Restart and first detection in a lackadaisical quantum walk with flat-band localization
Abstract
We study stochastic and sharp restart in a one-dimensional lackadaisical discrete-time quantum walk with self-loop weight ℓ. In the absence of restart, the dynamics has a flat band responsible for intrinsic localization and two dispersive bands supporting ballistic propagation. We compare two initially localized benchmark states: a flat-band-active state with finite flat-band overlap and a flat-band-dark state with zero flat-band overlap. For geometric stochastic restart with per-step restart probability q, the stationary mean-squared displacement scales as q−2 as q→0. In the same limit, the restart-site occupation probability approaches the restart-free intrinsic localized value for the flat-band-active state, whereas for the flat-band-dark state it vanishes as qln(1/q). For power-law restart, where pm∝ m−s is the probability that the waiting time to the next restart is m steps, a normalized stationary site-occupation distribution exists only for s>2, while the stationary absolute spatial moment of order p is finite only for s>p+2. In the regime 1<s≤2, at every fixed lattice site, the flat-band-active occupation converges to the intrinsic flat-band profile, while the flat-band-dark occupation tends to zero. We also consider monitored first detection with sharp restart, in which the walk is reinitialized after a fixed number r of consecutive unsuccessful measurements. For fixed r, the mean first-detected-passage time of the flat-band-active state exhibits a minimum at an intermediate self-loop weight, whereas the flat-band-dark state approaches a ballistic detection limit as ℓ→∞.
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BibTeX
@misc{das2026restart,
title = {{Restart and first detection in a lackadaisical quantum walk with flat-band localization}},
author = {Debraj Das},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08973},
eprint = {2609.08973},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08973},
}
RIS
TY - UNPB
TI - Restart and first detection in a lackadaisical quantum walk with flat-band localization
AU - Debraj Das
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08973
UR - https://arxiv.org/abs/2609.08973
AB - We study stochastic and sharp restart in a one-dimensional lackadaisical discrete-time quantum walk with self-loop weight ℓ. In the absence of restart, the dynamics has a flat band responsible for intrinsic localization and two dispersive bands supporting ballistic propagation. We compare two initially localized benchmark states: a flat-band-active state with finite flat-band overlap and a flat-band-dark state with zero flat-band overlap. For geometric stochastic restart with per-step restart probability q, the stationary mean-squared displacement scales as q^−2 as q→0. In the same limit, the restart-site occupation probability approaches the restart-free intrinsic localized value for the flat-band-active state, whereas for the flat-band-dark state it vanishes as qln(1/q). For power-law restart, where p_m∝ m^−s is the probability that the waiting time to the next restart is m steps, a normalized stationary site-occupation distribution exists only for s>2, while the stationary absolute spatial moment of order p is finite only for s>p+2. In the regime 1<s≤2, at every fixed lattice site, the flat-band-active occupation converges to the intrinsic flat-band profile, while the flat-band-dark occupation tends to zero. We also consider monitored first detection with sharp restart, in which the walk is reinitialized after a fixed number r of consecutive unsuccessful measurements. For fixed r, the mean first-detected-passage time of the flat-band-active state exhibits a minimum at an intermediate self-loop weight, whereas the flat-band-dark state approaches a ballistic detection limit as ℓ→∞.
ER -