Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs

Konno, Norio; Segawa, Etsuo; Štefaňák, Martin

doi:10.3390/sym13071169

Cited by 6 publications

(13 citation statements)

References 18 publications

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“…Interestingly, as we show, the common eigenstates for the eigenvalue -1 derived below are exactly the eigenstates for the eigenvalue -1 for the non-percolated version of the walk presented in [46], so none of the states is removed by percolation. Here we reuse the key idea of utilising fundamental cycles of a graph to construct these states as in [46] but we approach the construction in an alternative way.…”

Section: Quantum Walk Definition and Trapped Statessupporting

confidence: 54%

“…Let us now search for the common eigenstates, which allow the construction of the projector to the subspace of sr-trapped states and the calculation of the ATP for given initial states. Note that they form a subset of eigenstates of non-percolated CQWs derived in [46], since the original structure graph represents one of the possible configurations in (5). However, the common eigenstates have to fulfill (5) for all configurations of open edges, which leads to stricter conditions (6) and (7).…”

Section: Quantum Walk Definition and Trapped Statesmentioning

confidence: 99%

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Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks

Mareš,

Novotný,

Štefaňák

et al. 2022

Preprint

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Quantum walks exhibit properties without classical analogues. One of those is the phenomenon of asymptotic trapping -there can be non-zero probability of the quantum walker being localised in a finite part of the underlying graph indefinitely even though locally all directions of movement are assigned non-zero amplitudes at each step. We study quantum walks with the flip-flop shift operator and the Grover coin, where this effect has been identified previously. For the version of the walk further modified by a random dynamical disruption of the graph (percolated quantum walks) we provide a recipe for the construction of a complete basis of the subspace of trapped states allowing to determine the asymptotic probability of trapping for arbitrary finite connected simple graphs, thus significantly generalizing the previously known result restricted to planar 3-regular graphs. We show how the position of the source and sink together with the graph geometry and its modifications affect the excitation transport. This gives us a deep insight into processes where elongation or addition of dead-end subgraphs may surprisingly result in enhanced transport and we design graphs exhibiting this pronounced behavior. In some cases this even provides closed-form formulas for the asymptotic transport probability in dependence on some structure parameters of the graphs.

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Section: Quantum Walk Definition and Trapped Statessupporting

confidence: 54%

Section: Quantum Walk Definition and Trapped Statesmentioning

confidence: 99%

Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks

Mareš,

Novotný,

Štefaňák

et al. 2022

Preprint

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“…We now consider the remaining densities on the correct path in terms of knowledge that has been proven mathematically. The eigenstate of the time evolution operator of the quantum walk with sinks was constructed on the path between two-self loops [28]. This eigenstate is called the trapped state, and it is not absorbed by the sink.…”

Section: Discussionmentioning

confidence: 99%

“…This eigenstate is called the trapped state, and it is not absorbed by the sink. In the Grover walk, the eigenstates are constructed between two self-loops and also around the cycles [28].…”

Section: Discussionmentioning

confidence: 99%

“…The presented method is an application of the emergence of a trapped eigenstate on a network with sinks, and it provides an alternative to previously reported methods [18][19][20]. Although the mathematical foundation of this method was given by Konno, Segawa, and Štefa ňák [28], the results presented here are non-trivial because the interaction among multiple trapped eigenstates and the initial condition is generally difficult to characterize as of now.…”

Section: Introductionmentioning

confidence: 98%

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Maze Solving by a Quantum Walk with Sinks and Self-Loops: Numerical Analysis

Matsuoka

Yuki²,

Lavička³

et al. 2021

Symmetry

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Maze-solving by natural phenomena is a symbolic result of the autonomous optimization induced by a natural system. We present a method for finding the shortest path on a maze consisting of a bipartite graph using a discrete-time quantum walk, which is a toy model of many kinds of quantum systems. By evolving the amplitude distribution according to the quantum walk on a kind of network with sinks, which is the exit of the amplitude, the amplitude distribution remains eternally on the paths between two self-loops indicating the start and the goal of the maze. We performed a numerical analysis of some simple cases and found that the shortest paths were detected by the chain of the maximum trapped densities in most cases of bipartite graphs. The counterintuitive dependence of the convergence steps on the size of the structure of the network was observed in some cases, implying that the asymmetry of the network accelerates or decelerates the convergence process. The relation between the amplitude remaining and distance of the path is also discussed briefly.

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Survival probability of the Grover walk on the ladder graph

Segawa

Koyama

Konno

et al. 2023

J. Phys. A: Math. Theor.

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We provide a detailed analysis of the survival probability of the Grover walk on the ladder graph with an absorbing sink. This model was discussed in Mareš et al., Phys. Rev. A 101, 032113 (2020), as an example of counter-intuitive behaviour in quantum transport where it was found that the survival probability decreases with the length of the ladder L, despite the fact that the number of dark states increases. An orthonormal basis in the dark subspace is constructed, which allows us to derive a closed formula for the survival probability. It is shown that the course of the survival probability as a function of L can change from increasing and converging exponentially quickly to decreasing and converging like L-1 simply by attaching a loop to one of the corners of the ladder. The interplay between the initial state and the graph conﬁguration is investigated.

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Relation between Quantum Walks with Tails and Quantum Walks with Sinks on Finite Graphs

Cited by 6 publications

References 18 publications

Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks

Key graph properties affecting transport efficiency of flip-flop Grover percolated quantum walks

Maze Solving by a Quantum Walk with Sinks and Self-Loops: Numerical Analysis

Survival probability of the Grover walk on the ladder graph

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