Distributed synthesis of arbitrary graph states in quantum networks via rank-two GF(2) reduction
Existing schemes for synthesizing graph states in quantum networks are essentially edge-by-edge constructions, so quantities such as the time-slot depth and the resource overhead grow significantly with the edge density of the target graph. This paper proposes a new method. Exploiting the mathematical equivalence between joint Pauli-X measurements and graph pivot operations, we formulate graph state synthesis as a rank-2 reduction process of a difference matrix over GF(2), and give an upper bound floor(N/2) on the number of steps for synthesizing an arbitrary N-node graph state, independent of the edge density of the target graph state. At the physical level, the joint Pauli-X measurement of each step is mapped to a dual-star concurrent distribution. We model the protocol on Waxman physical topologies with fiber attenuation and give a heuristic algorithm, and evaluate it against a strengthened Steiner baseline through Monte Carlo experiments. The experimental results show that our protocol is superior in time-slot depth almost everywhere. The entanglement resource overhead, the total number of CZ gates, and the number of Pauli measurements drop below the baseline near edge density p approximately 0.3, and are superior across the board thereafter. The denser the target graph state, the more significant the advantage.