# Topological code

## Description

Stub.

## 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 \(\Omega(L)\) [1], 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 [2].

## 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
- 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 [3][4][5].

## 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 [6].
- 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 [7].
- Monitored random-circuit code — Topological order can be generated in 2D monitored random circuits [8].
- 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 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 [9]. 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, 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
- [2]
- Nathanan Tantivasadakarn, Ruben Verresen, and Ashvin Vishwanath, “The Shortest Route to Non-Abelian Topological Order on a Quantum Processor”. 2209.03964
- [3]
- 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
- [4]
- R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010). DOI; 1002.2816
- [5]
- Alexander Kirillov Jr, “String-net model of Turaev-Viro invariants”. 1106.6033
- [6]
- D. Aasen et al., “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020). DOI; 2002.05166
- [7]
- 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
- [8]
- 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
- [9]
- Michael Freedman and Matthew B. Hastings, “Building manifolds from quantum codes”. 2012.02249

## Page edit log

- Victor V. Albert (2022-09-15) — most recent
- Victor V. Albert (2022-06-05)
- Victor V. Albert (2022-01-05)

## Zoo code information

## Cite as:

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