Generators and Relations for Real Stabilizer Operators

TL;DR

This paper introduces a unique normal form for real stabilizer circuits, providing a finite set of relations to systematically rewrite any such circuit into this form, enhancing understanding and manipulation of real Clifford operators.

Contribution

It presents a novel normal form for real stabilizer circuits and establishes a finite set of relations for circuit simplification and standardization.

Findings

Every real stabilizer operator has a unique normal form.

A finite set of relations suffices to rewrite any real stabilizer circuit to its normal form.

The normal form facilitates systematic analysis of real stabilizer operators.

Abstract

Real stabilizer operators, which are also known as real Clifford operators, are generated, through composition and tensor product, by the Hadamard gate, the Pauli Z gate, and the controlled-Z gate. We introduce a normal form for real stabilizer circuits and show that every real stabilizer operator admits a unique normal form. Moreover, we give a finite set of relations that suffice to rewrite any real stabilizer circuit to its normal form.