Optimal Query Complexity for Ground-State Preparation

2026-09-28

AI Summary

This study addresses the open problem of determining the optimal query complexity for quantum ground state preparation under a known spectral gap threshold. By integrating block encoding, spectral filtering, and amplitude amplification techniques, this work proposes two quantum algorithms that achieve optimal query complexities in both expected and worst-case scenarios, respectively. Furthermore, it optimizes the sequential composition strategy of spectral filters within amplitude amplification. The primary contribution lies in providing the first proof of optimal query complexity that matches established lower bounds, thereby confirming theoretically optimal query efficiency and its unsurpassability under specific conditions. Ultimately, this research delivers a complete complexity characterization for quantum ground state preparation.

Abstract

We determine the optimal query complexity of ground-state preparation to trace-distance error $\varepsilon$ when an energy threshold in the spectral gap is known. Let $U_H$ be an $\alpha$-block-encoding of a Hamiltonian with unique ground state $|\psi_0\rangle$, and suppose $|\langle\psi_0|U_I|0\rangle|\ge\gamma$ for a state-preparation oracle $U_I$. The threshold lies at least $\Delta/2$ above the ground-state energy and at least $\Delta/2$ below every excited-state energy. We give two algorithms that prepare a state within trace distance $\varepsilon$ of the ground state. One uses $O((\alpha/\Delta)(\gamma^{-1}+\log(1/\varepsilon)))$ calls to $U_H$ in expectation; the other uses $O((\alpha/(\gamma\Delta))\log(1/\varepsilon))$ calls to $U_H$ in the worst case. We prove a lower bound matching the expected query count; the corresponding worst-case lower bound follows from Somma and de Wolf [SdW26]. The respective bounds on calls to $U_I$ are $O(1/\gamma)$ in expectation and $O(\gamma^{-1}\log(1/\varepsilon))$ in the worst case. On $(N+1)$-dimensional systems, these $U_I$ bounds are also optimal when the expected or worst-case count of $U_H$ calls, respectively, is $o((\alpha/\Delta)\sqrt N)$. Both algorithms use a constant-accuracy spectral filter to construct a purifier, which we then sequentially compose during amplitude amplification to prepare a state with constant overlap with the ground state. The expected-query algorithm repeats the preparation followed by one high-accuracy spectral filter until success. The worst-case algorithm uses filters of increasing accuracy and limits the total number of queries.

Problem

• ground-state preparation
• query complexity
• trace distance
• spectral gap
• Hamiltonian

Innovation

• ground-state preparation
• query complexity
• spectral filter
• amplitude amplification
• block-encoding