Topological code

Description

Stub.

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

Abelian topological code
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 [1][2][3].

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 [4].
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 [5].
Monitored random-circuit code — Topological order can be generated in 2D monitored random circuits [6].
Quantum low-density parity-check (QLDPC) code — Topological codes are not generally defined using Pauli strings. However, for appropriate tesselations, the codespace forms a ground-state subspace of a geometrically local Hamiltonain. 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 [7]. 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.

[1]: 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
[2]: R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010). DOI; 1002.2816
[3]: Alexander Kirillov Jr, “String-net model of Turaev-Viro invariants”. 1106.6033
[4]: D. Aasen et al., “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020). DOI; 2002.05166
[5]: 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
[6]: A. Lavasani, Y. Alavirad, and M. Barkeshli, “Topological Order and Criticality in (2+1)D Monitored Random Quantum Circuits”, Physical Review Letters 127, (2021). DOI; 2011.06595
[7]: Michael Freedman and Matthew B. Hastings, “Building manifolds from quantum codes”. 2012.02249

“Topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/topological