Topological code[1] 

Description

A code whose codewords form the ground-state or low-energy subspace of a (typically geometrically local) code Hamiltonian realizing a topological phase. A topological phase may be bosonic or fermionic, i.e., constructed out of underlying subsystems whose operators commute or anti-commute with each other, respectively. Unless otherwise noted, the phases discussed are bosonic.

2D topological codes

The physical Hilbert space of 2D topological codes consists of \(n\) subsystems which lie on edges, vertices, or faces of a tesselation of a 2D surface \(\Sigma^2\). 2D topological order requires weight-four Hamiltonian terms, i.e., it cannot be stabilized via weight-two or weight-three terms on nearly Euclidean geometries of qubits or qutrits [2–4].

For subsystems with finite local dimension, topological phases are defined by their anyons [5–8], which are local bulk excitations of the code Hamiltonian defined on a lattice; see Refs. [9–11] for more rigorous formulations.

Anyons are created in pairs by local operators, and two anyons lie in the same superselection sector if a local operator can convert one anyon into the other. Each superselection sector is assumed to be labeled by one anyon type, and local operators cannot change superselection sectors.

Anyons can braid with themselves, with their exchange statistics (a.k.a. topological spin) defined by phases \(\theta(a)\in U(1)\) obtained by exchanging two anyons of each type \(a\). They can also braid with each other, a process defined by braiding relations \(B(a,b)\) for an anyon pair \(a,b\). An anyon theory is called non-modular or pre-modular if there exists an anyon \(a\) that braids trivially with all anyons.

Anyons \(a\) and \(b\) can also fuse with each other, meaning that one considers both anyons as one anyon \(ab\) and decomposes \(ab\) into the anyons representing each superselection sector according to the anyons' fusion rules. For example, two anyons \(a,b\) may fuse to the trivial (i.e., vacuum) anyon \(1\), \(ab=1\), meaning that the composite excitation \(ab\) is indistinguishable from the case of no excitation.

The exchange statistics and fusion rules of anyons cannot be arbitrary and have to satisfy certain consistency relations. Admissible exchange and fusion data are characterized by a unitary braided fusion category.

Each anyon \(a\) has a quantum dimension \(d_a\) associated with it. The quantum dimensions add up to the total quantum dimension \(D\), \begin{align} \sum_{a}d_{a}^{2}=D^{2}~. \tag*{(1)}\end{align} These "dimensions" do not correspond to dimensions of vector spaces and may not be integer-valued.

An anyon theory that does not admit gapped boundaries (when put on a manifold with boundaries) is called chiral; otherwise, it is non-chiral or gapped. Chiral topological phases admit a nonzero value of the chiral central charge \(c_{-}\). A generalization [1] of the Gauss-Milgram sum rule for an anyon theory \(A\) admitting \(|A|\) anyon types, \begin{align} \frac{1}{\sqrt{|A|}}\sum_{a\in A}d_{a}^{2}\theta_{a}=De^{i\frac{2\pi}{8}c_{-}}~, \tag*{(2)}\end{align} relates the chiral central charge (modulo 8) to the exchange statistics and quantum dimensions.

Rate

The logical dimension \(K\) of 2D topological codes described by unitary modular fusion categories depends on the type of manifold \(\Sigma^2\) that is tesselated to form the many-body system. For closed orientable manifolds [12,13], \begin{align} K=\sum_{a\in A}\left(d_{a}/D\right)^{\chi(\Sigma^{2})}~, \tag*{(3)}\end{align} and a generalization of the formula to the non-orientable case can be found in Ref. [14].

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 \(\Omega(L)\) [15], 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 non-Abelian topological orders with a Lagrangian subgroup [16].

Gates

Ising anyon braiding and fusion were studied in a phenomenological model that was the first to study error correction with non-Abelian anyons [17].

Notes

Ref. [19–21][18; Appx. F] for introductions to topological phases.See AnyonWiki for lists of categories relevant to anyons.There is no 2D bosonic topological order at nonzero temperature [22].

Parent

  • Block quantum code — Topological codes are block codes because an infinite family of tensor-product Hilbert spaces is required to formally define a phase of matter.

Children

  • String-net code — String-net codes can be realized using Levin-Wen model Hamiltonians, which realize 2D topological phases based on unitary fusion categories [23–25]. These models realize local topological order (LTO) [26].
  • Walker-Wang model code — Walker-Wang model codes can be realized using Walker-Wang model Hamiltonians, which realize 3D topological phases based on unitary braided fusion categories.
  • Abelian topological code

Cousins

  • Hamiltonian-based code — Codespace of a topological code is typically the ground-state or low-energy subspace of a geometrically local Hamiltonian admitting a topological phase. Logical qubits can also be created via lattice defects or by appropriately scheduling measurements of gauge generators (see Floquet codes).
  • Monitored random-circuit code — Topological order can be generated in 2D monitored random circuits [27].
  • 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 [28]. Constant-rate QLDPC codes have no thermodynamic phase transitions [29].
  • Local Haar-random circuit qubit code — Local Haar-random codewords, like topological codewords, are locally indistinguishable [30].
  • Eigenstate thermalization hypothesis (ETH) code — ETH codewords, like topological codewords, are locally indistinguishable.
  • 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.
  • Fracton stabilizer code — Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted. Fracton phases can be understood as topological defect networks, meaning that they can be described in the language of topological quantum field theory with defects [31,32].

References

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A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
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D. Aharonov and L. Eldar, “On the complexity of Commuting Local Hamiltonians, and tight conditions for Topological Order in such systems”, (2011) arXiv:1102.0770
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D. Aharonov, O. Kenneth, and I. Vigdorovich, “On the Complexity of Two Dimensional Commuting Local Hamiltonians”, (2018) arXiv:1803.02213 DOI
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L. Biedenharn et al., “The Ancestry of the ‘Anyon’”, Physics Today 43, 90 (1990) DOI
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M. Barkeshli et al., “Reflection and Time Reversal Symmetry Enriched Topological Phases of Matter: Path Integrals, Non-orientable Manifolds, and Anomalies”, Communications in Mathematical Physics 374, 1021 (2019) arXiv:1612.07792 DOI
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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) arXiv:quant-ph/0603121 DOI
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N. Tantivasadakarn, R. Verresen, and A. Vishwanath, “Shortest Route to Non-Abelian Topological Order on a Quantum Processor”, Physical Review Letters 131, (2023) arXiv:2209.03964 DOI
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C. G. Brell et al., “Thermalization, Error Correction, and Memory Lifetime for Ising Anyon Systems”, Physical Review X 4, (2014) arXiv:1311.0019 DOI
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A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
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C. Jones et al., “Local topological order and boundary algebras”, (2023) arXiv:2307.12552
[27]
A. Lavasani, Y. Alavirad, and M. Barkeshli, “Topological Order and Criticality in (2+1)D Monitored Random Quantum Circuits”, Physical Review Letters 127, (2021) arXiv:2011.06595 DOI
[28]
M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
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Y. Hong, J. Guo, and A. Lucas, “Quantum memory at nonzero temperature in a thermodynamically trivial system”, (2024) arXiv:2403.10599
[30]
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
[31]
D. Aasen et al., “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020) arXiv:2002.05166 DOI
[32]
Z. Song et al., “Topological Defect Network Representations of Fracton Stabilizer Codes”, PRX Quantum 4, (2023) arXiv:2112.14717 DOI
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“Topological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/topological
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@incollection{eczoo_topological, title={Topological code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/topological} }
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