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2D lattice stabilizer code

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. [11–13].

Code Capacity Threshold

Noise thresholds can be formulated as anyon condensation transitions in a topological field theory [14], generalizing the mapping of the effect of noise on a code state to a statistical mechanical model [7,15–17]. 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 \(\mathcal{E}(|\psi\rangle\langle\psi|)\) [14,18–21].

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.

Member of code lists

Primary Hierarchy

Quantum Domain
Quantum code
Approximate operator-algebra QECC
Operator-algebra QECC (OAQECC)
Quantum error-correcting code (QECC)
Hamiltonian-based code
Commuting-projector Hamiltonian code
Stabilizer codeHamiltonian-based QECC Quantum
Quantum low-weight check (QLWC) codeQECC Quantum
Quantum LDPC (QLDPC) codeQECC Quantum
Quasi-cyclic QLDPC (QC-QLDPC) codeQECC Quantum
Lattice stabilizer code
Parents
Lattice stabilizer code
2D lattice stabilizer code
Children
Compactified \(\mathbb{R}\) gauge theory code
Analog surface code
\(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons GKP code
GKP-surface code
Square-lattice cluster-state code
Bivariate bicycle (BB) code
Bivariate bicycle codes are defined on 2D lattices with periodic boundary conditions, and versions with open boundary conditions have been investigated [5,30]. Bivariate bicycle codes are on par with the surface code in terms of threshold, but admit a much higher ancilla-added encoding rate at the expense of having non-geometrically local weight-six check operators.
Twist-defect color code
Twist-defect surface codeKitaev surface
Abelian TQD stabilizer codeCDSC Kitaev surface
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 [31]. Abelian TQD codes need not be translationally invariant and can realize multiple topological phases on one lattice.
Galois-qudit color code
Galois-qudit surface codeKitaev surface

References

[1]
J. Haah, “Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices”, Journal of Mathematical Physics 62, (2021) arXiv:1812.11193 DOI
[2]
S. Bravyi and M. Vyalyi, “Commutative version of the k-local Hamiltonian problem and common eigenspace problem”, (2004) arXiv:quant-ph/0308021
[3]
D. Aharonov and L. Eldar, “On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems”, (2011) arXiv:1102.0770
[4]
D. Aharonov, O. Kenneth, and I. Vigdorovich, “On the Complexity of Two Dimensional Commuting Local Hamiltonians”, (2018) arXiv:1803.02213 DOI
[5]
Z. Liang, B. Yang, J. T. Iosue, and Y.-A. Chen, “Operator algebra and algorithmic construction of boundaries and defects in (2+1)D topological Pauli stabilizer codes”, (2024) arXiv:2410.11942
[6]
G. Duclos-Cianci and D. Poulin, “A renormalization group decoding algorithm for topological quantum codes”, (2010) arXiv:1006.1362
[7]
C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions 8, 269 (2021) arXiv:1809.10704 DOI
[8]
E. Lake, S. Balasubramanian, and S. Choi, “Exact Quantum Algorithms for Quantum Phase Recognition: Renormalization Group and Error Correction”, (2023) arXiv:2211.09803
[9]
W. Zhong, O. Shtanko, and R. Movassagh, “Advantage of Quantum Neural Networks as Quantum Information Decoders”, (2024) arXiv:2401.06300
[10]
A. Lavasani and S. Vijay, “The Stability of Gapped Quantum Matter and Error-Correction with Adiabatic Noise”, (2024) arXiv:2402.14906
[11]
A. Kay, “Nonequilibrium Reliability of Quantum Memories”, Physical Review Letters 102, (2009) arXiv:0807.0287 DOI
[12]
F. Pastawski, A. Kay, N. Schuch, and I. Cirac, “Limitations of Passive Protection of Quantum Information”, (2009) arXiv:0911.3843
[13]
A. Kay, “Capabilities of a Perturbed Toric Code as a Quantum Memory”, Physical Review Letters 107, (2011) arXiv:1107.3940 DOI
[14]
Y. Bao, R. Fan, A. Vishwanath, and E. Altman, “Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions”, (2023) arXiv:2301.05687
[15]
E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[16]
A. A. Kovalev and L. P. Pryadko, “Fault tolerance of quantum low-density parity check codes with sublinear distance scaling”, Physical Review A 87, (2013) arXiv:1208.2317 DOI
[17]
A. A. Kovalev and L. P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”, (2014) arXiv:1311.7688
[18]
J. Y. Lee, C.-M. Jian, and C. Xu, “Quantum Criticality Under Decoherence or Weak Measurement”, PRX Quantum 4, (2023) arXiv:2301.05238 DOI
[19]
R. Fan, Y. Bao, E. Altman, and A. Vishwanath, “Diagnostics of Mixed-State Topological Order and Breakdown of Quantum Memory”, PRX Quantum 5, (2024) arXiv:2301.05689 DOI
[20]
Y.-H. Chen and T. Grover, “Separability Transitions in Topological States Induced by Local Decoherence”, Physical Review Letters 132, (2024) arXiv:2309.11879 DOI
[21]
K. Su, Z. Yang, and C.-M. Jian, “Tapestry of dualities in decohered quantum error correction codes”, Physical Review B 110, (2024) arXiv:2401.17359 DOI
[22]
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
[23]
H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014) arXiv:1107.2707 DOI
[24]
J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
[25]
Z. Liang, Y. Xu, J. T. Iosue, and Y.-A. Chen, “Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions”, PRX Quantum 5, (2024) arXiv:2312.11170 DOI
[26]
T. Schuster, N. Tantivasadakarn, A. Vishwanath, and N. Y. Yao, “A holographic view of topological stabilizer codes”, (2023) arXiv:2312.04617
[27]
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
[28]
S. Bravyi and B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes”, New Journal of Physics 11, 043029 (2009) arXiv:0810.1983 DOI
[29]
O. Landon-Cardinal and D. Poulin, “Local Topological Order Inhibits Thermal Stability in 2D”, Physical Review Letters 110, (2013) arXiv:1209.5750 DOI
[30]
J. N. Eberhardt, F. R. F. Pereira, and V. Steffan, “Pruning qLDPC codes: Towards bivariate bicycle codes with open boundary conditions”, (2024) arXiv:2412.04181
[31]
T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
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Zoo Code ID: 2d_stabilizer

Cite as:
“2D lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_stabilizer
BibTeX:
@incollection{eczoo_2d_stabilizer, title={2D lattice stabilizer code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/2d_stabilizer} }
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“2D lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/2d_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/stabilizer/lattice/2d_stabilizer.yml.