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.

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

Child

  • Three-fermion (3F) Walker-Wang model code — When treated as ground states of the code Hamiltonian, 3F Walker-Wang model code states realize a 3D time-reversal SPT order [5]. The 3F Walker-Wang QCA encoder [6,7] can be extended to SPTs in higher dimensions based on an exact bosonization duality [3].

Cousins

  • Binary code — SPT orders may be used for encoding classical information [8].
  • 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 [9,10].
  • Symmetry-protected self-correcting quantum code — Symmetry-protected self-correcting quantum codes can realize symmetry-protected topological phases [11]. Metastable subspaces of certain many-body Lindbladians can arise due to symmetry relations in the low-lying excitations [12].
  • Lattice stabilizer code — Lattice CSS codes in \(D\) dimensions can be converted in SPT Hamiltonians in one less dimension [13].
  • Cluster-state code — Cluster states defined on various lattices are representatives of SPT phases, and states realizing these phases can be resources for MBQC. In 1D, cluster states are examples of SPT phases with global symmetries [10,14–17] and enable MBQC on a single qubit [18,19]. The square-lattice cluster state, which is the prototypical resource for universal MBQC [18,19], and other 2D cluster states [20–22] have SPT order protected by subsystem symmetries [20,23,24]. States like AKLT states and SPT fixed-point states can be efficiently converted into cluster states using local measurements and subsequently used as resources for MBQC [15,25–29]. In 3D, cluster states belong to SPT phases protected by higher-form symmetries [30] and enable universal fault-tolerant MBQC [31]. A cluster-like state, or a state that is in the same SPT phase as a cluster state, can be prepared in finite time [32]. Cluster states can be created on various lattices [33].
  • Raussendorf-Bravyi-Harrington (RBH) cluster-state code — In 3D, cluster states belong to SPT phases protected by higher-form symmetries [30] and enable universal fault-tolerant MBQC [31].
  • Square-lattice cluster-state code — The square-lattice cluster state, which is the prototypical resource for universal MBQC [18,19], and other 2D cluster states [20–22] have SPT order protected by subsystem symmetries [20,23,24].
  • 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 [34]. Transversal \(S\) gate on color codes on general 3-manifolds corresponds to a higher-form symmetry [34].
  • Chen-Hsin invertible-order code — Instances of the Chen-Hsin invertible-order code realize beyond-group-cohomology SPTs [35].
  • 3D subsystem color code — Different stabilizer Hamiltonians of the 3D subsystem color code correspond to different SPTs, one of which describes the RBH model [13].
  • Fracton stabilizer code — CSS fracton codes can be converted in 2D fractal-like SPT Hamiltonians [13].
  • Magnon code — Magnon codewords [36] are associated with 1D SPT orders [37–40].
  • Valence-bond-solid (VBS) code — VBS codewords [41] are associated with 1D SPT orders [37–40].

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, E. Berg, A. M. Turner, and M. Oshikawa, “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]
F. J. Burnell, X. Chen, L. Fidkowski, and A. Vishwanath, “Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order”, Physical Review B 90, (2014) arXiv:1302.7072 DOI
[6]
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
[7]
J. Haah, “Topological phases of unitary dynamics: Classification in Clifford category”, (2024) arXiv:2205.09141
[8]
B. Zeng and D.-L. Zhou, “Topological and Error-Correcting Properties for Symmetry-Protected Topological Order”, (2014) arXiv:1407.3413
[9]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[10]
X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders in interacting bosonic systems”, (2013) arXiv:1301.0861
[11]
S. Roberts and S. D. Bartlett, “Symmetry-Protected Self-Correcting Quantum Memories”, Physical Review X 10, (2020) arXiv:1805.01474 DOI
[12]
M. J. Kastoryano, L. B. Kristensen, C.-F. Chen, and A. Gilyén, “A little bit of self-correction”, (2024) arXiv:2408.14970
[13]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[14]
R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states”, Physical Review A 68, (2003) arXiv:quant-ph/0301052 DOI
[15]
A. Miyake, “Quantum computational capability of a 2D valence bond solid phase”, Annals of Physics 326, 1656 (2011) arXiv:1009.3491 DOI
[16]
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
[17]
D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, “Symmetry-Protected Phases for Measurement-Based Quantum Computation”, Physical Review Letters 108, (2012) arXiv:1201.4877 DOI
[18]
R. Raussendorf, D. Browne, and H. Briegel, “The one-way quantum computer--a non-network model of quantum computation”, Journal of Modern Optics 49, 1299 (2002) arXiv:quant-ph/0108118 DOI
[19]
R. Raussendorf and H. J. Briegel, “A One-Way Quantum Computer”, Physical Review Letters 86, 5188 (2001) DOI
[20]
D. T. Stephen, H. P. Nautrup, J. Bermejo-Vega, J. Eisert, and R. Raussendorf, “Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter”, Quantum 3, 142 (2019) arXiv:1806.08780 DOI
[21]
T. Devakul and D. J. Williamson, “Universal quantum computation using fractal symmetry-protected cluster phases”, Physical Review A 98, (2018) arXiv:1806.04663 DOI
[22]
A. K. Daniel, R. N. Alexander, and A. Miyake, “Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices”, Quantum 4, 228 (2020) arXiv:1907.13279 DOI
[23]
Y. You, T. Devakul, F. J. Burnell, and S. L. Sondhi, “Subsystem symmetry protected topological order”, Physical Review B 98, (2018) arXiv:1803.02369 DOI
[24]
R. Raussendorf, C. Okay, D.-S. Wang, D. T. Stephen, and H. P. Nautrup, “Computationally Universal Phase of Quantum Matter”, Physical Review Letters 122, (2019) arXiv:1803.00095 DOI
[25]
X. Chen, R. Duan, Z. Ji, and B. Zeng, “Quantum State Reduction for Universal Measurement Based Computation”, Physical Review Letters 105, (2010) arXiv:1002.1567 DOI
[26]
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
[27]
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
[28]
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
[29]
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
[30]
S. Roberts, B. Yoshida, A. Kubica, and S. D. Bartlett, “Symmetry-protected topological order at nonzero temperature”, Physical Review A 96, (2017) arXiv:1611.05450 DOI
[31]
R. Raussendorf, J. Harrington, and K. Goyal, “Topological fault-tolerance in cluster state quantum computation”, New Journal of Physics 9, 199 (2007) arXiv:quant-ph/0703143 DOI
[32]
N. Tantivasadakarn and A. Vishwanath, “Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps”, Physical Review Letters 129, (2022) arXiv:2107.04019 DOI
[33]
M. Newman, L. A. de Castro, and K. R. Brown, “Generating Fault-Tolerant Cluster States from Crystal Structures”, Quantum 4, 295 (2020) arXiv:1909.11817 DOI
[34]
G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2024) arXiv:2310.16982
[35]
Y.-A. Chen and P.-S. Hsin, “Exactly solvable lattice Hamiltonians and gravitational anomalies”, SciPost Physics 14, (2023) arXiv:2110.14644 DOI
[36]
M. Gschwendtner, R. König, B. Şahinoğlu, and E. Tang, “Quantum error-detection at low energies”, Journal of High Energy Physics 2019, (2019) arXiv:1902.02115 DOI
[37]
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
[38]
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
[39]
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
[40]
X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders and the group cohomology of their symmetry group”, Physical Review B 87, (2013) arXiv:1106.4772 DOI
[41]
D.-S. Wang, G. Zhu, C. Okay, and R. Laflamme, “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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“Symmetry-protected topological (SPT) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/spt

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