Nearly-frustration-free ground state preparation

TL;DR

This paper introduces a quantum algorithm for preparing ground states of a specific subclass of Hamiltonians called nearly-frustration-free, achieving better dependence on the spectral gap than previous methods, with applications to excited states.

Contribution

The paper presents a new quantum ground state preparation algorithm with improved gap dependence for nearly-frustration-free Hamiltonians, including excited state preparation extensions.

Findings

Algorithm scales as δ^{y/2-1}, better than previous methods.

Provides examples of physically motivated Hamiltonians in the subclass.

Extension allows excited state preparation with similar speedup.

Abstract

Solving for quantum ground states is important for understanding the properties of quantum many-body systems, and quantum computers are potentially well-suited for solving for quantum ground states. Recent work has presented a nearly optimal scheme that prepares ground states on a quantum computer for completely generic Hamiltonians, whose query complexity scales as $\delta^{-1}$, i.e. inversely with their normalized gap. Here we consider instead the ground state preparation problem restricted to a special subset of Hamiltonians, which includes those which we term "nearly-frustration-free": the class of Hamiltonians for which the ground state energy of their block-encoded and hence normalized Hamiltonian $\alpha^{-1}H$ is within $\delta^y$ of -1, where $\delta$ is the spectral gap of $\alpha^{-1}H$ and $0 \leq y \leq 1$. For this subclass, we describe an algorithm whose dependence on the gap is asymptotically better, scaling as $\delta^{y/2-1}$, and show that this new dependence is optimal up to factors of $\log \delta$. In addition, we give examples of physically motivated Hamiltonians which live in this subclass. Finally, we describe an extension of this method which allows the preparation of excited states both for generic Hamiltonians as well as, at a similar speedup as the ground state case, for those which are nearly frustration-free.