# Shorter stabilizer circuits via Bruhat decomposition and quantum circuit transformations

## TL;DR

This paper introduces a new, shorter layered implementation of stabilizer circuits using Bruhat decomposition, reducing circuit depth and complexity, and providing a normal form that is optimal in Hadamard gates and parameters.

## Contribution

It presents a novel decomposition method for stabilizer circuits that shortens circuit depth and simplifies implementation, improving previous approaches with Bruhat decomposition and folding techniques.

## Key findings

- Reduced the 11-stage to 7-stage stabilizer circuit decomposition.
- Developed a two-qubit gate depth-$(14n{-}4)$ implementation in LNN architecture.
- Established a new normal form for stabilizer circuits that is optimal in Hadamard gates.

## Abstract

In this paper we improve the layered implementation of arbitrary stabilizer circuits introduced by Aaronson and Gottesman in Phys. Rev. A 70(052328), 2004: to obtain a general stabilizer circuit, we reduce their $11$-stage computation -H-C-P-C-P-C-H-P-C-P-C- over the gate set consisting of Hadamard, Controlled-NOT, and Phase gates, into a $7$-stage computation of the form -C-CZ-P-H-P-CZ-C-. We show arguments in support of using -CZ- stages over the -C- stages: not only the use of -CZ- stages allows a shorter layered expression, but -CZ- stages are simpler and appear to be easier to implement compared to the -C- stages. Based on this decomposition, we develop a two-qubit gate depth-$(14n{-}4)$ implementation of stabilizer circuits over the gate library $\{$H, P, CNOT$\}$, executable in the Linear Nearest Neighbor (LNN) architecture, improving best previously known depth-$25n$ circuit, also executable in the LNN architecture. Our constructions rely on Bruhat decomposition of the symplectic group and on folding arbitrarily long sequences of the form $($-P-C-$)^m$ into a 3-stage computation -P-CZ-C-. Our results include the reduction of the $11$-stage decomposition -H-C-P-C-P-C-H-P-C-P-C- into a $9$-stage decomposition of the form -C-P-C-P-H-C-P-C-P-. This reduction is based on the Bruhat decomposition of the symplectic group. This result also implies a new normal form for stabilizer circuits. We show that a circuit in this normal form is optimal in the number of Hadamard gates used. We also show that the normal form has an asymptotically optimal number of parameters.