Quantum Circuit Implementation of Two Matrix Product Operations and Elementary Column Transformations

TL;DR

This paper develops efficient quantum algorithms for key matrix operations, including Hadamard and Kronecker products and column transformations, with optimized circuit depths and reduced quantum resource usage.

Contribution

It introduces novel quantum circuit schemes for matrix operations that outperform traditional methods in efficiency and resource consumption.

Findings

Quantum schemes achieve O(1) depth for Kronecker product.

Hadamard product circuit depth is O(max(m,n)).

Column transformations are optimized with lower quantum gate usage.

Abstract

This paper focuses on quantum algorithms for three key matrix operations: Hadamard (Schur) product, Kronecker (tensor) product, and elementary column transformations each. By designing specific unitary transformations and auxiliary quantum measurement, efficient quantum schemes with circuit diagrams are proposed. Their computational depths are: O(1) for Kronecker product; O(max(m,n)) for Hadamard product (linked to matrix dimensions); and O(m) for elementary column transformations of (2^n X 2^m) matrices (dependent only on column count).Notably, compared to traditional column transformation via matrix transposition and row transformations, this scheme reduces computation steps and quantum gate usage, lowering quantum computing energy costs.