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that for the spin-coupling scheme, related analysis has been performed in [21]. 

The physical origin of disorder may be two-fold: (i) fabrication imperfections of the particular system realiz- ing the spin chain, and (ii) noise of the external controls, which is assumed to vary slowly enough on the time-scale of state transfer t$_{out}$ , as is typically the case in most experimental situations pertaining to coupled quantum dots [22], superconducting qubits [23] or atoms [24]. We shall distinguish diagonal and off-diagonal disorder. The diagonal disorder corresponds to random on-site energies, or equivalently the local magnetic fields h$_{j}$ , normally dis- tributed around 〈 h$_{j}$ 〉 = 0 with variance σ 2 h (without loss of generality, we assume that the energies of the first and the last sites of the chain are exempt from disor- der, h$_{1}$ = h$_{N}$ = 0, since otherwise the state | ψ 〉 would dephase even before and after the transfer). The off- diagonal disorder introduces randomness in the inter-site coupling strengths J$_{j}$ → J$_{j}$ (1 + δJ$_{j}$ ) where δJ$_{j}$ are nor- mally distributed around 〈 δJ$_{j}$ 〉 = 0 with variance σ 2 $_{J}$. Consistently with the above description, we will treat the disorder as static during each realization of the nu- merical experiment for the particular protocol, but com- pletely uncorrelated between different realizations. The results presented below are obtained by averaging over many (typically 1000) independent realizations. 

Figure 1 compares the single excitation transfer for the three protocols (a), (b) and (c) in ideal and disordered 

spin chains of length N = 25. In the noiseless chain we have perfect transfer | A$_{N}$ ( t$_{out}$ ) | 2 = 1, while in the pres- ence of diagonal and off-diagonal disorder characterized by standard deviations σ$_{h}$ = σ$_{J}$ = 0 . 15 J$_{max}$ , the aver- aged transfer probabilities are reduced to 〈| A$_{N}$ ( t$_{out}$ ) | $^{2}$〉 ≃ 0 . 2, 0 . 42 and 0 . 96 for the cases of (a), (b) and (c), re- spectively. Thus, among the three transfer protocols, the sequential swap scheme is the most susceptible to noise, especially to the off-diagonal disorder which leads to de- viations of the subsequent pulse areas from the required value of π ; in this particular example, the off-diagonal disorder alone is responsible for at least 70% reduction of the transfer probability. The spin-coupling scheme is somewhat more robust with respect to noise, with both diagonal and off-diagonal disorder comparably contribut- ing to the reduction of the transfer probability by about 20% and 40%, respectively. Finally, the adiabatic trans- fer scheme is very tolerant to noise, as far as the transfer probability is concerned, but is quite slow; in fact it can tolerate even more disorder at the expense of slowing it further down (equivalent to increasing C ). 

The probability of excitation transfer alone is not enough to fully characterize the state transfer, since, e.g., | A$_{N}$ ( t$_{out}$ ) | 2 = 1 but completely random phase φ amounts to classical information transfer only, and the resulting fidelity for quantum state transfer is merely F = 0 . 66. We will therefore quantify the performance of the sys- tem subject to varying level of noise using fidelity 〈 F 〉 

