Absorption-Based Qubit Estimation in Discrete-Time Quantum Walks

TL;DR

The paper tackles coin-state estimation in a discrete-time quantum walk with a single absorbing boundary by deriving the escape probability $P_E(\alpha,\beta;M)$ through a spectral approach and analyzing its Fisher information. It demonstrates a complementary information structure: near boundaries primarily encodes the population angle $\alpha$, while more distant boundaries expose the phase $\beta$, with two boundary placements yielding a full-rank Fisher information and tight joint Cramér–Rao bounds for a binary readout. The authors compare classical FI to the single-copy quantum FI, showing that a tomography-free absorption readout can approach quantum-limited precision when data from two appropriately chosen boundaries are combined. They also outline an integrated-photonics implementation with an on-chip sink, highlighting substantial reductions in configuration count versus full mode-resolved qubit tomography. Collectively, the work identifies absorption in quantum walks as a simple, scalable metrological primitive for coin-state estimation in photonic platforms.

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 coin state and the boundary position, and their corresponding classical Fisher information for a simple absorption readout. Comparing with the single-copy quantum Fisher information shows a clear complementarity: near boundaries carry broad information about the population angle of the coin, whereas moderate or distant boundaries reveal phase-sensitive regions. Because a single boundary probes only one information direction, combining two boundary placements yields a full-rank Fisher matrix and tight joint Cramér--Rao bounds, while retaining a binary, tomography-free measurement. We outline an integrated-photonics implementation in which an on-chip sink realizes the absorber and estimate a substantial reduction in configuration count compared to mode-resolved qubit tomography. These results identify absorption in quantum walks as a simple and scalable primitive for coin-state metrology.

Figures (4)

• Figure 1: Schematic representation of the method of images. The real walker (solid circle) starts at $j=0$, while a mirror walker (open circle) is placed at $j=2(M-1)$. Both walkers are symmetrically positioned around the boundary site, each at a distance $d=M-1$. The superposition of their wave functions ensures that the amplitude $L(M-1,t)$ vanishes at the boundary for all times.
• Figure 2: Escape probability $P_E(\alpha,\beta;M)$ as a function of the Bloch angles $(\alpha,\beta)$ for barrier positions (a)–(c) $M=1$, (d)–(f) $M=2$, and (g)–(i) $M\to\infty$. The first column shows the surface $P_E(\alpha,\beta;M)$ for each case, while the second and third columns display one-dimensional cuts of these surfaces for $\alpha,\beta = 0$ (black), $\pi/2$ (red), and $\pi$ (blue). Note that in panel (c) the curves for $\alpha=0$ and $\alpha=\pi$ coincide.
• Figure 3: Fisher information $F_\alpha$ (left column) and $F_\beta$ (right column) as functions of the initial qubit ($\alpha$,$\beta$) for the barrier positions (a)–(b) $M=1$ and (c)–(d) $M\rightarrow\infty$.
• Figure 4: Efficiency of absorption readout relative to the quantum limit. Left column: $\eta_{\alpha}(\alpha,\beta;M)=F_{\alpha}/H_{\alpha}$; right column: $\eta_{\beta}(\alpha,\beta;M)=F_{\beta}/H_{\beta}$. Top, middle, and bottom rows correspond to boundary placements $M=1$, $M=2$, and $M\to\infty$, respectively. Axes: $\alpha\in[0,\pi]$ (horizontal), $\beta\in[0,2\pi]$ (vertical). Color encodes per-trial efficiency (dimensionless), capped at the 99th percentile to avoid saturation near singular sets where $P_E(1-P_E)\to 0$. For $M=1$, $\eta_{\alpha}$ is broadly high except near $\alpha=\pi/2$, while $\eta_{\beta}$ is largely suppressed. As $M$ increases, phase-sensitive "hot spots" emerge in $\eta_{\beta}$ (notably around $\alpha\approx\pi/2$ and $\beta\approx\pi/2,3\pi/2$), and $\eta_{\alpha}$ becomes more structured. These complementary patterns motivate combining two boundary placements to obtain well-conditioned joint estimation of $(\alpha,\beta)$ and tighter Cramér--Rao bounds at fixed $N$.