# Subsystem surface code[1]

## Description

Subsystem version of the surface code defined on a square lattice with qubits placed at every vertex and center of everry edge.

For example, a \([[3L^2,2,L]]\) family has weight-six \(X,Z\)-type stabilizers supported on two of the four triangles of each plaquette.

## Fault Tolerance

Gauge fixing and changing the order in which check operators are measured yields a fault-tolerant decoder [2].

## Code Capacity Threshold

Independent \(X,Z\) noise: the threshold under ML decoding corresponds to the value of a critical point of the two-dimensional hexagonal-lattice random-bond Ising model (RBIM) on the Nishimori line [1,3], calculated to be around \(7\%\) in Ref. [4].

## Threshold

\(0.81\%\) threshold for circuit-level depolarizing noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [2], improving the \(0.67\%\) threshold for standard measurement schedules [1].\(2.22\%\) threshold for circuit-level infinitely biased noise under a variant of MWPM and using gauge-fixing and specific measurement schedules [2], improving the \(0.52\%\) threshold with standarn measurement schedules.

## Notes

See [5; Sec. III.C3] for an exposition.

## Parents

## Cousins

- Kitaev surface code
- Asymmetric quantum code — Subsystem surface codes perform well against biased circuit-level noise [2].
- Subsystem rotated surface code

## References

- [1]
- S. Bravyi et al., “Subsystem surface codes with three-qubit check operators”, (2013) arXiv:1207.1443
- [2]
- O. Higgott and N. P. Breuckmann, “Subsystem Codes with High Thresholds by Gauge Fixing and Reduced Qubit Overhead”, Physical Review X 11, (2021) arXiv:2010.09626 DOI
- [3]
- H. Nishimori, “Geometry-Induced Phase Transition in the ±JIsing Model”, Journal of the Physical Society of Japan 55, 3305 (1986) DOI
- [4]
- S. L. A. de Queiroz, “Multicritical point of Ising spin glasses on triangular and honeycomb lattices”, Physical Review B 73, (2006) arXiv:cond-mat/0510816 DOI
- [5]
- B. M. Terhal, “Quantum error correction for quantum memories”, Reviews of Modern Physics 87, 307 (2015) arXiv:1302.3428 DOI

## Page edit log

- Victor V. Albert (2022-10-11) — most recent

## Cite as:

“Subsystem surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/subsystem_surface