In this paper, we continue a line of research initiated by Basu, Brakensiek, and Putterman [2026] studying the sparsifiability of Hamiltonians. We focus particularly on the sparsifiability of the widely-studied Quantum Cut (QC) Hamiltonians. Our main result is that in an $n$-qubit system, any $n$-qubit QC Hamiltonian can be sparsified to $\widetilde{O}(n /\varepsilon^2)$ many terms while preserving the energy of every state up to a factor of $1 \pm \varepsilon$. Our result can be interpreted as giving an importance sampling scheme for the edges of an arbitrary graph $G$ such that the \emph{Kikuchi} graph at level $\ell$ of the sampled graph is a spectral approximation to the Kikuchi graph of $G$. Importantly, the \emph{same} sampling scheme works simultaneously for all $\ell$. The natural approach of leverage score sampling, analyzed via matrix concentration inequalities, yields a polynomially worse bound in our setting because the underlying matrices have dimension $\sim 2^n$. Instead, our approach relies on decomposing the action of these matrices into invariant subspaces. Then, by using an operator-valued inequality of Alon and Kozma [Ann. Henri Poincaré, 2020], itself building on an \emph{octopus inequality} of Caputo, Liggett, and Richthammer [J. AMS, 2010], we extend our sparsification technique to all expander graphs. We then invoke expander decomposition to extend our sparsifier to all graphs.

翻译：本文延续了Basu、Brakensiek与Putterman[2026]关于哈密顿量可稀疏化性的研究路线，重点聚焦于广泛研究的量子割（QC）哈密顿量的稀疏化问题。我们的主要结论是：在$n$量子比特系统中，任意$n$量子比特QC哈密顿量均可稀疏化为$\widetilde{O}(n /\varepsilon^2)$项，同时保持每个态的能量在$1 \pm \varepsilon$因子范围内。该结果可理解为给出了任意图$G$边的重要性采样方案，使得采样图的$\ell$级*Kikuchi*图是原图$G$的Kikuchi图的谱近似。关键突破在于，*同一*采样方案对所有$\ell$级同时有效。基于矩阵浓度不等式分析的杠杆分数采样自然方法在本场景中仅能得到多项式级更差的结果，这是因为底层矩阵维度高达$\sim 2^n$。为此，我们采用将矩阵作用分解为不变子空间的方法。继而通过运用Alon与Kozma[Ann. Henri Poincaré, 2020]基于Caputo、Liggett及Richthammer[J. AMS, 2010]的*章鱼不等式*建立的算子值不等式，将稀疏化技术推广至所有扩张图。最终借助扩张图分解，将稀疏化器扩展至任意图。