${\sf QMA}={\sf QMA}_1$ with an infinite counter
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arXiv
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| Format: | Preprint |
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2025
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| author | Jeffery, Stacey Witteveen, Freek |
| author_facet | Jeffery, Stacey Witteveen, Freek |
| contents | 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 $r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15551 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | ${\sf QMA}={\sf QMA}_1$ with an infinite counter Jeffery, Stacey Witteveen, Freek Quantum Physics 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 $r$. |
| title | ${\sf QMA}={\sf QMA}_1$ with an infinite counter |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2506.15551 |