Efficient quantum circuits for dense and non-unitary operators

TL;DR

This paper introduces resource-efficient quantum circuits for implementing circulant, Toeplitz, Hankel, and block circulant matrices, including their inverses and products, expanding quantum capabilities for non-sparse, non-unitary operators.

Contribution

It presents novel quantum circuit designs that reduce resource requirements for implementing a broad class of non-unitary, non-sparse operators like circulant matrices.

Findings

Quantum circuits for circulant operators are more resource-efficient.

Extensions to Toeplitz, Hankel, and block circulant matrices are demonstrated.

Algorithms for inverses and products of circulant operators are provided.

Abstract

Circulant matrices are an important family of operators, which have a wide range of applications in science and engineering related fields. They are in general non-sparse and non-unitary. In this paper, we present efficient quantum circuits to implement circulant operators using fewer resources and with lower complexity than existing methods. Moreover, our quantum circuits can be readily extended to the implementation of Toeplitz, Hankel, and block circulant matrices. Efficient quantum algorithms to implement the inverses and products of circulant operators are also provided.