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Quantum codes on a lattice with boundary
S. B. Bravyi, A. Yu. Kitaev
TL;DR
The paper presents local-check additive quantum codes on finite lattices with alternating x- and z-boundaries. It characterizes logical operators through relative homology, showing how boundary topology determines encoded qubits and logical operations.
Problem
Toric-code locality and topology are extended from closed surfaces to finite lattices with boundaries of two types.
Source: pp. 1–2Method
The construction assigns edge qubits and vertex/face stabilizers, then identifies logical operators with relative homology classes of the lattice and its dual.
Source: pp. 2, 4Results
Relative homology groups characterize the logical-operator algebra, while alternating boundary pieces determine the number and form of encoded logical qubits.
Source: pp. 4–5Takeaways & Limitations
Boundary conditions provide a topological description of encoding and support two corresponding rigid boundary types in the quantum-order picture.
Source: pp. 2, 5, 6
Abstract
from arXiv · showhide
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.