Subsystem surface codes

Subsystem surface codes decompose topological surface-code stabilizers into local low-weight gauge measurements and protect only the logical subsystem invariant under gauge freedom. A canonical square-lattice construction uses weight-three triangular gauge operators, trading extra gauge degrees of freedom for lower measurement weight.

Data qubits occupy vertex-, edge-, or face-related positions; X/Z triangle operators generate the gauge group and products of neighboring triangles form its stabilizer center. Triangle coloring, boundary truncation, and gauge-to-stabilizer products are required construction data.

Logical operators are strings connecting suitable boundaries or representing nontrivial homology, modulo gauge equivalence. Distance grows with lattice linear size, while some low-weight physical operators affect only the unprotected gauge subsystem and are not logical errors.

The values n, k, gauge count r, and d depend on planar or toric layout and boundaries. Resource comparisons should report data qubits, weight-three gauge measurements, reconstructed stabilizers, and rounds rather than reuse standard surface-code formulas.

Logical operations still rely primarily on topological deformation, lattice surgery, Pauli frames, and injected resources. Gauge fixing can switch measurement sets and assist operations, but low-weight gauge checks do not provide a universal transversal gate set.

Raw observations are noisy gauge outcomes that can be multiplied into stabilizer detectors or consumed directly by a joint decoder using gauge redundancy. Boundary correlations and temporal measurement faults belong in the matching or factor graph.

Lower measurement weight can reduce ancilla fault propagation, while gauge multiplication introduces correlated syndrome noise. Circuit-level thresholds depend on triangle schedule, hook direction, rounds, and decoder rather than topological distance alone.

Weight-three checks are attractive for connectivity-limited hardware, though the layout may need more data or ancilla resources. Experimental mapping should verify couplers for every triangle, parallel conflicts, reset, and gauge-outcome processing.

It shares topological logical structure with surface codes and subsystem-gauge ideas with Bacon-Shor and compass codes. Gauge fixing or stabilizer centers can relate constructions, but distinct layouts need not be locally Clifford-equivalent.

The name covers several layouts. An exact instance needs gauge generators, stabilizer center, boundaries, qubit order, and [[n,k,r,d]]; matching only [[n,k,d]] omits the gauge structure that determines the measurement route.