An Entropy-Governed Speedup for Quantum Algorithms on Local Hamiltonians

TL;DR

This paper introduces a faster quantum algorithm for estimating low-energy states of local Hamiltonians, leveraging entropy considerations to surpass previous bounds for certain state classes.

Contribution

The authors develop an improved quantum algorithm that efficiently estimates energies of depth-$d$ states, surpassing prior bounds for general cases and matching previous guarantees for specific classes.

Findings

Faster quantum algorithm for low-energy estimation of local Hamiltonians.

Achieves the same energy guarantees as previous work for depth-$d$ ground states.

Provides insights into the entanglement structure of states with efficient classical descriptions.

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.