Topological phases and quantum computation

TL;DR

Kitaev's notes develop a unified view of topological phases as robust quantum resources for computation. By analyzing exactly solvable 1D and 2D models—the Majorana chain, the toric code, and the honeycomb model—the paper demonstrates how boundary zero modes, anyonic excitations, and chiral edge states yield protected degeneracies and fault-tolerant edge transport. It establishes a detailed correspondence between lattice models and gauge theories, shows how topological order survives (or transitions under perturbations), and highlights the role of spectral invariants such as Chern numbers in determining edge physics. The work lays foundational concepts for topological quantum computation, including fault-tolerant qubits via Majorana modes and robust anyonic statistics for information encoding and processing.

Abstract

This is a collection of lecture notes from three lectures given by Alexei Kitaev at the 2008 Les Houches summer school "Exact methods in low-dimensional physics and quantum computing." They provide a pedagogical introduction to topological phenomena in 1-D superconductors and in the 2-D topological phases of the toric code and honeycomb model.

Figures (12)

• Figure 1: Spectrum of a physical qubit system.
• Figure 2: Majorana chain representation of 1-d superconductor. Each boxed pair of Majoranas corresponds to one site of the original fermionic chain.
• Figure 3: A piece of the toric code. The spins live on the edges of the square lattice. The spins adjacent to a star operator $A_s$ and a plaquette operator $B_p$ are shown.
• Figure 4: Large cycles on the torus.
• Figure 5: Electric and magnetic path operators.