Dephasing-time statistics of disordered Kitaev chains: the Oke scaling relation, noise speed and an optimal wire length

Majorana qubits built from Kitaev chains lose phase coherence when noise shifts the energy splitting δE of their Majorana pair. Real devices differ from one another through frozen, spatially correlated disorder, and they are also exposed to moving disorder (charge noise) whose switching rates span many decades. We ask how the frozen disorder sets the wire-to-wire statistics of the dephasing time T2, how the answer depends on the speed of the noise, and what it implies for the choice between long and short wires. (i) The Oke scaling relation carries over from the splitting to the dephasing time: Var(ln T2) grows by J²K per site for slow noise and 4J²K for fast noise (motional narrowing); in between, T2 ∝ δE^(−α), and for 1/f noise α rises from 1 to 2 as the wire gets longer. (ii) Noise acting on the whole wire dephases the qubit about √N times faster than site-by-site noise of the same strength. (iii) Combining dephasing with quasiparticle-pair creation gives each wire an optimal length N*, whose spread across wires follows std(N*) ≈ ξJ√(KN*). (iv) Confining disorder to part of a wire raises the median T2 by the Jensen gap, independently of where it is placed, but 22–40% of wires end up worse than a clean wire. The results are derived at Δ = t, tested numerically at Δ = 0.6t, for three correlation shapes and at two chemical potentials, and reproduce published limits.