Continuous-Time Quantum Walks on Directed Bipartite Graphs (1606.00992v3)

Abstract: This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter $\alpha$. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes $N$, time $t$ and the model parameter $\alpha$. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerical simulations of quantum walks on circulant graphs show complex interference phenomena and how complete suppression of transport is achieved near $\alpha=\pi/2$. By proving two mirror symmetries around $\alpha=0$ and $\pi/2$ we show that these quantum walks have a period of $\pi$ in $\alpha$. We show that undirected edges lose their effect on the quantum walk at $\alpha=\pi/2$ and present non-bipartite graphs that exhibit suppression of transport. Finally, we analytically compute the Hamiltonians of quantum walks on the directed ring graph.

• The paper investigates continuous-time quantum walks on directed bipartite graphs, demonstrating that probability transport can be suppressed by tuning a specific parameter.
• Analytical solutions for quantum walks on star and circulant graphs are provided, offering insights into modulation across different graph structures.
• Key symmetries are discovered that establish a periodicity for the tuning parameter, allowing precise control over quantum transport characteristics.

Continuous-Time Quantum Walks on Directed Bipartite Graphs

The paper "Continuous-Time Quantum Walks on Directed Bipartite Graphs" presents a detailed investigation into the behavior of continuous-time quantum walks on directed bipartite graphs. Utilizing the adjacency matrix of these graphs and incorporating a tunable parameter, the authors derive conditions under which quantum walks can effectively suppress probability transport between graph partitions. This research bridges the structure of directed networks and quantum computing, promising insights into quantum information transport.

Key Findings

The paper underscores several important phenomena in the domain of quantum walks on directed graphs:

1. Suppression of Transport: The research highlights that complete suppression of probability transport can be achieved on bipartite graphs by tuning a model parameter, denoted as $\alpha$, specifically near $\alpha = \pi/2$. This result is significant, as it suggests a method to control quantum state propagation in these systems.
2. Analytical Solutions for Specific Graphs: The authors provide analytic solutions for star and circulant graph configurations. These solutions are generalized for any number of nodes $N$, time $t$, and the parameter $\alpha$. Such analytical insights are crucial for understanding how quantum walks can be modulated across different graph structures.
3. Symmetries and Periodicity: By proving two mirror symmetries around $\alpha=0$ and $\alpha=\pi/2$, the paper establishes a periodicity of $\pi$ in $\alpha$. This result is essential for extending the utility of $\alpha$ beyond a mere scaling factor, highlighting its potential as a knob for tuning quantum transport characteristics.
4. Behavior of Directed vs. Undirected Graphs: It is shown that at $\alpha=\pi/2$, the influence of undirected edges vanishes in these quantum systems. Also, the paper presents non-bipartite graphs that display similar suppression of transport, broadening the applicability of the findings.

Practical and Theoretical Implications

The implications of this research are particularly relevant in the context of quantum computing and related fields:

• Quantum Information Processing: The ability to control transport suppression in quantum networks has implications for designing more efficient quantum information processing systems. The results can guide the creation of networks where entanglement is maintained while unwanted transport channels are effectively secluded.
• Material Science Applications: Since many materials such as crystal structures and graphene can be modeled as bipartite graphs, these findings could assist in tuning energy distributions in quantum materials, leading to enhanced functionalities in nanodevices and potentially improving technologies like quantum sensors and solar cells.

Speculation on Future Research

Future research could aim at:

• Expanding Graph Classes: Exploring other classes of graphs or extending the model to weighted graphs could unearth new phenomena related to quantum walks.
• Experimental Validation: Moving towards experimental setups that can validate these theoretical predictions would solidify the practical implications of this work.
• Algorithm Development: Given the role of quantum walks in algorithm design, leveraging these findings could lead to more efficient quantum algorithms, particularly for tasks that involve graph traversal and network analysis.

In summary, this paper provides foundational insights into the mechanics of continuous-time quantum walks on directed bipartite graphs. The ability to analytically solve for specific cases and control quantum transport presents profound implications both theoretically and in practical quantum computing and physics applications.