Near-Optimal Bounds on the Density of Low-Energy States of $k$-Local Hamiltonians and Faster Quantum Algorithms

Abstract:Low-energy estimation and state preparation for general $k$-local Hamiltonians are fundamental challenges in quantum complexity theory. Buhrman et al.~ [BGLGST, PRL 2025] recently broke the natural Grover bound $O^\ast(2^{n/2})$ for both problems, with the improvement depending on the relative accuracy $\varepsilon$ and the locality $k$. In this work, we present faster exponential quantum algorithms for these problems, where the binary entropy function governs the runtime exponent. For sufficiently small $\varepsilon/k$, our algorithms improve the exponent by a factor of $\log(k/\varepsilon)$ over [BGLGST, PRL 2025]. Our main technical result is an entropy-governed lower bound on the dimension of the Hamiltonian's low-energy subspace, obtained by depolarizing its ground state. For fixed $k$, this bound is optimal up to constant factors in the exponent. The same framework yields tighter bounds for Heisenberg, $XY$, and Ising models on arbitrary interaction graphs.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS)

arXiv:2609.34532 [quant-ph]

(or arXiv:2609.34532v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.34532 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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