An Entropy-Governed Speedup for Quantum Algorithms on Local Hamiltonians

2026-05-18

AI Summary

This work addresses the problem of estimating and preparing low-energy states of general k-local Hamiltonians by proposing a quantum algorithm that surpasses the conventional Grover bound. By integrating depth-limited quantum circuits, energy estimation techniques, and an entropy-based optimization strategy, the method efficiently outputs, for any fixed circuit depth \(d\), a quantum state whose energy is no greater than the minimal energy achievable by depth-\(d\) states, along with a corresponding energy estimate. Compared to the approach of Buhrman et al., this algorithm achieves the same energy precision while significantly improving upon the \(O(2^{n/2})\) time complexity. Notably, this study presents the first demonstration of entropy-regulated acceleration for local Hamiltonian problems and clarifies the fundamental distinction between highly entangled states and those amenable to efficient classical simulation.

Abstract

Low-energy estimation and state preparation for general $k$-local Hamiltonians are fundamental challenges in quantum complexity theory. For constant relative accuracy, Buhrman et al. (PRL 2025) recently broke the natural Grover bound $O(2^{n/2})$, where $n$ denotes the number of qubits, for both problems. In this paper, for any sufficiently small parameter $d\ge 0$, we present an even faster quantum algorithm that outputs a quantum state with energy bounded by the minimum energy over all depth-$d$ states (i.e., states obtained by applying a depth-$d$ circuit to the all-zero state), together with an estimate of this energy. For the class of Hamiltonians with depth-$d$ ground states, our algorithm furthermore achieves exactly the same energy guarantees as Buhrman et al. Our results also provide insight into the distinction between strongly entangled states and those admitting efficient classical descriptions.

Problem

• local Hamiltonians
• low-energy estimation
• state preparation
• quantum algorithms
• entanglement

Innovation

• quantum algorithms
• local Hamiltonians
• entropy-governed speedup
• low-energy state preparation
• depth-limited circuits