# Topological Qubits as Carriers of Quantum Information in Optics
## 4 Multiports and Winding Number  
Recall that quantum optical beams of photons can be divided by beam splitters (BS)--producing superposition states of paths for photons in ordinary space--and by their polarized versions (PBS), which selectively direct them in beams according to polarization characteristics. Ordinary beam splitters can be combined and/or modified to form more complex linear optical systems which elicit specific additional beam properties including winding number. Standard beam splitters produce two output beams from one input beam, with no backward propagation back out to the input direction. Basic multiport generalizations of standard beamsplitters allow for more than two output beams, but only allow _uni-directional_ net movement of photons--i.e., these quantum systems preclude reversals of beam direction inside themselves. Recently, a generalized multiport (MBS) has been proposed [49] and experimentally demonstrated [50] that allows a reversal of beam direction. These directionally unbiased multiports can be viewed as optical scattering vertices and thus can be combined into networks that give rise to scattering-based quantum walks on graph structures [51; 52; 53]. In such unbiased multiports, the beamsplitter input/output ports can be attached at their internal vertices, as shown in the inset of Figure 7 left. Adding a phase shifter allows the output of the multiport to be altered, so that different phase shifts at the vertices influence the interference between photon paths; when sufficiently small (see [49]), such a unit is described by an \(n\times n\) unitary transition matrix \(\hat{U}\) whose rows and columns correspond to the input and output states at the ports--rows and columns correspond to the ports \(A\), \(B\), \(C\). When the internal phase shifts at all three mirror units are the same, one has  
\[\hat{U}(\theta)=\frac{e^{i\theta}}{2+ie^{i\theta}}\,\left( \begin{array}{ccc}1& ie^{-i\theta}-1&ie^{-i\theta}-1\\ ie^{-i\theta}-1&1&ie^{-i\theta}-1\\ ie^{-i\theta}-1&ie^{-i\theta}-1&1\end{array} \right), \tag{14}\]  
where \(\theta\) is the total phase shift at each mirror unit (comprising the mirror and phase plate). Consider the optical units shown on the left in Figure 8[51; 52; 53] each of which corresponds to an implementation of the abstract diamond graph on the right. Each diamond graph unit consists of a pair of three-point scattering vertices connected at two edges, with an additional phase shift on one connecting edge. Each of the optical units contains a pair of three-ports; each graph edge corresponds to an optical path and each multiport provides a physical implementation of an abstract three-point scattering vertex. Let the multiports be very small relative to the distance between them, so they can be treated as approximately pointlike in scale; each unit cell contains four multiports and two phase shifters. The phase shifts \(\phi_{a}\) and \(\phi_{b}\) in the two diamonds may be different from each other. In the configuration of units shown in Figure 9, the phase plates rotate the polarization by \(90^{\circ}\). A time unit \(T\) is defined as the time to go from one diamond graph to the next. Photons present in the areas labeled \(a\) and \(b\), bounded by the phase plates, represent two substates at each lattice site. When \(\phi_{a}\
eq\phi_{b}\), the two triangle graphs inside each cell will have different transmittance; take the transmission probabilities corresponding to the two graphs \(T_{a}=|t_{a}|^{2}\) and \(T_{b}=|t_{b}|^{2}\) then. The polarization state flips upon each passage through a phase plate allows the \(a\) and \(b\) states to be easily distinguished. Photons can be introduced to and released from this system via optical switches and circulators [18]. Figure 8: The diamond graph (**left**), consisting of two three-edge scattering vertices and a phase shift, has been used theoretically to implement scattering-based quantum walks [51; 52; 53].