# nLab topological quantum computation

Contents

### Context

#### Constructivism, Realizability, Computability

intuitionistic mathematics

### Computability

#### Topological physics

Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.

General theory:

# Contents

## Idea

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:

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

### Extended TQC?

It is interesting to note that:

Here

This means that while every individual loop in $Conf_N(\mathbb{R}^3)$ 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 $Conf_N(\mathbb{R}^2)$). Such structure would be reflected by extended TQFT.

## References

### 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):

Textbook accounts:

Review:

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)

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):

Review:

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:

• Xia Gu, Babak Haghighat, Yihua Liu, Ising- and Fibonacci-Anyons from KZ-equations $[$arXiv:2112.07195$]$

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

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

Introduction and review:

Realization of Fibonacci anyons on quasicrystal-states:

• Marcelo Amaral, David Chester, Fang Fang, Klee Irwin, Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing, Symmetry 14 9 (2022) 1780 $[$arXiv:2207.08928, doi:10.3390/sym14091780$]$

Realization on supersymmetric spin chains:

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

### 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):

Review:

### Anyons in topological superconductors

On anyon-excitations in topological superconductors.

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)

Review:

Further development:

• Meng Cheng, Victor Galitski, Sankar Das Sarma, Non-adiabatic Effects in the Braiding of Non-Abelian Anyons in Topological Superconductors, Phys. Rev. B 84, 104529 (2011) (arXiv:1106.2549)

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)

Last revised on January 17, 2023 at 15:19:55. See the history of this page for a list of all contributions to it.