Universal Z-only Correction for Distributing Arbitrary Graph States in Quantum Networks

Abstract

Distributing arbitrary graph states across quantum networks is a central challenge for modular quantum computing and measurement-based quantum communication. I present a protocol converting |E| elementary two-qubit resource states -- one per edge of the target graph G = (V,E) -- into the distributed graph state |G>, using only single-qubit operations, measurements, and classical communication, and reaching arbitrary topologies, not just the GHZ class of prior walk-based schemes. The central result is a universal correction theorem: for any graph and any measurement outcome, the local correction C_v = Z_v^{g_v}, with g_v the XOR of the far-side outcomes on edges at v, restores the state to |G> -- one formula for all topologies, no case analysis. The elementary step comes from the coined discrete-time quantum walk with the position-permuting shift replaced by a diagonal conditional phase (CZ) gate: the phase quantum walk (PQW). Formulas are verified exhaustively on 22 connected graphs -- all up to four vertices, and five-vertex graphs with at most six edges -- over 32,212 outcomes at F = 1.0. Exact closed-form fidelities under independent depolarising and phase damping noise on the resource qubitsfollow, F*_dep = prod_v [1 + (1 - 4p/3)^{deg v}]/2 and F*_pd = prod_v [1 +(1 - p)^{(deg v)/2}]/2: the noise budget is set by the degree sequence of the target graph, not by its edge count. On ibm_kingston (IBM Heron r2, CZ-native) direct fidelity estimation certifies F = 0.815(3) for |L_4>, 0.776(3) for |GHZ_4>, and 0.733(3) for |C_4> in a single native-layout batch. I also establish a necessary condition for native (SWAP-free) execution at coupling girth g: the target graph must have girth at least g/3 and maximum degree at most 3. On heavy-hex (g = 12) this forbids triangles and places |L_4>, |C_4>, and |K_4> in a strict embeddability hierarchy -- flexible, rigid, and provably impossible.

26 pages, 8 figures