Also known as Checkerboard code, Medial surface code, Rectified surface code.

## Description

Variant of the surface code defined on a square lattice that has been rotated 45 degrees such that qubits are on vertices, and both \(X\)- and \(Z\)-type check operators occupy plaquettes in an alternating checkerboard pattern.

Stabilizer generators for this code are shown in Figure I.

## Protection

The \([[L^2,1,L]]\) variant [1] includes the \([[9,1,3]]\) surface-17 code, named as such because 8 ancilla qubits are used for check operator measurements alongside the 9 physical qubits.

## Decoding

Only certain syndrome extraction schedules are distance-preserving [4].Local neural-network using 3D convolutions, combined with a separate global decoder [5].Iterative CNOT decoder [6].

## Fault Tolerance

A particular choice of CNOT gates during syndrome extraction is required to avoid hook errors and be fault-tolerant to syndrome qubit errors [4,7,8].

## Threshold

Thresholds for various amounts of erasure, Pauli, and measurement noise are known [9].

## Parents

- Kitaev surface code — The lattice of the rotated surface code can be obtained by taking the medial graph of the surface code lattice (treated as a graph) and applying a similar procedure to construct the check operators [1,10][11; Fig. 8]. Applying the quantum Tanner transformation to the surface code yields the rotated surface code [12,13]. The rotated surface code presents certain savings over the original surface code [14].
- Quantum Tanner code — Applying the quantum Tanner transformation to the surface code yields the rotated surface code [12,13].
- Hierarchical code — Hierarchical codes are concatenations of constant-rate QLDPC codes with rotated surface codes.
- Yoked surface code — Yoked surface codes are concatenations of QMDPC codes with rotated surface codes.

## Children

## Cousins

- Hypergraph product (HGP) code — Rotated code can be obtained from hypergraph product of two cyclic binary cyclic codes with palindromic generator polynomial ([3], Ex. 7).
- Heavy-hexagon code — A rotated surface code can be mapped onto a heavy square lattice, resulting in a code similar to the heavy-hexagon code [15].
- Concatenated cat code — Cat codes have been concatenated with rotated surface codes [16].
- GKP-surface code — GKP codes have been concatenated with rotated surface codes [17–21].
- Tensor-network code — A tensor-network based modification of the rotated surface code improves performance against depolarizing noise by \(\approx 2\%\) [22].
- \([[4,2,2]]\) Four-qubit code — The subcodes \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\) [23], \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [24], \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [25], and \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) [26] of the \([[4,2,2]]\) code are small planar rotated surface codes.
- Ball-Verstraete-Cirac (BVC) code — An appropriately chosen stabilizer generator set for the BVC code contains the stabilizers of the rotated surface code [27].
- 3D surface code — There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [28].
- XZZX surface code — The XZZX code is obtained from the rotated surface code by applying Hadamard gates on a subset of qubits such that \(XXXX\) and \(ZZZZ\) generators are both mapped to \(XZXZ\). Both rotated and XZZX codes offer improved performance over the original surface code for biased noise [29].
- Compass code — The surface-density compass code family interpolates between Bacon-Shor codes and rotated surface codes.
- Subsystem rotated surface code

## References

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- Derby, Charles. Compact fermion to qubit mappings for quantum simulation. Diss. UCL (University College London), 2023.
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## Page edit log

- Eric Huang (2024-03-14) — most recent
- Marcus P da Silva (2023-03-20)
- Victor V. Albert (2022-07-30)

## Cite as:

“Rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/rotated_surface