Gaussian Elimination versus Greedy Methods for the Synthesis of Linear Reversible Circuits

TL;DR

This paper introduces optimized Gaussian elimination and improved greedy algorithms for synthesizing linear reversible circuits, achieving better scalability and reduced circuit complexity for quantum computing applications.

Contribution

It presents a new Gaussian elimination-based algorithm and enhancements to greedy methods, advancing the state-of-the-art in reversible circuit synthesis.

Findings

Gaussian elimination performs best for large problem sizes (n > 150).

Greedy methods are effective for small problem sizes (n < 30).

Significant reduction in CNOT count and circuit depth on benchmarks.

Abstract

Linear reversible circuits represent a subclass of reversible circuits with many applications in quantum computing. These circuits can be efficiently simulated by classical computers and their size is polynomially bounded by the number of qubits, making them a good candidate to deploy efficient methods to reduce computational costs. We propose a new algorithm for synthesizing any linear reversible operator by using an optimized version of the Gaussian elimination algorithm coupled with a tuned LU factorization. We also improve the scalability of purely greedy methods. Overall, on random operators, our algorithms improve the state-of-the-art methods for specific ranges of problem sizes: the custom Gaussian elimination algorithm provides the best results for large problem sizes (n > 150) while the purely greedy methods provide quasi optimal results when n < 30. On a benchmark of reversible functions, we manage to significantly reduce the CNOT count and the depth of the circuit while keeping other metrics of importance (T-count, T-depth) as low as possible.