Optimal quantum transport on a ring via locally monitored chiral quantum walks

TL;DR

This work shows that combining a locally monitored detection protocol with a chiral phase in a quantum walk on a ring significantly boosts excitation transfer by lifting dark states and reducing destructive interference. The authors develop a dual analytic approach: a Perron–Frobenius spectral analysis of the non-unitary step and a dark-state characterization, which together identify robust optimal parameters for detection probability. The optimal detection period satisfies a threshold $\tau^*\approx 1.58$ (with $\tau^*\to\pi/2^+$ as $N$ grows), while the optimal phase depends on parity ($\phi_{\rm opt}=0$ for even $N$, $\phi_{\rm opt}=\pm\pi/(2N)$ for odd $N$). Finite-time effects are quantified via the asymptotic time scale $t_{\rm as}\sim N^3$; PF-guided parameters provide near-optimal performance at practical times, offering a general route to enhance transport in monitored quantum systems.

Abstract

In purely coherent transport on finite networks, destructive interference can significantly suppress transfer probabilities, which can only reach high values through careful fine-tuning of the evolution time or tailored initial-state preparations. We address this issue by investigating excitation transfer on a ring, modeling it as a locally monitored continuous-time chiral quantum walk. Chirality, introduced through time-reversal symmetry breaking, imparts a directional bias to the coherent dynamics and can lift dark states. Local monitoring, implemented via stroboscopic projective measurements at the target site, provides a practical detection protocol without requiring fine-tuning of the evolution time. By analyzing the interplay between chirality and measurement frequency, we identify optimal conditions for maximizing the asymptotic detection probability. The optimization of this transfer protocol relies on the spectral properties of the Perron-Frobenius operator, which capture the asymptotic non-unitary dynamics, and on the analysis of dark states. Our approach offers a general framework for enhancing quantum transport in monitored systems.

Figures (9)

• Figure 1: Schematic illustration of the excitation transfer, modeled as a locally monitored chiral quantum walk under a stroboscopic detection protocol, investigated in this work.
• Figure 2: Detection probability as a function of the phase $\phi$ and the detection period $\tau$ at fixed total observation time $T=200$. Density plots $P_{\rm det}(\phi,\tau)$ (a) for $N=20$ and $\delta=N/2$ and (b) for $N=21$ and $\delta=(N-1)/2$. (c,d) Curves $P_{\rm det}(\phi)$ at given $\tau$ for system size $N$ and detection site $\delta$ as in (a,b), respectively.
• Figure 3: Eigenvalues of the Perron-Frobenius operator \ref{['eq:PF_operator']} on the unit circle for two representative values of the detection period, (a) $\tau<\tau^*$ and (b) $\tau>\tau^*$ at $\phi=\pi/2N$, $\delta=10$. (c) Density plot of the largest-modulus PF eigenvalue $|\mu_{\rm PF}|$ as a function of $\phi$ and $\tau$. (d) Curve $|\mu_{\rm PF}|$ as a function of $\tau$ at fixed $\phi=\pi/2N$. (e) Curve $|\mu_{\rm PF}|$ as a function of $\phi$ at fixed $\tau=1.53$. Other parameters are: $N=21$, $\delta=10$.
• Figure 4: Comparison between the Perron-Frobenius (PF) optimal parameters $(\phi_{\rm PF},\tau_{\rm PF})$, obtained by minimizing $\vert \mu_{\rm PF} \vert$, and the numerically-optimal parameters $(\phi_{\rm opt},\tau_{\rm opt})$, obtained by maximizing the detection probability, as functions of odd $N$ at finite observation time $T=200$. While the optimal phase is $\phi_{\rm opt}=\phi_{\rm PF}$ regardless of $N$, the optimal detection periods deviate from each other as $N$ increases: $\tau_{\rm PF} \to \pi/2$ whereas $\tau_{\rm opt}$ decreases.
• Figure 5: Detection probability as a function of $\tau$ evaluated at fixed $\phi=\phi_{\rm opt}$ and $T=200$ for various $N$. (a) Small $N$ for which the system reaches its asymptotic behavior within the observation time $T$. (b) High $N$ for which $T\ll t_{\mathrm{as}}$. The reference value $\tau=\pi/2$ is shown as vertical line. The parameters used are $\delta=N/2$ and $\phi_{\rm opt}=0$ for even $N$, $\delta=(N-1)/2$ and $\phi_{\rm opt}=\pi/2N$ for odd $N$.