Star and Plaquette Operators

The toric code's stabilizers come in two geometric flavours: star operators $A_v$ at vertices and plaquette operators $B_p$ at faces. For these to define a valid stabilizer code they must all mutually commute. This lesson examines the operators closely and proves their commutation comes for free from a simple counting fact: any star and any plaquette overlap on an even number of qubits.

The two operators

With one qubit per edge of the square lattice, define for each vertex $v$ and face $p$ :

A_v = \prod_{e \,\ni\, v} X_e \;=\; X_{e_1} X_{e_2} X_{e_3} X_{e_4}, \qquad B_p = \prod_{e \,\in\, \partial p} Z_e \;=\; Z_{f_1} Z_{f_2} Z_{f_3} Z_{f_4} .

On the bulk of the lattice each vertex has four incident edges and each face has four bounding edges, so both operators are weight-4. They are Hermitian and square to the identity, $A_v^2 = B_p^2 = I$ , hence their eigenvalues are $\pm 1$ .

Commutation from even overlap

Two Pauli strings either commute or anticommute. For tensor products built only from $X$ 's and $Z$ 's, the rule is local and clean. On any single qubit:

$X$ commutes with $X$ , and $Z$ commutes with $Z$ (same operator).
$X$ commutes with $I$ , $Z$ commutes with $I$ (one is trivial).
$X$ and $Z$ anticommute: $XZ = -ZX$ .

So $A_v$ and $B_p$ pick up a factor of $-1$ for each qubit on which one acts with $X$ and the other with $Z$ — i.e. for each edge in the overlap of the star's support and the plaquette's support. Across the whole string,

A_v B_p = (-1)^{\,|\,\mathrm{supp}(A_v)\,\cap\,\mathrm{supp}(B_p)\,|}\, B_p A_v .

They commute precisely when the overlap is even.

The overlap is always 0 or 2

Now the geometry. A vertex $v$ and a face $p$ on the square lattice can be positioned in only two relevant ways:

Not touching. If $v$ is not a corner of $p$ , no edge is both incident to $v$ and on $\partial p$ . Overlap $= 0$ .
Adjacent. If $v$ is one of the four corners of $p$ , then exactly the two edges of $p$ that meet at that corner are incident to $v$ . Overlap $= 2$ .

In both cases the overlap is even, so $(-1)^{\text{overlap}} = +1$ and

[A_v, B_p] = 0 \quad \text{for all } v, p .

Stars commute with stars (all- $X$ ) and plaquettes with plaquettes (all- $Z$ ) trivially. Therefore the full set $\{A_v, B_p\}$ generates an abelian group — a legitimate stabilizer group.

Try it

Confirm the commutation numerically. For an adjacent star/plaquette pair, count the shared edges and return the commutation sign $(-1)^{\text{overlap}}$ in $A_v B_p = s\, B_p A_v$ . A value of $+1$ certifies that the stabilizers commute.

Run your code to see the quantum state.