in every column of the array, and a multiport realizing the Grover matrix $G$ at every other position. The connection topology required to implement a walk on the hypercube is such that in the binary representation we have $e_d=0\ldots1\ldots0$ with 1 being at the $d$’th position, i.e.$e_d=2^d$. See Fig. \[fig:port-example\]b for a schematic example, when $\xtg=001$.

The above described scheme to realize quantum walks in an array of multiports using as many columns as the number of iterations of $U$ can be reduced to only a single column. To do this, one simply needs to connect the output ports back to the appropriate input ports of the destination multiport in the same column. This feed-back setup is similar to the one introduced in Ref. [@Kosik2005Scattering-mode].

Uniform decay {#sec:uniform}
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We begin our analysis of the effect of errors on the quantum walk search algorithm by concentrating on photon losses. In an optical network, photon losses are usually present due to imperfect optical elements. An efficient model for linear loss is to introduce fictitious beam-splitters with transmittances corresponding to the effective transmission rate (see Fig. \[fig:loss-scheme\]).

![Schematic illustration of the photon loss model being used. The losses suffered by each output 