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.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.

quant-pharXiv:2511.12872Physica ScriptaNov 18, 20250 citations

Multi-target quantum walk search on Johnson graph

Pulak Ranjan Giri

arXiv:2510.04424Physics Letters AJun 19, 20253 citations

Coined Quantum Walks on Graphs

Renato Portugal

Quantum science and technologyJan 1, 20181 citation

Quantum walk search on Johnson graphs

Thomas G. Wong

Abstract The Johnson graph J ( n , k ) is defined by n symbols, where vertices are k -element subsets of the symbols, and vertices are adjacent if they differ in exactly one symbol. In particular, J ( n , 1 ) is the complete graph K n , and J ( n , 2 ) is the strongly regular triangular graph T n , both of which are known to support fast spatial search by continuous-time quantum walk. In this paper, we prove that J ( n , 3 ) , which is the n -tetrahedral graph, also supports fast search. In the process, we show that a change of basis is needed for degenerate perturbation theory to accurately describe the dynamics. This method can also be applied to general Johnson graphs J ( n , k ) with fixed k .Abstract The Johnson graph J ( n , k ) is defined by n symbols, where vertices are k -element subsets of the symbols, and vertices are adjacent if they differ in exactly one symbol. In particular, J ( n , 1 ) is the complete graph K n , and J ( n , 2 ) is the strongly regular triangular graph T n , both of which are known to support fast spatial search by continuous-time quantum walk. In this paper, we prove that J ( n , 3 ) , which is the n -tetrahedral graph, also supports fast search. In the process, we show that a change of basis is needed for degenerate perturbation theory to accurately describe the dynamics. This method can also be applied to general Johnson graphs J ( n , k ) with fixed k .

arXiv:1601.04212Journal of Physics A Mathematical and TheoreticalApr 12, 201623 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>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.