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−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.