# Clifford recompilation for faster classical simulation of quantum circuits

## TL;DR

This paper introduces an improved Monte Carlo algorithm for simulating quantum circuits classically, significantly reducing runtime by utilizing circuit recompilation and Clifford decompositions, aiding quantum device validation.

## Contribution

It presents a novel recompilation routine based on Pauli operators that enhances randomized sparsification, reducing simulation runtime and enabling faster validation of quantum circuits.

## Key findings

- Reduces simulation runtime by a factor of  in certain cases
- Provides a method for estimating circuit amplitudes with faster runtime
- Facilitates optimization of circuit implementations over gate sets

## Abstract

Simulating quantum circuits classically is an important area of research in quantum information, with applications in computational complexity and validation of quantum devices. One of the state-of-the-art simulators, that of Bravyi et al, utilizes a randomized sparsification technique to approximate the output state of a quantum circuit by a stabilizer sum with a reduced number of terms. In this paper, we describe an improved Monte Carlo algorithm for performing randomized sparsification. This algorithm reduces the runtime of computing the approximate state by the factor $\ell/m$, where $\ell$ and $m$ are respectively the total and non-Clifford gate counts. The main technique is a circuit recompilation routine based on manipulating exponentiated Pauli operators. The recompilation routine also facilitates numerical search for Clifford decompositions of products of gates, which can further reduce the runtime in certain cases. It may additionally lead to a framework for optimizing circuit implementations over a gate set, reducing the overhead for state-injection in fault-tolerant implementations. We provide a concise exposition of randomized sparsification, and describe how to use it to estimate circuit amplitudes in a way which can be generalized to a broader class of gates and states. This latter method can be used to obtain additive error estimates of circuit probabilities with a faster runtime than the full techniques of Bravyi et al. Such estimates are useful for validating near-term quantum devices provided that the target probability is not exponentially small.