Symmetry-protected topological (SPT) code[1,2] 

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.

Notes

Review on generalized (i.e., non-tensor-product) symmetries [3].

Parent

Cousins

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]
J. McGreevy, “Generalized Symmetries in Condensed Matter”, Annual Review of Condensed Matter Physics 14, 57 (2023) arXiv:2204.03045 DOI
[4]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[5]
X. Chen et al., “Symmetry protected topological orders in interacting bosonic systems”, (2013) arXiv:1301.0861
[6]
S. Roberts and S. D. Bartlett, “Symmetry-Protected Self-Correcting Quantum Memories”, Physical Review X 10, (2020) arXiv:1805.01474 DOI
[7]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[8]
A. Miyake, “Quantum computational capability of a 2D valence bond solid phase”, Annals of Physics 326, 1656 (2011) arXiv:1009.3491 DOI
[9]
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
[10]
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
[11]
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
[12]
D. V. Else et al., “Symmetry-Protected Phases for Measurement-Based Quantum Computation”, Physical Review Letters 108, (2012) arXiv:1201.4877 DOI
[13]
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
[14]
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
[15]
Y.-A. Chen and P.-S. Hsin, “Exactly solvable lattice Hamiltonians and gravitational anomalies”, SciPost Physics 14, (2023) arXiv:2110.14644 DOI
[16]
M. Gschwendtner et al., “Quantum error-detection at low energies”, Journal of High Energy Physics 2019, (2019) arXiv:1902.02115 DOI
[17]
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
[18]
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
[19]
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
[20]
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
[21]
D.-S. Wang et al., “Quasi-exact quantum computation”, Physical Review Research 2, (2020) arXiv:1910.00038 DOI
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Zoo Code ID: spt

Cite as:
“Symmetry-protected topological (SPT) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/spt
BibTeX:
@incollection{eczoo_spt, title={Symmetry-protected topological (SPT) code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/spt} }
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Cite as:

“Symmetry-protected topological (SPT) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/spt

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/topological/spt.yml.