mode are represented by fictitious beam-splitters with transmittivities $\eta_d$. The beam-splitters incorporate the combined effect of imperfections of the multiport devices, and effects influencing the state during propagation between the multiports (e.g. scattering and absorption).[]{data-label="fig:loss-scheme"}](fig2.eps)

The simplest case is when all arms of the multiports are characterized by the same linear loss rate $\eta$. The operator describing the effect of decay on a single excitation density operator can then be expressed as $$\mD (\varrho) = \eta^2 \varrho + (1-\eta^2)\vrho.
\label{eq:homloss_op}$$ The total evolution of the system after one iteration may be written as $\varrho \to \mD(U\varrho U^{\dag})$. It is important to note that with the introduction of this error, the original Hilbert space $\cHG$ of one-photon excitations must be extended by the addition of the vacuum state ${\left|0\right>}$. The action of the SQRW evolution operator $U$ on the extended Hilbert space follows from the property $U{\left|0\right>}={\left|0\right>}$. Due to the nature of Eq. (\[eq:homloss\_op\]) and the extension of $U$, one can see that the order of applying the unitary time step and the error operator $\mD$ can be interchanged. Therefore, over $t$ steps the state of the system undergoes the transformation $$\varrho \to \eta^{2t} U^t \varrho U^{\dag t}+ (1-\eta^{2t})\vrho = \mD^t(U^t
