Surface code

Surface codes are topological CSS stabilizer codes defined on two-dimensional cellulations. In the standard edge-qubit presentation, vertex stars are $X$-type checks and face boundaries are $Z$-type checks. Periodic boundaries give the toric code, while planar patches with rough and smooth boundaries commonly encode one logical qubit.

Pauli errors form chains on the lattice and syndrome defects occur at their endpoints. Recovery chooses a chain with the same boundary. A logical failure occurs when error and recovery together form a nontrivial relative-homology class, so distance is the weight of the shortest nontrivial logical string.

Logical Paulis are nontrivial strings connecting appropriate boundaries. Fault-tolerant Clifford operations are commonly implemented by code deformation, defects, lattice surgery, and joint patch measurements. Universal computation additionally requires a non-Clifford resource such as state injection and magic-state distillation.

With ideal measurements, defects can be decoded using minimum-weight perfect matching, union-find, tensor-network, or related graph methods. With noisy measurements, decoding operates on a spacetime detector graph whose weights depend on the circuit and noise model. Correlations, leakage, erasures, and bias change the appropriate decoder and threshold.

Weight-four checks are commonly measured with ancillas and ordered two-qubit gates. Gate order controls hook-error orientation; a poor schedule can reduce circuit-level effective distance below nominal distance. Repeated rounds distinguish data and measurement faults, while reset, leakage, crosstalk, and boundary checks require explicit modeling.

Surface-code patches have been used for repeated correction, distance scaling, and real-time decoding on several gate-model platforms. Results must distinguish stabilizer cycling, logical memory, lattice surgery, and a complete universal-computation path. Atlas $d=3,5,7$ points encode structural parameters, not device-specific logical error rates.

The toric code is the periodic-boundary surface-code specialization. XZZX codes alter local Pauli structure for biased noise, while subsystem surface codes trade gauge freedom for lower-weight measurements. Surface codes also connect to homological codes, lattice surgery, and topological memories.

When comparing surface-code results, check patch geometry, whether $n$ counts data qubits only, code versus circuit distance, syndrome rounds, noise model, decoder, and boundary conventions. Atlas exposes QECDB and QECirc links only for verified finite-instance matches.