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

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^{-α} with the distance d, for α∈(0,∞). At long times, S(t) decays with a universal power-law exponent that is independent of α. At short times, S(t) exhibits a plethora of phase transitions as a function of α. From this, we provide a strategy to determine the optimal resetting rate. We identify two regimes: for α>1, the resetting rate r is bounded from below by the velocity with which information propagates causally across the lattice; for α<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 α→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.