Many Hamiltonians Are Sparsifiable

2026-05-04

AI Summary

This work investigates the sparsification of many-body quantum Hamiltonians—specifically, drastically reducing the number of terms while approximately preserving expectation values over arbitrary quantum states. The authors propose a sampling-and-weighting-based sparse approximation method, leveraging operator rank and locality analysis to construct high-fidelity sparse representations. Both theoretical analysis and numerical experiments demonstrate that canonical Hamiltonians—including Pauli strings, high-rank random operators, and Quantum SAT instances—exhibit strong intrinsic sparsity, challenging the conventional belief that quantum systems are inherently difficult to sparsify and revealing they are often more compressible than their classical counterparts. The resulting sparse representations require significantly fewer than $n^r$ terms, thereby enabling efficient streaming algorithms for problems such as quantum Max-Cut.

Abstract

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.

Problem

• Hamiltonian sparsification
• quantum Hamiltonians
• sparsifiability
• local Hamiltonians
• quantum systems

Innovation

• Hamiltonian sparsification
• quantum Hamiltonians
• r-local operators
• spectral approximation
• quantum streaming algorithms