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].
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 [4,6,7].
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 [1,8][9; Fig. 8]. Applying the quantum Tanner transformation to the surface code yields the rotated surface code [10,11].
- Quantum Tanner code — Applying the quantum Tanner transformation to the surface code yields the rotated surface code [10,11].
- 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 [12].
- Concatenated cat code — Cat codes have been concatenated with rotated surface codes [13].
- GKP-surface code — GKP codes have been concatenated with rotated surface codes [14–18].
- Quantum Lego code — A tensor-network based modification of the rotated surface code improves performance against depolarizing noise by \(\sim 2\%\) [19].
- \([[4,2,2]]\) Four-qubit code — The subcodes \(\{|\overline{10}\rangle,|\overline{11}\rangle\}\) [20], \(\{|\overline{00}\rangle,|\overline{10}\rangle\}\) [21], \(\{|\overline{00}\rangle,|\overline{01}\rangle\}\) [22], and \(\{|\overline{00}\rangle,|\overline{11}\rangle\}\) [23] 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 [24].
- 3D surface code — There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [25].
- 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 [26].
- 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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- Y. Tomita and K. M. Svore, “Low-distance surface codes under realistic quantum noise”, Physical Review A 90, (2014) arXiv:1404.3747 DOI
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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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- Anthony Leverrier, Mapping the toric code to the rotated toric code, 2022.
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- C. Chamberland et al., “Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits”, Physical Review X 10, (2020) arXiv:1907.09528 DOI
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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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- K. Fukui et al., “High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction”, Physical Review X 8, (2018) arXiv:1712.00294 DOI
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- K. Noh and C. Chamberland, “Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code”, Physical Review A 101, (2020) arXiv:1908.03579 DOI
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- M. V. Larsen et al., “Fault-Tolerant Continuous-Variable Measurement-based Quantum Computation Architecture”, PRX Quantum 2, (2021) arXiv:2101.03014 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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- A. Erhard et al., “Entangling logical qubits with lattice surgery”, Nature 589, 220 (2021) arXiv:2006.03071 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
- [22]
- “Exponential suppression of bit or phase errors with cyclic error correction”, Nature 595, 383 (2021) arXiv:2102.06132 DOI
- [23]
- J. F. Marques et al., “Logical-qubit operations in an error-detecting surface code”, Nature Physics 18, 80 (2021) arXiv:2102.13071 DOI
- [24]
- Derby, Charles. Compact fermion to qubit mappings for quantum simulation. Diss. UCL (University College London), 2023.
- [25]
- E. Huang et al., “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
- [26]
- 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