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

2026-09-28

AI Summary

This study addresses the computational complexity of estimating and preparing low-energy states of general k-local Hamiltonians by proposing a fast exponential quantum algorithm that surpasses the Grover bound. Methodologically, it derives an entropy-dominated lower bound on the low-energy subspace dimension via ground-state depolarization, integrating techniques from quantum complexity theory, entropy analysis, and universal graph constraint derivation. The results demonstrate that the algorithm's runtime is governed by the binary entropy function, achieving optimal exponents for fixed k. When ε/k is sufficiently small, the exponential factor improves upon recent work by a logarithmic term log(k/ε). Furthermore, tighter bounds are derived for the Heisenberg, XY, and Ising models.

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.

Problem

• k-local Hamiltonians
• low-energy estimation
• state preparation
• quantum complexity theory

Innovation

• k-local Hamiltonians
• quantum algorithms
• binary entropy function
• low-energy subspace
• depolarization