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arXiv:2511.12872 · Physica Scripta

Pulsation of quantum walk between two arbitrary graphs with weakly connected bridge

Taisuke Hosaka, Etsuo Segawa

quant-phmath-phSubmitted Nov 17, 2025• Announced Nov 18, 2025• 0 citations• Revised May 4, 2026 (v2)
v1 · Nov 17, 2025v2 · May 4, 2026
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Abstract

We consider the Grover walk on a finite graph composed of two arbitrary simple graphs connected by one edge, referred to as a bridge. The parameter $ε>0$ assigned at the bridge represents the strength of connectivity: if $ε=0$, then the graph is completely separated. We show that for sufficiently small values of $ε$, a phenomenon called pulsation occurs. The pulsation is characterized by the periodic transfer of the quantum walker between the two graphs. An asymptotic expression with respect to small $ε$ for the probability of finding the walker on either of the two graphs is derived. This expression reveals that the pulsation depends solely on the number of edges in each graph, regardless of their structure. In addition, we obtain that the quantum walker is transferred periodically between the two graphs, with a period of order $O(ε^{-1/2})$. Furthermore, when the number of edges of two graphs is equal, the quantum walker is almost completely transferred.

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•Apr 26, 2012•2 citations
We consider the Grover walk on a finite graph composed of two arbitrary simple graphs connected by one edge, referred to as a bridge. The parameter ε>0ε>0ε>0 assigned at the bridge represents the strength of connectivity: if ε=0ε=0ε=0, then the graph is completely separated. We show that for sufficiently small values of εεε, a phenomenon called pulsation occurs. The pulsation is characterized by the periodic transfer of the quantum walker between the two graphs. An asymptotic expression with respect to small εεε for the probability of finding the walker on either of the two graphs is derived. This expression reveals that the pulsation depends solely on the number of edges in each graph, regardless of their structure. In addition, we obtain that the quantum walker is transferred periodically between the two graphs, with a period of order O(ε−1/2)O(ε^{-1/2})O(ε−1/2). Furthermore, when the number of edges of two graphs is equal, the quantum walker is almost completely transferred.
We study discrete-time quantum walks on weighted graphs, where every edge in an edge cut set is assigned a small weight ε>0ε>0ε>0. The parameter εεε represents the strength of the connections through the cut edges: as ε→0ε\to 0ε→0, these connections vanish, and the graph decomposes into the connected components obtained by removing the cut edges. We refer to these weighted cut edges as {\it weak edges}. We show that, for sufficiently small εεε and on the time scale t=Θ(ε−1/2)t=Θ(ε^{-1/2})t=Θ(ε−1/2), the finding probabilities between the connected components are asymptotically described by a continuous-time wave equation on a reduced graph, whose vertices represent the connected components and whose edges represent the weak edges connecting them. The wave equation is governed by a symmetric weighted Laplacian determined by the reduced graph and the number of arcs contained in the components. Consequently, finding probabilities of leading-term are independent of the detailed internal structures of the components. We further show that this wave equation has the same form as Newton's equation of motion for a classical spring-mass system on the reduced graph.
Abstract We consider the Grover walk model on a connected finite graph with two infinite length tails and we set an -infinite external source from one of the tails as the initial state. We show that for any connected internal graph, a stationary state exists, moreover a perfect transmission to the opposite tail always occurs in the long time limit. We also show that the lower bound of the norm of the stationary measure restricted to the internal graph is proportional to the number of edges of this graph. Furthermore when we add more tails (e.g. r -tails) to the internal graph, then we find that from the temporal and spatial global view point, the scattering to each tail in the long time limit coincides with the local one-step scattering manner of the Grover walk at a vertex whose degree is .
Quantum walks exhibit properties without classical analogues. One of those is the phenomenon of asymptotic trapping---there can be nonzero probability of the quantum walker being localized in a finite part of the underlying graph indefinitely even though locally all directions of movement are assigned nonzero 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.
Search and state transfer between hubs, i.e., fully connected vertices, on otherwise arbitrary connected graph is investigated. Motivated by a recent result of Razzoli et al. [J. Phys. A: Math. Theor. 55, 265303 (2022)] on the universality of hubs in continuous-time quantum walks and spatial search, we extend the investigation to state transfer and also to the discrete-time case. We show that the continuous-time quantum walk allows for perfect state transfer between multiple hubs if the numbers of senders and receivers are close. Turning to the discrete-time case, we show that the search for hubs is successful provided that the initial state is locally modified to account for a degree of each individual vertex. Concerning state transfer using discrete-time quantum walk, it is shown that between a single sender and a single receiver one can transfer two orthogonal states in the same run time. Hence, it is possible to transfer an arbitrary quantum state of a qubit between two hubs. In addition, if the sender and the receiver know each other location, another linearly independent state can be transferred, allowing for exchange of a qutrit state. Finally, we consider the case of transfer between multiple senders and receivers. In this case we cannot transfer specific quantum states. Nevertheless, quantum walker can be transferred with high probability in two regimes---either when there is a similar number of senders and receivers, which is the same as for the continuous-time quantum walk, or when the number of receivers is considerably larger than the number of senders. Our investigation is based on dimensional reduction utilizing the invariant subspaces of the respective evolutions and the fact that for the appropriate choice of the loop weights the problem can be reduced to the complete graph with loops.
We completely classify connected Cayley graphs over finite abelian groups that exhibit perfect state transfer in Grover walks. First, we show that a Cayley graph over any finite group exhibiting perfect state transfer has valency 111, 222, or 444. For valency 444, we prove that an abelian Cayley graph Cay⁡(Γ,S)\operatorname{Cay}(Γ,S)Cay(Γ,S) exhibits perfect state transfer if and only if S={a,−a,a+h,−a+h}S=\{a,-a,a+h,-a+h\}S={a,−a,a+h,−a+h}, where hhh has order 222, and 4∤∣Γ∣/24\nmid |Γ|/24∤∣Γ∣/2. Combining this characterization with the valency-one and valency-two cases, we explicitly determine all pairs (Γ,S)(Γ,S)(Γ,S) for which Cay⁡(Γ,S)\operatorname{Cay}(Γ,S)Cay(Γ,S) exhibits perfect state transfer. Our results extend the classification of circulant graphs of valency at most four obtained by Kubota and Yoshino [Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks, \emph{J. Comb. Theory Ser. A.}, 216 (2025) 106064] to Cayley graphs over arbitrary finite abelian groups and to all valencies.
We present three families of graphs which admit quantum walks with interesting dynamics either in the continuous time walk, or in the discrete time walk for appropriate selections of coin and initial conditions. These dynamics are either periodicity, perfect state transfer, or very high fidelity state transfer. These families are modifications of families known not to exhibit periodicity or perfect state transfer in general. The robustness of the dynamics is tested by varying the initial state and by adding decoherence. The perfect state transfer is found to be very robust to small disturbances. The families of graphs found to exhibit useful properties are generalisations of small graphs. By varying the strength of edges in each family, interpolation between the families can be carried out, enabling further understanding of the factors affecting perfect state transfer. The perfect state transfer in these families is found to be destroyed by this variation, indicating that the transport strongly depends on the graph structure. A brief summary of other small graphs investigated but found not to exhibit such properties is included for completeness.