State transfer in discrete-time quantum walks via projected transition matrices

• URL: http://arxiv.org/abs/2411.05560v1
• Date: Fri, 08 Nov 2024 13:35:27 GMT
• Title: State transfer in discrete-time quantum walks via projected transition matrices
• Authors: Krystal Guo, Vincent Schmeits,
• Abstract summary: We consider state transfer in quantum walks by using methods. We define peak state transfer as the highest state transfer that could be achieved between an initial and a target state.
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• Abstract: In this paper, we consider state transfer in quantum walks by using combinatorial methods. We generalize perfect state transfer in two-reflection discrete-time quantum walks to a notion that we call peak state transfer; we define peak state transfer as the highest state transfer that could be achieved between an initial and a target state under unitary evolution, even when perfect state transfer is unattainable. We give a characterization of peak state transfer that is easy to apply and that allows us to fully characterize peak state transfer in the arc-reversal (Grover) walk on various families of graphs, including strongly regular graphs and incidence graphs of block designs (starting at a point). In addition, we provide many examples of peak state transfer, including an infinite family where the amount of peak state transfer goes to $1$ as the number of vertices grows. We further demonstrate that peak state transfer properties extend to infinite families of graphs generated by vertex blow-ups, and we characterize periodicity in the vertex-face walk on toroidal grids. In our analysis, we make extensive use of the spectral decomposition of a matrix that is obtained by projecting the transition matrix down onto a subspace. Though we are motivated by a problem in quantum computing, we identify several open problems that are purely combinatorial, arising from the spectral conditions required for peak state transfer in discrete-time quantum walks.

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