Quantum state transfer and periodicity in discrete-time quantum walks under non--Markovian dephasing noise

TL;DR

The paper tackles quantum state transfer and periodicity in discrete-time quantum walks under non-Markovian dephasing noise, employing the Grover coin on diverse graph topologies. By generalizing RTN and modified non-Markovian OUN channels to arbitrary dimensions via Weyl operators, it analyzes fidelity dynamics and revival behavior across path, cycle, star, and complete bipartite graphs. The results reveal topology-specific robustness and memory-induced revivals, offering guidance for noise-resilient quantum information tasks and DTQW-based algorithms in realistic environments. These insights inform design choices for quantum networks and highlight potential practical stability of state transfer under certain non-Markovian regimes.

Abstract

In quantum communication, quantum state transfer from one location to another in a quantum network plays a prominent role, where the impact of noise could be crucial. The idea of state transfer can be fruitfully associated with quantum walk on graphs. We investigate the consequences of non-Markovian quantum noises on periodicity and state transfer induced by a discrete-time quantum walk on graphs, governed by the Grover coin operator. Different bipartite graphs, such as the path graph, cycle graph, star graph, and complete bipartite graph, present periodicity and state transfer in a discrete-time quantum walk depending on the topology of the graph. We investigate the effect of quantum non-Markovian dephasing noises, particularly quantum non-Markovian Random Telegraph Noise (RTN) and modified non-Markovian Ornstein-Uhlenbeck Noise (OUN) on state transfer and periodicity. We demonstrate how the RTN and OUN noises allow state transfer and periodicity for a finite number of steps in a quantum walk. Our investigation brings out the feasibility of state transfer in a noisy environment.

Figures (10)

• Figure 1: Examples of simple graphs, which are used in this article.
• Figure 2: We consider a star graph $G$ with four vertices and three undirected edges, in sub-figure \ref{['star_4']}. In sub-figure \ref{['directed star_4']}, we assign two opposite orientations on every edge for converting it to a directed graph $\overrightarrow{G}$ having six directed edges. Corresponding to the directed edges, we assign a state vector from the computational basis of $\mathcal{H}^{(6)}$. The state vectors corresponding to the directed edges are marked in sub-figure \ref{['directed_star_with_edge_states']}.
• Figure 3: State-transfer on the path graph $P_{5}$ under non-Markovian RTN and modified non-Markovian OUN noise. The RTN parameters are set to $a = 0.1$, $\gamma = 0.01$, while the OUN parameters are $\lambda = 1$, $\gamma = 0.05$.
• Figure 4: Periodicity on the path graph $P_{5}$ under non-Markovian RTN and modified non-Markovian OUN noise. The RTN parameters are set to $a = 0.1$, $\gamma = 0.01$, while the OUN parameters are $\lambda = 1$, $\gamma = 0.05$.
• Figure 5: State transfer on the cycle graph $C_{6}$ under non-Markovian RTN and non-Markovian OUN noise. The channel parameters are the same as for the path graph.