## 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