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arXiv:2604.02169

The Phase Quantum Walk: A Unified Framework for Graph State Distribution in Quantum Networks

Soumyojyoti Dutta

quant-phSubmitted Apr 2, 2026• Announced Apr 3, 2026• 0 citations• Revised Apr 12, 2026 (v2)
v1 · Apr 2, 2026v2 · Apr 12, 2026v3 · Apr 14, 2026v4 · May 8, 2026v5 · May 16, 2026v6 · Aug 11, 2026v7 · Sep 11, 2026v8 · Oct 26, 2026
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Abstract

Distributing arbitrary graph states across quantum networks is a central challenge for modular quantum computing and measurement-based quantum communication. I introduce the phase quantum walk (PQW), a discrete-time quantum walk in which the conventional position-permuting shift operator is replaced by a diagonal conditional phase (CZ) gate, enabling distribution of arbitrary graph states -- not merely GHZ states -- from elementary two-qubit resources. The Byproduct Lemma shows that each walk step teleports edge entanglement with a correctable Pauli byproduct; the Coin Invariance Theorem proves that the optimal fidelity F*(C,E) = F*(H,E) for all unitary coins C and noise channels E, with closed-form expressions F_dep = (1 - 3p/4)^k and F_pd = ((1 + sqrt(1-p))/2)^k. A universal correction theorem proves that for any graph G = (V,E) and any measurement outcome, the local correction C_v = Z_v^{g_v} where g_v is the XOR of far-side resource outcomes restores the distributed state to |G> -- a single formula covering all graph topologies without case analysis, verified across 18 graph topologies at F = 1.0. Hardware validation on ibm_marrakesh (IBM Heron r2, CZ-native) yields F_cl = 0.924 for |GHZ4> and 0.922 for |L4>, statistically identical, providing the first experimental confirmation that fidelity is independent of graph topology as predicted by the Coin Invariance Theorem

Comments
8 pages, 7 figures

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