Optimal number of parametrized rotations and Hadamard gates in parametrized Clifford circuits with non-repeated parameters

TL;DR

This paper introduces an efficient algorithm to optimize the number of parametrized rotations and Hadamard gates in Clifford circuits, improving complexity and proving optimality for certain circuit classes.

Contribution

It presents a new, more efficient algorithm for merging rotations in quantum circuits and establishes its optimality for specific parametrized Clifford circuits.

Findings

Algorithm reduces non-Clifford and parametrized rotation gates effectively.

Proves optimality of the merging procedure for certain circuit types.

Improves complexity over previous methods for circuits with few Hadamard gates.

Abstract

We present an efficient algorithm to reduce the number of non-Clifford gates in quantum circuits and the number of parametrized rotations in parametrized quantum circuits. The method consists in finding rotations that can be merged into a single rotation gate. This approach has already been considered before and is used as a pre-processing procedure in many optimization algorithms, notably for optimizing the number of Hadamard gates or the number of $T$ gates in Clifford$+T$ circuits. Our algorithm has a better complexity than similar methods and is particularly efficient for circuits with a low number of internal Hadamard gates. Furthermore, we show that this approach is optimal for parametrized circuits composed of Clifford gates and parametrized rotations with non-repeated parameters. For the same type of parametrized quantum circuits, we also prove that a previous procedure optimizing the number of Hadamard gates and internal Hadamard gates is optimal. This procedure is notably used in our low-complexity algorithm for optimally reducing the number of parametrized rotations.