QMA = QMA1 with an infinite counter

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 $\it{doubly}$ exponentially close to 1, i.e. ${\sf QMA}={\sf QMA}(1-2^{-2^r},2^{-r})$ for any polynomial $r$.

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