Classical verification of quantum circuits containing few basis changes

TL;DR

This paper presents a classical verification method for a specific class of quantum circuits with limited basis changes, enabling efficient checking of outcomes with high probability, which is significant for quantum advantage demonstrations.

Contribution

It introduces a polynomial-time classical verification approach for quantum circuits with at most two basis changes, covering the second level of the Fourier Hierarchy.

Findings

Verification is possible for outcomes with probability at least inverse polynomial.

The method uses random sampling of computational paths.

Verification is efficient and bounded-error for certain quantum circuits.

Abstract

We consider the task of verifying the correctness of quantum computation for a restricted class of circuits which contain at most two basis changes. This contains circuits giving rise to the second level of the Fourier Hierarchy, the lowest level for which there is an established quantum advantage. We show that, when the circuit has an outcome with probability at least the inverse of some polynomial in the circuit size, the outcome can be checked in polynomial time with bounded error by a completely classical verifier. This verification procedure is based on random sampling of computational paths and is only possible given knowledge of the likely outcome.