FIG. 1. (a) The energy spectra of | 00 〉 → | 00 〉 and | 11 〉 → | 10 〉 operations of the adiabatic CNOT gate in the absence of noise. The solid grey diamonds show the results of direct diagonalisation of the H$_{0}$ . (b) Examples of the spectra for | 00 〉 → | 00 〉 and | 11 〉 → | 10 〉 operations of the CNOT gate with a GOE random matrix noise term in the Hamiltonian with amplitude ϵ = 0 . 1. For illustrative purposes, different line styles are used to denote the different energy levels and the third plot in each of the subfigures shows an enlarged view of a region of the | 11 〉 → | 10 〉 spectrum containing a number of crossings. mechanism is given by P$_{LZS}$ = exp$^{}$  − Δ 2 min |〈 n | ZH$_{b}$ | n + 1 〉| ∣ ∣ ˙ λ ( t ) ∣ ∣   (6) 7