State Transfer in Discrete-Time Quantum Walks via Projected Transition Matrices
Abstract
In this paper, we analyze 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 can be achieved between an initial and a target state under unitary evolution, even when perfect state transfer is unattainable. We give a spectral characterization of peak state transfer 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 (assuming that the walk starts at a point of the design). In addition, we provide many examples of peak state transfer, including an infinite family where the amount of peak state transfer tends 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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BibTeX
@article{guo2026state,
title = {{State Transfer in Discrete-Time Quantum Walks via Projected Transition Matrices}},
author = {Krystal Guo and Vincent Schmeits},
journal = {The Electronic Journal of Combinatorics},
year = {2026},
month = oct,
doi = {10.37236/13764},
url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p4},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - State Transfer in Discrete-Time Quantum Walks via Projected Transition Matrices
AU - Krystal Guo
AU - Vincent Schmeits
PY - 2026
DA - 2026/10/09
JO - The Electronic Journal of Combinatorics
DO - 10.37236/13764
UR - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v33i4p4
AB - In this paper, we analyze 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 can be achieved between an initial and a target state under unitary evolution, even when perfect state transfer is unattainable. We give a spectral characterization of peak state transfer 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 (assuming that the walk starts at a point of the design). In addition, we provide many examples of peak state transfer, including an infinite family where the amount of peak state transfer tends 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.
SN - 1077-8926
ER -