# Running Measurement Protocol for the Quantum First-detection problem

## TL;DR

This paper studies how a moving quantum detector affects the detection probabilities of a quantum walker, revealing a phase transition and modifications to the Zeno effect with implications for quantum search strategies.

## Contribution

It introduces a novel approach to quantum first-detection problems by analyzing a moving detector, mapping it to a stationary case, and uncovering a dynamical phase transition at a critical sampling time.

## Key findings

- Detection probability exhibits a phase transition at critical sampling time τ.
- Power-law decay with exponent 10/3 at the critical τ.
- Moving detector significantly alters the Zeno effect in quantum walks.

## Abstract

The problem of the detection statistics of a quantum walker has received increasing interest, connected as it is to the problem of quantum search. We investigate the effect of employing a moving detector, using a projective measurement approach with fixed sampling time $\tau$, with the detector moving right before every detection attempt. For a tight-binding quantum walk on the line, the moving detector allows one to target a specific range of group velocities of the walker, qualitatively modifying the behavior of the quantum first-detection probabilities. We map the problem to that of a stationary detector with a modified unitary evolution operator and use established methods for the solution of that problem to study the first-detection statistics for a moving detector on a finite ring and on an infinite 1D lattice. On the line, the system exhibits a dynamical phase transition at a critical value of $\tau$, from a state where detection decreases exponentially in time and the total detection is very small, to a state with power-law decay and a significantly higher probability to detect the particle. The exponent describing the power-law decay of the detection probability at this critical $\tau$ is 10/3, as opposed to 3 for every larger $\tau$. In addition, the moving detector strongly modifies the Zeno effect.