Classical simulation of quantum circuits using fewer Gaussian eliminations

Abstract

A compelling way to quantify the separation between classical and quantum computing is to determine how many $T$-gate magic states, $t$, a classical computer must simulate to calculate the probability of a universal quantum circuit's output. Unfortunately, efforts to determine the minimum number of stabilizer state inner products necessary to decompose $T$-gate magic states (${\ensuremath{\chi}}_{t}$) have proven intractable past $t=7$. By using a phase space formalism based on Wootters' discrete Weyl operator basis over a finite field, we develop an algebraic approach to determining ${\ensuremath{\chi}}_{t}$ for single-Pauli measurements. This allows us to extend the bounds on ${\ensuremath{\chi}}_{t}$ to $t=14$ for qutrits, effectively increasing the space searched by $>{10}^{{10}^{4}}$. Our results show that by using such phase space methods it is possible to validate noisy intermediate-scale quantum circuits of larger size than previously thought possible.