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arXiv:2205.09408 · Physical review. E

Perturbed graphs achieve unit transport efficiency without environmental noise

Simone Cavazzoni, Luca Razzoli, Paolo Bordone, Matteo G. A. Paris

Published Aug 22, 2022• 10 citations
arXivHTMLPDF

Abstract

Coherent transport of an excitation through a network corresponds to continuous-time quantum walk on a graph, and the transport properties of the system may be radically different depending on the graph and on the initial state. The transport efficiency, i.e., the integrated probability of trapping at a certain vertex, is a measure of the success rate of the transfer process. Purely coherent quantum transport is known to be less efficient than the observed excitation transport, e.g., in biological systems, and there is evidence that environmental noise is indeed crucial for excitation transport. At variance with this picture, we here address purely coherent transport on highly symmetric graphs, and show analytically that it is possible to enhance the transport efficiency without environmental noise, i.e., using only a minimal perturbation of the graph. In particular, we show that adding an extra weight to one or two edges, depending on whether the initial state is localized or in a superposition of two vertex states, breaks the inherent symmetries of the graph and may be sufficient to achieve unit transport efficiency. We also briefly discuss the conditions to obtain a null transport efficiency, i.e., to avoid trapping.

DOI
10.1103/physreve.106.024118

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Coherent transport of an excitation through a network corresponds to continuous-time quantum walk on a graph, and the transport properties of the system may be radically different depending on the graph and on the initial state. The transport efficiency, i.e., the integrated probability of trapping at a certain vertex, is a measure of the success rate of the transfer process. Purely coherent quantum transport is known to be less efficient than the observed excitation transport, e.g., in biological systems, and there is evidence that environmental noise is indeed crucial for excitation transport. At variance with this picture, we here address purely coherent transport on highly symmetric graphs, and show analytically that it is possible to enhance the transport efficiency without environmental noise, i.e., using only a minimal perturbation of the graph. In particular, we show that adding an extra weight to one or two edges, depending on whether the initial state is localized or in a superposition of two vertex states, breaks the inherent symmetries of the graph and may be sufficient to achieve unit transport efficiency. We also briefly discuss the conditions to obtain a null transport efficiency, i.e., to avoid trapping.
Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex that is accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but, in general, they are uncorrelated.
Coherent transport of excitations along chains of coupled quantum systems represents an interesting problem with a number of applications ranging from quantum optics to solar cell technology. A convenient tool for studying such processes are quantum walks. They allow us to determine all the process features in a quantitative way. We study the survival probability and the transport efficiency on a simple, highly symmetric graph represented by a ring. The propagation of excitation is modeled by a discrete-time (coined) quantum walk. For a two-state quantum walk, where the excitation (walker) has to leave its actual position to the neighboring sites, the survival probability decays exponentially and the transport efficiency is unity. The decay rate of the survival probability can be estimated using the leading eigenvalue of the evolution operator. However, if the excitation is allowed to stay at its present position, i.e. the propagation is modeled by a lazy quantum walk, then part of the wave-packet can be trapped in the vicinity of the origin and never reaches the sink. In such a case, the survival probability does not vanish and the excitation transport is not efficient. The dependency of the transport efficiency on the initial state is determined. Nevertheless, we show that for some lazy quantum walks dynamical, percolations of the ring eliminate the trapping effect and efficient excitation transport can be achieved.
We investigate quantum state transfer on a class of bipartite graphs, namely the butterfly graphs, within the framework of discrete-time quantum walks. These graphs facilitate the construction of scalable quantum networks that enable communication between a sender and a receiver via perfect state transfer. Our analysis demonstrates that state transfer occurs across different butterfly graphs, thereby extending the known families of networks that support high-fidelity quantum state transfer. In addition to the ideal noiseless dynamics, we further investigate the robustness of quantum state transfer in the presence of non-Markovian environmental noise, specifically, random telegraph noise, modified Ornstein-Uhlenbeck noise, which are examples of unital noise and non-Markovian amplitude damping noise, non-unital noise. These noise models capture different types of system-environment interactions and memory effects that influence the coherence of the quantum walk. These findings contribute to the theoretical understanding of how butterfly graph constructions influence quantum transport phenomena.
Abstract Continuous-time quantum walks (CTQWs) provide a versatile framework for exploring quantum transport on graphs. In this work, we investigate how the introduction of edge-weight modulation at a single vertex can suppress its occupation probability. We show that when the edges connected to the root vertex are enhanced by a factor J , the probability of detecting the walker at this vertex decays as 1/J21/J^21/J2 1 / J 2 , provided the initial state has no components on the vertex itself or its nearest neighbors. We derive the full eigenvalue and eigenvector structure of this system, revealing that the suppression arises from the decoupling of two symmetric line subgraphs and the destructive interference of higher-order contributions. The analysis is extended to tree graphs, where we demonstrate the same scaling behavior and identify the role of local graph geometry in controlling vertex probabilities. These results suggest edge-weight modulation as a mechanism for manipulating transport pathways in CTQWs, with potential applications in quantum information transfer and state engineering, and may serve as a probe of decoherence effects in open quantum systems.
The task of finding an element in an unstructured database is known as spatial search and can be expressed as a quantum-walk evolution on a graph. In this article, we modify the usual search problem by adding an extra trapping vertex to the graph, which is only connected to the target element. We study the transfer efficiency of the walker to a trapping site, using the search problem as a case study. Thus, our model offers no computational advantage for the search problem, but focuses on information transport in an open environment with a search Hamiltonian. The walker evolution is a mix between classical and quantum-walk search dynamics. The balance between unitary and nonunitary dynamics is tuned with a parameter, and we numerically show that, depending on the graph topology and the connectivity of the target element, this hybrid approach can outperform a purely classical or quantum evolution for reaching the trapping site. We show that this behavior is only observed in the presence of an extra trapping site, and that, depending on the topology and a tunable parameter controlling the strength of the oracle, a hybrid regime composed of 90% coherent dynamics can lead to either the highest or worst transfer efficiency to the trapping site. We also relate the performance of a hybrid regime to the entropy's decay rate. As the introduction of nonunitary operations may be considered noise, we interpret this phenomenon as a noise-assisted quantum evolution.
Quantum walks are accepted as a generic model for quantum transport. The character of the transport crucially depends on the properties of the walk like its geometry and the driving coin. We demonstrate that increasing transport distance between source and target or adding redundant branches to the actual graph may surprisingly result in a significant enhancement of transport efficiency. We explain analytically the observed nonclassical effects using the concept of trapped states for several intriguing geometries, including the ladder graph, the Cayley tree, and its modifications.