Observation of Criticality-Enhanced Quantum Sensing in Nonunitary Quantum Walks

TL;DR

The paper demonstrates criticality-enhanced quantum sensing in a non-Hermitian photonic quantum walk, achieving superlinear scaling of sensitivity near two distinct gap-closure transitions. By engineering a domain-wall NH Hamiltonian, the authors observe enhanced quantum Fisher information in transient dynamics and validate it with Bayesian estimation, establishing a direct link between spectral gap closures (point and line gaps) and metrological gain. The work provides a practical, scalable photonic platform for sensing bulk NH parameters and elucidates multiple mechanisms—skin-effect-related localization changes and alternative gap-closure pathways—that enable quantum-enhanced precision under non-unitary evolution.

Abstract

Quantum physics enables parameter estimation with precisions beyond the capability of classical sensors. Quantum criticality is a key resource for this quantum-enhanced sensing, but experimental realization has been challenging due to the complexity of ground-state preparation and the long time required to reach the steady state near criticality. Here, we experimentally demonstrate critical enhancement in a non-Hermitian topological system using a photonic quantum walk setup. Our system supports two distinct phase transitions at which enhanced sensitivity is observed even at transient times for which the Bayesian inference shows excellent estimation and precision. It is a direct demonstration of criticality-enhanced scaling laws with non-unitary dynamics.

Figures (11)

• Figure 1: Schematic of phase transition and model. (a) With point gap, the complex bulk spectrum has a loop-like structure when the parameter $\theta$ is away from the critical value $\theta_c$. Correspondingly, the cumulative site population $p_j$ of all the eigenstates are edge-localized in a finite system (skin effect). At criticality, the spectral loop collapses and skin effect vanishes. (b) Presence of line gap supports topologically non-trivial edge states $\ket{\psi}$, localized at the edge of a finite system. At line gap closing, the parts of the spectrum join and the edge state shifts to a delocalized bulk state. (c) The domain wall system used in our experiment for the non-unitary photonic quantum walk. The left and right regions have different coin state rotations.
• Figure 2: Theoretical analysis. (a-d) Sensing at point gap closing. The fixed parameters are $\gamma {=} 0.3$ and $\theta_1^L {=} \theta_2^R {=} 0.9 \pi$. (a) Point gap shown for $\theta_2^L {=} 0.05 \pi$ closes at the critical value $\theta_2^L {=} 0.1 \pi$. (inset) The quadratic scaling of maximum QFI of the steady state $F^Q_{\rm max}$ with system size $N$ and absence of scaling away from criticality at $\theta_2^L {=} 0.15 \pi$. (b) QFI after $t$ steps of the quantum walker, at and away from criticality. (c) QFI and CFI of $\ket{\Psi_N}$ obtained after $(N{-}1)/2$ time-steps. (inset) QFI and CFI of the steady state. (d) Scaling of QFI of $\ket{\Psi_N}$ with $N$ (numerical fit $N^b$) near criticality and away from it ($\theta_2^L {=} 0.15 \pi, \theta_2^L {=} 0.2 \pi$). (e-h) Sensing at line gap closing. The fixed parameters are $\gamma {=} 0.3$ and $\theta_1^L {=} \theta_2^R {=} 0.05 \pi$. (e) Line gap shown for $\theta_2^L {=} 0.8 \pi$ with dotted lines closes at the critical value $\theta_2^L {=} 0.779 \pi$. (inset) The quadratic scaling of $F^Q_{\rm max}$ with $N$ and absence of scaling away from criticality at $\theta_2^L {=} 0.15 \pi$ for the steady state. (f) QFI evolution, at and away from criticality. (g) QFI and CFI of $\ket{\Psi_N}$. (inset) QFI and CFI of the steady state. (h) Scaling of QFI of $\ket{\Psi_N}$ near criticality and away from it ($\theta_2^L {=} 0.8 \pi, 0.65 \pi$).
• Figure 3: Experimental setup. A pulsed laser beam is attenuated to the single-photon level. The photon polarization is then initialized in the desired state with a PBS and a HWP. It then enters a quantum walk interferometric network, where the loss operator $\Gamma$, coin operator $R$, and shift operator $S$ are implemented by the PPBS, two HWPs, and the BD, respectively. Finally, the photons are detected by APD.
• Figure 4: Experimental results for criticality-enhanced sensing. (a-b) Point gap closing case with $\theta_1^{L}{=}\theta_2^{R}{=}0.9 \pi$, $\theta_2^{L}{=}\theta_1^{R}$ and initial state $|\Psi_0\rangle{=}|0\rangle{\otimes}|V\rangle$. (a) CFI as a function of $\theta_2^{L}$. Maximum CFI $F^C_{\rm max}$ is observed near criticality, at $\theta_2^{L} {=} 0.11\pi$. (b) Scaling of $F^C_{\rm max}$ with $N$ and CFI evaluated at $\theta_2^{L}{=}0.2\pi$, away from criticality. (c-d) Line gap closing case with $\theta_1^{L}{=}\theta_2^{R}{=}0.05 \pi$, $\theta_2^{L}{=}\theta_1^{R}$ and $|\Psi_0\rangle{=}|0\rangle {\otimes} \left(|H\rangle {-} |V\rangle \right)/\sqrt{2}$. (c) CFI as a function of $\theta_2^{L}$. $F^C_{\rm max}$ is observed near criticality, at $\theta_2^{L}{=}0.74\pi$. (d) Scaling of $F^C_{\rm max}$ with $N$ and CFI at $\theta_2^{L}{=}0.8\pi$, away from criticality. All error bars indicate statistical uncertainties, estimated for Poissonian counting statistics (see SM Supp).
• Figure 5: Bayesian estimation from the experimental data. (a-b) Point gap closing. (a) Posterior updates with measurement repetition numbers $M$. (b) Estimated value $\theta_{\rm est}$ for $\theta_2^L$ versus the experimentally set value $\theta_{\rm exp}$ with $M {=} 25000$. (inset) Precision of estimation in terms of standard deviation $\delta \theta$ and its lower bound given by the QFI (solid line). (c-d) Line gap closing. (c) Posterior updates with $M$. (d) Estimation $\theta_{\rm est}$ for $\theta_2^L$ versus $\theta_{\rm exp}$ with $M {=} 25000$. (inset) Precision $\delta \theta$ and its lower bound given by the QFI (solid line). Other relevant parameters are same as those in Fig. \ref{['fig:exp']}.