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quantumcomputer. Timepointsdownwardsinthisdiagram. |
of the elementary braiding matrices (Sec. V.B) and an |
Thiscomputerhastwoqubitscomposedoffouranyonseach, |
explanation of how we can perform quantum computa- where the ellipses group the anyons into qubits. Some braid- |
tion with Fibonacci anyons (Sec. V.C). Section V.D il- ingisperformedwiththeanyons,thentheanyonsarefusedto |
lustrates how braids which approximate desired opera- measurethestateofthequbits. Thelightgrey,inert,anyons |
tions can be formed. Section VI covers the details of the do not participate in any non-trivial braiding, and could po- |
tentially be deployed for error correction. |
AJLalgorithm,includingtheHadamardtest(Sec.VI.B) |
that can be performed on a quantum computer. Section |
VI.C contains a discussion on how non-Abelian anyons |
1. Non-Abelian Anyons and Qubits |
could be used to exactly calculate the magnitude of the |
Jones polynomial. Intermediate results demonstrating |
the rate of convergence of braids approximating matri- Anyons are a type of particle which can exist in two- |
ces and the Hadamard test are presented in Sec. V.D.3 dimensional quantum systems (Wilczek, 1982). When |
and Sec. VI.B.3, respectively. Finally, our simulation of two anyons are exchanged, the states of those particles |
thetopologicalquantumcomputerispresentedinSection maybesubjectedtoanarbitraryphaseshift(forAbelian |
VIII.Wealsoprovideaqualitativesummaryofthemain anyons) or even a unitary operation (for non-Abelian |
points of this work in Section II for ease of reference. anyons)(BrennenandPachos,2008;Pachos,2012). This |
is unlike the bosons and fermions which constitute regu- |
lar three-dimensional particles, where the particle states |
undergo a multiplication by 1 or 1, respectively, upon |
II. OVERVIEW − |
particle exchange. For non-Abelian anyons, exchanging |
of particles can perform significant changes to the state |
A. Principles of Topological Quantum Computation |
of the system, which can be used to perform quantum |
computation. |
A quantum computer uses the principles of quantum |
The state of a system of anyons is defined by the |
mechanics to manipulate a quantum state in such a way |
anyons produced by fusing those anyons together, with |
as to perform a useful computation. A topological quan- |
each possible set of fusion outcomes representing one |
tum computer uses quantum states which are encoded |
state in the Hilbert space of the quantum system of |
by the topology of the system rather than in any local |
anyons. ThedimensionofthisHilbertspace,orthenum- |
properties. |
ber of different possible fusion outcomes, grows by a fac- |
There are three fundamental steps in performing a |
torcalledthequantumdimensionwhenmoreanyonsare |
topological quantum computation, illustrated in Fig. 1. |
added, on average and in the limit of many anyons. For |
Abelian anyons, because each fusion gives a definite out- |
1. Creating qubits from non-Abelian anyons. |
come, the quantum dimension is 1, because adding more |
anyonsdoesnotaddmorepossiblefusionoutcomes. Non- |
2. Moving the anyons around—‘braiding’ them—to Abelian anyons have a quantum dimension greater than |
perform a computation. 1. The quantum dimension does not need to be an inte- |
ger, or even rational number (Trebst et al., 2008). |
3. Measuring the state of the anyons by fusion. A qubit is a quantum system which can be in two |
possible states, and forms the basic unit of most quan- |
Eachofthesestepsisdiscussedinfurtherdetailbelow. tum computers (Nakahara, 2012). Multiple qubits are |
4 |
brought together to form a register of qubits. For topo- for later use during quantum computation. Here we rely |
logical quantum computers, each qubit is composed of a on the simpler exhaustive search method, which is ade- |
number of anyons. In the Fibonacci model, a qubit can quateforfirst-orderapproximationsofasmallnumberof |
be constructed from four Fibonacci anyons, Fig. 1, with quantum gates. |
zero net overall ‘charge’ or ‘spin’ (i.e. the four anyons |
willannihilate whenall ofthem arefused) (Brennen and |
Pachos, 2008). As such, the first step in performing a 3. Measuring Anyons |
topological quantum computation is to create anyons to |
form a register of qubits. After the computation is complete, it is necessary to |
For the sake of concreteness, we focus on the model of measure the state of the system. This is performed by |
Fibonacci anyons. However, the concepts explored are fusing two of the anyons in each qubit and observing the |
directlyapplicabletogenericnon-Abeliananyonmodels. outcomeofeachfusion. Eachsetoffusionoutcomescor- |
respondstoauniquebasisstate(Pachos,2012). Because |
theanyonsareaquantumsystem,theprobabilityofeach |
2. Braiding Anyons set of fusion outcomes is determined by the amplitudes |
of the basis states in the quantum system. |
Exchanging two non-Abelian anyons performs a uni- The state of the system after the braiding encodes the |
tary operation on the quantum state, which can change result of the computation. However, the full state can- |
the relative phases and probability densities of the basis not be measured directly. As such, it is often necessary |
states corresponding to each fusion outcome. toperformrepeatedidenticalcomputationsandmeasure- |
The anyons exist in two-dimensional space. Consider ments to statistically determine the probability distribu- |
a 2+1 dimensional space, where the third dimension tion and thus the state of the system. However, due to |
is time. The worldlines that thread through the time the embarrassing parallelism of such repeated measure- |
dimension as the anyons move around each other are ments,thistaskcanbecompletedefficientlyandsimulta- |
strands which are braided, as in Fig. 1. Hence, exchang- neouslybydeployingmultiplecopiesofthesamesystem. |
ing anyons is referred to as braiding, because the opera- |
tion braids their worldlines. Furthermore, the operation |
performed on the quantum state is dependent solely on 4. Physical Realization |
thetopologyofthebraid, meaningthatthebraidcanbe |
stretched and deformed in almost any manner but still A variety of physical systems have been suggested for |
performthesameoperation. Thistopologicalrobustness implementing topological quantum computation using |
provides the key advantage of topological quantum com- non-Abelian anyons (Nayak et al., 2008; Sarma et al., |
putersoverotherquantumcomputers,whichistolerance 2015). Hence, complementing the generic but abstract |
to errors from local perturbations (Kitaev, 2003; Nayak notion of anyons, braiding their worldlines, and their |
et al., 2008). eventual fusion as illustrated in Fig. 1, it may be use- |
Bybraidinganyonswithinaqubit,theprobabilitiesof fultohaveaconcretementalpictureofthephysicalenti- |
the fusion outcomes within that qubit can be changed. tiesandprocessescomprisingsuchatopologicalquantum |
This puts the qubits into a superposition of states. By computer. For this purpose, we may choose to consider |
braiding anyons between two qubits, the states of the theanyonstobe(quasiparticlesassociatedwith)vortices |
qubits in general become dependent on each other, such nucleated in a quasi-two-dimensional superfluid. Such |
that it is not possible to measure the state of one qubit vortices are the quantum mechanical counterpart to the |
without affecting the other qubit. Thus performing a familiar bathtub vortices and are ubiquitous in quantum |
braidwhichliterallyentanglestwoqubitswillalsoinduce liquids including superfluid helium-4 (Fonda et al., 2014; |
quantum entanglement between those two qubits. Yarmchuket al.,1979),superfluidhelium-3(Auttiet al., |
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