Topological code[1] 

Description

A code whose codewords form the ground-state or low-energy subspace of a 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 \(\Sigma^2\).

For subsystems with finite local dimension, topological phases are defined by their anyons [2–5], which are local bulk excitations of the code Hamiltonian defined on a lattice; see Refs. [6–8] 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 [9,10], \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. [11].

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 \(\Omega(L)\) [12], irrespective of whether the ground state admits Abelian or nonabelian anyonic excitations. However, only a finite-depth circuit and one round of measurements is required for nonabelian topological orders with a Lagrangian subgroup [13].

Notes

Ref. [15–17][14; Appx. F] for introductions to topological phases.

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 [18–20].
  • 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 — All Abelian topological orders can be realized as modular-qudit subsystem stabilizer codes [21]. Nonabelian topological orders are purported not to be realizable with Pauli stabilizer codes [22].
  • Galois-qudit topological code

Cousins

  • Hamiltonian-based code — Codespace if 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).
  • 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 [23].
  • Monitored random-circuit code — Topological order can be generated in 2D monitored random circuits [24].
  • Good QLDPC code — Chain complexes describing some good QLDPC codes can be 'lifted' into higher-dimensional manifolds admitting some notion of geometric locality [25]. Applying this procedure to good QLDPC codes yiels geometrically local \([[n,n^{1-2/D},n^{1-1/D}]]\) codes in \(D\) spatial dimensions, up to corrections poly-logarithmic in \(n\) [26].
  • 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 [25]. 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.
  • Local Haar-random circuit qubit code — Local Haar-random codewords, like topological codewords, are locally indistinguishable [27].
  • 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.
  • Eigenstate thermalization hypothesis (ETH) code — ETH codewords, like topological codewords, are locally indistinguishable.

References

[1]
A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
[2]
J. M. Leinaas and J. Myrheim, “On the theory of identical particles”, Il Nuovo Cimento B Series 11 37, 1 (1977) DOI
[3]
G. A. Goldin, R. Menikoff, and D. H. Sharp, “Particle statistics from induced representations of a local current group”, Journal of Mathematical Physics 21, 650 (1980) DOI
[4]
F. Wilczek, “Quantum Mechanics of Fractional-Spin Particles”, Physical Review Letters 49, 957 (1982) DOI
[5]
L. Biedenharn et al., “The Ancestry of the ‘Anyon’”, Physics Today 43, 90 (1990) DOI
[6]
S. Doplicher, R. Haag, and J. E. Roberts, “Local observables and particle statistics I”, Communications in Mathematical Physics 23, 199 (1971) DOI
[7]
S. Doplicher, R. Haag, and J. E. Roberts, “Local observables and particle statistics II”, Communications in Mathematical Physics 35, 49 (1974) DOI
[8]
M. Cha, P. Naaijkens, and B. Nachtergaele, “On the Stability of Charges in Infinite Quantum Spin Systems”, Communications in Mathematical Physics 373, 219 (2019) arXiv:1804.03203 DOI
[9]
E. Witten, “Quantum field theory and the Jones polynomial”, Communications in Mathematical Physics 121, 351 (1989) DOI
[10]
G. Moore and N. Seiberg, “Classical and quantum conformal field theory”, Communications in Mathematical Physics 123, 177 (1989) DOI
[11]
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
[12]
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
[13]
N. Tantivasadakarn, R. Verresen, and A. Vishwanath, “The Shortest Route to Non-Abelian Topological Order on a Quantum Processor”, (2022) arXiv:2209.03964
[14]
A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
[15]
P. H. Bonderson, Non-Abelian Anyons and Interferometry, California Institute of Technology, 2007 DOI
[16]
C. Nayak et al., “Non-Abelian anyons and topological quantum computation”, Reviews of Modern Physics 80, 1083 (2008) arXiv:0707.1889 DOI
[17]
S. Sachdev, Quantum Phases of Matter (Cambridge University Press, 2023) DOI
[18]
M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005) arXiv:cond-mat/0404617 DOI
[19]
R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010) arXiv:1002.2816 DOI
[20]
A. Kirillov Jr, “String-net model of Turaev-Viro invariants”, (2011) arXiv:1106.6033
[21]
T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
[22]
A. C. Potter and R. Vasseur, “Symmetry constraints on many-body localization”, Physical Review B 94, (2016) arXiv:1605.03601 DOI
[23]
D. Aasen et al., “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020) arXiv:2002.05166 DOI
[24]
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
[25]
M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
[26]
E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755
[27]
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
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Zoo Code ID: topological

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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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