Perfect state transfer on quotient graphs

Abstract

We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/\pi, under any equitable partition \pi, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs \Box_{k} G_{k}/\pi_{k} is isomorphic to the quotient graph \Box_{k} G_{k}/\pi, for some equitable partition \pi. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk.

Generation and Robustness of Quantum Entanglement in Spin Graphs

“…3. We note that for the presented quotient graph, Bachmann et al introduced a lift-and-quotient reduction [22], allowing for a further simplification of the graph. The top structure from Fig.…”

“…We use the methods of graph partitioning from Refs. [21,22] to simplify such graph into a quotient graph and a quotient linear chain. In Sect.…”

Perfect quantum state transfer on diamond fractal graphs

“…In addition to its fundamental interest, this means that perfect quantum state transfer also has potential applications to the design of sub-protocols for quantum information and quantum computation [3][4][5]. A number of one dimensional cases, when perfect transmission can be achieved, have been found in some XX chains with inhomogeneous couplings, see [2,3,[6][7][8][9][10][11][12][13][14][15], and references therein]. These models have the advantage that the perfect transfer can be done without the need for active control.…”

Unique square property, equitable partitions, and product-like graphs

“…Equitable partitions of graphs [5,6] were originally introduced as a means of simplifying the computation of graph spectra [20] and walks on graphs [8]. A series of recent results on so-called perfect state transfer revealed a close connection between equitable partitions of the vertex set of G, the corresponding quotient graphs, and the Cartesian product structure of G [2,7].…”