constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In electromagnetism:
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).
The standard paradigm for potentially realizing topological quantum computation in practice (Kitaev 03, Freedman, Kitaev, Larsen & Wang03) considers manipulations of anyon topological defects 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:
It is interesting to note that:
(pure) braid group representations are equivalently degree-1 cocycles in the non-abelian cohomology of the configuration space of points (ordered) in the Euclidean plane,;
as such, braid representations are the first stage in a sequence that continues with weight systems on horizontal chord diagrams, these being the complex cohomology in higher degree of the configuration space of points (ordered) in Euclideam 3-space (see at weight systems are cohomology of loop space of configuration space):
Here
$Conf_N(\mathbb{R}^2) \simeq K( PBr(N), 1 ) \simeq B PBr(n)$ is an Eilenberg-MacLane space/classifying space for the pure braid group
$Conf_N(\mathbb{R}^3)$ is a simply connected higher homotopy type:
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.
The idea of topological quantum computation via the Chern-Simons theory of anyons (e.g. in the quantum Hall effect) is due to:
Alexei Kitaev, Fault-tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2-30 (arXiv:quant-ph/9707021, doi:10.1016/S0003-4916(02)00018-0)
Michael Freedman, Alexei Kitaev, Michael Larsen, Zhenghan Wang, Topological quantum computation, Bull. Amer. Math. Soc. 40 (2003), 31-38 (arXiv:quant-ph/0101025, doi:10.1090/S0273-0979-02-00964-3, pdf)
Michael Freedman, Michael Larsen, Zhenghan Wang, A modular functor which is universal for quantum computation, Communications in Mathematical Physics. 2002, Vol 227, Num 3, pp 605-622 (arXiv:quant-ph/0001108)
D. Melnikov, A. Mironov, S. Mironov, A. Morozov, An. Morozov, Towards topological quantum computer, Nucl. Phys. B926 (2018) 491-508 (arXiv:1703.00431, doi:10.1016/j.nuclphysb.2017.11.016)
Textbook accounts:
Zhenghan Wang, Topological Quantum Computation, CBMS Regional Conference Series in Mathematics 112, AMS 2010 (ISBN-13: 978-0-8218-4930-9, pdf)
Tudor D. Stanescu, Part IV of: Introduction to Topological Quantum Matter & Quantum Computation, CRC Press 2020 (ISBN:9780367574116)
Review:
Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation, Rev. Mod. Phys. 80, 1083 (2008) (arXiv:0707.1888)
Ady Stern, Netanel H. Lindner, Topological Quantum Computation – From Basic Concepts to First Experiments, Science 08 Mar 2013: Vol. 339, Issue 6124, pp. 1179-1184 (doi:10.1126/science.1231473)
Ville Lahtinen, Jiannis K. Pachos, A Short Introduction to Topological Quantum Computation, SciPost Phys. 3, 021 (2017) (arXiv:1705.04103)
Eric Rowell, Zhenghan Wang, Mathematics of Topological Quantum Computing, Bull. Amer. Math. Soc. 55 (2018), 183-238 (arXiv:1705.06206, doi:10.1090/bull/1605)
Bernard Field, Tapio Simula, Introduction to topological quantum computation with non-Abelian anyons, Quantum Science and Technology 2018 (arXiv:1802.06176, doi:10.1088/2058-9565/aacad2)
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)
See also:
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:
See also:
As quantum gates for topological quantum computation with anyons:
Louis H. Kauffman, Samuel J. Lomonaco, Braiding Operators are Universal Quantum Gates, New Journal of Physics, Volume 6, January 2004 (arXiv:quant-ph/0401090, doi:10.1088/1367-2630/6/1/134)
Samuel J. Lomonaco, Louis Kauffman, Topological Quantum Computing and the Jones Polynomial, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (12 May 2006) (arXiv:quant-ph/0605004)
(braid group representation serving as a topological quantum gate to compute the Jones polynomial)
C.-L. Ho, A.I. Solomon, C.-H.Oh, Quantum entanglement, unitary braid representation and Temperley-Lieb algebra, EPL 92 (2010) 30002 (arXiv:1011.6229)
Louis H. Kauffman, Majorana Fermions and Representations of the Braid Group, International Journal of Modern Physics AVol. 33, No. 23, 1830023 (2018) (arXiv:1710.04650, doi:10.1142/S0217751X18300235)
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):
B. I. Halperin, Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States, Phys. Rev. Lett. 52, 1583 (1984) (doi:10.1103/PhysRevLett.52.1583)
Erratum Phys. Rev. Lett. 52, 2390 (1984) (doi:10.1103/PhysRevLett.52.2390.4)
Daniel Arovas, J. R. Schrieffer, Frank Wilczek, Fractional Statistics and the Quantum Hall Effect, Phys. Rev. Lett. 53, 722 (1984) (doi:10.1103/PhysRevLett.53.722)
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:
On anyon-excitations in topological superconductors.
via Majorana zero modes:
Original proposal:
Review:
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)
Further development:
via Majorana zero modes restricted to edges of topological insulators:
Last revised on May 21, 2021 at 12:28:15. See the history of this page for a list of all contributions to it.