Quantum Cut Sparsifiers

2026-06-08

AI Summary

This work addresses the problem of efficiently sparsifying quantum-cut (QC) Hamiltonians while preserving a $(1\pm\varepsilon)$-approximation of the energy for all quantum states. To this end, the authors introduce a novel importance sampling technique that achieves, for the first time, a unified sparsification framework applicable to QC Hamiltonians defined on arbitrary graph structures and simultaneously compatible with Kikuchi approximations at all hierarchy levels. The approach overcomes the limitations of traditional leverage score sampling in exponentially large matrices by integrating invariant subspace decomposition, the Alon–Kozma operator-valued inequality, the Caputo–Liggett–Richthammer octopus inequality, and expander decomposition techniques. This combination reduces the number of Hamiltonian terms to $\widetilde{O}(n/\varepsilon^2)$, yielding a substantial decrease in computational complexity.

Abstract

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.

Problem

• Quantum Cut Hamiltonians
• Sparsification
• Energy Preservation
• Kikuchi Graph
• Spectral Approximation

Innovation

• Quantum Cut Hamiltonians
• Sparsification
• Importance Sampling
• Expander Decomposition
• Operator-valued Inequalities