Topological code




Topological order cannot be stabilized via weight-two or weight-three stabilizer generators on nearly Euclideam geometries of qubits or qutrits [1][2].


The unitary circuit depth required to initialize in a general topologically ordered state using geometrically local gates on an \(L\times L\) lattice is \(\Omega(L)\) [3], irrespective of whether the ground state admits Abelian or non-Abelian anyonic excitations. However, only a finite-depth circuit and one round of measurements is required for nonabelian topological orders with a Lagrangian subgroup [4].


  • Hamiltonian-based code — Codespace is either the ground-state or low-energy subspace of a geometrically local Hamiltonian admitting a topological phase.


  • Abelian topological code
  • Galois-qudit topological code
  • Generalized color code — Quantum-double code Hamiltonians admit topological phases associated with finite groups \(G\).
  • Kitaev honeycomb code — Kitaev honeycomb codes utilizes the non-Abelian Ising-anyon topological order phase of the Kitaev honeycomb model [5]. This phase is also known as a \(p+ip\) superconducting phase [6]. The Kitaev honeycomb model also admits an abelian \(\mathbb{Z}_2\) topological order.
  • Quantum-double code — Quantum-double code Hamiltonians admit topological phases associated with finite groups \(G\).
  • String-net code — String-net codes can be realized using Levin-Wen model Hamiltonians, which realize various topological phases [7][8][9].


  • Eigenstate thermalization hypothesis (ETH) code — ETH codewords, like topological codewords, are locally indistinguishable.
  • Fracton code — Fracton phases can be understood as topological defect networks, meaning that they can be described in the language of topological quantum field theory [10].
  • Fusion-based quantum computing (FBQC) code — Arbitrary topological codes can be created using FBQC, as can topological features such as defects and boundaries, by modifying fusion measurements or adding single qubit measurements.
  • Local Haar-random circuit code — Local Haar-random codewords, like topological codewords, are locally indistinguishable [11].
  • Monitored random-circuit code — Topological order can be generated in 2D monitored random circuits [12].
  • Quantum low-density parity-check (QLDPC) code — Topological codes are not generally defined using Pauli strings. However, for appropriate tesselations, the codespace is the ground-state subspace of a geometrically local Hamiltonian. In this sense, topological codes are QLDPC codes. On the other hand, chain complexes describing some QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [13]. This opens up the possibility that some QLDPC codes, despite not being geometrically local, can in fact be associated with a geometrically local theory described by a category.


S. Bravyi and M. Vyalyi, “Commutative version of the k-local Hamiltonian problem and common eigenspace problem”. quant-ph/0308021
Dorit Aharonov and Lior Eldar, “On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems”. 1102.0770
S. Bravyi, M. B. Hastings, and F. Verstraete, “Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order”, Physical Review Letters 97, (2006). DOI; quant-ph/0603121
Nathanan Tantivasadakarn, Ruben Verresen, and Ashvin Vishwanath, “The Shortest Route to Non-Abelian Topological Order on a Quantum Processor”. 2209.03964
A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006). DOI; cond-mat/0506438
F. J. Burnell and C. Nayak, “SU(2) slave fermion solution of the Kitaev honeycomb lattice model”, Physical Review B 84, (2011). DOI; 1104.5485
M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005). DOI; cond-mat/0404617
R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010). DOI; 1002.2816
Alexander Kirillov Jr, “String-net model of Turaev-Viro invariants”. 1106.6033
D. Aasen et al., “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020). DOI; 2002.05166
F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, “Local Random Quantum Circuits are Approximate Polynomial-Designs”, Communications in Mathematical Physics 346, 397 (2016). DOI
A. Lavasani, Y. Alavirad, and M. Barkeshli, “Topological Order and Criticality in <mml:math xmlns:mml="" display="inline"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mi mathvariant="normal">D</mml:mi></mml:mrow></mml:math> Monitored Random Quantum Circuits”, Physical Review Letters 127, (2021). DOI; 2011.06595
Michael Freedman and Matthew B. Hastings, “Building manifolds from quantum codes”. 2012.02249
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“Topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_topological, title={Topological code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.