# Phase space simulation method for quantum computation with magic states on qubits

## TL;DR

This paper introduces a classical simulation method for qubit quantum systems using quasiprobability distributions, enabling efficient simulation of certain quantum computations and extending previous results to all finite dimensions.

## Contribution

It generalizes simulation techniques to all finite dimensions, including qubits, and introduces a robustness measure for simulation cost, surpassing stabilizer-based methods.

## Key findings

- Efficient classical simulation for non-negative quasiprobability states.
- Extension of simulation to negative quasiprobability distributions with amplitude estimation.
- Identification of states outside the stabilizer polytope that are still efficiently simulable.

## Abstract

We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et al., New J. Phys. 14, 113011 (2012)] applying only to odd dimension. Our approach is more general than previous ones based on mixtures of stabilizer states. Namely, all mixtures of stabilizer states can be efficiently simulated, but for any number of qubits there also exist efficiently simulable states outside the stabilizer polytope. Further, our simulation method extends to negative quasiprobability distributions, where it provides amplitude estimation. The simulation cost is then proportional to a robustness measure squared. For all quantum states, this robustness is smaller than or equal to robustness of magic.