Hamiltonian Locality Testing via Trotterized Postselection
John Kallaugher, Daniel Liang
TL;DR
This paper advances the theory of tolerant Hamiltonian locality testing by analyzing the time-evolution oracle access model. It introduces a technique called Trotterized postselection to suppress nonlocal terms and approximates evolution under the nonlocal component $H_{>k}$, enabling locality testing with forward-time evolution. The authors prove a new upper bound on the required total evolution time, $O\left(\sqrt{\frac{\varepsilon_2}{(\varepsilon_2-\varepsilon_1)^5}}\right)$ (up to log factors), and a matching lower bound $\Omega\left(\frac{1}{\varepsilon_2-\varepsilon_1}\right)$, establishing near-optimality in the forward-evolution setting. Moreover, they show that permitting reverse-time evolution yields a tight $O\left(\frac{1}{\varepsilon_2-\varepsilon_1}\right)$ algorithm, indicating the essential role of time-reversal in achieving linear scaling. The work synthesizes Bell-basis Pauli-spectrum sampling with a controlled Elitzur–Vaidman/Bomb tester-inspired protocol to bound locality-distance via the nonlocal Pauli content, and provides a framework that extends to various distance measures in the tolerant regime.
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 $Ω\left(\frac{1}{\varepsilon_2-\varepsilon_1}\right)$ 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.