Optimal Hadamard gate count for Clifford$+T$ synthesis of Pauli rotations sequences

TL;DR

This paper introduces an algorithm to minimize Hadamard gates in Clifford+T circuits, significantly reducing T-counts and resource overhead in fault-tolerant quantum computing.

Contribution

It presents a novel algorithm for synthesizing Pauli rotation sequences with minimal Hadamard gates and optimally reducing Hadamard gates between the first and last T gates.

Findings

Reduces the number of Hadamard gates in quantum circuits.

Enhances T-count reduction by pre-processing Hadamard gate minimization.

Improves resource efficiency in fault-tolerant quantum computation.

Abstract

The Clifford$+T$ gate set is commonly used to perform universal quantum computation. In such setup the $T$ gate is typically much more expensive to implement in a fault-tolerant way than Clifford gates. To improve the feasibility of fault-tolerant quantum computing it is then crucial to minimize the number of $T$ gates. Many algorithms, yielding effective results, have been designed to address this problem. It has been demonstrated that performing a pre-processing step consisting of reducing the number of Hadamard gates in the circuit can help to exploit the full potential of these algorithms and thereby lead to a substantial $T$-count reduction. Moreover, minimizing the number of Hadamard gates also restrains the number of additional qubits and operations resulting from the gadgetization of Hadamard gates, a procedure used by some compilers to further reduce the number of $T$ gates. In this work we tackle the Hadamard gate reduction problem, and propose an algorithm for synthesizing a sequence of $\pi/4$ Pauli rotations with a minimal number of Hadamard gates. Based on this result, we present an algorithm which optimally minimizes the number of Hadamard gates lying between the first and the last $T$ gate of the circuit.