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

Description

Rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it.

Admits generators \(\{ZZZII,IIZZZ,XIXXI,IXXIX\} \).

Gates

Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs [1].

Fault Tolerance

Fault-tolerant implementation of the Clifford group based on transversal gates and SWAPs [1].

Cousins

  • \([[7,1,3]]\) Steane code— The \([[5,1,2]]\) rotated surface code can be obtained by morphing the Steane code [1].
  • Twist-defect surface code— The \([[5,1,2]]\) rotated surface code is a genon code on a sphere, with the missing external stabilizer forming the back of the sphere. More generally, any surface code with a single boundary component can be interpreted this way [2].

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
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. 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. 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.
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[5,1,2]]\) rotated surface code

References

[1]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[2]
S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
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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.), 2024. https://errorcorrectionzoo.org/c/stab_5_1_2
BibTeX:
@incollection{eczoo_stab_5_1_2, title={\([[5,1,2]]\) rotated surface code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/stab_5_1_2} }
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Permanent link:
https://errorcorrectionzoo.org/c/stab_5_1_2

Cite as:

“\([[5,1,2]]\) rotated surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/stab_5_1_2

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