We study the problem of Hamiltonian sparsification: given a parameter $\varepsilon \in (0,1)$ and an $n$-qubit Hamiltonian $H$ which is the sum of $r$-local positive semi-definite (PSD) terms $H_1, \dots H_m$, our goal is to compute a sparse set $L \subseteq [m]$, along with weights $w: L \rightarrow \mathbb{R}_{\geq 0}$ such that for every state $|ψ\rangle\in \mathbb{C}^{2^n}$, $$ \sum_{i \in L} w(i) \langle ψ| H_i | ψ\rangle \in (1 \pm ε) \sum_{i = 1}^m \langle ψ| H_i | ψ\rangle $$. When the set $L$ is significantly smaller than $m$, this reduces the number of terms in the underlying system, while still ensuring that the behavior of the system is essentially unchanged. We show that many Hamiltonians indeed are sparsifiable to a number of terms much smaller than $n^r$, including: (a) Hamiltonians where each term is an $r$-local Pauli string, (b) Hamiltonians where each term is an $r$-local random operator of rank $R$, for $R \geq 2^{r-1}+1$, and (c) Hamiltonians where each term is an arbitrary $r$-local operator of rank $\geq 2^r -1$ (a.k.a. Quantum SAT). Taken together, our results show that the sparsifiability of Hamiltonians is a robust phenomenon, contrary to prevailing belief (see for instance, Aharonov-Zhou ITCS 2019, QIP 2019). Our results find applications, for instance, to better (semi-)streaming algorithms for quantum Max-Cut, answering a question left open by Kallaugher and Parekh (FOCS 2022). In fact, our results even codify that quantum systems are often easier to sparsify than their classical counterparts.

翻译：我们研究哈密顿量稀疏化问题：给定参数ε∈(0,1)和一个n量子比特哈密顿量H，它由m个r-局域正半定(PSD)项H₁,…,Hm构成，目标是计算一个稀疏子集L⊆[m]及其权重w:L→ℝ≥₀，使得对任意状态|ψ⟩∈ℂ²ⁿ，有∑_{i∈L} w(i)⟨ψ|H_i|ψ⟩∈(1±ε)∑_{i=1}^m⟨ψ|H_i|ψ⟩。当L远小于m时，这减少了底层系统中的项数，同时确保系统行为基本不变。我们证明许多哈密顿量确实可稀疏化为远少于nʳ个项，包括：(a) 每个项为r-局域泡利串的哈密顿量，(b) 每个项为秩R≥2ʳ⁻¹+1的r-局域随机算子的哈密顿量，以及(c) 每个项为秩≥2ʳ-1的任意r-局域算子（即量子SAT）的哈密顿量。综合来看，这些结果表明哈密顿量的可稀疏化性是一种稳健现象，与普遍认知相反（参见Aharonov-Zhou ITCS 2019, QIP 2019）。我们的结果有实际应用，例如可改进量子Max-Cut的（半）流算法，回答了Kallaugher和Parekh（FOCS 2022）遗留的问题。事实上，我们的结果甚至表明量子系统往往比经典系统更易稀疏化。