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B. KnotInvariants 7 1. QuantumCircuit 33
C. BraidsandClosures 9 2. HadamardTestintheAJLAlgorithm 34
3. ConvergenceoftheHadamardTest 35
IV. BasicsofConventionalQuantumComputation 10 C. AnExactAlgorithm 36
A. Qubits 10
B. QuantumGates 11 VII. IntermediateSummary 37
C. StateMeasurement 11
D. QuantumCircuitDiagrams 12 VIII. NumericalImplementation 38
E. Errors 12 A. SimulatorCode 38
B. SimulationofAJLAlgorithm 40
V. TopologicalQuantumComputing 12 1. GeneralProcedure 40
A. Anyons 12 2. PositiveHopfLink 43
1. FibonacciAnyons 13 3. NegativeHopfLink 43
B. Braiding 14 4. LeftTrefoil 44
1. TheFMove 14 5. RightTrefoil 44
2. TheRMove 15 6. Figure-EightKnot 44
3. BraidMatrices 16 C. Discussion 45
C. UsingFibonacciAnyonsforComputing 19
1. TopologicalQubits 20 IX. Conclusions 48
2. ComputationbyBraiding 20
3. Measurement 21 Acknowledgments 49
D. CompilingBraidsforComputation 21
1. ErrorMetrics 22 References 49
2
I. INTRODUCTION 1998;LossandDiVincenzo,1998;Reillyetal.,2008),the
energy levels of an ion (Cirac and Zoller, 1995; Leibfried
Theexponentialgrowthobservedoverthepastdecades et al., 2003), optical modes containing one photon (Knill
in information processing capacity of digital computers, et al., 2001), or superconducting Josephson junctions
and as quantified by Moore’s law, is unsustainable and (Shnirman et al., 1997), topological quantum computers
will eventually be complemented or surpassed by quan- encode information using global, topological properties
tum technologies (Kauffman and Lomonaco, 2007; Mil- of a quantum system, which are resilient to local per-
burn, 1996). Quantum computing is a field of much in- turbations (Bombin and Martin-Delgado, 2008; Bombin
terest because it promises to outperform regular, clas- and Martin-Delgado, 2011; Kitaev, 2003; Nayak et al.,
sical computing for many otherwise intractable prob- 2008;PachosandSimon,2014). Thesetopologicalquan-
lems. While classical computers perform Boolean op- tum computers can be implemented using non-Abelian
erations on a register of bits, quantum computers per- anyons, which are quasiparticles in two-dimensional sys-
formunitaryoperationsonanexponentiallylargevector tems which exhibit exotic exchange statistics, beyond a
space, typically composed from many quantum bits, or simple phase change (Pachos, 2012). Considering the
qubits (Galindo and Martín-Delgado, 2002; Nakahara, anyons in 2+1 dimensions (where the third dimension
2012; Nielsen and Chuang, 2010). Using this exponen- is time), the motion of these anyons traces worldlines in
tially large computation space, it is possible, at least in this 2+1 dimensional space, and exchanging the anyons
principle,forquantumcomputerstoefficientlysolveclas- results in braiding the worldlines (Brennen and Pachos,
sically difficult problems such as prime factorisation of 2008). Exchanging non-Abelian anyons results in a uni-
large numbers (Shor, 1994) or the simulation of complex tary operation determined solely by the topology of this
quantum systems (Feynman, 1982; Lloyd, 1996). braid, and for certain models of anyon, such as the Fi-
Anotherexampleofaclassicallyhardalgorithm,which bonaccimodel,itispossibletoreproduceanyunitaryop-
canbenefitfromquantumcomputation,isthedetermina- erationtoarbitraryaccuracybychoosingtherightbraid
tionoftheJonespolynomialofknots(Jones,1985). The to perform, making them universal for quantum compu-
Jones polynomial is a knot invariant with connections to tation (Nayak et al., 2008; Preskill, 2004). Because the
topological quantum field theory (Freedman, 1998; Wit- operations are determined by topology alone, they are
ten, 1989) and other knot-like systems. It is also, in farmoreresistanttodecoherenceanderrors. Thismakes
general, exponentially difficult to compute by classical topological quantum computers an area of significant in-
means. However, a quantum algorithm developed by terest and investment (Collins, 2006b; Gibney, 2016).
Aharonov, Jones and Landau (AJL) (Aharonov et al., In the case of topological quantum computers made
2009) can be used to efficiently estimate the value of the fromFibonaccianyons,compilingmoreusefuloperations
Jones polynomial at the roots of unity, by first reduc- from the elementary braiding operations available with
ing the problem to finding the diagonal elements of the Fibonacci anyons (Bonesteel et al., 2005, 2007; Carna-
product of certain matrices. The resource of nonclassi- han et al., 2016; Freedman and Wang, 2007; Hormozi
cal correlations required in such evaluation of the Jones et al., 2007; Kliuchnikov et al., 2014; Simon et al., 2006;
polynomial (Shor and Jordan, 2008) may be quantified Xu and Wan, 2008), and testing of various error cor-
byquantumdiscord(DattaandShaji,2011;Dattaet al., rection codes for Fibonacci anyon-based quantum com-
2008; Modi et al., 2012; Zurek, 2003). puters (Burton et al., 2017; Feng, 2015; Wootton et al.,
Most implementations of a quantum computer are 2014), as well as simulation of the physics involved with
highly susceptible to errors. A major source of error in Fibonacci anyons (Ayeni et al., 2016) have been investi-
quantumcomputationisdecoherence, causedbyinterac- gated. Therehasalsobeenconsiderablestudyintocandi-
tions between the quantum state and the environment, date physical systems which could contain non-Abelian
whichcausesuncontrolledrandomnessinthesystem(Pa- anyons. Most notable candidate for finding Fibonacci
chos, 2012; Zurek, 2003). Local perturbations can also anyons is the fractional quantum Hall effect at ν =12/5
cause errors in many quantum systems, as can imperfec- (Ardonne and Schoutens, 2007; Bonderson et al., 2006;
tions in the execution of quantum operations (Preskill, Brennen and Pachos, 2008; Mong et al., 2017; Nayak
1997). Thisresultsinnotableoverheadsdevotedtoerror et al., 2008; Rezayi and Read, 2009; Sarma et al., 2006;
correctionschemes,whichonlyworkincomputerswitha Stern,2008;Trebstetal.,2008;Wuetal.,2014),although
sufficientlylowbasicerrorrate,whichmakesimplement- othercandidatesexist(BrennenandPachos,2008;Ðurić
ing such a quantum computer very difficult. et al., 2017; Cooper et al., 2001; Fendley et al., 2013).
Onewaytomitigatetheeffectoftheseerrorsisinusing Meanwhile, significant effort is directed toward finding
topological quantum computing (Collins, 2006a; Freed- Ising anyons in nanowires hosting Majorana zero modes
man, 1998; Kitaev, 2003; Nayak et al., 2008; Pachos, (Alicea, 2012; Sarma et al., 2015; Zhang et al., 2018)
2012;Stanescu,2017;Wang,2010). Incontrasttolocally In this work, we have explicitly carried out a quantum
encoding information and computation using, for exam- algorithm, specifically the AJL algorithm, by simulating
ple, the spin of an electron (Castelvecchi, 2018; Kane, the braiding of Fibonacci anyons. In doing so, we have
3
demonstratedfromfirstprincipleshowFibonaccianyons qubit qubit
can be used for quantum computation, and provided an 1. Quantummemory
explicit recipe for the actions that would need to be per- (anyonqubits)
formed on a system of Fibonacci anyons to perform such
computations. Wehavealsopresentedandperformedan
exact algorithm, which demonstrates the direct connec-
tion between Fibonacci and Ising anyons and the value
2. Computation
of the Jones polynomial at a specific point. (anyonbraiding)
In Section III, we review the relevant components of
knot theory and topology, including the definition of
knots(Sec.III.A),braids(Sec.III.C)andtheJonespoly-
nomial (Sec. III.B). Section IV provides a brief review
of conventional quantum computation. In Section V, 3. Measurement
(anyonfusion)
we cover the theoretical basis for the Fibonacci anyon
topological computer starting with a discussion on Fi-
FIG. 1 A demonstration of braiding anyons in a topological
bonacci anyons (Sec. V.A), followed by the derivation