of the column index as a discrete time index, and drop it as an explicit label of modes. Thus a time-evolution $U=SC$ can be generated by the propagation through columns of multiports.

A quantum walk can be realized in terms of the basis defined using the destination indices, and we shall term it “standard basis” through this section. First, we recall that an $n$-multiport can be fully characterized by an SU($n$) transformation matrix $\mathbf{C}$. The effect of such multiport on single excitation states ${\left|\psi\right>}\in\cHC$ is given by the formula, $${\left|\psi\right>}=\sum_{d=0}^{n-1} a_d {\left|d\right>} \to \sum_{d,k=0}^{n-1}
  C_{dk} a_k {\left|d\right>}, 
\label{eq:C}$$ where ${\left|d\right>}$ denotes the single photon state with the photon being in the $d$ mode, i.e. ${\left|d\right>} = {\left|0\right>}_0 \ldots {\left|1\right>}_d \ldots
{\left|0\right>}_{n-1}$. We note, that a multiport with any particular transformation matrix $\mathbf{C}$ can be realized in a laboratory [@Reck_prl73]. To simplify calculations it may be beneficial to choose an indexing of input and output ports such that the connections required to realize the graph $G$ can be made in such way that each input port has the same index as the corresponding source output port. Therefore the label $d$ can stay unique during “propagation.” We emphasize that this is not 