Absorption-based qubit estimation in discrete-time quantum walks

TL;DR

This paper presents an analytical method for estimating the initial state of a quantum coin in discrete-time quantum walks using absorption boundaries, revealing how boundary placement affects information gain and proposing a practical photonic implementation.

Contribution

It introduces a spectral approach to quantify information from absorption boundaries and demonstrates how combining boundary positions enables full state estimation without mode-resolved tomography.

Findings

Closed-form expressions for escape probability and Fisher information.

Boundary placement influences the information about different state parameters.

Combining two boundaries achieves full-rank Fisher information for joint estimation.

Abstract

We investigate state estimation in discrete-time quantum walks with a single absorbing boundary. Using a spectral approach, we obtain closed expressions for the escape probability as a function of the initial coin state and the boundary position, together with the corresponding classical Fisher information for a binary absorption readout. Comparison with the single-copy quantum Fisher information reveals a clear complementarity: near boundaries carry broad information about the polar (Bloch-sphere) angle of the coin state, whereas moderate or distant boundaries reveal phase-sensitive regions. Because a single boundary probes only one information direction, combining two boundary placements yields, generically, a full-rank Fisher matrix and tight joint Cram\'er-Rao bounds while retaining a binary measurement without mode-resolved tomography. We also discuss a restricted-readout photonic implementation in which an on-chip sink realizes the absorber, and we frame the resulting advantage as a potential reduction in measurement-setting and reconfiguration overhead for low-dimensional parameter estimation tasks in architectures where direct projective access to the coin is unavailable. Our results show that absorption in quantum walks defines an analytically tractable restricted-access primitive for coin-state estimation.