nLab topological quantum computation



Constructivism, Realizability, Computability

Topological physics



In topological quantum computation one aims to make use of quantum systems described by topological quantum field theory for quantum computation, the idea being that the defining invariance of TQFTs under small deformations implements an intrinsic fault tolerance of the quantum computer against noise and decoherence (see also at quantum error correction).

On Anyons in 2+1 d

The standard paradigm for potentially realizing topological quantum computation in practice (Kitaev 03, Freedman, Kitaev, Larsen & Wang03) considers adiabatic braiding of defect anyons in effectively 2-dimensional quantum materials, such as in the quantum Hall effect and effectively described by some kind of Chern-Simons theory/Reshetikhin-Turaev theory:

Adapted from Rowell Wang 18, Rouabah 20

Here topological quantum gates are encoded by braid group-elements and are executed by actions through braid representations on the space of quantum states:

From Sati-Schreiber 21
From Lahtinen Pachos 17

Extended TQC?

It is interesting to note that:

from Sati-Schreiber 2021


This means that while every individual loop in Conf N( 3)Conf_N(\mathbb{R}^3) is homotopically trivial (all “braid-gates” are equivalent) there is now non-trivial structure in higher-dimensional deformation families of braids (which is absent in Conf N( 2)Conf_N(\mathbb{R}^2)). Such structure would be reflected by extended TQFT.


Need for topological protection

Highlighting the need for topological stabilization mechanisms:

  • Jay Sau, A Roadmap for a Scalable Topological Quantum Computer, Physics 10 68 (2017)

    “small machines are unlikely to uncover truly macroscopic quantum phenomena, which have no classical analogs. This will likely require a scalable approach to quantum computation. […] based on […] topological quantum computation (TQC) as envisioned by Alexei Kitaev and Michael Freedman […] The central idea of TQC is to encode qubits into states of topological phases of matter. Qubits encoded in such states are expected to be topologically protected, or robust, against the ‘prying eyes’ of the environment, which are believed to be the bane of conventional quantum computation.”

  • Sankar Das Sarma, Quantum computing has a hype problem, MIT Technology Review (March 2022)

    “The qubit systems we have today are a tremendous scientific achievement, but they take us no closer to having a quantum computer that can solve a problem that anybody cares about. [][\cdots] What is missing is the breakthrough [][\cdots] bypassing quantum error correction by using far-more-stable qubits, in an approach called topological quantum computing.”

Topological quantum computation with anyons

The idea of topological quantum computation via a Chern-Simons theory with anyon braiding defects is due to:

and via a Dijkgraaf-Witten theory (like Chern-Simons theory but with discrete gauge group):



Focus on abelian anyons:

Realization in experiment:

  • Daniel Nigg, Markus Mueller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel A. Martin-Delgado, Rainer Blatt,

    Experimental Quantum Computations on a Topologically Encoded Qubit, Science 18 Jul 2014: Vol. 345, Issue 6194, pp. 302-305 (arXiv:1403.5426, doi:10.1126/science.1253742)

    (for quantum error correction)

Simulation of Ising anyons in a lattice of ordinary superconducting qbits:

  • T. Andersen et al. Observation of non-Abelian exchange statistics on a superconducting processor [[arXiv:2210.10255]]

Braid group representations (as topological quantum gates)

On linear representations of braid groups (see also at braid group statistics and interpretation as quantum gates in topological quantum computation):


in relation to modular tensor categories:

  • Colleen Delaney, Lecture notes on modular tensor categories and braid group representations, 2019 (pdf, pdf)

Braid representations from the monodromy of the Knizhnik-Zamolodchikov connection on bundles of conformal blocks over configuration spaces of points:

and understood in terms of anyon statistics:

Braid representations seen inside the topological K-theory of the braid group‘s classifying space:

See also:

  • R. B. Zhang, Braid group representations arising from quantum supergroups with arbitrary qq and link polynomials, Journal of Mathematical Physics 33, 3918 (1992) (doi:10.1063/1.529840)

As quantum gates for topological quantum computation with anyons:

Introduction and review:

Realization of Fibonacci anyons on quasicrystal-states:

Realization on supersymmetric spin chains:

  • Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, Topological Quantum Computation on Supersymmetric Spin Chains [[arXiv:2209.03822]]

See also:

Compilation to braid gate circuits

On approximating (cf. the Solovay-Kitaev theorem) given quantum gates by (i.e. compiling them to) cicuits of anyon braid gates (generally considered for su(2)-anyons and here mostly for universal Fibonacci anyons, to some extent also for non-universal Majorana anyons):

Approximating all topological quantum gates by just the weaves among all braids:

Anyons in the quantum Hall liquids

References on anyon-excitations (satisfying braid group statistics) in the quantum Hall effect (for more on the application to topological quantum computation see the references there):

The prediction of abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in 1-dimensional linear representations of the braid group):

The original discussion of non-abelian anyon-excitations in the quantum Hall effect (i.e. satisfying braid group statistics in higher dimensional linear representations of the braid group, related to modular tensor categories):


Anyons in topological superconductors

On anyon-excitations in topological superconductors.

via Majorana zero modes:

Original proposal:

  • Nicholas Read, Dmitry Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect, Phys. Rev. B61:10267, 2000 (arXiv:cond-mat/9906453)


  • Sankar Das Sarma, Michael Freedman, Chetan Nayak, Majorana Zero Modes and Topological Quantum Computation, npj Quantum Information 1, 15001 (2015) (nature:npjqi20151)

  • Nur R. Ayukaryana, Mohammad H. Fauzi, Eddwi H. Hasdeo, The quest and hope of Majorana zero modes in topological superconductor for fault-tolerant quantum computing: an introductory overview (arXiv:2009.07764)

  • Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors [arXiv:2301.11614]

Further developments:

via Majorana zero modes restricted to edges of topological insulators:

  • Biao Lian, Xiao-Qi Sun, Abolhassan Vaezi, Xiao-Liang Qi, and Shou-Cheng Zhang, Topological quantum computation based on chiral Majorana fermions, PNAS October 23, 2018 115 (43) 10938-10942; first published October 8, 2018 (doi:10.1073/pnas.1810003115)

See also:

  • Yusuke Masaki, Takeshi Mizushima, Muneto Nitta, Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors [[arXiv:2301.11614]]

Last revised on March 7, 2023 at 13:03:10. See the history of this page for a list of all contributions to it.