Surface-code checks and logical operators

Take an L-by-L square lattice with periodic boundaries in both directions and at least three cells per side. Each horizontal or vertical edge carries one data qubit. At a vertex v, the check $A_v$ multiplies X on the four incident edges. Around a plaquette p, $B_p$ multiplies Z on its four boundary edges. All vertex pairs and all plaquette pairs commute because they use one Pauli type.

A vertex and plaquette share either zero or two edges. In the shared case, two local X-Z anticommutations cancel, so every mixed pair also commutes. The code is the common positive check eigenspace. A single Z on an edge flips the two vertex checks at its endpoints but leaves all plaquette checks unchanged. A single X similarly flips the neighboring plaquette checks, described as endpoints on the dual lattice. A Y contains both effects.

There are twice L squared edges and L squared checks of each type. However, multiplying every vertex check gives identity because each edge appears twice. The plaquette checks have the analogous single relation. These are the only relations within their respective types on this connected torus. Consequently the number of independent checks is two L squared minus two, leaving two encoded qubits. Counting every printed check as independent would incorrectly erase the global information.

A product of neighboring plaquette checks cancels internal edges and leaves a contractible Z boundary. Such a loop is a stabilizer. A Z loop winding once around a periodic direction has no vertex endpoints and hence zero syndrome, but it cannot be obtained as a boundary of plaquettes. It implements a logical Z. A dual X cycle crossing it exactly once anticommutes with it, establishing a logical conjugate pair; the other torus direction supplies the second pair.

A multi-edge Z error chain has negative vertex checks only at its endpoints: each internal vertex touches two erroneous edges and sees even parity. An ideal decoder chooses a correction chain with the same endpoints. Their combined binary chain is closed. Recovery succeeds when it is a boundary of plaquettes, meaning its homology class over $\mathbb Z_2$ is trivial, even if it contains several loops. Nonzero winding parity in either periodic direction instead leaves a logical operation. Matching a syndrome is therefore necessary but insufficient for successful recovery.

The shortest noncontractible cycle uses L edges, giving distance L. With perfect checks this guarantees correction of any data Pauli error below half that length, using an appropriate decoder. It does not guarantee correction of that many faulty circuit locations: one faulty ancilla can spread errors, and a wrong check outcome can invent endpoints. Repeated rounds and a spacetime decoder are needed when measurement is unreliable. Local check geometry enables a scalable architecture only together with those additional assumptions.