The unitary circuit depth required to initialize in a general topologically ordered state using geometrically local gates on an \(L\times L\) lattice is \(\Omega(L)\) , 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 .
- 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
- 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 .
- 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 .
- 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 .
- Monitored random-circuit code — Topological order can be generated in 2D monitored random circuits .
- 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 . 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.
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Zoo code information
“Topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/topological