Despite the fundamental role the Quantum Satisfiability (QSAT) problem has played in quantum complexity theory, a central question remains open: At which local dimension does the complexity of QSAT transition from "easy" to "hard"? Here, we study QSAT with each constraint acting on a $k$-dimensional and $l$-dimensional qudit pair, denoted $(k,l)$-QSAT. Our first main result shows that, surprisingly, QSAT on qubits can remain $\mathsf{QMA}_1$-hard, in that $(2,5)$-QSAT is $\mathsf{QMA}_1$-complete. In contrast, $2$-SAT on qubits is well-known to be poly-time solvable [Bravyi, 2006]. Our second main result proves that $(3,d)$-QSAT on the 1D line with $d\in O(1)$ is also $\mathsf{QMA}_1$-hard. Finally, we initiate the study of 1D $(2,d)$-QSAT by giving a frustration-free 1D Hamiltonian with a unique, entangled ground state. Our first result uses a direct embedding, combining a novel clock construction with the 2D circuit-to-Hamiltonian construction of [Gosset, Nagaj, 2013]. Of note is a new simplified and analytic proof for the latter (as opposed to a partially numeric proof in [GN13]). This exploits Unitary Labelled Graphs [Bausch, Cubitt, Ozols, 2017] together with a new "Nullspace Connection Lemma", allowing us to break low energy analyses into small patches of projectors, and to improve the soundness analysis of [GN13] from $\Omega(1/T^6)$ to $\Omega(1/T^2)$, for $T$ the number of gates. Our second result goes via black-box reduction: Given an arbitrary 1D Hamiltonian $H$ on $d'$-dimensional qudits, we show how to embed it into an effective null-space of a 1D $(3,d)$-QSAT instance, for $d\in O(1)$. Our approach may be viewed as a weaker notion of "simulation" (\`a la [Bravyi, Hastings 2017], [Cubitt, Montanaro, Piddock 2018]). As far as we are aware, this gives the first "black-box simulation"-based $\mathsf{QMA}_1$-hardness result, i.e. for frustration-free Hamiltonians.

翻译：尽管量子可满足性(QSAT)问题在量子复杂性理论中扮演着基础性角色，但一个核心问题仍然悬而未决：QSAT的复杂性在何种局域维度下从"简单"转变为"困难"？本文研究每个约束作用于$k$维与$l$维量子比特对上的QSAT问题，记为$(k,l)$-QSAT。首个主要结果表明，令人意外的是，量子比特上的QSAT可保持$\mathsf{QMA}_1$-困难性，即$(2,5)$-QSAT是$\mathsf{QMA}_1$-完全的。相比之下，量子比特上的2-SAT众所周知可在多项式时间内求解[Bravyi, 2006]。第二个主要结果证明，一维线上满足$d\in O(1)$的$(3,d)$-QSAT同样是$\mathsf{QMA}_1$-困难的。最后，我们通过构造具有唯一纠缠基态的无阻挫一维哈密顿量，开创了对一维$(2,d)$-QSAT的研究。第一个结果采用直接嵌入方法，将新型时钟构造与[Gosset, Nagaj, 2013]的二维电路到哈密顿量构造相结合。值得关注的是，我们为后者给出了新的简化解析证明（区别于[GN13]的部分数值证明）。该证明利用幺正标记图[Bausch, Cubitt, Ozols, 2017]与新的"零空间连接引理"，将低能分析分解为投影算子的局部补丁，并将[GN13]的可靠度分析从$\Omega(1/T^6)$改进为$\Omega(1/T^2)$（$T$为门数量）。第二个结果通过黑盒归约实现：给定$d'$维量子比特上的任意一维哈密顿量$H$，我们展示了如何将其嵌入到满足$d\in O(1)$的一维$(3,d)$-QSAT实例的有效零空间中。我们的方法可视为一种更弱意义上的"模拟"（参照[Bravyi, Hastings 2017]、[Cubitt, Montanaro, Piddock 2018]）。据我们所知，这是首个基于"黑盒模拟"的$\mathsf{QMA}_1$-困难性结果，即针对无阻挫哈密顿量的情况。