Uniformity testing when you have the source code

TL;DR

This paper develops improved quantum algorithms for testing whether a distribution generated by a circuit is uniform or matches a known distribution, using source code access, with bounds that are conjectured to be optimal.

Contribution

The authors present enhanced quantum algorithms for uniformity and identity testing with better upper bounds, advancing the efficiency of property testing with source code access.

Findings

Improved upper bound for uniformity testing: $O( ext{min}igrace{d^{1/3}/ ext{epsilon}^{4/3}, d^{1/2}/ ext{epsilon}igrace})$

Enhanced algorithms for identity testing with tighter complexity bounds

Conjecture that the new bounds are optimal for the problem

Abstract

We study quantum algorithms for verifying properties of the output probability distribution of a classical or quantum circuit, given access to the source code that generates the distribution. We consider the basic task of uniformity testing, which is to decide if the output distribution is uniform on $[d]$ or $\epsilon$-far from uniform in total variation distance. More generally, we consider identity testing, which is the task of deciding if the output distribution equals a known hypothesis distribution, or is $\epsilon$-far from it. For both problems, the previous best known upper bound was $O(\min\{d^{1/3}/\epsilon^{2},d^{1/2}/\epsilon\})$. Here we improve the upper bound to $O(\min\{d^{1/3}/\epsilon^{4/3}, d^{1/2}/\epsilon\})$, which we conjecture is optimal.