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

TL;DR

The paper introduces the phase quantum walk (PQW), a novel quantum walk framework that enables efficient distribution of arbitrary graph states in quantum networks with universal correction, verified analytically and on hardware.

Contribution

It presents the PQW framework with a universal correction method, allowing distribution of any graph state topology from elementary resources, advancing quantum network state distribution.

Findings

Universal correction theorem enables perfect graph state distribution.

Analytical fidelity models match hardware experimental results.

Validated on IBM hardware with high fidelity for GHZ4 and L4 states.

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).