Anyons and Topological Quantum Computation Presented by: Jonah Sobel (Dated: Thursday, 3/18/21) I. INTRODUCTION When restricted to two spatial dimensions, particles behave with fewer restrictions than normal fermions and bosons in three spatial dimensions. Emergent phenomena known as quasiparticles can occur in strange circumstances, in which a microscopically complex sys- tem behaves as if it contained weakly interacting particles in a vacuum. One such example is known as an anyon, a quasiparticle that emerges in two spatial dimensions. Anyons are of particular interest as they have been theoretically shown to solve a major problem in the implementation of quantum computing. A quantum computer created from anyons is known as a topological quantum computer, and while researchers have yet to provide experimental (and undisputed) evidence of non-abelian anyons, recent promising findings offer hope that topological quantum computation may be realized soon enough. Anyons were first theorized in 1977 by Leinaas and Myrheim [1] as a two dimensional ex- ception to the boson-fermion particle divide. Frank Wilczek was the first to explore the properties of anyons in depth and published two papers on anyons in 1982 [2]. Then, in 1985 it was predicted that the particles that exist in fractional quantum hall effect states would be anyons, and later confirmed by Wilczek. Moore, Read, and Wen showed that non-abelian anyons could be realized in fractional quantum hall effect states [3]. Then, Alexei Kitaev showed that these non-abelian anyons could be used to build a topological quantum com- puter. More recently, two independent teams of scientists, one in the US and one in France, announced experimental evidence of Abelian anyons in 2020 [4, 5]. II. WHAT IS AN ANYON? Bosons and fermions are distinguished from one another by their spin numbers. Bosons possess integer spins, whereas fermions possess half-integer spins. This leads to important symmetry properties: under interchange of identical particles, the wave function of bosons 2 is symmetric: j 1 2i = j 2 1i whereas that of fermions is anti-symmetric: j 1 2i = −| 2 1i In two dimensions, anyons obey fractional statistics. As described by Nayak and Simon [6], when two identical particles each make a counterclockwise half-revolution about each other, the particles return to their original wave function with an adopted phase of eiθ: iθ j 1 2i = e j 2 1i Similarly, when they make a clockwise half-revolution, their wavefunction adopts a phase of e−iθ. Directions of clockwise and counterclockwise are only meaningful and clearly defined in two spatial dimensions. This is why this scheme would not make sense in three or more dimensions. A unique consequence of these statistics is that when anyons are exchanged twice in the same direction, their wavefunction picks up a phase of e2iθ. This is also commonly written with θ = 2πs, where s is the spin quantum number: 2s j 1 2i = (−1) j 2 1i In 1988, J¨urgFr¨ohlich demonstrated that, according to the spin-statistics theorem for particle exchange, it was possible for the particle exchange to be non-abelian. This is achieved when the system exhibits degeneracy (so that multiple distinct states of the system have the same configuration of particles). According to Trebst, exchanging the particles contributes not just a phase change, but sends the system into a different state with the same particle configuration. This means that particle exchange corresponds to a linear transformation on the subspace of degenerate states [7]. Without degeneracy, the subspace is one-dimensional, and so all linear transformations commute (because they are essentially just multiplications by a phase). With degeneracy, this subspace has higher dimension and so these linear transformations will not always commute. III. TOPOLOGICAL QUANTUM COMPUTING One of the biggest problems that quantum computing faces is that of decoherence and stability. A quantum system is said to be coherent as long as there is a definite phase 3 relation between states (this is necessary for quantum computing). Every system is loosely coupled with the state of its environment, and so decoherence can be understood to be the loss of information from a quantum system to its surrounding environment. If a system were perfectly isolated from any external stimuli, it would, in theory, maintain coherence indefi- nitely. Decoherence cannot be undone, as it is effectively non-unitary. Therefore, it would be impossible to measure or manipulate a state that is isolated as such, and so decoherence seems to be a necessary component of this scheme. However, decoherence poses a major challenge to achieving quantum supremacy as many important quantum algorithms require much longer coherence times than are currently achievable. This is why, today, there is a large effort focused on minimizing (or at least controlling) decoherence. A traditional quantum computer relies on trapped quantum particles { these particles are easily disturbed, and small cumulative perturbations quickly cause these states to deco- here. A topological quantum computer instead uses quantum braids, which are much more stable and resistant to decoherence. The same perturbations do not affect the topological properties of the quantum braids. Like fermions, anyons cannot occupy the same state, so their world lines cannot intersect or merge. This allows their paths in 3 (2 spatial + 1 time) dimensional spacetime to form stable braids. When braided, the quantum state of the anyonic system is transformed only according to the topological class of the trajectories of the anyons. So, small errors introduced in the trajectories will not affect the information stored in the state of the system. A common example of this process uses Fibonacci anyons, as detailed by Trebst [7]. Fibonacci anyons are the simplest non-abelian anyons. In a system of anyons, the ground state of the system (or vacuum) is denoted by 1. This trivial particle is its own anti-particle. In order to have a non-trivial system of anyons, we need at least one more particle type. The Fibonacci anyonic system is the simplest such system, and so it has one additional non- trivial anyon of type τ. These anyons are called Fibonacci anyons, and they are also their own anti-particles. Anyons are combined in a process similar to the addition of quantum spins to form a new total spin { this is called the fusion of anyons. We may also think of fusion as a measurement, because the resulting anyons may be of several different types, 4 each with its own probability. According to Trebst, given an anyon X, if the fusion of X with any other anyon produces an anyon of the same type, then X is said to be Abelian. The trivial anyon of type 1 is Abelian, because 1 ⊗ x = x for any type x. For our Fibonacci system of anyons, the fusion rules are as follows: 1. 1 ⊗ 1 = 1 2. 1 ⊗ τ = τ ⊗ 1 = τ 3. τ ⊗ τ = 1 ⊕ τ where ⊗ is the same tensor product as for 1=2-spin particles, and ⊕ is the direct sum and represents the two possible fusion channels [6]. When extended to three anyons: τ ⊗ τ ⊗ τ = τ ⊗ (1 ⊕ τ) = τ ⊗ 1 ⊕ τ ⊗ τ = τ ⊕ 1 ⊕ τ = 1 ⊕ 2τ The fusion of three anyons yields a final state with total charge τ in 2 ways or 1 in one way. So, we use 3 states to define our basis. According to Trebst, these fusion paths represent an orthonormal basis of the degenerate ground-state manifold. In the Fibonacci 1 system, when two τ anyons fuse, the probability of seeing 1 is p0 = '2 and the probability p 1 1+ 5 of seeing τ is p1 = ' , where ' = 2 (the Golden Ratio). When two anyons are exchanged in the plane, there are two possible exchanges which are not topologically equivalent. The convention is to move the right-hand anyon in front of and across the left-hand anyon; this is called right-handed braiding. The inverse of this is called left-handed braiding. This is depicted in the following image: FIG. 1. Here, each line represents the world line of an anyon. Time flows upwards. The image on the left depicts right-handed braiding and the image on the right depicts left-handed braiding (unfortunately formatted diagram). [7] 5 Now, Bonesteel discusses how to encode these anyons as superpositions of 0 or 1 [8]. First, we limit our basis to a 2-D Hilbert space. So, we only consider 2 states with total charge τ. Here, we group the two leftmost anyons into the control group and call the rightmost anyon the noncomputational anyon. We let j0i be when the control group has a total charge equal to 1, and j1i when the control group has a total charge equal to τ. Basis states are j(•; •)0i and j(•; •)1i, where (•; •)a describes two anyons with total q-spin a. Q-spin is a q-deformed spin quantum number [8]. When a third anyon is added, the Hilbert space is 3 dimensional and is spanned by j((•; •)0; •)1i, j((•; •)1; •)1i, and j((•; •)0; •)0i. For quantum computing, take triplets of anyons with total q-spin 1, and let the logical qubit states be j0Li = j((•; •)0; •)1i and j1Li = j((•; •)1; •)1i. Therefore, the state j((•; •)0; •)0i with total q-spin 0 is noncomputational. These qubits are measured by determining the q-spin of the two leftmost anyons. However, error may be introduced through transitions to noncomputational states, called leakage errors. To swap the two computational anyons, apply the gate σ1: 0 1 e−i4π=5 0 0 B C B −i2π=5 C σ1 = B 0 −e 0 C (1) @ A 0 0 −e−i2π=5 To swap the rightmost computational anyon with the noncomputational anyon, apply the p 5−1 gate σ2, where τ = 2 is the inverse golden ratio: 0 p 1 −τe−iπ=5 −i τe−iπ=10 0 B p C B −iπ=10 C σ2 = B−i τe −τ 0 C (2) @ A 0 0 −e−i2π=5 The upper 2 × 2 section acts on the computational anyons and are used to perform qubit rotations whereas the lower right element acts as a phase on the noncomputational qubit.