Universal Quantum Computation with Gapped Boundaries

This Letter discusses topological quantum computation with gapped boundaries of two-dimensional topological phases. Systematic methods are presented to encode quantum information topologically using gapped boundaries, and to perform topologically protected operations on this encoding. In particular, we introduce a new and general computational primitive of topological charge measurement and present a symmetry-protected implementation of this primitive. Throughout the Letter, a concrete physical example, the ${\mathbb{Z}}_3$ toric code [$\mathfrak{D}({\mathbb{Z}}_3)$], is discussed. For this example, we have a qutrit encoding and an abstract universal gate set. Physically, gapped boundaries of $\mathfrak{D}({\mathbb{Z}}_3)$ can be realized in bilayer fractional quantum Hall $1/3$ systems. If a practical implementation is found for the required topological charge measurement, these boundaries will give rise to a direct physical realization of a universal quantum computer based on a purely Abelian topological phase.

Figure 1

Ground state for $n$ gapped boundaries ${\mathcal{A}}_i$ on a plane and total charge vacuum. All edges are directed to point downward.

Figure 2

Braiding of two gapped boundaries (${\sigma }_2^2$). Solid lines indicate tunneling operators from the basis vectors (i.e., not motion of the holes), while dotted lines indicate how the holes move in the braiding process.

Figure 3

Topological charge projection ($n=2$).

Figure 4

Braid for the $\wedge {\sigma }_3^z$ gate.

Figure 5

Short circuit (generalizing Ref. [7]) to use three SUM gates between $1+m+\overline{m}$ and $1+e+\overline{e}$ qutrits to implement a SUM gate between $1+e+\overline{e}$ qutrits. All entangling gates drawn are ${\mathrm{SUM}}_3$.