Description
Lattice stabilizer code in two spatial dimensions.
Any prime-qudit code can be converted to several copies of the prime-qudit 2D surface code (i.e., \(\mathbb{Z}_2\) topological order) along with some trivial codes [1]. Any 2D topological order requires weight-four Hamiltonian terms, i.e., it cannot be stabilized via weight-two or weight-three terms on 2D lattices of qubits or qutrits [2–4].
There is a numerical procedure to construct all possible boundaries and twist defects associated with a given lattice code that admits a particular topological phase [5].
Decoding
Renormalization group (RG) decoder [6].Tensor-network based decoder for 2D codes subject to correlated noise [7].Standard stabilizer-based error correction can be performed even in the presence of perturbations to the codespace [8–10]; see also Refs. [11,12].
Code Capacity Threshold
Noise thresholds can be formulated as anyon condensation transitions in a topological field theory [13], generalizing the mapping of the effect of noise on a code state to a statistical mechanical model [7,14–16]. Namely, the noise threshold for a noise channel \(\cal{E}\) acting on a 2D stabilizer state \(|\psi\rangle\) can be obtained from the properties of the resulting (mixed) state \(\mathcal{E}(|\psi\rangle\langle\psi|)\) [13,17–20].
Parent
Children
- Twist-defect color code
- Twist-defect surface code
- Abelian TQD stabilizer code — All Abelian TQD codes can be realized as modular-qudit lattice stabilizer codes by starting with an Abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and condensing certain bosonic anyons [21]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.
- Galois-qudit color code
- Galois-qudit surface code
Cousins
- Kitaev surface code — Translation-invariant 2D qubit lattice stabilizer codes are equivalent to several copies of the Kitaev surface code via a local constant-depth Clifford circuit [22–24]. There exists an algorithm with which one can determine the fusion and braiding rules of a 2D translationally invariant qubit code, and decompose the given code into copies of the surface code [25].
- Abelian quantum-double stabilizer code — Translation-invariant 2D prime-qudit lattice stabilizer codes are equivalent to several copies of the prime-qudit surface code and a trivial code via a local constant-depth Clifford circuit [1].
- Holographic code — 2D lattice stabilizer codes admit a bulk-boundary correspondence similar to that of holographic codes, namely, the boundary Hilbert space of the former cannot be realized via local degrees of freedom [26].
- Quantum LDPC (QLDPC) code — Chain complexes describing QLDPC codes can be converted to 2D lattice stabilizer codes [27].
- Self-correcting quantum code — 2D stabilizer codes [28] and encodings of frustration-free code Hamiltonians [29] admit only constant-energy excitations, and so do not have an energy barrier.
- \([[144,12,12]]\) gross code — Boundary operators on the gross code correspond to anyons of 8 copies of the surface code [5].
References
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- D. Aharonov, O. Kenneth, and I. Vigdorovich, “On the Complexity of Two Dimensional Commuting Local Hamiltonians”, (2018) arXiv:1803.02213 DOI
- [5]
- Z. Liang et al., “Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes”, (2024) arXiv:2410.11942
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- G. Duclos-Cianci and D. Poulin, “A renormalization group decoding algorithm for topological quantum codes”, (2010) arXiv:1006.1362
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- Y.-H. Chen and T. Grover, “Separability Transitions in Topological States Induced by Local Decoherence”, Physical Review Letters 132, (2024) arXiv:2309.11879 DOI
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- T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
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- H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012) arXiv:1103.4606 DOI
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- H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014) arXiv:1107.2707 DOI
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- J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
- [25]
- Z. Liang et al., “Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions”, PRX Quantum 5, (2024) arXiv:2312.11170 DOI
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- T. Schuster et al., “A holographic view of topological stabilizer codes”, (2023) arXiv:2312.04617
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- X. Li, T.-C. Lin, and M.-H. Hsieh, “Transform Arbitrary Good Quantum LDPC Codes into Good Geometrically Local Codes in Any Dimension”, (2024) arXiv:2408.01769
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Page edit log
- Victor V. Albert (2024-01-27) — most recent
Cite as:
“2D lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_stabilizer