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Introduction to topological quantum computation with non-Abelian anyons
Bernard Field and Tapio Simula
School of Physics and Astronomy,
Monash University, Victoria 3800,
Australia
(Dated: April 23, 2018)
Topological quantum computers promise a fault tolerant means to perform quantum
computation. Topologicalquantumcomputersuseparticleswithexoticexchangestatis-
ticscallednon-Abeliananyons,andthesimplestanyonmodelwhichallowsforuniversal
quantum computation by particle exchange or braiding alone is the Fibonacci anyon
model. Oneclassicallyhardproblemthatcanbesolvedefficientlyusingquantumcom-
8102 putationisfindingthevalueoftheJonespolynomialofknotsatrootsofunity. Weaim
to provide a pedagogical, self-contained, review of topological quantum computation
with Fibonacci anyons, from the braiding statistics and matrices to the layout of such
a computer and the compiling of braids to perform specific operations. Then we use
a simulation of a topological quantum computer to explicitly demonstrate a quantum
rpA computationusingFibonaccianyons, evaluatingtheJonespolynomialofaselectionof
simpleknots. Inadditiontosimulatingamodularcircuit-stylequantumalgorithm,we
also show how the magnitude of the Jones polynomial at specific points could be ob-
tained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally
02
suitedforaproofofconceptdemonstrationofatopologicalquantumcomputer.
]hp-tnauq[
Keywords: Aharonov-Jones-Landaualgorithm;Kauffmanbracketpolynomial;braid;Fibonaccianyons;fusion;
Hadamardtest;Isinganyons;Jonespolynomial;knot;link;Majoranazeromode;non-Abelianvortex;quantum
circuit;quantumcomputer;quantumdimension;superfluid;topologicalquantumcomputing;topologicalqubit
CONTENTS 2. CompilingSingleQubitBraids 23
3. ConvergenceofSingleQubitBraids 23
I. Introduction 2 4. AlternativestoExhaustiveSearch 25
5. CompilingTwoQubitBraids 25
2v67160.2081:viXra II. Overview 3 E. SimulatingGenericQuantumAlgorithms 26
A. PrinciplesofTopologicalQuantumComputation 3
1. Non-AbelianAnyonsandQubits 3 VI. TopologicalQuantumAlgorithm 28
2. BraidingAnyons 4 A. TheAJLAlgorithm 28
3. MeasuringAnyons 4 1. UnitaryRepresentationoftheBraidGroup 30
4. PhysicalRealization 4 2. TheMarkovTrace 30
B. SimulationofaTopologicalQuantumComputer 5 3. PlatClosures 31
4. AnExample 31
III. TopologyandKnotTheory 6 5. AJLMatrices 33
A. Knots 6 B. HadamardTest 33