Time complexity of a monitored quantum search with resetting

TL;DR

The paper analyzes a monitored quantum search with periodic resetting by a continuous-time quantum walk on a complete graph where the target is continuously monitored by a detector. It derives a non-Hermitian effective Hamiltonian $\hat{H}_{\rm eff}^{(s)}$ and the no-click probability $P(t)$ via a biorthogonal eigenbasis, then decomposes the search fidelity into competing contributions and optimizes resetting to minimize the no-click probability. By mapping the parameter dependence of the eigenvalues and overlaps to scaling exponents, it identifies multiple regimes where the search time scales as $\tau=\Theta(N^{\alpha})$ with $\alpha$ varying across regimes, including sub-Grover performance away from an exceptional point and Grover-like scaling at the exceptional point. It further shows that, although resetting can alter asymptotic scaling, accounting for the physical resources required to implement measurements preserves Grover’s optimal bound in the overall query complexity, highlighting the role of non-unitary dynamics and measurement costs in quantum search performance.

Abstract

Searching a database is a central task in computer science and is paradigmatic of transport and optimization problems in physics. For an unstructured search, Grover's algorithm predicts a quadratic speedup, with the search time $τ(N)=Θ(\sqrt{N})$ and $N$ the database size. Numerical studies suggest that the time complexity can change in the presence of feedback, injecting information during the search. Here, we determine the time complexity of the quantum analog of a randomized algorithm, which implements feedback in a simple form. The search is a continuous-time quantum walk on a complete graph, where the target is continuously monitored by a detector. Additionally, the quantum state is reset if the detector does not click within a specified time interval. This yields a non-unitary, non-Markovian dynamics. We optimize the search time as a function of the hopping amplitude, detection rate, and resetting rate, and identify the conditions under which time complexity could outperform Grover's scaling. The overall search time does not violate Grover's optimality bound when including the time budget of the physical implementation of the measurement. For databases of finite sizes monitoring can warrant rapid convergence and provides a promising avenue for fault-tolerant quantum searches.

Figures (13)

• Figure 1: (a) Search dynamics with resetting. A feedback mechanism reinitializes the state to $\hat{\varrho}_0$ if the detector monitoring the target has not clicked in the time interval $T$. (b) Dynamics: For $t\in (mT, (m+1)T)$, the dynamics is governed by the CPTP map $\Lambda_{t_m'}[\hat{\varrho}_0]$ with $t_m'=t-mT$. At times $t= mT$ the protocol either terminates upon a click or otherwise restarts from state $\hat{\varrho}_0$. (c) The search is a continuous-time quantum walk on a globally connected graph with a detector continuously monitoring the target $\vert w\rangle$, corresponding to the orange-colored vertex.
• Figure 2: Color plot of the exponent $\alpha$ as a function of $\bar{r}$ and $s$ of the parametrization in Eq. \ref{['eq:scaling_rs']} for (a) the monitored search and (b) for the full resetting protocol (with optimized resetting time $T$) The value $\alpha=1/2$ of Grover's time complexity is indicated by the blue dashed line and by the blue star. White regions have exponent $\alpha>1$. Dotted lines separate different scaling regimes, labeled (B), (C), and (D), see Table \ref{['tab:scaling_exponent']}. In the resetted search, in regime (C) the exponent of the time-complexity is now $\alpha=s+1$. It remains unchanged in regime (D). The symbols indicate the points studied numerically in Fig. \ref{['fig:3']}.
• Figure 3: (a) Scaling of the time $\tau$ of the monitored search with the database size $N$. The symbols of the four curves correspond to the values of $\bar{r}, s$ of the symbols in Fig. \ref{['fig:2']}(a). The dotted line marks $N_i^*$ (here for $\mathrm{d}t_0=0.01$) where the predictions of the model become invalid. (b) Comparison between the scaling of the search times $\tau$ (monitored search) and $\tau_R$ (resetting) as a function of $N$. The data is numerical and determined for the squared symbol of Fig. \ref{['fig:2']}(b). The shaded region demarcates the areas where $N>N^*$. The blue line gives $\tau_G=\Theta(\sqrt{N})$. In all plots, the underlying dashed lines show the analytic scaling, demonstrating quantitative agreement between numerical and analytical results. See data_non_hermitian_search.
• Figure S1: Dependence of the two eigenvalues $\lambda_{\pm}$ on the scaled monitoring rate $\kappa\sqrt{N}$ with $\gamma=\gamma^{\mathrm{EP}}$. (a) Real parts $\mathrm{Re}(\lambda_{\pm})$. (b) Imaginary parts $\mathrm{Im}(\lambda_{\pm})$. The vertical dashed line at $\kappa=\kappa^{\mathrm{EP}}\approx2/\sqrt{N}$ marks the exceptional point where the eigenvalues coalesce; for larger $\kappa$ they split along the imaginary axis. We take $N=100$.
• Figure S2: Dependence of the two eigenvalues $\lambda_{\pm}$ on the scaled hopping rate $\gamma N$ with $\kappa=\kappa^{\mathrm{EP}}$. (a) Real parts $\mathrm{Re}(\lambda_{\pm})$. (b) Imaginary parts $\mathrm{Im}(\lambda_{\pm})$. The vertical dashed line at $\gamma=\gamma^{\mathrm{EP}}\approx1/N$ marks the exceptional point where the eigenvalues coalesce. We take $N=100$.