Z ( H$_{b}$ + Z − $^{1}$H$_{0}$ + Z − $^{1}$δh (1)). We assume that H (1) has a non-degenerate, well spaced en- ergy spectrum. Note that the initial conditions contain all the information about the final Hamiltonian H$_{0}$ . The initial noise term δh ( λ = 1) will be a random matrix drawn from the GOE with amplitude ϵ . Throughout the paper, we take µ = − 0 . 1, Z − 1 = 0 . 1 and τ = 0 . 1. Figure 1(a) shows the energy spectra of the CNOT gate acting on the | 00 〉 and | 11 〉 com- putational basis states. In both cases, the results from the eigenvalue dynamics simulations agree with the results of direct diagonalisation of H$_{0}$ to four significant figures. In both of the spectra, there is an abundance of level crossings which arise because of the symmetries Degeneracies between the ground and first excited states occur in the | 11 〉 → | 10 〉 op- eration, which will result in a success probability of zero for this ideal case. This case can be viewed as an example of AQC with an “improper” choice of initial configuration, i.e. where [ H$_{0}$, H$_{b}$ ] = 0. It is clear that the addition of any type of perturbation (e.g. noise) will break these symmetries, therefore splitting the degeneracies and resulting in a finite success probability. The results of the addition of noise are shown in plot Fig. 1(b) and it is evident that there are now no degeneracies in either of the spectra. The spectra for the | 01 〉 → | 01 〉 and | 10 〉 → | 11 〉 operations were found to show similar trends to the | 11 〉 → | 10 〉 , in that they also exhibited a degeneracy between the ground and first excited states in the absence of noise. To properly characterise the effects of noise on the adiabatic quantum algorithm for the CNOT gate it is necessary to look at its effects on the probability of level occupation and in particular the success probability of the algorithm (i.e. P ( n = 0; λ ( t ) = 0 | n = 0; λ ( t ) = 1)). During the evolution of an adiabatic quantum computer the main mechanism by which the system can tunnel from one state to another is Landau-Zener-St¨ uckelberg tunneling [8–10]. This occurs when the separation of two adjacent energy levels is at a local minimum (i.e. at an avoided or level crossing) and the probability of excitation from | n 〉 to | n + 1 〉 via this IV. ENERGY SPECTRA of H ( λ ( t )). V. THE EFFECTS OF NOISE ON LEVEL OCCUPATION STATISTICS 6