Causality, localization, and universality of monitored quantum walks with long-range hopping

Abstract

A powerful strategy to accelerate quantum-walk-based search algorithms leverages on resetting protocols, where a detector monitors a target site and the evolution of the walker is restarted if no detection occurs within a fixed time interval. The optimal resetting rate can be extracted from the time evolution of the probability $S(t)$ that the detector has not clicked up to time $t$. We analyze $S(t)$ for a quantum walk on a one-dimensional lattice when the coupling between sites decays algebraically as $d^{-\alpha}$ with the distance $d$, for $\alpha\in(0,\infty)$. At long times, $S(t)$ decays with a universal power-law exponent that is independent of $\alpha$. At short times, $S(t)$ exhibits a plethora of phase transitions as a function of $\alpha$. From this, we provide a strategy to determine the optimal resetting rate. We identify two regimes: for $\alpha>1$, the resetting rate $r$ is bounded from below by the velocity with which information propagates causally across the lattice; for $\alpha<1$, instead, the long-range hopping tends to localize the walker: The optimal resetting rate depends on the size of the lattice and diverges as $\alpha\to 0$. Our strategy directly connects local measurement outcomes with the global dynamics encoded in $S(t)$. We derive simple models explaining our numerical results, shedding light on the interplay of long-range coherent dynamics, symmetries, and local quantum measurement processes in determining equilibrium. Our findings offer experimentally testable predictions and provide new physical insights on optimizing quantum search through resetting.