Fast Amplification of QMA

TL;DR

This paper introduces a faster quantum amplification method for QMA verification gaps, improves acceptance probability amplification in special cases, and simplifies witness search techniques, advancing quantum complexity theory.

Contribution

It presents a quadratically faster amplification technique for QMA verification gaps, extends amplification to perfect acceptance cases, and simplifies existing witness search methods.

Findings

Exponential amplification of QMA verification gap achieved

Amplification to perfect acceptance probability in certain cases

Simplified filter-state method for QMA witness search

Abstract

Given a verifier circuit for a problem in QMA, we show how to exponentially amplify the gap between its acceptance probabilities in the `yes' and `no' cases, with a method that is quadratically faster than the procedure given by Marriott and Watrous. Our construction is natively quantum, based on the analogy of a product of two reflections and a quantum walk. Second, in some special cases we show how to amplify the acceptance probability for good witnesses to 1, making a step towards the proof that QMA with one-sided error is equal to QMA. Finally, we simplify the filter-state method to search for QMA witnesses by Poulin and Wocjan.