Hamiltonian Locality Testing via Trotterized Postselection

TL;DR

This paper establishes tight bounds for Hamiltonian locality testing using Trotterized postselection, improving understanding of the minimal evolution time needed for accurate testing without reverse evolution.

Contribution

It provides the first tight bounds for Hamiltonian locality testing, including algorithms that do not require reverse evolution and matching lower bounds.

Findings

Upper bound of O(√(ε₂/(ε₂-ε₁)^5)) for evolution time

Lower bound of Ω(1/(ε₂-ε₁)) for algorithms without reverse evolution

Matching upper bound when reverse evolution is allowed

Abstract

The (tolerant) Hamiltonian locality testing problem, introduced in [Bluhm, Caro,Oufkir `24], is to determine whether a Hamiltonian $H$ is $\varepsilon_1$-close to being $k$-local (i.e. can be written as the sum of weight-$k$ Pauli operators) or $\varepsilon_2$-far from any $k$-local Hamiltonian, given access to its time evolution operator and using as little total evolution time as possible, with distance typically defined by the normalized Frobenius norm. We give the tightest known bounds for this problem, proving an $\text{O}\left(\sqrt{\frac{\varepsilon_2}{(\varepsilon_2-\varepsilon_1)^5}}\right)$ evolution time upper bound and an $\Omega\left(\frac{1}{\varepsilon_2-\varepsilon_1}\right)$ lower bound. Our algorithm does not require reverse time evolution or controlled application of the time evolution operator, although our lower bound applies to algorithms using either tool. Furthermore, we show that if we are allowed reverse time evolution, this lower bound is tight, giving a matching $\text{O}\left(\frac{1}{\varepsilon_2-\varepsilon_1}\right)$ evolution time algorithm.