in the search algorithm.

To implement the scattering quantum random walk on an $n$ regular graph of $N$ nodes, identical $n$-multiports [@Jex-optcomm117; @Zukovski_pra55] are arranged in columns each containing $N$ multiports. The columns are enumerated from left to right, and each row is assigned a number sequentially. The initial state enters on the input ports of multiports in the leftmost column. The output and input ports of multiports of neighbouring columns $j$ and $j+$ are then indexed suitably and connected according to the graph $G$.

For the formal description of quantum walks on arrays of multiports, we propose to label every mode by the row index and input port index of its *destination* multiport. We note that an equally good labelling can be defined using the row index and output port index of the *source* multiport. To describe single excitation states, we use the notation ${\left|d,x\right>}$ where the input port index of the destination multiport is $d=0,1,\ldots,n-1$, and the row index is $x=0,1,\ldots,N-1$. Thus the total Hilbert space can effectively be separated into some product space $\cHC \otimes \cHG$. To be precise, the additional label $j$ would be necessary to identify in which column the multiport is, however, we think 