An Introduction to Topological Quantum Codes

TL;DR

Topological quantum codes exploit locality to store quantum information in global, nonlocal degrees of freedom, enabling fault-tolerant operation in two-dimensional lattices with local stabilizers. The chapter develops surface codes and color codes, linking logical operators to nontrivial homology classes and showing that decoding reduces to homology-informed strategies with a finite error threshold (≈0.11) for uncorrelated noise; in color codes, transversal Clifford gates and even transversal P (in certain geometries) broaden the fault-tolerant gate set. A unifying homological framework underpins both code families, with planar realizations achieved via boundaries and a deep connection to phase transitions in related Ising models. These insights illuminate practical routes to scalable quantum memory and computation while highlighting the interplay between topology, geometry, and quantum error correction.

Abstract

This is the chapter \emph{Topological Codes} of the book \emph{Quantum Error Correction}, edited by Daniel A. Lidar and Todd A. Brun, Cambridge University Press, New York, 2013. http://www.cambridge.org/us/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/quantum-error-correction

Figures (22)

• Figure 1: Closed curves in a torus can be the boundary of a region, like curve $a$. However, curve $b$ is not the boundary of a region. The difference cannot be detected by looking only at a local region such as the one marked with a dotted line.
• Figure 2: In 2D local codes qubits are arranged in a 2D array. The support of any stabilizer generators must be contained in a box of a fixed size, here a $3\times 3$ box. Circles represent qubits and darker ones form the support of a generator.
• Figure 3: From the topological perspective, orientable closed surfaces are classified by the genus $g$ or number of handles. These are the three with lower genus: the sphere, the sphere with a handle or torus, and the sphere with two handles or 2-torus.
• Figure 4: Several closed curves in a torus. Curve $a$ is the boundary of region A, so it is homologically trivial. Curves $b$ and $c$ form the boundary of region $B$, and thus are homologically equivalent. Curve $d$ is homologically nontrivial and not equivalent to $c$.
• Figure 5: The action of boundary operators. $\partial_2$ maps a set of faces to the set of edges that form its boundary. $\partial_1$ maps a set of edges to to the set of vertices where an odd number of these edges meet.