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 a robust form of quantum error correction.
One approach to potentially realizing topological quantum computation in practice is via anyons in the quantum Hall effect and effectively described by some kind of Chern-Simons theory.
For the time being see at quantum computation for more.
The idea of topological quantum computation via the Chern-Simons theory of anyons 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)
Further discussion:
Samuel J. Lomonaco Jr., 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)
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)
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)
Review:
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)
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 February 17, 2021 at 00:53:52. See the history of this page for a list of all contributions to it.