Shorter Stabilizer Circuits via Bruhat Decomposition and Quantum Circuit Transformations

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 implement a general stabilizer circuit, we reduce their 11-stage computation -H-C-P-C-P-C-H-P-C-P-Cover 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 depth25n 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-) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> into a three-stage computation -P-CZ-C-. Our results include the reduction of the 11-stage decomposition -H-C-P-C-PC-H-P-C-P-Cinto a 9-stage decomposition of the form -C-P-C-PH-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.