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

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. This single structural change enables distribution of arbitrary graph states -- not merely GHZ or star-topology states -- from elementary two-qubit resources shared between adjacent network nodes. The Byproduct Lemma shows that each walk step teleports one edge of entanglement with a correctable Pauli byproduct. A universal correction theorem establishes that a single local Z correction at each node, computed from the XOR of neighboring measurement outcomes, restores the distributed state to the target graph state for any graph topology and any measurement outcome -- no case analysis required. Correction formulas are verified analytically for 18 graph topologies with up to 4096 measurement outcomes, all at fidelity F = 1.0. Closed-form fidelity expressions under depolarizing and phase damping noise are derived and verified, scaling as (1 - 2p/3)^k and ((1 + sqrt(1-p))/2)^k respectively, where k is the number of resource qubits. Hardware validation on ibm_marrakesh (IBM Heron r2, 156 qubits, CZ-native) confirms topology-independent fidelity: protocols distributing GHZ4 and L4 states both using k = 6 resource qubits yield Bhattacharyya fidelities of 0.924 and 0.922, statistically identical and consistent with the analytical noise model. Stabilizer measurements on the corrected data qubits confirm distributed graph-state structure with fidelity lower bounds of 0.787 (GHZ4) and 0.759 (L4).