# Efficient classical simulation of Clifford circuits with nonstabilizer input states

## TL;DR

This paper presents efficient classical algorithms for approximating output probabilities of Clifford circuits with nonstabilizer inputs, especially when inputs are mixed or pure nonstabilizer states, under certain restrictions.

## Contribution

It introduces new algorithms that efficiently approximate output probabilities for Clifford circuits with nonstabilizer inputs, expanding classical simulation capabilities.

## Key findings

- Efficient approximation algorithms for mixed input states.
- Approximation algorithms for pure nonstabilizer product states with restrictions.
- Applications to Clifford circuits with magic states, PBC, and IQP circuits.

## Abstract

We investigate the problem of evaluating the output probabilities of Clifford circuits with nonstabilizer product input states. First, we consider the case when the input state is mixed, and give an efficient classical algorithm to approximate the output probabilities, with respect to the $l_1$ norm, of a large fraction of Clifford circuits. The running time of our algorithm decreases as the inputs become more mixed. Second, we consider the case when the input state is a pure nonstabilizer product state, and show that a similar efficient algorithm exists to approximate the output probabilities, when a suitable restriction is placed on the number of qubits measured. This restriction depends on a magic monotone that we call the Pauli rank. We apply our results to give an efficient output probability approximation algorithm for some restricted quantum computation models, such as Clifford circuits with solely magic state inputs (CM), Pauli-based computation (PBC) and instantaneous quantum polynomial time (IQP) circuits.