The Toric Code

Kitaev's toric code is the foundational topological code. It lives on a square lattice wrapped onto a torus, with one qubit on every edge. Two families of mutually commuting, geometrically local stabilizers define a code space, and the global topology of the torus endows that space with exactly $k=2$ protected logical qubits. This lesson sets up the lattice, the operators, and the bookkeeping.

The lattice and where the qubits live

Take an $L \times L$ square lattice with periodic boundary conditions in both directions — a discrete torus. Place one physical qubit on each edge of the lattice. An $L \times L$ torus has $L^2$ vertices, $L^2$ plaquettes (faces), and $2L^2$ edges, so

n = 2L^2 \quad \text{physical qubits}.

(Euler's formula $V - E + F = 0$ for the torus is satisfied: $L^2 - 2L^2 + L^2 = 0$ .)

Two kinds of stabilizer

For each vertex $v$ we define a star operator $A_v$ , the product of Pauli- $X$ over the four edges meeting at $v$ ; for each plaquette $p$ we define a plaquette operator $B_p$ , the product of Pauli- $Z$ over the four edges bordering $p$ :

A_v = \prod_{e \,\ni\, v} X_e, \qquad B_p = \prod_{e \,\in\, \partial p} Z_e .

Each acts on exactly four qubits — the checks are local. The next lesson studies these operators in detail; here we just need that they all commute. Two stars or two plaquettes obviously commute (disjoint, or all- $X$ / all- $Z$ ). A star $A_v$ and a plaquette $B_p$ share either $0$ or $2$ edges; since $X$ and $Z$ anticommute, sharing an even number of qubits makes the operators commute. Hence $\{A_v, B_p\}$ generate an abelian stabilizer group.

The toric-code Hamiltonian makes the code space the ground space:

H = -\sum_v A_v - \sum_p B_p .

Every term is $\pm 1$ ; the ground states saturate all of them at $+1$ , i.e. they are exactly the stabilized states.

The two redundancies

The $L^2$ star operators are not independent: their product over all vertices is the identity, because every edge touches exactly two vertices and so each $X_e$ appears twice:

\prod_v A_v = I .

Likewise $\prod_p B_p = I$ . Each family therefore contributes only $L^2 - 1$ independent generators, for a total of

n - k = 2(L^2 - 1) = 2L^2 - 2 \quad \text{independent stabilizers}.

Counting the logical qubits

The number of encoded qubits follows immediately:

k = n - (n-k) = 2L^2 - (2L^2 - 2) = 2 .

The toric code encodes two logical qubits, independent of $L$ . This is no accident of counting: the two logical qubits are tied to the two independent noncontractible cycles of the torus (its first homology $H_1(T^2) = \mathbb{Z}^2$ ), as the lesson on logical operators will make precise.

Summary of parameters

For the $L \times L$ toric code:

[[\,n, k, d\,]] = [[\,2L^2,\; 2,\; L\,]],

where the distance $d = L$ is the length of the shortest noncontractible loop (derived two lessons from now). Growing $L$ leaves the checks four-local while increasing the protection — the hallmark of a topological code.