Topological Quantum Computation

QUANTUM COMPUTING (CS682) PROJECT REPORT 1 Topological Quantum Computation Vatshal Srivastav, 14790 Mentor: Prof Dr. Rajat Mittal, Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur, India. Abstract To build a quantum computer we need to first physically realize the fundamental unit of information which is qubit. Here, a possibility is introduce that exploits the topological property of phase of matter, which is being used to construct the idea of fault tolerant quantum computing. These topological phases of matter have been experimentally observed in fractional quantum Hall effects as quasi-particle excitations. They exhibit distinct statistics different than bosons and fermions in terms of exchange of particles. I. INTRODUCTION As components of electronics are getting smaller and we are approaching to the limit where quantum effects become counter intuitive, one can have this question, either quantum computation is a problem or an opportunity? Manin (1980), Feynman (1982, 1986), Deutsch (1985) and Shor (1994) favored the latter. From a physicist s perspective, if we have a quantum computer at our disposal then we will be able to bolster various problems in physics that need thorough numerical evaluation. For e.g. simulating quantum many body Hamiltonians on a classical computer takes quite a lot time and precision is also questionable, while a quantum computer offers exponential time difference and with better precision. As we will be able to manipulate atoms at quantum level, that may give more insights about various unexplained quantum effects such as strongly correlated systems, turbulence etc. II. BUILDING A QUANTUM COMPUTER There are two from the many recent approaches proposed to realize qubit experimentally, one is through optical and other is the nuclear magnetic resonance. In optical approach, entangled photons are utilized for qubits and quantum operations are done through various optical apparatus. But the main problem with it is that photons do not interact with each other and that s why they are very difficult to entangle which in principle is also a very inefficient process. There were some methods proposed to conduct in cavity with high quality factor but due to the lack of its feasibility, construction for an optical quantum computer on full scale is still implausible. Another incentive proposed was nuclear magnetic resonance (NMR), where we work with the spin angular momentum of atom, through density matrix formalism. For any computing, be it classical or quantum, we should be able to address our information. In quantum computing, we do the addressing by specifying the spatial locations. In NMR, we re working with ensemble of spins and specifying spatial informations here is not possible because in sample itself it is rapidly moving. III. OBSTACLES Even if we somehow manage to construct a suitable quantum computer, there are still some inevitable threats that are needed to be taken care of in any computation may it classical or quantum. These are the errors due to various sources. Here we re going to discuss three of the main sources and the proposed solutions with there restrictions. First source of error is due to measurement. In classical computation, one make several copies of the result and the remove redundancies by checking against these copies. Whereas in quantum computation, limitation posed by nocloning theorem, i.e. multiple copies of any state is implausible, which stop us to make any error measurement in intermediate computation. In contrast, If we try to measure a quantum state to check an error in between, then we will collapse wave function, consequently destroying our valuable information. Furthermore, errors can be both continuous (a 0 + be iθ 1 ) and discrete(flip of 0 to 1 ). In spite of these difficulties, error correction is possible in quantum computation. Peter Shor first proposed a quantum analog of error correction code by storing information of one qubit over other nine qubits which are highly entangled. The Shor code itself is very involved, rather discussion on a small model of error correction code will be enough to promote the idea here. For instance, given that the spin flip rate is very low (important condition) and 0 000 and 1 111, then we can identify errors by checking the spins of all the qubits. Suppose that our initial state is in ψ = α 000 + β 111. if the first spin has flipped abruptly in the intermediate stage, then our spins are in the state ψ r = α 100 + β 011. We can detect this error with out measuring any qubit, i.e. without collapsing it to any state. We will check whether the first spin is the same as the other two. This evaluation is known as syndrome diagnosis, which is proceeded by four projectors: P 0 = 000 000 + 111 111 P 1 = 100 100 + 011 011 P 2 = 010 010 + 101 101 P 3 = 001 001 + 110 110, from where we can obtain, ψr P 0 ψ r = 0 ψr P 1 ψ r = 1 ψr P 2 ψ r = 0

QUANTUM COMPUTING (CS682) PROJECT REPORT 2 ψr P 3 ψ r = 0, clearly, P 1 corresponds to our error, thus we will correct the error by flipping the first spin, given that during diagnosing none of the other spin flips erroneously (i.e. basic spin flip rate is low). This quantum error correction code is known as bit-flip code. However, the error correction code all along had one stringent condition which is the low rate of spin-flips. Which means that more errors could then occur during error correction, and the whole procedure will fail. This is an extremely strict condition and it is currently unclear if local qubit-based quantum computation can ever be made fault-tolerant though error correction protocols. Other class of errors which are caused due to the interaction with environment are known as random errors. Our initial state which was pure superposition gets entangled with environment, thus with limits us to proceed with further computations. This phenomena is known as decoherence. For example, we have an initial state for quantum computer ψ = 0 and the environment stat is in the E 0. The composite state for the whole system will be; ψ = 0 E 0. as the combined state time evolve to a state ψ r, which is now entangled with thee environment, ψ r = α 0 E0 + β 1 E1 where E 1 is another state contribution from the environment, which need not be orthogonal to E 0, but here for simplicity I have taken E 0 E 1 = 0, now we can represent ψr in density matrix form, ρ = ψ r ψr and then trace out the environment (partial tracing), ( ) α 2 0 ρ o = T r env (ρ) = 0 β 2 where in the ρ r there is non-zero probability to observe state other than 0, which means, our state which was pure initially, got mixed with the superposition of states, resulting the loss of the information to the environment. There are certain error correcting codes proposed for dealing with issue, where the store the qubit information to highly entangled ancillary qubits, but quantum states are very fragile. And the extent of the interactions due to environment is good enough to exploit its fragility. The last source of error for the discussion is the known as unitary errors. These errors account for the precision in operating the quantum states. For example, I want to rotate my state to 90 degrees, instead I end up rotating it to 90.01 degrees. These errors arises due to the approximate modeling of the unitary for the gates experimentally. These errors may be fatal to our computation but they aid us to channel our vision to build a better quantum computer. The two primary goals need to be fulfilled for constructing a fault-tolerant quantum computer are; We need to isolate our initial state somehow to prevent the loss of information We need to manipulate our initial state precisely to minimize the measurement and unitary errors Now, the billion-dollar question arises, that how can we achieve to do both simultaneously? Answer to this question lies in the mathematical idea topology, which was experimentally observed in exotic phases of matter. Topology being study of shapes, plays far beyond role in physics by distinguishing prominent categories of particles (fermions and bosons) to one specific class. We will further see its role in our discussion, but before we need to set a dichotomy in categories of particles through quantum mechanics. IV. QUANTUM STATISTICS OF PARTICLES This section will be a paradigm shift for some part of the report. Here, I ll discuss the understanding of exchange statistics in quantum mechanics and using it I will introduce a new possibility of statistics that is a promising incentive for our purpose in quantum computation. Quantum mechanics have categorized particles into two parts in three spatial dimensions, prominently known as fermions (follows Fermi-Dirac statistics0 and bosons (follows Bose-Einstein statistics). This dichotomy has been established through the understanding of phases and statistics under exchange of particles. Fortunately, the notion of exchange is well defined as it is motion of particles with respect to each other. Mathematically, the easiest way to understand this concept is through path integrals. To briefly recollect, path integral is in quantum mechanics, is the probability amplitude to go from one space-time point to another, and is given by; A = e is allpaths where S = Ldt is defined as the action for the particular trajectory. In simple words, quantum mechanically, we need to interfere all possible paths between the initial and final spacetime points. For this discussion we need to focus the Ldt part on the trajectory for exchange of particles. Using this information. we will observe aberrant contrast in 3D and 2D situation. A. In 3D (spatial dimensions) Fortunately, the term exchange statistics refers to the induction of a phase in N-particle wave function after an exchange. In 3D (three spatial dimensions and one time), exchange two particles is topologically equivalent to winding one particle around another. Which (only in 3D) consequently give rise to the argument that topologically, two exchanges are equivalent to no exchange. This argument becomes more lucid if we refer to an example, where I have two identical particles that are kept apart from each other at fixed distance r, Fig.[1]. r 0, implies that for exchange, they are not going through each other, hence my origin is a singularity. There trajectories can be visualize on a sphere of diameter r. The configuration given in figure (a) let s say is our original wave function ψ(x 1, x 2 ).

QUANTUM COMPUTING (CS682) PROJECT REPORT 3 Fig. 1: Paths in three dimensional space with fixed radius In configuration (b), my particles have been exchanged and due to this my wave function will be; ψ(x 1, x 2 ) ηψ(x 2, x 1 ) where η is the phase that is been induced due to the exchange of the particles. Now similarly, in configuration (c), there is again an exchange is made, to bring back the particles at their original position, now the wave function will become; ηψ(x 2, x 1 ) η 2 ψ(x 1, x 2 ) now we can visualize the trajectory take by these particles for the whole process as the loop mapped on the sphere. Now recalling the crucial point made while introducing the path integral formalism, S = Ldt, where the integral is on the contour of the trajectory taken. Notice, in 3D, even if the origin is a singularity, it still remains a simple connected domain, which means any contour integral of an analytic function around it, is always going to be zero, hence Ldt = 0, for two exchanges (one complete loop). This establishes the fact that the wave function in configuration (a) and (c) are equivalent. Thus, η 2 = 1, η = ±1 where +1 corresponds to bosonic systems and 1 corresponds to fermionic. Hence in three spatial dimensions, there is only two possible statistics possible that is Fermi-Dirac statistics (fermions) and Bose-Einstein statistics (bosons). The Hilbert space in 3D, is only spanned by these two statistics. There statistics can be categorized in permutation group, where the total number of elements is N! for an N particle system. Every permutation corresponds to an exchange between two sites. The valeus of η can only be 1 and -1, which means that only 0 and π are the possible phases in 3D. While in 2D, there is striking difference than in 3D, which will be more clear after the next argument. B. In 2D (spatial dimensions) In 2D, the two particles are confined in a plane, where my origin is still a singularity. Refer Fig.[2], where my original (configuration (a)) wave function for two particle system is ψ(x 1, x 2 ), and for one exchange the wave function will have a phase factor η, like before. So, in configuration (b), my wave function will become ηψ(x 2, x 1 ). And similarly in configuration (c), where particles are again exchange in order to bring them back to their original positions, my wave function is written as η 2 ψ(x 1, x 2 ). The key point here is, given that my origin is a singularity, this converts 2D plane to a multi-connected domain, where by definition, a contour integral of any analytic function around origin is non zero ( Ldt 0). This is best visualized through tying knots. In 3D, we can easily tie any knots, even if any singularity is present, because we have the freedom to dodge the plane of singularity to complete the knot. While in 2D, we don t have that freedom as we are confined to a 2d plane itself. So every time I tie knot around my singularity (exchange), I create a new configuration of knots. This leads to the interpretation that the configuration (c) is not equivalent to my original wave function, which gives my phase factor, η 2 ±1 while exchange is an unitary transformation, it does not change the norm of our wave function, hence η = e iθ where θ ɛ (0, π). Hence, there exists a different statistics other than known ones (fermionic and bosonic), defined through a statistical phase angle θ. The particles, which obey these strange statistics are called anyons, because any possible value of θ is valid, except 0 and π. Where 0 corresponds to the bosonic case and π corresponds to the fermionic. Another point here to note is, unlike fermions and bosons, anyons do not belongs to the permutation group. Since they are different in terms of their topological property, anyons have been categorized in two types on the base of their statistics, one is Abelian-statistics which is discussed here to draw the contrast, meaning there ll be an +θ phase for an anti-clockwise exchange ( θ for clockwise) and two arbitrary exchange will commute (e i(θ1+θ2) = e i(θ2+θ1). The other type (in which

QUANTUM COMPUTING (CS682) PROJECT REPORT 4 Fig. 2: Paths in two dimensional space with fixed radius we are concerned) is more interesting is Non-Abelian anyons which follows braid statistics. V. BRAID STATISTICS There is a difference between the phase of the wave function obtained by exchanging the quantum numbers (e.g. spin or energy level) than the phase obtained through adiabatic transport of the particles. Under the former definition, the η 2, that is phase after two exchanges is always unity, whereas the latter definition has many possibilities. Mathematically, the former one classifies the particles under the permutation group P N, and the second one categorizes particles under the braid group B N. The permutation group P N is formed by performing all the possible permutations of N objects with group multiplication defined as successive permutations and group inverse as undoing permutation. There are N! possible permutations for given N particles (bosons or fermions). And we know from our discussion, that doing permuting two particles twice brings the system to its original state. One say) is evolved to a new configuration under the particles exchange to some time t = T. Each history of an exchange becomes braid. Fig. 4: Group Multiplication Since we have defined them as group, they have there generators, group operation and inverse. Given an N particles system, we define σ i as the generators of groupb N, given 0 < i N 1, where σ i describes the exchange of i th particle with (i + 1) th particle, in anti-clockwise sense, so that clockwise exchange is now are inverse (σ 1 i ), e.g. for B 4 generators are shown in Fig.[3]. The group operation is defined as the vertical stacking of one trajectory over another, that is one trajectory follows the other in time as shown in Fig.[4]. It is easy to visualize that (σ i σ 1 i ) = σ 0, defined as identity element of the group, Fig.[5]. Fig. 3: Three generators of braid group B 4 the other hand, given that we are exchanging the particles adiabatically, one can visualize the complete process as paths in space-time with time being the vertical axis and space being the horizontal. These trajectories in space-time axes are known as world lines. The particles can circle around each other in spatial plane and are seen there time-evolved version. Formally, the configuration of N particles at time t = 0 (let s Fig. 5: Product of σ 1 and σ 1 1 One can use these properties and derive (σ 1 ) n σ 0. Which basically means by permuting twice my state will not come back to its original configuration, consequently means that B N have infinite elements in it. These generator further follows

QUANTUM COMPUTING (CS682) PROJECT REPORT 5 two basic relations, given as; σ i σ j = σ j σ i, i j 2 σ j σ j+1 σ j = σ j+1 σ j σ j+1 the first one establishes the non-abelian statistics and the second one is known as Yang-Baxter relation. These relation can be visualize through given Fig.[6]. Fig. 6: (a) commutation relation (b) Yang-Baxter relation This representation of braid groups is first given by Artin, and is easy to see that these quasi-particles (anyons) have the capability to store memory of their exchange over the time period. This is an exquisite property, since till now every quantum state only has the memory of the previous unitary operation made. Anyons are capable of doing so through their topological trajectories and the knots formed during the exchange. Since the description of anyons are only limited to two dimensional space, taking it to any other geometry which is topologically non-equivalent will eventually break the braid statistics. That s why anyonic system are topologically protected. Topological phases of matter are quite interesting from the perspective of condensed matter physics. There are two different types of topological phases: one is called symmetry protected topological phases (SPT) and other is topological phases (TP) (displaying (intrinsic) topological order ). Some of the quick examples are topological insulators and topological superconductors (SPT), quantum Hall states and various spin liquids, quantum topological liquids (TP). Anyonic systems are observed as quasi-particle excitations (bulk gaps) in fractional quantum Hall states which are example topological phases. Topological phases are stable under arbitrary perturbations. Which means, we cannot connect this phase to a trivial one without going through a phase transition (quantum phase transition). The intuition here is that these phases consist long range of entanglement in thee ground state. In simple words, in our system, at far-away sites can be strongly entangled with one another), and we cannot undo such correlation by mere local perturbations. This long range entanglement can cause many interesting physical consequences. In particular such topological phases have very low energy excitations. Unlike many phases which form lattice in this low energy region, these phases form an incompressible liquid, where fermi-levels are now become Landau levels. We call any phase topological on the base of two concepts: The mathematical formalism of these phases is topology On basic level, the system behaves differently on a different topology Having discussed briefly about topological phases of matter, we need to deal with many anyons system. We will find later on in the discussion, that the many anyons system can be modeled as our qubits from the physical events of fusion. VI. FUSION OF ANYONS To solve many body system, our first attempt is to approximate is as a single particle. In practical situations, we have to deal with many particle system. And it is natural for anyonic case as well. We wish to bring two anyons close together while all others are far away. Then the two anyon system can be approximated as a single particle whose quantum numbers are obtained by combining individual quantum numbers, including the topological quantum numbers, of the two-particle system. These fusion rules are the physical results of Toric code model. Here, in this report, I will use only the result, as the derivation do not concern us. For Abelian anyons, of statistics π/m particles, the fusion rule is (I am going to use it as result): n 2 π m k2 π m = (n + k)2 π m where we will use a b to denote a fused with b. For e.g. for an anyon of π/m statistic and other with π/m is fused together, the result has statistic θ = 0. It is convenient to call it a trivial particle or vacuum. We will often denote trivial particle by 1. However, for non-abelian anyons, fusion rule is complicated as compared to Abelian anyons. As with ordinary quantum numbers, there might be different possibilities of combining quantum numbers (e.g. two spin half particles could combine to form either spin-0 or spin-1). We call these different possibilities as different fusion channels, which are given as φ a φ b = Nabφ c c c which represents that when an anyon of species a fuses with one of the species b, the result can be an anyon of species c if Nab c 0. For Abelian anyons there exist only c for which Nab c 0, while for non-abelian anyons, there is at least one a, b such that there multiple fusion channels c with Nab c 0. Now, if a system of non-abelian anyons are fusion in nontrivial fashion, these fusion can be group together, then a basis

QUANTUM COMPUTING (CS682) PROJECT REPORT 6 for the two-dimensional space is given by the state in which a particular fusion is happening. These states will further span my Hilbert space. We can also have different order of fusion, that will only change the basis states, not my Hilbert space. Thus, this change of basis can be parameterized by a unitary matrix, which is called F -matrix, which is defined as (for two particle): Ψ a = F ab Ψ b If [ we include indices to specify the 4 particle system then F ijk] l, where i, j and k are the species that are ultimately fused to species of l. For e.g if i and j fuse to a, which further fuses with k to give topological charge l, the different allowed a define the basis. Another possibility can be that if j and k fuse to b and then fuses with i to give l, this define another basis. Hence, F -matrix is the unitary transformation connecting these different basis. The non-abelian anyon ground state is heavily degenerate. These degenerate state space transform into each other under braiding. However, fusion channel for two anyons cannot be changed by braiding since their total topological charge can be measured along a loop containing both. They must braid with a third particle in order to change the fusion channel. As a result, when two anyons fuse in a particular channel, (or superposition of different channels) are braided with each other, then my wave-function has a multiplication by phase. This phase resulting from a counter-clockwise exchange of anyons a and b which fuse to an anyon of type c is called Rab c. In order to define the braiding statistics of a system of anyon, the requirements are to specify, Particle species Fusion rules, N c ab F -matrices, and R-matrices For further reading, these mathematical ideas are very well structured in conformal field theory (CFT). As quasi-particles obeying non-abelian statistics were first considered under the context of conformal field theory by Moore and Seiber, 1988, 1989 and in the context of Chern-Simons theory by Witten, 1989. These properties of non-abelian anyons make them convincing choice for use of quantum computing. Fig. 7: Bratteli diagram for Fibonacci Anyons Fibonacci number Fib(n). Therefore the quantum dimension of the τ particle is the golden mean, d τ = (1 + 5)/2. It may seem that a dimension parameter is not an integer. Hence, this is an aberrant property of non-abelian anyons. We know that non-abelian anyons have heavily degenerate ground state. the dimension of these degenerate states are proportional to the number of quasi-particles (which is obvious). It turns out that for M a quasi-particles of type a, the dimension of the ground state for large M a is roughly d M 2 a, where d a is the number which depends on the species of particle a. This looks analogous to the definition of tensor product in Hilbert space, and for this reason this is know as quantum dimension of quasi-particles of type a. For Abelian anyons we have one-dimensional ground state, irrespective of the number of quasi-particles in it, for them d a = 1. It has been proven that Fibonacci anyons model is the simplest model capable to perform universal quantum computation. It is useful to study this model in detail as many of its results can be generalize for other class of non-abelian anyons. A. Structure of Hilbert Space VII. FIBONACCI ANYONS- SIMPLEST MODEL FOR NON-ABELIAN ANYONS Fibonacci anyons are the simplest example of non-abelian statistics as it is also apt for universal quantum computation. In this model, there are only two quasi-particles, the identity 1 which corresponds to vacuum, and another is the single non-trivial anyon usually called τ. There is a single nontrivial fusion rule in it, τ τ = 1 + τ which results in Bratteli diagram given in Fig.[7]. The simplicity of this model lies in the result that any cluster of quasiparticles can fuse only to 1 or τ. The reason it is known as Fibonacci anyons is because the dimension of the Hilbert space with n anyons is given by the Fig. 8: The three possible states of three Fibonacci anyons. The quantum number of an individual is τ. In the parenthesis and ellipse notation (middle), each particle is shown as a black dot, and each pair of parenthesis or ellipse around a group of particles is labeled at the lower right with the total quantum number associated with the fusion of that group. Analogously in the fusion tree notation (right) we group particles as described by the branching of the tree, and each line is labeled with the quantum number corresponding to the fusion of all the particles in the branches above it.

QUANTUM COMPUTING (CS682) PROJECT REPORT 7 A given state in the Hilbert space is describe through a fusion tree Fig.[8]. Using fusion rule defined for Fibonacci anyons, one can read the fusion tree for example, the two τ particles can fuse to form 1 or τ. Hence, they may fuse in one of the orthogonal degenerate states. These two states are denoted as (, ) 1 and (, )τ, where I have use to represent each particle. When a third particle is added then, from the fusion rule the state (, ) 1, it must fuse to form τ, denoting this state by ((, ) 1 ), ) τ 0. But if the third particle is fused with the other state i.e. (, ) τ, then the outcome can have two possibility which is either τ or 1. The state with outcome τ i.e. ((, ) τ ), ) τ is denoted as 1 and ((, )τ ), ) 1 as N. (the notations will be discussed further). Thus we have a three dimensional Hilbert space for three particles. One can change the order of fusion, although Hilbert space will be invariant under this change, the basis which is spanning it will now be changed. This basis change as described in previous section VI. is given by F -matrix, and for Fibonacci anyons it is easy to see that, (, (, ) τ ) 1 = ((, )τ ), ) 1 In either of the fusion order, there is only single state that has total topological charge 1. However, the remaining two states of three particle transform non-trivially under the exchange of fusion order. We can write a change of basis using F -matrix as [ (, (, ) i ) k = F τττ k ]ij ((, ) j), ) k j where i, j and k take the values of the field 1 or τ. From above two results we can derive, F τττ 1 = I, ( ) ( ) Fτ τττ F11 F = 1τ φ 1 φ 1 = F τ1 F ττ φ 1 φ 1 Where φ is the golden mean. these F -matrices can be calculated through some of these similar identities and the condition of being unitary. these conditions completely fix the Fibonacci F -matrix up to a gauge freedom in the definition of the basis states. The complete calculation of F -matrices involves a consistency condition known as pentagon equation (Fuchs, 1992; Gomez et al. 1996; Moore and Seiberg, 1988, 1989). this condition states that one should make changes of basis of four particles in several possible ways and get the same result in the end. Here I will utilize the result instead of deriving it from the scratch. B. Braiding Fibonacci Anyons As introduced before, for non-abelian anyons, adiabatically braiding around each other results in a unitary operation on the degenerate ground state. In section V. we have discussed the Artin s representation for the braid statistics. Here we want to determine the matrix representation of braid statistics i.e. which unitary operation results from the which braid. We have to first determined the phase accumulation for the full exchange for the two particles. Using the results of topological spins and parafermions theory, we know that anticlockwise exchange of two anyons is 4π/5 if they fuse to 1 or 2π/5 if they fuse to τ. We thus determine the ˆR-matrix, ˆR (, ) 1 = e 4πi/5 (, ) 1 ˆR (, ) τ = e 2πi/5 (, ) τ i.e. R 1 ττ = e 4πi/5 and R τ ττ = e 2πi/5. Using R-matrices Fig. 9: The elementary braid operations σ 1 and σ 2 on three particles along with F -matrices we calculate the matrix representation of every braid in that group. For example taken, where there are three particles in my system, which belongs to B 3, and σ 1, σ 2 are the generators for this particular group. σ 1 s representation can be easily determined using above equations, ρ(σ 1 ) = e 4πi/5 0 0 0 e 2πi/5 0 0 0 e 2πi/5 0 1 N Calculating the effect of σ 2 is bit complicated. First we need to make a basis change in order to know how the two rightmost anyons fused, then we make the rotation using ˆR, and then undo the basis change. Hence, we can write symbolically, ρ(σ 2 ) = F 1 ˆRF where ˆR rotates the two right most particle. Let us see what happens to 0 under these transformation. We can write 0 = F 11 (, (, ) 1 ) τ + F τ1 (, (, ) τ ) τ, then rotating the two rightmost particles will result, ˆRF 0 = e 4πi/5 F 11 (, (, ) 1 ) τ e 2πi/5 F τ1 (, (, ) τ ) τ Then we will apply F 1, which will give me my ρ(σ 2 ), e πi/5 /φ ie πi/10 / φ 0 ρ(σ 2 ) = ie πi/10 / φ 1/φ 0 0 0 e 2πi/5 Since, σ 1 and σ 2 are the generators of respective braid group, and can create any arbitrary braids on three strands, we can use these matrices to determine the unitary operation of any braid structure on three strands.

QUANTUM COMPUTING (CS682) PROJECT REPORT 8 C. Computing with Fibonacci Anyons Having set the formalism for the matrix representation of braid groups, now we need to know how to use them in computation. First we need to realize our qubits in terms of these particles. One may suggest two particle system for the qubit such as if both the particle fuses to 1 then it will be 0 and if the fuses to τ then it will be labeled as 1. While this is reasonably natural looking qubits, it turns out that they are not convenient for computations. As we have discussed before, braiding two particles around each other will not change their topological charge. To solve it, we need third particle in the system, and it turns out it can be represented as qubit. Thus, we represent the two states of the qubits the 0 and 1 states as shown in Fig.8. The N state is the additional state which is non-computational state, which means in the end of the computation there is no amplitude in this stage. If there is non-zero amplitude in this state, then this is an indicator of leakage error. One can observe the structure of ρ(σ 1 ) and ρ(σ 2 ) is block diagonal, which means braiding matrices will never the non-computational state N with the qubits. Therefore, braiding the three particles gives us a way to do single qubit operations with no leakage. Freedman et al. 2002 proved that there exists a braid that corresponds to a unitary operation arbitrarily close to any desired operation. More the accuracy would be, longer the braid will become. Now the problem boils down to find corresponding braid for the desired unitary operations. This problem is apparently complicated. One can simply use brute force to find all the possible braids (on three strands) up to a certain length. This approach works for short length braids, but fails for long cases because problem of braid finding grows exponentially in the length of the braid. Solovay and Kitaev proposed an algorithm which allowed to put short braids together to form a long braid close to desired unitary operation. Till now we have dealt with single qubit system, let us now imagine we have multiple qubits, each encoded with three particles. Now finding a braid close to the desired unitary is much more difficult task now. The system now has six quasi-particles and my Hilbert space is now 13 dimensional. Searching for a result is extremely hard even on a powerful classical computer. Therefore, this problem is tackled through divide and conquer strategy, by building up two qubit gates using simple braid. For e.g. in the two-qubit system, the first qubit is the control (let s say) and the second one is the target. Now if the pair in control qubit have the quantum number 1 ( 0 ), then any amount of braid will only given an Abelian phase to it (since 1 is vacuum thus, it is topologically equivalent to move around nothing). However, if the quantum number of control pair is τ ( 1 ), then we can visualize this pair as a single particle, and we can cause some non-trivial rotation around this pair. Hence, we have created a controlled rotation gate, where the rotation is equivalent to exchange two particles in the target qubit only if the control state is in 1. The resulting two-qubit controlled gate is an illustration of the property of universal quantum computation of Fibonacci anyons. Several versions of controlled gates, NOT gates have been already designed using braids. There are many other non-abelian models, which are not related to Fibonacci anyons, but still possess the same foundations. In all of those cases, Hilbert space can be understood through fusion rules, F -matrices and ˆR-matrices. one can always encode qubit information through manipulating the quantum numbers of some group of particles. REFERENCES [1] Colin P. Williams, Quantum computing and Quantum Communications, Lecture Notes in Computer Science [2] Jonathan A. Jones, Quantum Computing and Nuclear Magnetic Resonance [3] Sumathi Rao, Introduction to abelians and non-abelians anyons [4] Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computations