A longstanding open problem in quantum complexity theory is whether Quantum Merlin-Arthur (QMA), the quantum analog of nondeterministic polynomial time, is equal to ${\mathrm{QMA}}_1$, its one-sided error variant. We show that $\mathrm{QMA}={\mathrm{QMA}}^{\infty }={\mathrm{QMA}}_1^{\infty }$, where ${\mathrm{QMA}}_1^{\infty }$ is like ${\mathrm{QMA}}_1$, 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, $\mathrm{QMA}={\mathrm{QMA}}^{\infty }$ means such an infinite register does not increase the power of QMA, but does imply perfect completeness. By truncating our construction to finite dimensions, we get a QMA-amplifier that only amplifies completeness, not soundness, but does so in significantly less time than previous 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 QMA has completeness doubly exponentially close to 1, i.e., $\mathrm{QMA}=\mathrm{QMA}(1-2^{-2^r},2^{-r})$ for any polynomial $r$.

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• Quantum computation
• Computational complexity