Topological code

Description

Stub.

Protection

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

Encoding

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].

Parent

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

Children

  • 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].

Cousins

  • 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.

References

[1]
S. Bravyi and M. Vyalyi, “Commutative version of the k-local Hamiltonian problem and common eigenspace problem”. quant-ph/0308021
[2]
Dorit Aharonov and Lior Eldar, “On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems”. 1102.0770
[3]
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
[4]
Nathanan Tantivasadakarn, Ruben Verresen, and Ashvin Vishwanath, “The Shortest Route to Non-Abelian Topological Order on a Quantum Processor”. 2209.03964
[5]
A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006). DOI; cond-mat/0506438
[6]
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
[7]
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
[8]
R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010). DOI; 1002.2816
[9]
Alexander Kirillov Jr, “String-net model of Turaev-Viro invariants”. 1106.6033
[10]
D. Aasen et al., “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020). DOI; 2002.05166
[11]
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
[12]
A. Lavasani, Y. Alavirad, and M. Barkeshli, “Topological Order and Criticality in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" 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
[13]
Michael Freedman and Matthew B. Hastings, “Building manifolds from quantum codes”. 2012.02249
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Zoo code information

Internal code ID: topological

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Zoo Code ID: topological

Cite as:
“Topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/topological
BibTeX:
@incollection{eczoo_topological, title={Topological code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/topological} }
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Cite as:

“Topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/topological

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/hamiltonian/topological.yml.