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arXiv:2609.39890

Quantum State Routing and Perfect State Transfer on Signed Graphs under Environmental Noise

Nur Mohammad Sanfui, Supriyo Dutta

quant-phcs.DMcs.ITmath.COSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

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.

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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 D2m,0,2nD_{2m,0,2n}D2m,0,2n​ and scalable glued binary trees. Furthermore, we analyse routing performance under realistic open-system amplitude and phase damping noise channels.
We investigate quantum state transfer on a class of bipartite graphs, namely the butterfly graphs, within the framework of discrete-time quantum walks. These graphs facilitate the construction of scalable quantum networks that enable communication between a sender and a receiver via perfect state transfer. Our analysis demonstrates that state transfer occurs across different butterfly graphs, thereby extending the known families of networks that support high-fidelity quantum state transfer. In addition to the ideal noiseless dynamics, we further investigate the robustness of quantum state transfer in the presence of non-Markovian environmental noise, specifically, random telegraph noise, modified Ornstein-Uhlenbeck noise, which are examples of unital noise and non-Markovian amplitude damping noise, non-unital noise. These noise models capture different types of system-environment interactions and memory effects that influence the coherence of the quantum walk. These findings contribute to the theoretical understanding of how butterfly graph constructions influence quantum transport phenomena.
We provide a brief survey of perfect state transfer in quantum walks on finite graphs. The ability to transfer a quantum state from one part of a quantum computer to another is a key ingredient of scalable architectures. Transfer through structures that do not require locally varying dynamic control simplifies the design and hence reduces the opportunities for errors to arise. Continuous time walks quantum walks on highly structured graphs exhibit perfect state transfer for the complete graph of size 2, the path of length 3, and the cycle of size 4. From these, larger graphs can be constructed, and the use of edge weights widens this set considerably. Discrete-time quantum walks have more flexibility through exploiting the coin degrees of freedom, but with the disadvantage that local control of the coin is required if the degree of the vertices varies. The closely related property of periodicity (exact return to the starting state) is also mentioned.
We show that a perfect state transfer of an arbitrary unknown two-qubit state can be achieved via a discrete-time quantum walk with various settings of coin flips and extend this method to the distribution of an arbitrary unknown multiqubit entangled state between every pair of sites in the multidimensional network. Furthermore, we study the routing of quantum information on this network in a quantum-walk architecture, which can be used as quantum information processors to communicate between separated qubits.
I present a protocol that converts |E| elementary two-qubit resource states, one per edge of a target graph G=(V,E), into the distributed graph state |G> using only operations local to each party and classical communication, reaching arbitrary graph topologies rather than the GHZ class of prior walk-based schemes. The elementary step is obtained from the coined discrete-time quantum walk by replacing the position-permuting shift with a diagonal CZ gate, the phase quantum walk (PQW); on each edge it realises the entanglement-assisted non-local CZ gate. Because every gate and correction is diagonal, all edges run in a single round with the local correction C_v = Z_v^{g_v}, verified exhaustively on 22 connected graphs (32,212 outcomes) at F = 1.0. The central result is an exact fidelity theory under independent Pauli noise on the resource qubits, with a separate channel on each. When the noise acts after the local gate, the fidelity factorises over vertices; for depolarising noise F* = prod_v [1 + (1 - 4p/3)^{deg v}]/2, with the degree sequence entering only at second order. When it acts on the stored pair, as link-generation, transmission and storage noise does, it becomes an edge-cover partition function, F* = 2^{-|V|} sum_{S subset V} (1 - 4p/3)^{2|E(S)|}, bounded above by the vertex product and with a leading-order loss coefficient 3/2 times larger. Dephasing gives the same fidelity at both placements. On ibm_kingston, direct fidelity estimation certifies F = 0.815(3) for |L_4>, 0.776(3) for |GHZ_4> and 0.733(3) for |C_4>. Native SWAP-free execution at coupling girth gamma requires target girth at least gamma/3 and maximum degree at most 3; on heavy-hex this forbids triangles, though |K_4>, being LC-equivalent to a star, remains reachable.
Perfect state transfer between two marked vertices of a graph by means of a discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that, for the coined quantum walks on a star graph and a complete graph with self-loops, perfect state transfer between the sender and receiver vertex is achieved for an arbitrary number of vertices NNN in O(N)O(\sqrt{N})O(N​) steps of the walk. Finally, we show that Szegedy's walk with queries on a complete graph allows for state transfer with unit fidelity in the limit of large NNN.
Robust quantum routing is essential for scalable quantum technologies. This paper investigates the resilience of routing protocols in network architectures designed for perfect, high-fidelity transfer of both classical and quantum information under ideal conditions. We encode information in the position of a quantum walker on a graph, modelling the routing of a generic qubit state from a single input to multiple (orthogonal) outputs. We analyse and assess routing performance in various regimes, evaluating their robustness against static and dynamical noise.