a necessary assumption for a proper definition of SQRW, but an important property that makes also easier to see that SQRWs are a superset of generalized CQRWs. This indexing of input and output ports for walks on a hypercube is depicted on Fig. \[fig:port-example\]a, with some of the actual connections illustrated for a (three dimensional) cube on Fig. \[fig:port-example\]b.

Considering an array of identical multiports, an arbitrary input state undergoes the transformation by the same matrix $\mathbf{C}$ for every $x$. Let the output port $d$ of multiport $x$ be connected to multiport $x\oplus e_{dx}$ in the next row. Thus the mode labelled by the source indices $d$ and $x$, is labelled by $d$ and $x\oplus
e_{dx}$ in terms of the destination indices. Therefore, effect of propagation in terms of our standard basis is written, $$\sum_{d,x} a_{dx} {\left|d,x\right>} \to \sum_{dkx} C_{dk} a_{kx} {\left|d,x
  \oplus e_{dx}\right>}.
\label{eq:SC}$$ Comparing this formula with Eqs. (\[eq:S\_gendef\]) and (\[eq:C\]) we see that this formula corresponds to a $U=SC=S(C_0\otimes\Id)$ transformation where $C_0$ is generated by the matrix $\mathbf{C}$. Due to the local nature of the realization of the coin operation, it is straight-forward to realize position dependent coin operations, such as the one required for the quantum 