Quantum Merlin-Arthur and proofs without relative phase

TL;DR

This paper investigates a variant of QMA where quantum proofs lack relative phase, revealing that the class's power varies significantly depending on the completeness and soundness parameters, with implications for quantum proof complexity.

Contribution

The paper characterizes the computational power of QMA+ with different parameters, showing it can be as powerful as NEXP or equivalent to QMA, depending on the gap settings.

Findings

QMA+ with certain gaps equals NEXP

QMA+ with other gaps equals QMA

Relative phase is crucial for quantum proof deception

Abstract

We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [arXiv:1410.2882]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some *other* constant gap is equal to QMA. One interpretation is that Merlin's ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) $\subseteq$ NEXP.