Topological Sensing in the Dynamics of Quantum Walks with Defects

Abstract

Topological quantum sensing leverages unique topological features to suppress noise and improve the precision of parameter estimation, emerging as a promising tool in both fundamental research and practical application. In this Letter, we propose a sensing protocol that exploits the dynamics of topological quantum walks incorporating localized defects. Unlike conventional schemes that rely on topological protection to suppress disorder and defects, our protocol harnesses the evolution time as a resource to enable precise estimation of the defect parameter. By utilizing topologically nontrivial properties of the quantum walks, the sensing precision can approach the Heisenberg limit. We further demonstrate the performance and robustness of the protocol through Bayesian estimation. Our results show that this approach maintains high precision over a broad range of parameters and exhibits strong robustness against disorder, offering a practical pathway for topologically enhanced quantum metrology.

Figures (9)

• Figure 1: (a) Schematic illustration of a split-step quantum walk (QW) with a localized defect. Driven by two shift and coin operators with parameters $\theta_1$ and $\theta_2$, a defect is introduced by changing the coin parameter $\theta_2$, at position 0, to $\theta_{02}$. (b) Topological phase diagram, characterized by a winding number as a function of coin parameters $(\theta_1, \theta_2)$. The red circle and the blue square denote two specific choices of $(\theta_1, \theta_2)$, corresponding to the topologically nontrivial and trivial phases, respectively. (c) Fisher information (FI) versus the evolution time and $\theta_1$, and $\theta_{2}$ is fixed to $0.75\pi$. The green dashed lines mark the boundary between different phases.
• Figure 2: The colored curves depict the dynamical evolution of the FI over steps with fixed parameters $\theta_2 = 0.75\pi$ and $\theta_{02} = -0.55\pi$. In panel (a), $\theta_1 = 0.90\pi$ is chosen within the topologically nontrivial (NTr) regime, while in panel (b), $\theta_1 = 0.05\pi$ corresponds to the topologically trivial (Tr) regime. The black solid curve represents the Heisenberg limit scaling. (c) and (d) show the variation of the FI under static disorder. The coin parameters are randomly chosen in the interval $[\theta_i-\pi/20, \theta_i+\pi/20]$ at each position. (e) and (f) show the variation of the FI under dynamic disorder. The coin parameters are randomly chosen in the interval $[\theta_i-\pi/20, \theta_i+\pi/20]$ at each step. The colored curves in (c-f) correspond to the data obtained by averaging over 10 different disorder configurations, with the red shaded area representing the corresponding standard deviation.
• Figure 3: FI under different values of the defect parameter $\theta_{02}$. (a) Topologically nontrivial case. (b) Topologically trivial case. The black solid lines represent the fitting curve corresponding to quadratic scaling proportional to $t^2$ (Heisenberg limit). The parameters are fixed to $\theta_2=0.75\pi$, with $\theta_1=0.90\pi$ in (a) and $\theta_1=0.05\pi$ in (b).
• Figure 4: Numerical results of Bayesian parameter estimation. The posterior distributions for sensing $\theta_{02}=-0.55\pi$ with uniform priors $\theta_{02}\in[-0.556\pi, -0.544\pi]$ in topologically nontrivial (a) and trivial (b) cases, respectively. (c) and (d) show the mean of the squared relative error $\Delta \theta_{02}^2$ as a function of steps in two cases. The black solid line represents the Heisenberg limit. The red curve shows the result for the topologically nontrivial case with fixed parameters $\theta_1=0.90\pi$ and $\theta_2=0.75\pi$, while the blue curve corresponds to the topologically trivial case with $\theta_1=0.05\pi$ and $\theta_2=0.75\pi$. (e) and (f) show the impact of static disorder to $\Delta \theta_{02}^2$. (g) and (h) show the impact of dynamic disorder. The colored curves in (e-h) correspond to the data obtained by averaging over 10 different disorder configurations, with the red shaded area representing the corresponding standard deviation.
• Figure S1: Energy bands and occupation probabilities (in the insets) of two defect-induced localized states with the fixed parameters $\theta_1 = 0.9\pi$, $\theta_2 = 0.75\pi$, and $\theta_{02} = -0.55\pi$ in (a), and $\theta_{02}=-\pi$ in (b). When $\theta_{02}$ approaches $-\pi$, the defect acts as a domain wall to interrupt the propagation, and the two localized states can be regarded as edge states.