstract

Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations. Using the "Sender-Receiver" model, we propose quantum algorithms for matrix operations such as matrix-vector product, matrix-matrix product, the sum of two matrices, and calculation of determinant and inverse of a matrix. We encode the matrix entries into the probability amplitudes of pure initial states of senders. After applying a proper unitary transformation to the complete quantum system, the desired result can be found in certain blocks of the receiver's density matrix. These quantum protocols can be used as subroutines in other quantum schemes. Furthermore, we present an alternative quantum algorithm for solving linear systems of equations.