Toric Code

Definition

The toric code (Kitaev, 1997) is the simplest exactly solvable model of topological-order: qubits on the links of a square lattice, with a commuting-projector Hamiltonian $H=-{\sum }_v{A}_v-{\sum }_p{B}_p,A_v={\prod }_{\ell \ni v}{\sigma }_{\ell }^x,B_p={\prod }_{\ell \in \partial p}{\sigma }_{\ell }^z,$ where $A_v$ (star) and $B_p$ (plaquette) all commute. The ground space is their common $+1$ eigenspace.

Its structure, solvable to the bone:

• Degeneracy $2^{2g}$ on a genus-$g$ surface (4 on the torus), labelled by the eigenvalues of non-contractible Wilson/‘t Hooft loops.
• Anyons: $e$ (star violation), $m$ (plaquette violation), and $\epsilon =e\times m$; $e$ and $m$ are mutual semions (braiding phase $-1$), $\epsilon$ is a fermion.
• String operators create anyon pairs at their ends; only topologically non-trivial (non-contractible) strings connect distinct ground states.

The toric code is exactly the ${\mathbb{Z}}_2$ Kogut–Susskind lattice gauge theory in its deconfined phase: $A_v=1$ is the Gauss law, $B_p$ the magnetic energy. It is also the ${\mathbb{Z}}_2$ Dijkgraaf–Witten theory and the ${\mathbb{Z}}_2$ string-net.

Role in Research

In the generalized-symmetries course the toric code is where confinement/deconfinement, generalized symmetry, and quantum information become one object: its 4-fold torus degeneracy is spontaneously broken 1-form symmetry, its anyons are the electric and magnetic charges of ${\mathbb{Z}}_2$ gauge theory, and its ground space is a quantum error-correcting code (the surface code). The perturbed toric code reproduces the Fradkin–Shenker phase diagram, tying Semester I to Semester II.

• As a code: logical operators are homology classes of $H_1(\Sigma ,{\mathbb{Z}}_2)$; code distance = shortest non-contractible loop; no local operator acts within the code space (Dennis–Kitaev–Landahl–Preskill).
• Topological entanglement entropy $\gamma =\ln 2$ is computed by an explicit Schmidt decomposition.

Relations

• topological-order — the toric code is its exactly solvable representative
• lattice-gauge-theory — the toric code is deconfined ${\mathbb{Z}}_2$ Kogut–Susskind theory
• higher-form-symmetries — its degeneracy is broken 1-form symmetry; anyons are charged objects
• string-net-condensation — the toric code is the ${\mathbb{Z}}_2$ string-net
• wilson-loop — the non-contractible loop operators that label the ground states

Papers

Kitaev, “Fault-tolerant quantum computation by anyons,” Annals Phys. 303 (2003) 2 [quant-ph/9707021]; Dennis–Kitaev–Landahl–Preskill, quant-ph/0110143 (the code). See the course bibliography.

Notes

• “Toric” refers to the torus used to expose the degeneracy; on a planar patch with boundaries the same model is the surface code, the leading platform for fault-tolerant quantum computing.
• The ${\mathbb{Z}}_N$ generalization (clock/shift qubits) and the non-abelian quantum double $D(G)$ extend the construction; $D(S_3)$ already has non-abelian anyons.