Quantum codes on a lattice with boundary

Abstract

A new type of local-check additive quantum code is presented. Qubits are associated with edges of a 2-dimensional lattice whereas the stabilizer operators correspond to the faces and the vertices. The boundary of the lattice consists of alternating pieces with two different types of boundary conditions. Logical operators are described in terms of relative homology groups.

Figures (4)

• Figure 1: Square lattices with a)$z$-boundary and b)$x$-boundary.
• Figure 2: A $2\times 3$ lattice with two pieces of $x$-boundary and two pieces of $z$-boundary. The free ends labeled by the same letter could be identified.
• Figure 3: The nontrivial relative homology class $[c_{12}]\in H_1(L,V,{\bf Z}_2)$ is shown by a solid line. The nontrivial element $[c^*_{12}]\in H_1(L^*,V^*,{\bf Z}_2)$ is shown by a dashed line.
• Figure 4: A lattice with $4+4$ pieces of boundary. Solid and dashed lines represent the relative cycles $c_i$ and $c^*_i$ which correspond to the logical operators $Y^z_i$ and $Y^x_i$, respectively.