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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 May 8, 2026 (v4)
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. We 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. 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 = \sum_{e \ni v} s_{e,\bar{v}} restores the distributed state to |G>. This gives a single correction formula covering all graph topologies without case analysis. Analytical correction formulas are verified for 18 topologies (up to 4096 outcomes) at F=1.0. Closed-form fidelity expressions F^*_{dep} = (1 - 2p/3)^k and F^*_{pd} = ((1 + \sqrt{1-p})/2)^k are derived and verified, where k is the number of resource qubits. Hardware validation on IBM Marrakesh (IBM Heron r2, CZ-native) yields Bhattacharyya fidelity F^*_{cl} = 0.924 for |GHZ4> and 0.922 for |L4>, statistically identical as predicted by the noise formulas for protocols using equal numbers of resource qubits (k=6 in both cases).

Comments
17 pages, 7 figures

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Distributing arbitrary graph states across quantum networks is a central challenge for modular quantum computing and measurement-based quantum communication. We 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. 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 = \sum_{e \ni v} s_{e,\bar{v}} restores the distributed state to |G>. This gives a single correction formula covering all graph topologies without case analysis. Analytical correction formulas are verified for 18 topologies (up to 4096 outcomes) at F=1.0. Closed-form fidelity expressions
F^*_{dep} = (1 - 2p/3)^k
and
F^*_{pd} = ((1 + \sqrt{1-p})/2)^k
are derived and verified, where k is the number of resource qubits. Hardware validation on IBM Marrakesh (IBM Heron r2, CZ-native) yields Bhattacharyya fidelity F^*_{cl} = 0.924 for |GHZ4> and 0.922 for |L4>, statistically identical as predicted by the noise formulas for protocols using equal numbers of resource qubits (k=6 in both cases).
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