Simulating quantum computation: how many "bits" for "it"?

TL;DR

This paper analyzes a classical simulation method for quantum computation using magic states, revealing that it requires tracking a surprisingly small amount of classical data, specifically about 2n^2 bits for n magic states.

Contribution

It demonstrates that the classical data needed for simulating quantum systems with magic states scales quadratically with the number of states, highlighting the efficiency of the simulation method.

Findings

The simulation method never involves negativity in quasiprobability functions.

The amount of classical data needed scales as 2n^2 + O(n) bits for n magic states.

The model remains probabilistic without requiring negativity, unlike other approaches.

Abstract

A recently introduced classical simulation method for universal quantum computation with magic states operates by repeated sampling from probability functions [M. Zurel et al. PRL 260404 (2020)]. This method is closely related to sampling algorithms based on Wigner functions, with the important distinction that Wigner functions can take negative values obstructing the sampling. Indeed, negativity in Wigner functions has been identified as a precondition for a quantum speed-up. However, in the present method of classical simulation, negativity of quasiprobability functions never arises. This model remains probabilistic for all quantum computations. In this paper, we analyze the amount of classical data that the simulation procedure must track. We find that this amount is small. Specifically, for any number $n$ of magic states, the number of bits that describe the quantum system at any given time is $2n^2+O(n)$.