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\([[5,1,2]]\) rotated surface code

Alternative Names: \([[5,1,2]]\) morphed Steane code, \([[5,1,2]]\) rotated toric code, \([[5,1,2]]\) genon code, \([[5,1,2]]\) holographic code, \([[5,1,2]]\) planar-perfect code.

Description

A rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it. This is the smallest code that implements a fault-tolerant logical \(S\) gate using a diagonal depth-one Clifford circuit [1].

A stabilizer tableau for the code is given by [2; ID 18] \begin{align} \begin{array}{ccccc} Z & Z & I & Z & I \\ I & I & Z & Z & Z \\ I & X & X & X & I \\ X & I & I & X & X \end{array}~. \tag*{(1)}\end{align} The code is depicted in Fig. I.

Figure I: Stabilizer generators of the \([[5,1,2]]\) rotated surface code. The 5 data qubits (circles) consist of 4 corner qubits and 1 center qubit. Each triangular region corresponds to a weight-three stabilizer generator. Red regions correspond to \(X\) operators while blue regions correspond to \(Z\) operators.

A non-CSS genon-code form of the same stabilizer group, local Clifford equivalent to the above via Hadamard on the four corner qubits, is [3] \begin{align} \begin{array}{ccccc} X & X & I & Z & I \\ I & I & X & Z & X \\ I & Z & Z & X & I \\ Z & I & I & X & Z \end{array}~. \tag*{(2)}\end{align} The missing external stabilizer \(YYYIY\) (product of all four generators) forms the back face when the code is viewed as a genon code on a sphere.

Gates

In the qubit order of the stabilizer tableau above, the physical action \(S_0 S_2 S_3^{\dagger} CZ_{1,4}\) implements the logical gate \(\bar{S}\) [4; Fig. 1].Fault-tolerant implementation of the single-qubit Clifford group [4; Fig. 1].

Fault Tolerance

Fault-tolerant implementation of the single-qubit Clifford group [4; Fig. 1].

Cousins

  • Twist-defect surface code— The \([[5,1,2]]\) rotated surface code is a genon code on a sphere, with the missing external \(Y\)-type stabilizer forming the back of the sphere. More generally, any surface code with a single boundary component can be interpreted this way [3].
  • \([[7,1,3]]\) Steane code— The \([[5,1,2]]\) morphed Steane code is obtained by morphing the Steane code on a region whose child code is a \([[4,2,2]]\) code [4; Fig. 1].
  • \([[4,2,2]]\) Four-qubit code— The \([[5,1,2]]\) morphed Steane code is obtained by morphing the Steane code on a region whose child code is a \([[4,2,2]]\) code [4; Fig. 1].
  • \([[10,2,3]]\) rotated toric code— The \([[10,2,3]]\) rotated toric code is the symplectic double (a.k.a. genus-one double cover) of the \([[5,1,2]]\) rotated surface code [3].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Homological code
Kitaev surface codeCDSC Twist-defect surface QLDPC Qubit CSS Generalized homological-product Lattice stabilizer Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum
Rotated surface codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Rotated surface code
\([[2^r+r-1,1,2]]\) morphed simplex codeCSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
The \([[5,1,2]]\) code is a specific instance of the \([[2^r+r-1,1,2]]\) morphed simplex codes with \(r=2\) [4; Fig. 1].
Surface-code-fragment (SCF) holographic codeQubit CSS Stabilizer Hamiltonian-based HQECC QECC Quantum
The \([[5,1,2]]\) rotated surface code is the smallest SCF holographic code [5]. The encoding of more general SCF holographic codes is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a planar-perfect tensor.
Planar-perfect-tensor codeQECC Quantum
The \([[5,1,2]]\) rotated surface code is the smallest SCF holographic code [5]. The encoding of more general SCF holographic codes is a holographic tensor network consisting of the encoding isometry for the \([[5,1,2]]\) rotated surface code, which is a planar-perfect tensor.
\([[5,1,2]]\) rotated surface code

References

[1]
V. V. Albert, private communication, 2026
[2]
Qiskit Community, “Qiskit QEC framework”, URL
[3]
S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
[4]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[5]
R. J. Harris, E. Coupe, N. A. McMahon, G. K. Brennen, and T. M. Stace, “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020) arXiv:2008.10206 DOI
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Zoo Code ID: stab_5_1_2

Cite as:
“\([[5,1,2]]\) rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_5_1_2, arXiv:2606.11484
BibTeX:
@incollection{eczoo_stab_5_1_2,
title={\([[5,1,2]]\) rotated surface code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/stab_5_1_2}
}
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Cite as:

“\([[5,1,2]]\) rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/stab_5_1_2, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/small_distance/small/5/stab_5_1_2/stab_5_1_2.yml.