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
- 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 [5,21]. 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 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 [22]. 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 [23–25]. 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 [26].
- 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 [27].
- Quantum LDPC (QLDPC) code — Chain complexes describing QLDPC codes can be converted to 2D lattice stabilizer codes [28].
- Self-correcting quantum code — 2D stabilizer codes [29] and encodings of frustration-free code Hamiltonians [30] admit only constant-energy excitations, and so do not have an energy barrier.
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]
- F. Pastawski, A. Kay, N. Schuch, and I. Cirac, “Limitations of Passive Protection of Quantum Information”, (2009) arXiv:0911.3843
- [12]
- A. Kay, “Capabilities of a Perturbed Toric Code as a Quantum Memory”, Physical Review Letters 107, (2011) arXiv:1107.3940 DOI
- [13]
- 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
- [14]
- E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [15]
- 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
- [16]
- A. A. Kovalev and L. P. Pryadko, “Spin glass reflection of the decoding transition for quantum error correcting codes”, (2014) arXiv:1311.7688
- [17]
- J. Y. Lee, C.-M. Jian, and C. Xu, “Quantum Criticality Under Decoherence or Weak Measurement”, PRX Quantum 4, (2023) arXiv:2301.05238 DOI
- [18]
- 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
- [19]
- Y.-H. Chen and T. Grover, “Separability Transitions in Topological States Induced by Local Decoherence”, Physical Review Letters 132, (2024) arXiv:2309.11879 DOI
- [20]
- 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
- [21]
- 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
- [22]
- 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
- [23]
- 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
- [24]
- H. Bombín, “Structure of 2D Topological Stabilizer Codes”, Communications in Mathematical Physics 327, 387 (2014) arXiv:1107.2707 DOI
- [25]
- J. Haah, “Algebraic Methods for Quantum Codes on Lattices”, Revista Colombiana de Matemáticas 50, 299 (2017) arXiv:1607.01387 DOI
- [26]
- 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
- [27]
- T. Schuster, N. Tantivasadakarn, A. Vishwanath, and N. Y. Yao, “A holographic view of topological stabilizer codes”, (2023) arXiv:2312.04617
- [28]
- 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
- [29]
- 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
- [30]
- O. Landon-Cardinal and D. Poulin, “Local Topological Order Inhibits Thermal Stability in 2D”, Physical Review Letters 110, (2013) arXiv:1209.5750 DOI
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