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Freedman-Meyer-Luo code[1]

Description

Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [2]. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries [3]. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology.

The Freedman-Meter-Luo code has been generalized to a family with rate of order \(O(1/\sqrt{\log n})\) and minimum distance of order \(\Omega(\sqrt{\log n})\) which supports fault-tolerant non-Clifford gates [4].

Protection

Four-dimensional manifolds with weak systolic freedom yield \([[n,2,\Omega(\sqrt{n \sqrt{\log n}})]]\) surface codes.

Rate

Codes held a 20-year record the best lower bound on asymptotic scaling of the minimum code distance, \(d=\Omega(\sqrt{n \sqrt{\log n}})\), broken by Ramanujan tensor-product codes.

Fault Tolerance

The Freedman-Meter-Luo code has been generalized to a family with rate of order \(O(1/\sqrt{\log n})\) and minimum distance of order \(\Omega(\sqrt{\log n})\) which supports fault-tolerant non-Clifford gates [4].

Notes

See thesis by Fetaya for pedagogical exposition [5].

Cousin

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 Qubit Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Homological code
Hyperbolic surface code
Parents
Hyperbolic surface code
Freedman-Meyer-Luo code

References

[1]
“Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002) DOI
[2]
M. H. Freedman, “Z\({}_{\text{2}}\)–Systolic-Freedom”, Geometry & Topology Monographs 113 (1999) DOI
[3]
E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, (2012) DOI
[4]
G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2024) arXiv:2310.16982
[5]
E. Fetaya, “Homological Error Correcting Codes and Systolic Geometry”, (2011) arXiv:1108.2886
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Zoo Code ID: freedman_meyer_luo

Cite as:
“Freedman-Meyer-Luo code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/freedman_meyer_luo
BibTeX:
@incollection{eczoo_freedman_meyer_luo, title={Freedman-Meyer-Luo code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/freedman_meyer_luo} }
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Permanent link:
https://errorcorrectionzoo.org/c/freedman_meyer_luo

Cite as:

“Freedman-Meyer-Luo code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/freedman_meyer_luo

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/freedman_meyer_luo.yml.