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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- C. Chamberland et al., “Techniques for combining fast local decoders with global decoders under circuit-level noise”, Quantum Science and Technology 8, 045011 (2023) arXiv:2208.01178 DOI
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- Nikolas P. Breuckmann, private communication, 2022
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- C. Chamberland et al., “Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes”, PRX Quantum 3, (2022) arXiv:2012.04108 DOI
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- M. Lin, C. Chamberland, and K. Noh, “Closest Lattice Point Decoding for Multimode Gottesman-Kitaev-Preskill Codes”, PRX Quantum 4, (2023) arXiv:2303.04702 DOI
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- T. Farrelly, D. K. Tuckett, and T. M. Stace, “Local tensor-network codes”, New Journal of Physics 24, 043015 (2022) arXiv:2109.11996 DOI
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- C. K. Andersen et al., “Repeated quantum error detection in a surface code”, Nature Physics 16, 875 (2020) arXiv:1912.09410 DOI
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- “Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
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- J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
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- Derby, Charles. Compact fermion to qubit mappings for quantum simulation. Diss. UCL (University College London), 2023.
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- E. Huang et al., “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
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- D. Forlivesi, L. Valentini, and M. Chiani, “Logical Error Rates of XZZX and Rotated Quantum Surface Codes”, (2023) arXiv:2312.17057
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