# Topological Qubits as Carriers of Quantum Information in Optics  
[PERSON]  
1 Quantum Communication and Measurement Laboratory, Department of Electrical and Computer Engineering and Division of Natural Science and Mathematics, Boston University, Boston, MA 02215, USA 1  
[PERSON]  
2 Department of Physics and Astronomy, Stonehill College, 320 Washington Street, Easton, MA 02357, USA; [PERSON]_ADDRESS]  
[PERSON]  
34 Department of Electrical and Computer Engineering and Photonics Center, Boston University, 8 Saint Mary's St., Boston, MA 02215, USA; [EMAIL_ADDRESS]  
November 6,  
\(t\) runs from \(0\) to \(1\). The winding number is therefore well described using the polar coordinates \(\mathbf{s}(t)=\{r(t),\phi(t)\}\) in the plane. The winding number \(\
u\in\mathbb{Z}\), can be given as  
\[\
u=\frac{\phi(1)-\phi(0)}{2\pi}. \tag{1}\]  
In optical contexts, the winding number might count the number of rotations of a polarization vector, the amount of orbital angular momentum present, or the number of times a phase rotates as the photon momentum circles the Brillioun zone of a spatially periodic system. Although this characterization can be generalized to include the winding number relative to any point \(p\) in the plane, the winding number about the origin is considered here without loss of generality; see Figure 1. Winding number is a topologically significant quantity, in that it is invariant under smooth deformations of the curve which don't involve crossing the origin. When the origin is crossed, \(\
u\) changes by a discrete integer amount. A wide range of physical systems whose descriptions have interesting topological aspects have been considered in optics, solid state physics, elementary particle physics, and elsewhere in the study of nature; a number of these can be considered via the winding number or its various generalizations, such as the Chern number [6]. In the quantum theoretical context in particular, quantum states with various values of winding number can be identified. Among the quantum applications is the characterization of distinct topological phases of materials and the topologically protected edge or boundary states between them [7; 8; 9; 10; 11; 12]. Long known in momentum space in solid state systems, such topological states have recently been studied in photonic quantum walk systems as well [13; 14; 15; 16]. In applications, the fact that the homotopy class of a curve does not depend on the the radial part of the parametrized curve \(\mathbf{s}(t)\) is of great value; for example, it means that physical quantities depending only on the winding number are very stable against the effects of small perturbations to the system. As a consequence of the lack of radial dependence, one can simply consider homotopy classes of maps from the circle to itself: any such curve is continuously deformable to one of maps of the circle, \(S^{1}\to S^{1}:S\mapsto S^{n}\) with multiplication being that on the complex unit circle. (Recall that the \(i\)-th homotopy group \(\pi(S^{n})\) captures the continuous mapping of the \(i\)-dimensional sphere \(S^{i}\) into the \(n\)-dimensional sphere \(S^{n}\).) This does not distinguish two maps that can be continuously deformed from one to the other; only equivalence classes of mappings are pertinent. The homotopy groups of spheres algebraically describe the spheres as topological spaces with an operation of addition defined on their equivalence classes, giving rise to an Abelian group structure. The homotopy classes of maps from a circle to a topological space form the _first homotopy group_, which is some subgroup of the group of integers \(\mathbb{Z}\) under addition; the winding number of a curve in the complex plane \(\mathbb{C}\) (the set of points \(x+iy\)) labels its homotopy class [1; 17]. Figure 1: The winding number \(\
u\in\mathbb{Z}\) of a continuous curve about some point \(p\) counts the number of times the curve encircles \(p\).