\varrho U^{\dag t}).$$

To simplify calculations, we introduce a linear (but non-unitary) operator to denote the effect of the noise operator $\mD$ on the search Hilbert space: $$D {\left|\psi\right>} = \eta {\left|\psi\right>}.$$ This operator is simply a multiplication with a number. It is obviously linear, however, for $\eta<1$ not unitary. The operator $D$ does not describe any coherence damping within the one-photon subspace, since it only uniformly decreases the amplitude of the computational states and introduces the vacuum. Since all final statistics are gathered from the search Hilbert space $\cHG$, it is possible to drop the vacuum from all calculations, and incorporate all information related to it into the norm of the remaining state. In other words, we can think of $DU$ as the time step operator, and relax the requirement of normalization. Using this notation, the effect of $t$ steps is very straight-forward to express: $${\left|\psi\right>} \to \eta^t U^{t} {\left|\psi\right>}.
\label{eq:homo_nstep}$$ This formula indicates that inclusion of the effect of uniform loss may be postponed until just before the final measurement. The losses, therefore, may simply be included in the detector efficiency (using an exponential function of the number of iterations).

Applying the above model of decay to the 