In the classical context, it is well known that, sometimes, if a search does not find its target, it is better to start the process anew. This is known as resetting. The quantum counterpart of resetting also indicates speeding up the detection process by eliminating the dark states, i.e., situations in which the particle avoids detection. In this work, we introduce the most probable position resetting (MPR) protocol, in which, at a given resetting step, resets are done with certain probabilities to the set of possible peak positions (where the probability of finding the particle is maximum) that could occur because of the previous resets and followed by uninterrupted unitary evolution, irrespective of which path was taken by the particle in previous steps. In a tight-binding lattice model, there exists a twofold degeneracy (left and right) of the positions of maximum probability. The survival probability with optimal restart rate approaches 0 (detection probability approaches 1) when the particle is reset with equal probability on both sides path independently. This protocol significantly reduces the optimal mean first-detected-passage time (FDT), and it performs better even if the detector is far apart compared to the usual resetting protocols in which the particle is brought back to the initial position. We propose a modified protocol, an adaptive two-stage MPR, by making the associated probabilities of going to the right and left a function of steps. In this protocol, we see a further reduction of the optimal mean FDT and improvement in the search process when the detector is far apart.
In the classical context, it is well known that, sometimes, if a search does not find its target, it is better to start the process anew. This is known as resetting. The quantum counterpart of resetting also indicates speeding up the detection process by eliminating the dark states, i.e., situations in which the particle avoids detection. In this work, we introduce the most probable position resetting (MPR) protocol, in which, at a given resetting step, resets are done with certain probabilities to the set of possible peak positions (where the probability of finding the particle is maximum) that could occur because of the previous resets and followed by uninterrupted unitary evolution, irrespective of which path was taken by the particle in previous steps. In a tight-binding lattice model, there exists a twofold degeneracy (left and right) of the positions of maximum probability. The survival probability with optimal restart rate approaches 0 (detection probability approaches 1) when the particle is reset with equal probability on both sides path independently. This protocol significantly reduces the optimal mean first-detected-passage time (FDT), and it performs better even if the detector is far apart compared to the usual resetting protocols in which the particle is brought back to the initial position. We propose a modified protocol, an adaptive two-stage MPR, by making the associated probabilities of going to the right and left a function of steps. In this protocol, we see a further reduction of the optimal mean FDT and improvement in the search process when the detector is far apart.
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.
Classical first-passage times under restart are used in a wide variety of models, yet the quantum version of the problem still misses key concepts. We study the quantum hitting time with restart using a monitored quantum walk. The restart strategy eliminates the problem of dark states, i.e., cases where the particle evades detection, while maintaining the ballistic propagation which is important for a fast search. We find profound effects of quantum oscillations on the restart problem, namely, a type of instability of the mean detection time, and optimal restart times that form staircases, with sudden drops as the rate of sampling is modified. In the absence of restart and in the Zeno limit, the detection of the walker is not possible, and we examine how restart overcomes this well-known problem, showing that the optimal restart time becomes insensitive to the sampling period.
We study a non-interacting quantum particle, moving on a one-dimensional lattice, which is subjected to repetitive measurements. We investigate the consequence when such motion is interrupted and restarted from the same initial configuration, known as the quantum resetting problem. We show that such systems can be described by the time evolution under certain time-dependent non-Hermitian Hamiltonians. We construct two such Hamiltonians and compare the results with the exact dynamics. Using this effective non-Hermitian description we evaluate the timescale of the survival probability as well as the optimal resetting time for the system.
We consider a scenario where we wish to bring a closed system of known Hilbert space dimension d S (the target), subject to an unknown Hamiltonian evolution, back to its quantum state at a past time t 0 .The target is out of our control: This means that we ignore both its free Hamiltonian and how the system interacts with other quantum systems we may use to influence it.Under these conditions, we prove that there exist protocols within the framework of nonrelativistic quantum physics that reset the target system to its exact quantum state at t 0 .Each "resetting protocol" is successful with nonzero probability for all possible free Hamiltonians and interaction unitaries, save a subset of zero measure.When the target is a qubit and the interaction is sampled from the Haar measure, the simplest resetting circuits have a significant average probability of success and their implementation is within reach of current quantum technologies.Finally, we find that, in case the resetting protocol fails, it is possible to run a further protocol that, if successful, undoes both the natural evolution of the target and the effects of the failed protocol over the latter.By chaining in this fashion several such protocols, one can substantially increase the overall probability of a successful resetting.
We investigate the dynamics of a quantum system subjected to a time-dependent and conditional resetting protocol. Namely, we ask what happens when the unitary evolution of the system is repeatedly interrupted at random time instants with an instantaneous reset to a specified set of reset configurations taking place with a probability that depends on the current configuration of the system at the instant of reset? Analyzing the protocol in the framework of the so-called tight-binding model describing the hopping of a quantum particle to nearest-neighbor sites in a one-dimensional open lattice, we obtain analytical results for the probability of finding the particle on the different sites of the lattice. We explore a variety of dynamical scenarios, including the one in which the resetting time intervals are sampled from an exponential as well as from a power-law distribution, and a setup that includes a Floquet-type Hamiltonian involving an external periodic forcing. Under exponential resetting, and in both the presence and absence of the external forcing, the system relaxes to a stationary state characterized by localization of the particle around the reset sites. The choice of the reset sites plays a defining role in dictating the relative probability of finding the particle at the reset sites as well as in determining the overall spatial profile of the site-occupation probability. Indeed, a simple choice can be engineered that makes the spatial profile highly asymmetric even when the bare dynamics does not involve the effect of any bias. Furthermore, analyzing the case of power-law resetting serves to demonstrate that the attainment of the stationary state in this quantum problem is not always evident and depends crucially on whether the distribution of reset time intervals has a finite or an infinite mean.
Even after decades of research, the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time τ, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We illustrate our results with tight-binding quantum walk models. We examine a closed system, i.e., a ring, and reveal the intricate influence of the sampling time τ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals, and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases, the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one-dimensional quantum walk, the probability of first detection decays like (time)^{(-3)} with superimposed oscillations, with exceptional behavior when the sampling period τ times the tunneling rate γ is a multiple of π/2. The amplitude of the power-law decay is suppressed as τ→0 due to the Zeno effect. Our work, an extended version of our previously published paper, predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like (time)^{-3/2}, as elucidated by Schrödinger in 1915.