Quantum Circuits for Sparse Isometries

TL;DR

This paper introduces a Householder reflection-based method for efficiently decomposing sparse isometries into quantum gates, reducing C-NOT gate counts and improving sparse state preparation.

Contribution

It presents a novel decomposition technique tailored for sparse isometries, enhancing efficiency over existing methods.

Findings

Significant reduction in C-NOT gates for sparse isometries

Effective in sparse state preparation scenarios

Classical complexity analysis supports practical applicability

Abstract

We consider the task of breaking down a quantum computation given as an isometry into C-NOTs and single-qubit gates, while keeping the number of C-NOT gates small. Although several decompositions are known for general isometries, here we focus on a method based on Householder reflections that adapts well in the case of sparse isometries. We show how to use this method to decompose an arbitrary isometry before illustrating that the method can lead to significant improvements in the case of sparse isometries. We also discuss the classical complexity of this method and illustrate its effectiveness in the case of sparse state preparation by applying it to randomly chosen sparse states.