Routing unknown quantum information across distributed communication networks requires autonomous, measurement-free mechanisms to prevent wave-function collapse. Szegedy quantum walks provide a mechanism for spatial state transport. The conventional walks on unweighted graphs suffer from severe back-reflection, spatial dispersion, and channel crosstalk. In this paper, we introduce a deterministic topological quantum routing architecture based on coined Szegedy quantum walks on edge-duplicated signed graphs. By treating edge signs as localized phase shifts within a balanced coin reflection, we enforce an exact zero back-scattering condition across routing nodes. We demonstrate deterministic Perfect State Transfer (PST) with unit fidelity at exact arrival times across fundamental archetypes, including the signed dumbbell switch $D_{2m,0,2n}$ and scalable glued binary trees. Furthermore, we analyse routing performance under realistic open-system amplitude and phase damping noise channels.Routing unknown quantum information across distributed communication networks requires autonomous, measurement-free mechanisms to prevent wave-function collapse. Szegedy quantum walks provide a mechanism for spatial state transport. The conventional walks on unweighted graphs suffer from severe back-reflection, spatial dispersion, and channel crosstalk. In this paper, we introduce a deterministic topological quantum routing architecture based on coined Szegedy quantum walks on edge-duplicated signed graphs. By treating edge signs as localized phase shifts within a balanced coin reflection, we enforce an exact zero back-scattering condition across routing nodes. We demonstrate deterministic Perfect State Transfer (PST) with unit fidelity at exact arrival times across fundamental archetypes, including the signed dumbbell switch $D_{2m,0,2n}$ and scalable glued binary trees. Furthermore, we analyse routing performance under realistic open-system amplitude and phase damping noise channels.

quant-pharXiv:2609.39890Oct 1, 20260 citations

Quantum Planar Graphs and Its Application

Rupkumar Mahapatra, Tofigh Allahviranloo, Smita A. Desai +2 more

Studies in computational intelligenceJan 1, 20252 citations

Perfect state transfer on Cayley graphs over dihedral groups

Xiwang Cao, Keqin Feng

Recently, there are extensive studies on perfect state transfer on graphs due to their significant applications in quantum information processing and quantum computations. However, most of the graphs previously investigated are abelian Cayley graphs. In this paper, we study perfect state transfer on Cayley graphs over dihedral groups. Using the representations of the dihedral group , we present some necessary and sufficient conditions for the Cayley graph to have a perfect state transfer between two distinct vertices for some connection set S. Based on these conditions, we show that cannot have PST if n is odd and S is conjugation-closed. For some even integers n, it is possible for to have PST, some concrete constructions are provided.Recently, there are extensive studies on perfect state transfer on graphs due to their significant applications in quantum information processing and quantum computations. However, most of the graphs previously investigated are abelian Cayley graphs. In this paper, we study perfect state transfer on Cayley graphs over dihedral groups. Using the representations of the dihedral group , we present some necessary and sufficient conditions for the Cayley graph to have a perfect state transfer between two distinct vertices for some connection set S. Based on these conditions, we show that cannot have PST if n is odd and S is conjugation-closed. For some even integers n, it is possible for to have PST, some concrete constructions are provided.

Linear and Multilinear AlgebraApr 3, 201928 citations

Quantum walks on finite and bounded infinite graphs

Chris D. Godsil, Steve Kirkland, Mohapatra, Sarojini +2 more

A weighted graph $G$ with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in $G$ . In this paper, we prove a fundamental result on equitable partitions of bounded weighted graphs with twin subgraphs and use this fact to construct finite and bounded infinite graphs with pair and plus state transfer with the adjacency matrix as a Hamiltonian. We show that for each $k \ge 3$ , (i) there are infinitely many connected unweighted graphs with maximum degree $k$ admitting pair state transfer at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$ , and (ii) there are infinitely many signed graphs with exactly one negative edge weight and whose underlying unweighted graphs have maximum degree $k$ admitting plus state transfer at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$ . Parallel results are proven for perfect state transfer between a plus state and a pair state, and for the existence of sedentary pair and plus states. We further prove that almost all connected unweighted finite planar graphs admit pair state transfer at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$ , and almost all connected unweighted finite planar graphs can be assigned a single negative edge weight resulting in plus state transfer, or perfect state transfer between a plus state and a pair state, at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$ . Analogous results are shown to hold for unweighted finite trees. Using blow-up graphs, Cayley graphs and graphs with tails, we construct new infinite families of (finite and infinite) unweighted graphs and signed graphs admitting pair or plus state transfer.A weighted graph $G$ with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in $G$. In this paper, we prove a fundamental result on equitable partitions of bounded weighted graphs with twin subgraphs and use this fact to construct finite and bounded infinite graphs with pair and plus state transfer with the adjacency matrix as a Hamiltonian. We show that for each $k \ge 3$, (i) there are infinitely many connected unweighted graphs with maximum degree $k$ admitting pair state transfer at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$, and (ii) there are infinitely many signed graphs with exactly one negative edge weight and whose underlying unweighted graphs have maximum degree $k$ admitting plus state transfer at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$. Parallel results are proven for perfect state transfer between a plus state and a pair state, and for the existence of sedentary pair and plus states. We further prove that almost all connected unweighted finite planar graphs admit pair state transfer at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$, and almost all connected unweighted finite planar graphs can be assigned a single negative edge weight resulting in plus state transfer, or perfect state transfer between a plus state and a pair state, at $τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}$. Analogous results are shown to hold for unweighted finite trees. Using blow-up graphs, Cayley graphs and graphs with tails, we construct new infinite families of (finite and infinite) unweighted graphs and signed graphs admitting pair or plus state transfer.

arXiv:2510.05306Oct 6, 20250 citations

D_{2m,0,2n}

and scalable glued binary trees. Furthermore, we analyse routing performance under realistic open-system amplitude and phase damping noise channels.

A weighted graph

G

with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in

G

. In this paper, we prove a fundamental result on equitable partitions of bounded weighted graphs with twin subgraphs and use this fact to construct finite and bounded infinite graphs with pair and plus state transfer with the adjacency matrix as a Hamiltonian. We show that for each

k \ge 3

, (i) there are infinitely many connected unweighted graphs with maximum degree

k

admitting pair state transfer at

τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}

, and (ii) there are infinitely many signed graphs with exactly one negative edge weight and whose underlying unweighted graphs have maximum degree

k

admitting plus state transfer at

τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}

. Parallel results are proven for perfect state transfer between a plus state and a pair state, and for the existence of sedentary pair and plus states. We further prove that almost all connected unweighted finite planar graphs admit pair state transfer at

τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}

, and almost all connected unweighted finite planar graphs can be assigned a single negative edge weight resulting in plus state transfer, or perfect state transfer between a plus state and a pair state, at

τ\in\{\fracπ{\sqrt{2}},\fracπ{2}\}

. Analogous results are shown to hold for unweighted finite trees. Using blow-up graphs, Cayley graphs and graphs with tails, we construct new infinite families of (finite and infinite) unweighted graphs and signed graphs admitting pair or plus state transfer.