Description
A code whose codewords form the ground-state or low-energy subspace of a code Hamiltonian realizing symmetry-protected topological (SPT) order.
Protection
SPT codes typically do not offer protection against generic errors, but can protect against noise that respects the underlying symmetry.
Encoding
Conjectured QCA encoder for SPTs defined by Stiefel-Whitney classes in arbitrary dimensions [3].
Notes
Review on generalized (i.e., non-tensor-product) symmetries [4].
Parent
- Topological code — SPT codes realize symmetry-protected topological phases.
Cousins
- Binary code — SPT orders may be used for encoding classical information [5].
- Twisted quantum double (TQD) code — A TQD code Hamiltonian can be obtained from a 2D particular SPT model by promoting the model's background gauge field to a dynamical gauge field. The same group and cocycle data classifies both 2D SPTs and TQDs [6,7].
- Symmetry-protected self-correcting quantum code — Symmetry-protected self-correcting quantum codes can realize symmetry-protected topological phases [8]. Metastable subspaces of certain many-body Lindbladians can arise due to symmetry relations in the low-lying excitations [9].
- Lattice stabilizer code — Lattice CSS codes in \(D\) dimensions can be converted in SPT Hamiltonians in one less dimension [10].
- Cluster-state code — States realizing various SPT phases are universal resources for MBQC [7,11–17]. A cluster-like state, or a state that is in the same SPT phase as a cluster state, can be prepared in finite time [18].
- 3D color code — Transversal action of \(T\) gates on color codes on general 3-manifolds realizes a \(CCZ\) gate on three logical qubits and is related to a topological invariant that is called the triple intersection number [19]. Transversal \(S\) gate on color codes on general 3-manifolds corresponds to a higher-form symmetry [19].
- Chen-Hsin invertible-order code — Instances of the Chen-Hsin invertible-order code realize beyond-group-cohomology SPTs [20].
- Three-fermion (3F) Walker-Wang model code — The 3F QCA encoder [21,22] can be extended to SPTs in higher dimensions based on an exact bosonization duality [3].
- 3D subsystem color code — Different stabilizer Hamiltonians of the 3D subsystem color code correspond to different SPTs, one of which describes the RBH model [10].
- Fracton stabilizer code — CSS fracton codes can be converted in 2D fractal-like SPT Hamiltonians [10].
- Magnon code — Magnon codewords [23] are associated with 1D SPT orders [24–27].
- Valence-bond-solid (VBS) code — VBS codewords [28] are associated with 1D SPT orders [24–27].
References
- [1]
- Z.-C. Gu and X.-G. Wen, “Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order”, Physical Review B 80, (2009) arXiv:0903.1069 DOI
- [2]
- F. Pollmann et al., “Symmetry protection of topological phases in one-dimensional quantum spin systems”, Physical Review B 85, (2012) arXiv:0909.4059 DOI
- [3]
- L. Fidkowski, J. Haah, and M. B. Hastings, “A QCA for every SPT”, (2024) arXiv:2407.07951
- [4]
- J. McGreevy, “Generalized Symmetries in Condensed Matter”, Annual Review of Condensed Matter Physics 14, 57 (2023) arXiv:2204.03045 DOI
- [5]
- B. Zeng and D.-L. Zhou, “Topological and Error-Correcting Properties for Symmetry-Protected Topological Order”, (2014) arXiv:1407.3413
- [6]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [7]
- X. Chen et al., “Symmetry protected topological orders in interacting bosonic systems”, (2013) arXiv:1301.0861
- [8]
- S. Roberts and S. D. Bartlett, “Symmetry-Protected Self-Correcting Quantum Memories”, Physical Review X 10, (2020) arXiv:1805.01474 DOI
- [9]
- M. J. Kastoryano et al., “A little bit of self-correction”, (2024) arXiv:2408.14970
- [10]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [11]
- A. Miyake, “Quantum computational capability of a 2D valence bond solid phase”, Annals of Physics 326, 1656 (2011) arXiv:1009.3491 DOI
- [12]
- T.-C. Wei, I. Affleck, and R. Raussendorf, “Two-dimensional Affleck-Kennedy-Lieb-Tasaki state on the honeycomb lattice is a universal resource for quantum computation”, Physical Review A 86, (2012) arXiv:1009.2840 DOI
- [13]
- W. Son, L. Amico, and V. Vedral, “Topological order in 1D Cluster state protected by symmetry”, Quantum Information Processing 11, 1961 (2011) arXiv:1111.7173 DOI
- [14]
- T.-C. Wei, I. Affleck, and R. Raussendorf, “Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource”, Physical Review Letters 106, (2011) arXiv:1102.5064 DOI
- [15]
- D. V. Else et al., “Symmetry-Protected Phases for Measurement-Based Quantum Computation”, Physical Review Letters 108, (2012) arXiv:1201.4877 DOI
- [16]
- T.-C. Wei, P. Haghnegahdar, and R. Raussendorf, “Hybrid valence-bond states for universal quantum computation”, Physical Review A 90, (2014) arXiv:1310.5100 DOI
- [17]
- H. P. Nautrup and T.-C. Wei, “Symmetry-protected topologically ordered states for universal quantum computation”, Physical Review A 92, (2015) arXiv:1509.02947 DOI
- [18]
- N. Tantivasadakarn and A. Vishwanath, “Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps”, Physical Review Letters 129, (2022) arXiv:2107.04019 DOI
- [19]
- G. Zhu et al., “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2023) arXiv:2310.16982
- [20]
- Y.-A. Chen and P.-S. Hsin, “Exactly solvable lattice Hamiltonians and gravitational anomalies”, SciPost Physics 14, (2023) arXiv:2110.14644 DOI
- [21]
- J. Haah, L. Fidkowski, and M. B. Hastings, “Nontrivial Quantum Cellular Automata in Higher Dimensions”, Communications in Mathematical Physics 398, 469 (2022) arXiv:1812.01625 DOI
- [22]
- J. Haah, “Topological phases of unitary dynamics: Classification in Clifford category”, (2024) arXiv:2205.09141
- [23]
- M. Gschwendtner et al., “Quantum error-detection at low energies”, Journal of High Energy Physics 2019, (2019) arXiv:1902.02115 DOI
- [24]
- X. Chen, Z.-C. Gu, and X.-G. Wen, “Classification of gapped symmetric phases in one-dimensional spin systems”, Physical Review B 83, (2011) arXiv:1008.3745 DOI
- [25]
- N. Schuch, D. Pérez-García, and I. Cirac, “Classifying quantum phases using matrix product states and projected entangled pair states”, Physical Review B 84, (2011) arXiv:1010.3732 DOI
- [26]
- X. Chen, Z.-C. Gu, and X.-G. Wen, “Complete classification of one-dimensional gapped quantum phases in interacting spin systems”, Physical Review B 84, (2011) arXiv:1103.3323 DOI
- [27]
- X. Chen et al., “Symmetry protected topological orders and the group cohomology of their symmetry group”, Physical Review B 87, (2013) arXiv:1106.4772 DOI
- [28]
- D.-S. Wang et al., “Quasi-exact quantum computation”, Physical Review Research 2, (2020) arXiv:1910.00038 DOI
Page edit log
- Victor V. Albert (2024-06-13) — most recent
Cite as:
“Symmetry-protected topological (SPT) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/spt