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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2411.02681 |
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| _version_ | 1866912319988039680 |
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| author | Rudolph, Dorian |
| author_facet | Rudolph, Dorian |
| contents | $\mathsf{QMA}_1$ is $\mathsf{QMA}$ with perfect completeness, i.e., the prover must accept with a probability of exactly $1$ in the YES-case. Whether $\mathsf{QMA}_1$ and $\mathsf{QMA}$ are equal is still a major open problem. It is not even known whether $\mathsf{QMA}_1$ has a universal gateset; Solovay-Kitaev does not apply due to perfect completeness. Hence, we do not generally know whether $\mathsf{QMA}_1^G=\mathsf{QMA}_1^{G'}$ (superscript denoting gateset), given two universal gatesets $G,G'$. In this paper, we make progress towards the gateset question by proving that for all $k\in\mathbb N$, the gateset $G_{2^k}$ (Amy et al., RC 2024) is universal for all gatesets in the cyclotomic field $\mathbb{Q}(ζ_{2^k}),ζ_{2^k}=e^{2πi/2^k}$, i.e. $\mathsf{QMA}_1^G\subseteq\mathsf{QMA}_1^{G_{2^k}}$ for all gatesets $G$ in $\mathbb{Q}(ζ_{2^k})$. For $\mathsf{BQP}_1$, we can even show that $G_2$ suffices for all $2^k$-th cyclotomic fields. We exhibit complete problems for all $\mathsf{QMA}_1^{G_{2^k}}$: Quantum $l$-SAT in $\mathbb{Q}(ζ_{2^k})$ is complete for $\mathsf{QMA}_1^{G_{2^k}}$ for all $l\ge4$, and $l=3$ if $k\ge3$, where quantum $l$-SAT is the problem of deciding whether a set of $l$-local Hamiltonians has a common ground state. Additionally, we give the first $\mathsf{QMA}_1$-complete $2$-local Hamiltonian problem: It is $\mathsf{QMA}_1^{G_{2^k}}$-complete (for $k\ge3$) to decide whether a given $2$-local Hamiltonian $H$ in $\mathbb{Q}(ζ_{2^k})$ has a nonempty nullspace. Our techniques also extend to sparse Hamiltonians, and so we can prove the first $\mathsf{QMA}_1(2)$-complete (i.e. $\mathsf{QMA}_1$ with two unentangled provers) Hamiltonian problem. Finally, we prove that the Gapped Clique Homology problem defined by King and Kohler (FOCS 2024) is $\mathsf{QMA}_1^{G_2}$-complete, and the Clique Homology problem without promise gap is PSPACE-complete. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards a universal gateset for $\mathsf{QMA}_1$ Rudolph, Dorian Quantum Physics Computational Complexity $\mathsf{QMA}_1$ is $\mathsf{QMA}$ with perfect completeness, i.e., the prover must accept with a probability of exactly $1$ in the YES-case. Whether $\mathsf{QMA}_1$ and $\mathsf{QMA}$ are equal is still a major open problem. It is not even known whether $\mathsf{QMA}_1$ has a universal gateset; Solovay-Kitaev does not apply due to perfect completeness. Hence, we do not generally know whether $\mathsf{QMA}_1^G=\mathsf{QMA}_1^{G'}$ (superscript denoting gateset), given two universal gatesets $G,G'$. In this paper, we make progress towards the gateset question by proving that for all $k\in\mathbb N$, the gateset $G_{2^k}$ (Amy et al., RC 2024) is universal for all gatesets in the cyclotomic field $\mathbb{Q}(ζ_{2^k}),ζ_{2^k}=e^{2πi/2^k}$, i.e. $\mathsf{QMA}_1^G\subseteq\mathsf{QMA}_1^{G_{2^k}}$ for all gatesets $G$ in $\mathbb{Q}(ζ_{2^k})$. For $\mathsf{BQP}_1$, we can even show that $G_2$ suffices for all $2^k$-th cyclotomic fields. We exhibit complete problems for all $\mathsf{QMA}_1^{G_{2^k}}$: Quantum $l$-SAT in $\mathbb{Q}(ζ_{2^k})$ is complete for $\mathsf{QMA}_1^{G_{2^k}}$ for all $l\ge4$, and $l=3$ if $k\ge3$, where quantum $l$-SAT is the problem of deciding whether a set of $l$-local Hamiltonians has a common ground state. Additionally, we give the first $\mathsf{QMA}_1$-complete $2$-local Hamiltonian problem: It is $\mathsf{QMA}_1^{G_{2^k}}$-complete (for $k\ge3$) to decide whether a given $2$-local Hamiltonian $H$ in $\mathbb{Q}(ζ_{2^k})$ has a nonempty nullspace. Our techniques also extend to sparse Hamiltonians, and so we can prove the first $\mathsf{QMA}_1(2)$-complete (i.e. $\mathsf{QMA}_1$ with two unentangled provers) Hamiltonian problem. Finally, we prove that the Gapped Clique Homology problem defined by King and Kohler (FOCS 2024) is $\mathsf{QMA}_1^{G_2}$-complete, and the Clique Homology problem without promise gap is PSPACE-complete. |
| title | Towards a universal gateset for $\mathsf{QMA}_1$ |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2411.02681 |