${\sf QMA}={\sf QMA}_1$ with an infinite counter

TL;DR

This paper proves that the quantum complexity class QMA equals its one-sided error variant with an infinite counter, showing perfect completeness and providing a more efficient amplification method.

Contribution

It introduces the concept of an infinite counter in QMA verifiers, demonstrating that this does not increase computational power but achieves perfect completeness and improved amplification.

Findings

QMA equals QMA with an infinite counter, i.e., QMA = QMA^∞.

Constructs a QMA amplifier with exponential completeness close to 1 in fewer steps.

Achieves doubly exponential completeness amplification with minimal verifier calls.

Abstract

A long-standing open problem in quantum complexity theory is whether ${\sf QMA}$, the quantum analogue of ${\sf NP}$, is equal to ${\sf QMA}_1$, its one-sided error variant. We show that ${\sf QMA}={\sf QMA}^{\infty}= {\sf QMA}_1^{\infty}$, where ${\sf QMA}_1^\infty$ is like ${\sf QMA}_1$, but the verifier has an infinite register, as part of their witness system, in which they can efficiently perform a shift (increment) operation. We call this register an ``infinite counter'', and compare it to a program counter in a Las Vegas algorithm. The result ${\sf QMA}={\sf QMA}^\infty$ means such an infinite register does not increase the power of ${\sf QMA}$, but does imply perfect completeness. By truncating our construction to finite dimensions, we get a ${\sf QMA}$-amplifier that only amplifies completeness, not soundness, but does so in significantly less time than previous ${\sf QMA}$ amplifiers. Our new construction achieves completeness $1-2^{-q}$ using $O(1)$ calls to each of the original verifier and its inverse, and $O(\log q)$ other gates, proving that ${\sf QMA}$ has completeness doubly exponentially close to 1, i.e. ${\sf QMA}={\sf QMA}(1-2^{-2^r},2^{-r})$ for any polynomial $r$.