walk search algorithm.

![a) Illustration of the labelling of input and output ports of multiports used for the realization of the walk on the $n$ dimensional hypercube. b) Schematic depiction of the setup of the SQRW implementation of the SKW algorithm for $n=3$, with the marked node being $\xtg=001$. For clarity, a unique line pattern is associated with each input (output) port index. []{data-label="fig:port-example"}](fig1.eps)

In particular, the SKW algorithm [@shenvi:052307] is based on the application of two distinct coin operators, e.g.

\[eq:std-coin-pair\] $$\begin{aligned}
C_0 &=& G, \\
C_1 &=& -\Id,\end{aligned}$$

where $G$ is the Grover inversion or diffusion operator $G:= -\Id + 2
|s^C\rangle\langle s^C|$, with $|s^C\rangle=1/\sqrt{n} \sum_{d=1}^{n}
{\left|d\right>}$ [@moore02quantum]. In the algorithm, the application of the two coin operators is conditioned on the result of oracle operator $\cO$. The oracle marks one $\xtg$ as target, hence the coin operator becomes conditioned on the node: $$C' = C_0 \otimes \Id + (C_1 - C_0)\otimes {\left|\xtg\right>}{\left<\xtg\right|}.
\label{eq:pert_coin}$$ When $n$ is large, the operator $U':=SC'$ can be regarded as a perturbed variation of $U=S(C_0\otimes\Id)$. The conditional transformation (\[eq:pert\_coin\]) is straight-forward to implement in the multiport network. For the two coins (\[eq:std-coin-pair\]) one has to use a simple phase shifter at position $\xtg$ 