The $k$-$\mathsf{XOR}$ problem is one of the most well-studied problems in classical complexity. We study a natural quantum analogue of $k$-$\mathsf{XOR}$, the problem of computing the ground energy of a certain subclass of structured local Hamiltonians, signed sums of $k$-local Pauli operators, which we refer to as $k$-$\mathsf{XOR}$ Hamiltonians. As an exhibition of the connection between this model and classical $k$-$\mathsf{XOR}$, we extend results on refuting $k$-$\mathsf{XOR}$ instances to the Hamiltonian setting by crafting a quantum variant of the Kikuchi matrix for CSP refutation, instead capturing ground energy optimization. As our main result, we show an $n^{O(\ell)}$-time classical spectral algorithm certifying ground energy at most $\frac{1}{2} + \varepsilon$ in (1) semirandom Hamiltonian $k$-$\mathsf{XOR}$ instances or (2) sums of Gaussian-signed $k$-local Paulis both with $O(n) \cdot \left(\frac{n}{\ell}\right)^{k/2-1} \log n /\varepsilon^4$ local terms, a tradeoff known as the refutation threshold. Additionally, we give evidence this tradeoff is tight in the semirandom regime via non-commutative Sum-of-Squares lower bounds embedding classical $k$-$\mathsf{XOR}$ instances as entirely classical Hamiltonians.

翻译：$k$-$\mathsf{XOR}$问题是经典计算复杂性理论中研究最为深入的问题之一。本文研究$k$-$\mathsf{XOR}$的一个自然量子类比：计算一类特定结构化局域哈密顿量的基态能量问题，这类哈密顿量由$k$-局域泡利算符的带符号和构成，我们称之为$k$-$\mathsf{XOR}$哈密顿量。为展示此模型与经典$k$-$\mathsf{XOR}$问题的关联，我们将约束可满足性问题反驳中关于$k$-$\mathsf{XOR}$实例的结果推广至哈密顿量框架，为此我们构建了Kikuchi矩阵的量子变体，该变体转而用于基态能量优化问题的分析。作为主要结果，我们提出一个$n^{O(\ell)}$时间复杂度的经典谱算法，该算法能够在两种情况下证明基态能量至多为$\frac{1}{2} + \varepsilon$：(1) 具有$O(n) \cdot \left(\frac{n}{\ell}\right)^{k/2-1} \log n /\varepsilon^4$个局域项的半随机$k$-$\mathsf{XOR}$哈密顿量实例；(2) 具有相同数量高斯符号$k$-局域泡利算符求和的实例，此数量权衡关系被称为反驳阈值。此外，我们通过将经典$k$-$\mathsf{XOR}$实例嵌入为完全经典的哈密顿量，基于非交换平方和方法的下限证明，为该权衡关系在半随机区域内的紧致性提供了证据。