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Part III Essay on Topological Quantum Computing by Philip Zupancic DAMTP, University of Cambridge supervised by Prof Richard Josza X = (Hormozi et al., 2007) Contents 1 Introduction 2 Topological States 2.1 A Phenomenology of Quantum Hall Physics . . . . . . . 2.2 Particle Statistics in 2D . . . . . . . . . . . . . . . . . . 2.3 The Braid Group Bn and its Representations . . . . . . 2.4 The Bigger Picture: Topological Quantum Field Theory 1 . . . . . . . . . . . . . . . . . . . . 2 2 4 6 8 3 Topological Quantum Computation with Anyons 10 3.1 Anyon Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Universality of Quantum Computation via Fibonacci Anyons . . 17 3.3 Fault Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Conclusion 23 References 25 1 Introduction Calculating the time evolution of an interacting quantum many-body system is an NP-hard problem (Kitaev et al., 2002). Turning this around, one could say: this physical system efficiently solves an NP-hard problem — its own evolution. If we could encode other problems onto such a quantum system, we would be able to speed-up computations considerably. This idea of quantum computation has been around for more than thirty years (Feynman, 1982). The complexity class associated with quantum circuits is BQP. While its relation to QMA and QMA’s classical analogue NP is an open question, certain quantum algorithms promise an exponential improvement over the best known classical ones (Nielsen and Chuang, 2010). A famous example is Shor’s factoring algorithm (Shor, 1994). The reason we still rely on public-key cryptography and do not design the structure of room temperature superconductors on a commercial quantum computer is that building such a device is very difficult. For this, we need to be able to encode information onto a physical qubit, let it interact with others in a precisely defined manner, store it, and read it. After solving these issues conceptually, there remains the problem of decoherence, i.e. interactions with the environment that cause a degradation of the qubit state and thus loss of information. As Asher Peres put it, “Quantum phenomena do not occur in Hilbert space. They occur in a laboratory.” There exist a number of error correcting codes that can work against the effects of decoherence (Shor, 1996; Kitaev et al., 2002; Nielsen and Chuang, 2010). Those are effective if decoherence is sufficiently weak, but they require a certain threshold fidelity of the quantum gates. Estimates of the required precision range from 10−6 (Knill et al., 1996) to 1/300 (Zalka, 1996), which is hard to achieve experimentally. A topological quantum computer follows a different approach. It “does not try to make [the system] noiseless, but instead make it deaf” (Sarma et al., 2006). The idea due to Kitaev (2003, submitted 1997) is to make the logical qubit inaccessible to local noise sources by hiding it in a topological Hilbert space, encoded as a homotopy invariant. Though presenting additional challenges, the still theoretical concept of a topological quantum computer relaxes the high requirements on low noise in physical systems for reliable quantum computation. Writing a short but non-trivial introduction to the vast topic of topological quantum computing is a challenging task, since general and specific aspects as well as the involved mathematical structures fill entire books. My goal is to write a coherent story, from the emergence of topological phases of matter in the form of anyons up to an investigation of their applicability for computation. The reader is expected to be knowledgeable in concepts of quantum computing on the level of an introductory class. Basic familiarity with quantum field theory might also be helpful. The essay is comprised of two major sections. In the first, I will motivate the concept of anyons, a kind of quasiparticle that obeys neither Bose nor Fermi statistics, from its emergence in a real physical system in the quantum Hall effect (2.1), before deriving some of their elementary properties from their symmetry group (2.2 and 2.3). The first section concludes with a brief outline of topological quantum field theories and their relation to the mathematics of knots, which 1 provides a bit of context to the behavior of anyons (2.4). The longer second section develops a particular anyon model that is simple enough to fit on a few pages and complex enough to be useful for quantum computing (3.1). How useful exactly is evaluated by considering the possibility to simulate a conventional quantum circuit with anyons and vice versa (3.2). Finally, the advantages of topological quantum computers over conventional ones are analyzed more thoroughly (3.3). 2 2.1 Topological States A Phenomenology of Quantum Hall Physics It is an empirical fact that all elementary and composite particles are either fermions or bosons. These labels distinguish between behaviors of identical particles under exchange. The wave function describing a system of indistinguishable bosons remains the same when any two particles are swapped while the wave function of a system of identical fermions acquires a minus sign. Properties and dynamics of such a system are insensitive to global phases, so the physical observables are not affected by this sign change. The behavior under an exchange operation — called the statistics — is linked to the spin of a particle by the spin-statistics theorem which states that bosons have integer spins (0, 1, ...), while fermions have half-integer spins (1/2, 3/2, ...). Protons and electrons carry a charge e — the elementary charge. Quarks carry fractional charges of 1/3e or −2/3e, but they are forced by color confinement into composite particles such as protons and neutrons, which have a total charge of an integer multiple of e. Thus there are no free particles with charges that are a fraction of e, and the elementary charge really deserves its name. However, this view was challenged at the end of the twentieth century by experiments on the quantum Hall effect. When an electric current I~ = Ix~ex is sent through a two-dimensional conduc~ = B0~ez applied, the Lorentz tor (x-y-plane) with a transverse magnetic field B ~ ~ force F = −e~v × B acts on the conducting electrons (where ~v = v0~ex denotes their velocity). This leads to a redistribution of charges along the y-axis and thus to a voltage difference Vy . Simple calculations reveal that Vy ∝ B0 and thus Vy ρxy = ∝ B0 (1) Ix where ρxy denotes the Hall resistivity. This is called the (classical) Hall effect after its discoverer Edwin Hall in 1879. About a hundred years later, Klaus von Klitzing turned to a two-dimensional electron gas (2DEG, which can be found in semiconductor devices) at low temperatures and exposed to high magnetic fields. New quantum physics was expected to be found in two-dimensional systems, i.e. systems where particles are confined to a length on the order of 10nm in one dimension (Fowler et al., 1966). Von Klitzing discovered a quantization of the Hall resistivity (Englert and von Klitzing, 1978): 1 h (2) ρxy = ν e2 2 Figure 1: Integer Quantum Hall effect. At low temperatures, the electrons in a 2DEG can only occupy states in narrow bands, the so-called Landau levels. The magnetic field strength takes the role of the chemical potential which defines the threshold up to which energy these bands are filled. When the chemical potential µ(B0 ) lies within a Landau level, this band is not filled completely and excitations can occur within this band. These excitations are necessary for longitudinal conductance and correspond to peaks in ρxx . The transverse conductance, however, is given by the number of filled bands. The Hall resistivity ρxy rises steeply when µ(B0 ) lies within a Landau level, while a chemical potential between the ν th and (ν + 1)th Landau levels gives rise to a plateau at h/(νe2 ) (Gross and Marx, 2008). Figure from von Klitzing (1986). where ν can be any integer (fig. 1). The discovery earned von Klitzing the Nobel Prize in Physics in 1985. A few years later, Daniel C. Tsui and Horst L. Störmer made a discovery that brought them a Nobel Prize (1998) and the community of physicists some riddles: in their experimental setup, ν did not only assume integer, but also fractional values, which is called the Fractional Quantum Hall Effect (FQHE) (Tsui et al., 1982; Willett et al., 1987). A measurement of the Hall resistivity is shown in fig. 2. The emergence of a fractional ν was explained partly by Robert B. Laughlin, who shares Tsui’s and Störmer’s Nobel Prize. In a nutshell, he explained that the FQHE is mediated by quasiparticles (collective excitations with particle-like character) with fractional charges. Even more surprisingly, these FQHE states were also shown to carry fractional spin and obey fractional statistics (Arovas et al., 1984; Laughlin, 1983; Wilczek, 1990). 3 Figure 2: Fractional Quantum Hall Effect. In the FQHE, maxima in the longitudinal resistivity ρxx occur at fractional values of ν with corresponding rises in the transverse resistivity ρxy . This effect is mediated by collective excitations of fractional charge (Gross and Marx, 2008). Figure from Willett et al. (1987). 2.2 Particle Statistics in 2D In quantum mechanics, identical particles are indistinguishable. Thus, the dynamics of a system of identical particles has to be invariant under an exchange of any number of those particles. This means that the Hamiltonian governing the evolution of the particles commutes with the swapping operator. In introductory classes to quantum mechanics, students learn that particles can be either fermions or bosons. The product wavefunction of two identical bosons is constant under the swapping operation, while the product wave function of two indistinguishable fermions changes sign: B/F Ψ1 B/F (x1 )Ψ2 B/F (x2 ) = ±Ψ1 B/F (x2 )Ψ2 (x1 ). (3) (The behavior of a system is insensitive to overall phases, so that the negative sign appearing after the fermion exchange does not affect the physics.) A particle exchange can be represented by a (counterclockwise) rotation (in space time) of both particles around their common center by π: SO(3) is the Lie Group of proper rotations in 3D and can thus be connected to 4 Figure 3: The Lie group SO(3) is isomorphic to a three-dimensional ball with radius π, where opposite points on the boundary are identified. A closed loop extending from one point on the surface to its antipodal point twice (• → ×, × → and identified with the top ) covers a rotation by 4π, but can be reduced to a point. π –π Figure 4: In SO(2), a closed loop connecting 0 and 4π cannot be reduced to a point. particle exchange. The fact that the swapping operation σ with B/F B/F B/F B/F σ Ψ1 (x1 )Ψ2 (x2 ) = Ψ1 (x2 )Ψ2 (x1 ) B/F B/F = ± Ψ1 (x1 )Ψ2 (x2 ) (4) only has the eigenvalues ±1 is due to the fact that the symmetry group SO(3) of a three-dimensional vector space has the simply connected covering group SU (2). SO(3) is parameterized by three angles, two of them defining a rotation axis and one the angle to be rotated by. The parameter space of the rotation axis is the 2-sphere S 2 while the rotation angle φ ∈ [−π, π] defines the radius of the sphere. Noticing that a rotation by π is the same as a rotation by −π, we conclude that the manifold underlying SO(3) is isomorphic to a ball of radius π in R3 with opposite points of the boundary identified (e.g. Manton, 2013). Now, a rotation by 2π is identified with the identity operator (no rotation at all), but the two operators are still not equivalent. This is to say that a curve connecting two antipodal group elements of SO(3) cannot be smoothly contracted to a point, though they are identified. A rotation by 4π, however, can be contracted to a point, so that a rotation by 4π is the identity operation (fig. 3). Thus, exchanging twice leads back to the original configuration, σ 2 (Ψ1 (x1 )Ψ2 (x2 )) = Ψ1 (x1 )Ψ2 (x2 ) (5) Eq. 4 follows directly. In two dimensions, however, things are different. SO(2), the Lie group of proper rotations in 2D, does not have a simply connected covering group. The underlying manifold is the interval (−π, π] with the ends of the range identified. Lacking the “extra dimensions”, no smooth transformation exists that could lift the discrete jump (fig. 4). Thus, in 2D the major constraint σ 2 = I on exchange statistics is lifted. We should clarify what we mean when we talk about particles in two dimensions. Clearly, our world is three-dimensional, and even if we confine the 5 movements of particles to a two-dimensional plane, they will still “live” in three dimensions. However, excitations in quantum many-body systems are usually collective modes that are localized enough to be considered particles — so called quasiparticles. These inherit the dimensionality of their environment, which can well be two-dimensional, as is the case with the 2D electron gas exhibiting fractional spins in the FQHE, which was mentioned earlier. What do we know about particle exchange in 2D? Restricting ourselves to one-dimensional representations of SO(2) for now, we have σ (Ψ1 (x1 )Ψ2 (x2 )) = Ψ1 (x2 )Ψ2 (x1 ) = eıθ Ψ1 (x1 )Ψ2 (x2 ) (6) where the statistical angle θ = 0 describes bos-ons, θ = π fermi-ons and any other value of θ describes any-ons. Rotations R by an angle φ are generated by angular momentum, ˆ R(φ) = eıJφ , (7) where Jˆ is the total angular momentum operator composed of the spins Ŝ of the individual particles, Jˆ = 2Ŝ. For the swapping operation we get σ (Ψ1 (x1 )Ψ2 (x2 )) = R(π) (Ψ1 (x1 )Ψ2 (x2 )) ˆ = eıπJ (Ψ1 (x1 )Ψ2 (x2 )) = eı2πs Ψ1 (x1 )Ψ2 (x2 ) (8) where s is the spin of the particle. Comparing eqns. 6 and 8, we see that 2πs = θ mod 2π. (9) We have seen in eq. 6 that θ = 0 (π) corresponds to bosons (fermions), so we recover the spin-statistics theorem stating that bosons have integer spins while fermions have half-integer spins. Furthermore, we notice that anyons can have any spin. As it turned out, the emergence of anyons is sufficient to explain some peaks in the FQHE (Laughlin, 1983). In particular, FQHE states with ν = 1/k can be ascribed to anyons with statistical angle θ = π/k, fractional spin s = 1/k and fractional charge 1/k (Arovas et al., 1984). Schemes that allow those anyons to condense into more elaborate quantized states give a satisfactory explanation of the other values of ν (Haldane, 1983; Halperin, 1984; Prange and Girvin, 1990). Since anyons carry fractional spin and angular momentum is a conserved quantity, single anyons cannot be created from vacuum or annihilate into it. They can, however, be created in anyon-antianyon pairs and fuse back together. Similar to fermions, which have a (−1)F superselection rule with F being the fermion number, anyons of statistical angle θ have a θ-superselection rule (Preskill, 2004). 2.3 The Braid Group Bn and its Representations The key difference between two and three dimensions is that the condition σ 2 = I is lifted in 2D, i.e. 6= 6 (10) As a consequence, σ ∈ / Pn where Pn denotes the permutation group of n elements. Instead, σ is an element of the braid group Bn . The name “braid group” makes sense intuitively if you imagine N indistinguishable anyons at positions (x1 , ..., xN ) at a time t = 0 and the same anyons at the positions (x1 , ..., xN ) at a time t = T . In between t = 0 and t = T , the world lines of the particles can wind around each other a number of times before ending at their final position, creating a braid: t=T t=0 x1 x2 x3 x4 x5 x6 x7 x8 The braid group is generated by elements σi , which exchange particles i and i + 1 in a counterclockwise manner: 7→ σi : i–2 i–1 i i+1 i+2 i+3 (11) i–2 i–1 i i+1 i+2 i+3 Under certain conditions (which will be discussed later), braiding is the only non-trivial operation acting on the anyons. The topological class of a braid does not depend on the specific particle trajectories, thus the final state is invariant under smooth transformations of paths, such as noise in a physical system: ∼ = i–2 i–1 i i+1 i+2 i+3 (12) i–2 i–1 i i+1 i+2 i+3 Eqn. 6 is an example of a one-dimensional representation of the braid group. Since the action is restricted to a complex phase, this representation is abelian, i.e. different swapping operators commute as their phases simply add up. But even this representation is infinite, since e.g. σin and σim are topologically distinct for any n 6= m. The braiding operator is given by R1−dim (σi ) : Ψ 7→ eıθi Ψ (13) with the multi-particle state Ψ = ψ(x1 )...ψ(xn ), where I assume that all particles are in the ground state ψ. However, if the particles’ ground state space is g-fold degenerate with a basis {ψ1 , .., ψg }, R can mediate between different states of this space and becomes a g × g matrix x1 R11 · · · R1g x1 .. · .. . .. R : ... 7→ ... (14) . . . xg Rg1 ··· Rgg xg Elements of this higher-dimensional representation generally do not commute. These non-abelian anyons give rise to interesting braiding behavior, which shall be investigated in the next section after a brief overview of the broader theory of topological quantum field theories. 7 2.4 The Bigger Picture: Topological Quantum Field Theory This subsection is intended to provide a little context to the anyons introduced before and the mathematical structures we will encounter in the following section. Anyons and their braiding characteristics are an emergent feature of topological phases of matter. These phases are subject and object of a topological quantum field theory (TQFT), i.e. a field theory in which transition amplitudes are completely determined by topological properties. In analogy to functional field integrals that emerge in non-topological field theories, which depend on space-time parameters such as velocities or distances, amplitudes in TQFT are of the kind Z Z = D{configurations}eıS[configuration] (15) where Z denotes the partition function and S is the action of a particular configuration. A configuration is a topological class of particle trajectories from the initial to the final state (Simon, 2012). One particular formulation of a TQFT in 2 + 1 dimensions (2 space dimensions plus time) is Chern-Simons theory, which is defined by the action integral Z 2 k tr a ∧ da + a ∧ a ∧ a (16) SCS [a] = 4π M 3 where a is the field of the theory, M is the underlying manifold and k is an integer defining the level of the theory. The Chern-Simons theory of level k has a gauge group SU (2)k . Here, SU (2)k denotes a “deformation” of SU (2) that has a level k cutoff, i.e. the only allowed quantum numbers are 0, 1/2, 1, ..., k/2 (SU (2) ≡ SU (2)∞ ). For k > 2, this group is non-abelian (Nayak et al., 2008). I will discuss a model associated with the SU (2)3 theory later, which is a candidate for topological quantum computation and is believed to be found in the ν = 12/5 FQHE state. To find out how amplitudes in a TQFT can be calculated, we need to take a look at the mathematical theory of knots. Knot theory is concerned with the classification of knots into topologically distinct groups. A knot invariant is a mathematical object that is different for topologically distinct knots and the same for those which can be transformed into each other by a diffeomorphism. Examples are the Jones polynomial (Jones et al., 1985) and the related but simpler Kauffman invariant (Kauffman, 1987). The Kauffman invariant is defined by the relations =A· + A−1 · (17a) =A· + A−1 · (17b) = −A2 − A−2 = d. (17c) where A is some number (Simon, 2012). (The second rule is just a rotated version of the first one.) These rules tell us to substitute crossings by arcs until 8 we are left with circles. To demonstrate them, we can take a look at a simplistic example: + A−1 · =A· = A A · + A−1 · " + A−1 A · #2 + A−1 = A A " + A−1 A + A−1 · #3 " #2 + A−1 = A Ad2 + A−1 d + A−1 Ad3 + A−1 d2 = d + d3 + A2 + A−2 d2 {z } | −d 2 = −A − A −2 , which should not come as a surprise since the pictured knot can be transformed into — and is thus equivalent to — a circle, in contrast to e.g. the distinct diagram (notice the orientation in the crossings) = −A8 − A4 − A−4 by a similar calculation. With each crossing, the number of terms doubles. It is apparent that knot invariants like this one are exponentially hard to compute (Jaeger et al., 1990). Fields Medalist Edward Witten (the only physicist who was ever awarded this prize) discovered that the amplitudes of a TQFT are really given by Jones polynomials describing the intertwining of the particle world lines (Witten, 1989). This important link between the unitary evolution of a many-body system on the one hand and topological classes of knots on the other hand should give a good motivation for using the braid group to simulate the unitary action of a quantum circuit. Matters of interest for topological quantum computing, such as the effect of braiding on a multi-particle state, can be derived from the full Chern-Simons 9 theory (Nayak et al., 2008). However, this will not be necessary for the simple example considered in the following section. 3 Topological Quantum Computation with Anyons The previous section gave an introduction to particular aspects of topological quantum field theories describing (non-abelian) anyons. However, our goal is to build a quantum computer, and this means being able to (DiVincenzo, 2000) 1. define physical qubits, 2. prepare a particular initial state of the multi-qubit system, 3. have long coherence times (compared to gate operation times), 4. implement a universal set of gates, and 5. measure the final state. A general topological quantum computation starts with creating a set of anyon-antianyon pairs from the vacuum as an initial state. This state is then acted on by unitary transformations that are implemented via braiding (since we have said that braiding is the only non-trivial operation acting on the Hilbert space of the anyons). The anyons are finally fused back together and it is measured if they annihilate to vacuum or not. In this section, I will fill in some details on those steps, in particular on the mapping of quantum information onto anyons, the universality of the braid group for quantum gates, state measurement through fusion, and fault tolerance of the setup. These topics cover the afore-mentioned DiVincenzo criteria and can be considered necessary conditions for universal quantum computing. 3.1 Anyon Models There is a number of physical systems that non-abelian anyons can (theoretically) live in, some of which will be mentioned later. In order to analyze computational models of anyons, however, the physical realization is not of primary importance. All we need to know are some general properties: • What types of anyons can occur in this theory? • What are the rules for creation/splitting and fusion? • What is the effect of braiding? The answers to those questions define an anyon model (or, more technically, a topological field theory in 2 + 1 dimensions as well as a unitary topological modular functor — see e.g. Freedman et al. (2002a) for a definition). The presentation in this section follows the structure in Preskill (2004). 10 Particle Types An anyon is a quasiparticle carrying a locally conserved (topological) charge, such as the fractional electric charge in the FQHE. The type of the anyon is a label associated with this charge. There can be any number of types {a, b, c, ...} in an anyon model, as long as two conditions are fulfilled: Firstly, the trivial anyon 1 must be in the set. It carries zero charge and has trivial quantum numbers, so that it can be associated with the vacuum state. Thus, it has trivial braiding effect and behaves under braiding like no anyon at all. Secondly, since the charge is conserved, for each type a there must be a conjugate type ā, the antiparticle, with opposite quantum numbers. Particles can be self-conjugate, e.g. 1 = 1̄. I will assume that our system allows just two particle types, denoted by 1 and τ . Fusion Fusion rules describe what happens when two anyons a and b are combined to form a new anyon c: X c a×b= Nab c (18) c c Nab c If = 0, a and b cannot fuse to c. On the other hand, if Nab > 0, they can c fuse to c, and if Nab = 1, the fusion channel is unique. I denote a |a, b; c, µi = b (19) µ c c as the fusion state of a and b fusing into c, where µ ∈ {1, ..., Nab }. These states c span the space Vab . (This should look familiar from representation theory: E.g. for irreducible representations d(j) of su(2) with spin j, we have j ⊗ j 0 = |j − j 0 | ⊕ ... ⊕ (j + j 0 ). (20) An example of a non-unique fusion channel in this picture is the product of two octet representations of SU (3) 8 ⊗ 8 = 1 ⊕ 8 ⊕ 8 ⊕ 10 ⊕ 10 ⊕ 27 (21) 8 so that N88 = 2 (e.g. Fuchs and Schweigert, 2003).) For the example model with anyons 1 and τ there shall be only one nontrivial fusion rule τ ×τ =1+τ (22) in addition to the trivial ones, 1 × a = a (which is trivial since 1 behaves like 1 c no particle at all), so e.g. Nτττ = Nτ1τ = 1, but N1τ = 0. Since Nab > 1 does not c occur for any a, b and c, the vector space Vab has dimension zero or one, and we can drop the index µ from its basis states, a b |a, b; c, µ = 1i ≡ |a, b; ci = . c 11 (23) This diagrammatic representation of fusion rules, with arrows following the direction of time describing particles, motivates to represent antiparticles by arrows in the opposite direction, similar to Feynman diagrams in quantum field theory. By moving the legs of the diagrams around, a b a c c ∼ = ∼ = c b (24) a b we get the identities c ∼ b ∼ Vab = Vac = Vcab ∼ = ... (25) This confirms that only pairs of opposite charge can be created from vacuum, a a ∼ since 1 × a = a, so that N1a = 1 and V1a = V1aa . The direct sum M c Vab (26) c is called topological Hilbert space of a pair (ab). Its dimension is ! M X c c dab = dim Vab = Nab . c (27) c If, for any a and b, dab > 1, i.e. the fusion is not unique, the anyon model is non-abelian. For the 1-τ -model, X dτ τ = Nτcτ = Nτ1τ + Nτττ = 1 + 1 = 2, (28) c∈{1,τ } so it is a non-abelian anyon model (and actually the simplest one). Multi-particle basis. When more than two particles fuse, we need to keep track of the order in which they do so. The total charge is an intrinsic property of a group of anyons and should not depend on this order, i.e. fusion is associative: (a × b) × c = a × (b × c). (29) For the fusion of three particles a, b and c with total charge d, we get the fusion space M M d ∼ e d ∼ d e0 Vabc Vab ⊗ Vec Vae (30) 0 ⊗ Vbc . = = e0 e Here the direct sum over e on the left-hand side describes the case in which a and b first fuse to e and then e and c fuse to d. The vector space has an orthonormal basis |(ab)c → d; ei ≡ |ab; ei ⊗ |ec; di. Because of the associativity of fusion, it is isomorphic to the direct sum over e0 on the right hand side of eq. 30, in which b and c fuse to e0 before e0 and a fuse to d. This latter vector space has an ONB |a(bc) → d; e0 i ≡ |ae0 ; di ⊗ |bc; e0 i. Since the two spaces are isomorphic, the basis elements can be transformed into each other by |(ab)c → d; ei = X d Fabc e0 12 e 0 e |a(bc) → d; e0 i (31) a b c a X = e d Fabc b c e 0 e’ e e0 d d d Fabc where is called the F-matrix in the literature. This scheme can be generalized to more particles. Having n anyons a1 , ..., an of total charge c ordered along a line, the standard basis shall be the one in which particles fuse from the left to the right, i.e. a1 × a2 → b1 , b1 × a3 → b2 , ..., bn−2 × an → c. The fusion space is given by M Vac1 ...an = Vab11a2 ⊗ Vbb12a3 ⊗ ... ⊗ Vbcn−2 an (32) b1 ,...,bn−2 with basis elements |a1 ...an → c; b1 ...bn−2 i ≡ |a1 a2 ; b1 i ⊗ |b1 a2 ; b2 i ⊗ ... ⊗ |bn−2 an ; ci a1 a2 = ... a3 b1 b 2 (33) an–1 an . ... bn–2 c This particular fusion ordering can be turned into any other by the action of the F -matrix. Again, the fusion result, i.e. the total charge, does not depend on the ordering, but the basis state does. There is no natural decomposition of the space Vac1 ...an into a tensor product of subsystems. Fibonacci anyons. Let us return to the 1-τ -model. The basis of the fusion c space Vτ...τ = Vτcn of dimension Nnc is labeled by n − 2 binary elements bi (eq. 33 with bi ∈ {1, τ }). However, since 1 × τ = τ , there cannot be two subsequent 1s in this “bitstring” b. If the first two anyons fuse to 1, then the remaining n − 2 c anyons can fuse to c in Nn−2 different ways, while if the first two anyons fuse c to τ , this τ can fuse with the remaining n − 2 anyons in Nn−1 different ways: τ τ n τ fusing τ τ ... b1 b 2 ... τ τ τ τ = c 1 τ τ ... τ ... (n–2) τ fusing τ + c τ τ τ b2 τ ... ... (n–1) τ fusing τ (34) c Thus, the dimension of the fusion space is given by the recursion relation c c Nnc = Nn−1 + Nn−2 (35) which defines the Fibonacci numbers Fn ∈ {0, 1, 1, 2, 3, 5, 8, 13, ...}. For this reason, the 1-τ -model of anyons is often called Fibonacci model. Braiding For two particles a and b with a total charge c, braiding is an isomorphism R: c c Vba → Vab 13 c c |ba; ci = Rab |ab; ci = eıθab |ab; ci a b a = (36) b c Rab c (37) c where the R-matrix is just a phase factor since we restrict ourselves to onec c µ0 dimensional fusion spaces here (but Rab could be any unitary matrix (Rab )µ if we had not dropped the index µ in eq. 19). The action of braiding on a pair of definite total charge is thus not particularly interesting. The situation is different for a multi-particle fusion state. Given a state |a1 ...an → c; b1 ...bn−2 i in the standard basis, we need to figure out the effect of an exchange of the j th with (j + 1)th particle. This can be achieved by applying the F -matrix to get to a basis in which the R-matrix is diagonal, i.e. a state where the j th and the (j +1)th do have definite total charge, applying R, and returning to the standard basis via F −1 : a b c a b F c a b a b c F –1 R d c d d d d Let’s take the fusion space Vacb and calculate the effect of exchanging b with c. The braiding operator B in the standard basis acts like X f d B |(ac)b → d; ei = Facb B |a(cb) → d; f i e f = X f d Rbc Facb f e |a(bc) → d; f i f = X g f f d (F −1 )dacb f Rbc Facb |(ab)c → d; f i e (38) f,g so that we get the B-matrix g −1 d g f f d d Babc = (F )acb f Rbc Facb . e e (39) Moore-Seiberg Polynomial Equations. In a general anyon model, the operators F , R and B need to be derived from the topological quantum field theory. However, the simple Fibonacci model is completely determined by two algebraic consistency equations, the Moore-Seiberg polynomial equations. These consistency conditions are identities between different paths that apply F-moves and R-moves to get from one fusion space to another (Trebst et al., 2009). They can best be understood diagrammatically (fig. 5). The pentagon equation (fig. 5a) says that the state X d 5 c 5 |12; ai |a3; bi |b4; 5i = F12c Fa34 b |1d; 5i |2c; di |34; ci (40) a c,d obtained via the upper path in the figure should equal the state X c 5 d b e d |12; ai |a3; bi |b4; 5i = F234 F1e4 b F123 a |1d; 5i |2c; di |34; ci , e c,d,e 14 (41) 1 a 4 3 a c final state 2 3 4 F F initial state 1 2 3 4 2 5 1 d b 5 c 5 F F 1 2 4 3 1 e b 2 4 3 e F d 5 5 (a) “Pentagon” equation. 1 2 3 1 F R 3 c 4 R a initial state 1 2 3 2 4 final state 1 2 3 c a 4 4 1 2 3 1 2 3 1 2 3 F F b = b 4 R b 4 4 (b) “Hexagon” equation. Figure 5: The Moore-Seiberg relations are identities between different paths of F moves and R-moves leading to the same final state. They act as consistency equations that completely determine the F - and R-matrices in the Fibonacci model (Trebst et al., 2009). which we get on the lower path. Since we expect all incoming anyons to be of kind τ , we get Fτ5τ c d a 5 Faτ τ c b = X Fτdτ τ c e Fτ5eτ d b Fτbτ τ e a (42) e τ τ τ The fusion spaces Vτ1τ τ , V1τ τ , Vτ 1τ and Vτ τ 1 are one-dimensional, i.e. there is a τ unique fusion channel, and thus we have some trivial F -matrices Fτ1τ τ = F1τ τ = τ τ Fτ 1τ = Fτ τ 1 = 1 (up to negligible phase factor). The solution to eq. 42 is then Fτ1τ τ = 1 on the space spanned by |(τ τ )τ → 1; τ i and −1 φ φ−1/2 τ Fτ τ τ = φ−1/2 −φ−1 15 (43) (44) √ on the space spanned by {|(τ τ )τ → τ ; 1i , |(τ τ )τ → τ ; τ i} where φ = (1 + 5)/2 is the golden ratio (Trebst et al., 2009). Similarly, the hexagon equation (fig. 5b) is based on the observation that X a 4 c |13; ai |a2; 4i = R13 (F132 )ca R23 |1c; 4i |23; ci (45) c (upper path) must be equivalent to X 4 4 4 |13; ai |a2; 4i = (F312 )ba Rb3 (F123 )cb |1c; 4i |23; ci (46) b (lower path). This leads to c Rτc τ (Fτττ τ )a Rτaτ = X c b (Fτττ τ )b Rττ b (Fτττ τ )a (47) b with the solution R= eı4π/5 0 0 −eı2π/5 (48) acting on the space spanned by {|τ τ → 1i , |τ τ → τ i} (Trebst et al., 2009). Qubit encoding. We know that braiding two anyons only adds a phase factor to the fusion state (eq. 48). However, the braiding operation gets non-trivial when more particles are added. An anyon triplet has three fusion states: |0i ≡ |(τ τ )τ → τ ; 1i, |1i ≡ |(τ τ )τ → τ ; τ i and |NCi ≡ |(τ τ )τ → 1; τ i (reminder: the last label in the ket describes the intermediate fusion result). These triplets can be used to encode qubits. The third anyon is labeled non-computational because it is not used for information storage (but it is essential for non-trivial braiding). The braid group of a triplet is generated by the elementary counterclockwise exchanges σ1 (swapping particles 1 and 2), σ2 (swapping particles 2 and 3) and their clockwise inverses. Their action on the fusion states in the basis {|0i , |1i , |NCi} is (in generalization of eqns. 44 and 48) −ı4π/5 0 e 0 |Ψi 0 −e−ı2π/5 0 σ1 : |Ψi 7→ R |Ψi = (49) −ı2π/5 0 0 −e and σ2 : |Ψi 7→ B |Ψi = F −1 RF |Ψi −e−ıπ/5 /φ √ −e−ıπ/10 / φ = 0 √ −e−ıπ/10 / φ 0 |Ψi −1/φ 0 −ı2π/5 0 −e (Nayak et al., 2008). 16 (50) Relevance of the Fibonacci model. The Fibonacci model is a neat example of a topological quantum field theory because it is very simple but still exhibits non-abelian statistics. But apart from that, it is also of importance because a close relative of it is believed to be found in the ν = 12/5 FQHE state. This state is described by the so called Z3 parafermion model, which has a gauge group SU (2)3 /U (1) at its core (Read and Rezayi, 1999). Again, SU (2)k is SU (2) with a level k cutoff, i.e. the only allowed quantum numbers are 0, 1/2, 1, ..., k/2, so that the SU (2) fusion rules are truncated, e.g. in SU (2)3 we have 1 × 1 = 0 + 1 while we get 1 × 1 = 0 + 1 + 2 in SU (2) ≡ SU (2)∞ . The SU (2)3 fusion rule should look familiar from the Fibonacci model. While SU (2)3 also contains the half-integer labels 1/2 and 3/2, the reduction to the (integer) Fibonacci subspace is unproblematic since the half-integer subspace is inaccessible from within the integer one anyway due to the fusion rules (Nayak et al., 2008). The ν = 12/5 FQHE state has been studied experimentally (Xia et al., 2004). However, there has been no direct evidence of the emergence of nonabelian statistics in this or any other FQHE state yet. 3.2 Universality of Quantum Computation via Fibonacci Anyons Now that the Fibonacci model has been introduced and the action of elementary braiding operations has been examined, it is time to prove that it is as powerful as the quantum circuit model. This will be done in two steps: First, I will show that any quantum circuit can be efficiently simulated using Fibonacci anyons, i.e. this kind of topological quantum computer is at least as powerful as nontopological ones. In a second step, I will demonstrate that the reverse is true as well, i.e. anyon braiding can be simulated on a conventional quantum computer. This leads to the result that the topological and the non-topological models for quantum computing are equally powerful. In order to show universality, we will need the Solovay-Kitaev theorem. The Solovay-Kitaev Theorem The definition of a universal quantum computer is that it can perform any unitary operation on the Hilbert space of its n qubits H = C2n . Obviously, a physical computer cannot have the uncountable set of SU (2n) operations as elementary building blocks at its disposal. Instead, the condition can be watered down slightly by only requiring that any operation U ∈ SU (2n) can be approximated to arbitrary precision by some countable (possibly finite) universal set G of transformations. More technically, let S be the product of l elements in G. S is an approximation to U if d(U, S) ≡ kU − Sk ≡ supkψk=1 k(U − S)ψk < . Saying that for any U and there exists such an S, is to say that the image of G is dense in SU (2n). However, the scaling of the length of the sequence S as a function of the precision, l(), is of paramount importance. As an example, take Grover’s search algorithm, which brings a quadratic speed up over classical algorithms (Nielsen and Chuang, 2010). Assuming that the m gates U1 , ..., Um it is comprised of are not in the set G and our target precision is , we can allow an error of /m per 17 gate. If the scaling of l was l = O(1/), which might be a reasonable guess, we would need a total of m · O(m/) = O(m2 /) gates. Thus, the advantage of the quantum algorithm over its classical counterpart would be lost. Luckily, this is not the case: Solovay-Kitaev Theorem. Let G be a finite set of elements in SU (d) containing its own inverses, such that the image of G is dense in SU (d), and let a desired accuracy be given. There exists a constant c such that for any U ∈ SU (d) there exists a finite sequence S of gates in G of length O(logc (1/)) with d(U, S) < (Dawson and Nielsen, 2005). In simpler terms, the number of gates needed to approximate a unitary transformation to precision scales polylogarithmically in 1/. The proof of the theorem, which can be found in the cited paper, consists of an iterative algorithm which explicitly constructs the element S (and, as a matter of fact, this algorithm runs on a classical computer in polylogarithmic time as well). (Quantum compiling is an active field of research. The value of c depends on both the compiling algorithm and the chosen basis set. For conventional quantum computing, the algorithm in Dawson and Nielsen (2005) gives c = 3.97. Harrow et al. (2002) have proven that c = 1 can theoretically be achieved for an optimally chosen basis set, though it is not clear how to construct such a set.) Simulating Quantum Circuits with Anyons The elementary braiding matrices σ1 and σ2 from the Fibonacci model generate a representation of the braid group that is indeed dense in SU (2) (Freedman et al., 2002b). (The proof is very technical and its mathematical framework goes beyond the scope of this introduction, so I will not outline it here, though it is an important link in the argument.) The instruction set G = {σ1 , σ2 , σ1−1 , σ2−1 } ∈ SU (2) thus fulfills the requirements of the Solovay-Kitaev theorem (the inverse operators correspond to clockwise quasiparticle exchanges), so that any one-qubit operations can be approximated efficiently by braiding operations. Since the dimension D = Fn of the topological Hilbert space of n Fibonacci anyons grows exponentially, this generalizes to multi-qubit systems. E.g. the Hilbert space of six anyons (corresponding to 2 qubits) has dimension F6 = 5. Similar to the one-qubit case, the braiding matrices turn out to be dense in SU (F6 ) = SU (5). This clearly includes SU (4), so that arbitrary two-qubit transformations can be applied (e.g. Nayak et al., 2008). It is well known that a generating set of one-qubit operations and a two-qubit entanglement gate such as the CNOT-gate are universal (Nielsen and Chuang, 2010). The Solovay-Kitaev algorithm can be applied to construct such a set from G = {σ1 , σ2 , σ1−1 , σ2−1 }. E.g., the NOT-gate 0 i 0 i 0 0 (51) 0 0 1 acting on (|0i , |1i , |NCi) is approximated to ≈ 8.5 × 10−4 by (Bonesteel et al., 2005) σ1−2 σ2−4 σ14 σ2−2 σ12 σ22 σ1−2 σ24 σ1−2 σ24 σ12 σ2−4 σ12 σ2−2 σ12 σ2−2 σ12 σ2−2 σ1−2 18 (52) 1 (53) (remember that an arbitrarily lower can be achieved by longer sequences). In this figure, the world lines of anyons 1 and 3 are “strained” so that it becomes apparent that this braid is effectively created by “weaving” anyon 2 through the world lines of the other two. In fact, it is a general theorem that any braid operation can be accomplished by weaving a single particle (Simon et al., 2006). This (K)NOT-gate can be extended in a very clever way. Weaving a τ -anyon around two others in the way described acts as a NOT-gate. Weaving a 1-anyon around the other two does not do anything, since a 1-anyon is like no particle at all. This means that the action of the braid depends on the particle type at anyon 2. Now, there are only τ -anyons in the system, but a pair of anyons can have a total charge of either 1 or τ . Thus, building this braid with a pair of particles in place of anyon 2, a controlled NOT operation is performed. In order to get to the original two-qubit CNOT gate, we need to take anyons 2 and 3 of the control qubit and braid them with anyons 2 and 3 of the target qubit. But since those anyons are no direct neighbors, we first need a braid that swaps the anyons around while acting trivially on their states. This can be accomplished by an “injection weave” (Bonesteel et al., 2005) σ23 σ1−2 σ2−4 σ12 σ24 σ12 σ2−2 σ1−2 σ2−4 σ1−4 σ2−2 σ14 σ22 σ1−2 σ22 σ12 σ2−2 σ13 1 (54) 1 1 1 that moves one particle down two positions while acting as 1 0 0 0 1 0 0 0 1 (55) with a precision of ≈ 1.5 × 10−3 . Combing the injection weave (eq. 54) with the NOT operation (eq. 52) and an inverse injection, modified trivially to braid two anyons instead of one, we get a fully functional two-qubit CNOT-gate with a total accuracy of at least ≈ 1.8 × 10−3 (again Bonesteel et al., 2005): 1 injection NOT inverse injection 1 Concluding, the elementary braiding operators σ1 and σ2 and their inverses constitute a universal set of unitary operations. Since any quantum circuit can be simulated by anyon braiding, the complexity class associated with its computational power is at least BQP. 19 Simulating Anyons with Quantum Circuits Now I will show that the topological and non-topological models of quantum computation are in fact equally powerful by illuminating how to simulate anyons with quantum circuits. The argument is a modified version of the proof in Preskill (2004) (which itself is inspired by Freedman et al. (2002a)). The mapping of anyons to qubits is a little less obvious than the other way round, since the topological Hilbert space of n anyons of total charge 1 (the total topological charge has to be 1 since all anyons are created in pairs from vacuum) M bn−2 1 1 Vτ...τ Vτbτ1 ⊗ Vbb12τ ⊗ ... ⊗ Vbn−3 ≡ τ ⊗ Vbn−2 τ b1 ,b2 ,...,bn−2 b M = =τ n−2 Vτbτ1 ⊗ Vbb12τ ⊗ ... ⊗ Vbn−3 τ , (56) b1 ,b2 ,...,bn−3 does not have a natural decomposition into a tensor product of subsystems. This is apparent e.g. from the dimension of this space, 1 dim Vτ...τ = Fn ∝ φn for large n (57) √ which assigns an irrational dimension (the golden ration φ = (1+ 5)/2 ≈ 1.618) to each anyon. The task will be to explain a mapping of anyons to qubits, to show how braiding can be implemented in the qubit space, and to get an idea of how fusion outcomes can be measured in the qubit basis. Encoding The braiding operation acting on the topological Hilbert space in eq. 56 swaps around labels, but does not change the general structure of the vector space. Each summand in eq. 56 consists of n − 2 fusion spaces O b d Vaτ ∈ Vcτ ≡ Hd , (58) c,d so that for each summand from eq. 56 ⊗(n−2) . Vτbτ1 ⊗ Vbb12τ ⊗ ... ⊗ Vbτn−3 τ ∈ (Hd ) (59) The dimension of Hd is given by d= X d Ncτ . (60) c,d Thus n − 2 d-qudits can encode an n-anyon state. For Fibonacci anyons, Hd is spanned by {|τ τ ; 1i , |τ τ ; τ i , |1τ ; τ i}, so d = 3. (A change to an m-qubit system is easily done via C2dd(n−2)/2e = C2m , i.e. m = dd(n − 2)/2e, but the qudit representation is more convenient for now.) Braiding The effect of braiding in the anyon standard basis is completely determined by the B-matrix: a τ τ a = c b X b Baτ τ d d c · τ τ d b 20 |aτ ; ci |cτ ; bi = | {z } X b Baτ τ d c d ∈(Hd ⊗Hd ) |aτ ; di |dτ ; bi | {z } (61) ∈(Hd ⊗Hd ) Thus, the action of braiding on the qudit space is given by a unitary transformation B : Hd ⊗ Hd = C2d → C2d , (62) so that braiding can be implemented by a d2 × d2 unitary matrix. Fusion To calculate the outcome of fusing two neighboring anyons in the context of a multi-anyon state in the standard basis, we need to apply the F -matrix: a τ τ a = c b Faτ τ d c d b |aτ ; ci |cτ ; bi = | {z } ∈(Hd ⊗Hd ) X X τ · τ d b b Faτ τ d b = Faτ τ d c |ad; bi |τ τ ; di | {z } ∈(Hd ⊗Hd ) 1 c b |a1; bi |τ τ ; 1i + Faτ τ τ c |aτ ; bi |τ τ ; τ i . (63) Fusion can be simulated in the qudit system by applying the F -matrix (another unitary d2 ×d2 -matrix), and performing a projective measurement on the second qubit in the {|τ τ ; 1i , |τ τ ; τ i} basis and sampling the probabilities for getting 1 and τ . In the previous section, I had demonstrated that anyonic quantum computing is at least as powerful as “conventional” quantum computing. Having in place a scheme for braiding and fusing anyons in a qubit representation, we see that the topological model of computation is also not more powerful than the circuit model. This completes the proof of the conjecture that both models dwell in the same complexity class BQP. 3.3 Fault Tolerance Now that we know how topological quantum computers work in general, and that they can be just as powerful as non-topological ones, it is time to return to the original motivation why people (Alexei Kitaev in particular, 2003) came up with the idea. As already stated in the introduction, “conventional” quantum computers suffer from decoherence, i.e. interaction with the environment that decreases the lifetime of qubits. There exist a number of error correcting codes that can work against the effects of decoherence (Shor, 1996; Nielsen and Chuang, 2010), including Kitaev’s toric code which also hides logical information in topological states (Kitaev et al., 2002). Those are effective, but they require a certain threshold fidelity of the quantum gates. Estimates of the required precision range from 10−6 (Knill et al., 1996) to 1/300 (Zalka, 1996), which is hard to achieve experimentally. Topological quantum computers on the other hand offer an alternative solution to the problem. Quantum information is not encoded in local properties of particles (such as spins), instead the information is stored in topological, i.e. 21 non-local, features of a system of particles. This leads to an essential immunity of the quantum information from local perturbations in the physical system — the associated Hilbert space is protected (Preskill, 2004). Errors are not only introduced by decoherence but also by inexact gate operation. Can we be sure that the braiding generator σ is a rotation by π and not π + δ, i.e. do particles return to their original position or is the winding number changed slightly? Another beautiful aspect of a topological quantum computer is that we do not need to care about this. Since pairs are created from vacuum and re-annihilate to it, all world lines are closed loops — all operations are truly topological. In a physical system with an energy gap ∆ above the degenerate ground state, excitations from the ground state are suppressed by the Boltzmann factor e−∆/(kT ) where k is the Boltzmann constant and T denotes temperature. Any local interactions (e.g. electron-phonon-interaction) has trivial matrix-elements within the ground state subspace (Nayak et al., 2008). Thus, at sufficiently low temperatures, local interactions and noise do not disturb the Hilbert space of the anyons. Thus the only non-trivial unitary evolution a system of anyons can undergo is by braiding. (So, clearly, a topological quantum computer should keep good track of its particle positions to avoid unintentional braiding.) But what about unintentional anyons existing in the system? Anyon-antianyon pairs can be created from vacuum by thermal fluctuations or virtually by quantum fluctuations. They can travel through the system and re-annihilate. These fluctuations are again suppressed by the Boltzmann factor e−∆/(kT ) for thermal processes and by e−∆t/h for virtual processes (temperature T , quasiparticle energy gap ∆, time t for which the virtual excitation exists). Taking a look at the parameters, a topological quantum computer should operate at low temperatures and on a physical system with a big energy gap to minimize thermal processes. Virtual processes however only depend on the distance l = vt traveled by the pair before re-annihilation (with some characteristic velocity v) (Nayak et al., 2008). When a particle-antiparticle pair bb is created close to the world line of a computational anyon pair aa0 , either thermally or virtually, there are two main processes by which the pair can interact with it (Nayak et al., 2008): • Firstly, b (b) could wind around a and re-annihilate with b (b) (fig. 6a). Since the pair bb is created from vacuum, its total charge is 1. Let the charge of the computational pair be j. Then, the total charge of the 4 particles upon creation (time t1 ) is also j. One virtual particle winds around one anyon and the two virtual particles annihilate, which means that their total charge is still 1. Thus, the charge j remains with the computational pair, so that the process has a trivial effect. • The alternative process is that a virtual pair is created next to a computational pair of charge j and one virtual particle annihilates a computational one while the other virtual particle replaces the lost anyon of the computational pair (fig. 6b). However, reasoning similar to the previous case reveals that the state of the new computational pair will be indistinguishable from the original state. This shows that any braiding and fusing involving a single computational anyon has a trivial effect on the topological Hilbert space. Braiding processes 22 j j t2 1 1 time 1 1 t1 (a) j (b) j Figure 6: The only virtual processes involving just a single computational pair do not cause any errors, since the topological charge of this pair is not affected. Processes involving more anyons are exponentially suppressed in the anyon spacing. Figure reproduced from Nayak et al. (2008). that do induce errors must act on at least two anyons, so that those created by quantum fluctuations occur with a probability ∝ e−2∆L/v where L is the spacing of the physical anyons in the system. Thus, errors of this kind are suppressed exponentially in the distance of anyons. (There is another advantage in having large quasiparticle spacings. The degeneracy of the ground state space that is necessary for non-abelian statistics is often an approximate degeneracy. For example, the degeneracy can be lifted by Coulomb or spin-spin interactions of the particles involved. Since these interactions are local, the degeneracy is better approximated for larger spacings (Nayak et al., 2008).) A further error source is disorder in the physical system which creates localized quasiparticles that can interfere with braiding operations (Nayak et al., 2008). This puts high constraints on the quality of the used material. 4 Conclusion I have given a short description of topological quantum computing by first demonstrating the emergence of anyons with non-abelian exchange statistics in two-dimensional systems, and then showing that a particular anyon model, the Fibonacci model, is capable of carrying out universal quantum computations. Because of its topological nature, an anyonic quantum computer is inherently insusceptible to decoherence and noise, which are the Achilles’ heel of conventional quantum computers. I have chosen to present the Fibonacci model because of its simplicity and (suspected) implementation in a FQHE state, but it is by far not unique. ChernSimons theories at other levels k > 2 can be treated in the same way by defining fusion rules and constructing R-matrices and F -matrices. Some, but not all, of these alternative models have a representation of the braid group which is dense in SU (n), which enables them to simulate quantum circuits (Nayak et al., 2008). Topological quantum computation is a relatively new field, having been started by Kitaev’s paper in 1997, and there are no devices yet that can actually carry it out. Furthermore, there is not even direct evidence of the existence of non-abelian anyons. But there are theoretical papers proposing a wide range of physical systems that could host those quasiparticles, including 23 but not restricted to the already discussed FQHE states, super conductors (Sau et al., 2010), rotating Bose-Einstein condensates (Viefers, 2008), atom lattices (Tewari et al., 2007) and Josephson-junction arrays (Ioffe et al., 2002). Nayak et al. (2008) includes an overview of some of these systems. Several experiments are being built to realize those proposals. Finding topological phases of matter is a non-trivial task on the theoretical side as well. Different models lack a common footing and there is no mathematical classification of topological phases (Nayak et al., 2008). Topological phases of matter are intriguing subjects and their application for quantum computing is a promising idea. The next years should bring forth interesting experimental insights into its feasibility. Anyon, anyon, where do you roam? Braid for a while before you go home. Though you’re condemned just to slide on a table, A life in 2D also means that you’re able To be of a type neither Fermi nor Bose And to know left from right — that’s a kick, I suppose. You and your buddy were made in a pair Then wandered around, braiding here, braiding there. You’ll fuse back together when braiding is through Well bid you adieu as you vanish from view. 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