, where Δ$_{min}$ is the minimum separation between x$_{n}$ and x$_{n}$$_{+1}$ . We assume that the system undergoes uniform evolution (i.e. | ˙ λ | = 1 $_{T}$, where T is the computation time) and that the system can initially be found in the ground state with certainty (i.e. P ( n = 0; λ ( t ) = 1) = 1). Given this information, it is possible to calculate the level occupations as a function of time by identifying all the avoided or level crossings in the spectrum and applying the LZS equation, (6) to them. Figure 2 shows the dependence of the average success probability on the computation speed at a range of noise amplitudes for the | 00 〉 → | 00 〉 and | 11 〉 → | 10 〉 operations. In general, the success probability scales polynomially with the speed, i.e. P ∝ T − γ with γ ≈ 1, before approaching unity asymptotically. The exponent is different to those found in [11] for Hamiltonians drawn from the GUE, however there is no reason to expect a single exponent to hold universally for all choices of H$_{0}$ . From a computational performance point of view, it is important to note that the scaling exponent is independent of the noise amplitude, ı.e. only a prefactor change. This prefactor change could actually be viewed as being beneficial as we can see that the success probability at a given computation speed increases linearly with the noise amplitude (also shown in Fig. 3). This occurs as Δ$_{min}$ increases proportionally with ϵ and hence this effect is more pronounced for the | 11 〉 → | 10 〉 operation where the existence of the ground state gap is purely due to noise-induced level splitting. As the | 01 〉 → | 01 〉 and | 10 〉 → | 11 〉 operations also have degenerate ground state gaps, their success probability curves behave in a very similar way to the plot shown in Fig. 2 for the | 11 〉 → | 10 〉 operation. The increase in success probability with the noise amplitude (at a specific speed) may not be wholly beneficial though, as it will come at the cost of the fidelity of the final state (i.e. F = |〈 0$_{ideal}$( λ = 0) | 0$_{noise}$( λ = 0) 〉| , where 0 ≤ F ≤ 1). This is because the noise fluctuations will drive the state away from the ideal (i.e. in the absence of noise) path of evolution. The dependence of the fidelity on noise amplitude is shown in Fig. 3. The fidelity of the final state is an important quantity which needs to be maximised to ensure that readout will yield the desired solution. For the three operations where the existence of the ground state gap is due to noise-induced level splitting, there will be a noise amplitude that offers the optimal compromise between the success probability and the fidelity at a given speed and this is shown by the intersections of the curves in Fig. 3. These results suggest that the conventional stipulation that [ H$_{0}$, H$_{b}$ ] = 0 is unnecessary in practical realisations, as when the effects of a generic noise source are taken in to consideration, any hidden symmetries in 8