Reduced quantum circuits for stabilizer states and graph states

TL;DR

This paper introduces a new normal form for stabilizer circuits, reducing two-qubit gate counts for graph states, with practical algorithms implemented in C and tested on classical and quantum computers.

Contribution

It proposes a novel normal form for stabilizer circuits based on subgroup structures, simplifying circuit design and reducing gate counts for graph states.

Findings

New normal form for stabilizer circuits with specific gate sequence

Reduced two-qubit gate count in graph state circuits

Algorithms implemented and tested on classical and quantum platforms

Abstract

We start by studying the subgroup structures underlying stabilizer circuits and we use our results to propose a new normal form for stabilizer circuits. This normal form is computed by induction using simple conjugation rules in the Clifford group. It has shape CX-CZ-P-H-CZ-P-H, where CX (resp. CZ) denotes a layer of $\cnot$ (resp. $\cz$) gates, P a layer of phase gates and H a layer of Hadamard gates. Then we consider a normal form for stabilizer states and we show how to reduce the two-qubit gate count in circuits implementing graph states. Finally we carry out a few numerical tests on classical and quantum computers in order to show the practical utility of our methods. All the algorithms described in the paper are implemented in the C language as a Linux command available on GitHub.