Compiling single-qubit braiding gate for Fibonacci anyons topological quantum computation

Harnessing non-abelian statistics of anyons to perform quantum computational tasks is getting closer to reality. While the existence of universal anyons by braiding alone such as the Fibonacci anyon is theoretically a possibility, accessible anyons with current technology all belong to a class that is called weakly integral---anyons whose squared quantum dimensions are integers. We analyze the computational power of the first non-abelian anyon system with only integral quantu...

The content of this thesis can be broadly summarised into two categories: first, I constructed modified numerical algorithms based on tensor networks to simulate systems of anyons in low dimensions, and second, I used those methods to study the topological phases the anyons form when they braid around one another. In the first phase of my thesis, I extended the anyonic tensor network algorithms, by incorporating U(1) symmetry to give a modified ansatz, Anyon-U(1) tensor netwo...

Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show tha...

In this article we present a pedagogical introduction of the main ideas and recent advances in the area of topological quantum computation. We give an overview of the concept of anyons and their exotic statistics, present various models that exhibit topological behavior, and we establish their relation to quantum computation. Possible directions for the physical realization of topological systems and the detection of anyonic behavior are elaborated.

The topological model for quantum computation is an inherently fault-tolerant model built on anyons in topological phases of matter. A key role is played by the braid group, and in this survey we focus on a selection of ways that the mathematical study of braids is crucial for the theory. We provide some brief historical context as well, emphasizing ways that braiding appears in physical contexts. We also briefly discuss the 3-dimensional generalization of braiding: motions...

We consider a two-dimensional spin system that exhibits abelian anyonic excitations. Manipulations of these excitations enable the construction of a quantum computational model. While the one-qubit gates are performed dynamically the model offers the advantage of having a two-qubit gate that is of topological nature. The transport and braiding of anyons on the lattice can be performed adiabatically enjoying the robust characteristics of geometrical evolutions. The same contro...

We give a general proof for the existence and realizability of Clifford gates in the Ising topological quantum computer. We show that all quantum gates that can be implemented by braiding of Ising anyons are Clifford gates. We find that the braiding gates for two qubits exhaust the entire two-qubit Clifford group. Analyzing the structure of the Clifford group for n \geq 3 qubits we prove that the the image of the braid group is a non-trivial subgroup of the Clifford group so ...

It is an open question if there are leakage-free entangling Fibonacci braiding gates. We provide evidence to the conjecture for the negative in this paper. We also found a much simpler protocol to generate approximately leakage-free entangling Fibonacci braiding gates than existing algorithms in the literature.

We show that quasicrystals exhibit anyonic behavior that can be used for topological quantum computing. In particular, we study a correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of a class of quasicrystals, which includes the one dimensional Fibonacci chain and the two dimensional Penrose tiling. A possible encoding on tiling spaces of topological quantum information processing is also discussed.

Topological quantum computation with Fibonacci anyons relies on the possibility of efficiently generating unitary transformations upon pseudoparticles braiding. The crucial fact that such set of braids has a dense image in the unitary operations space is well known; in addition, the Solovay-Kitaev algorithm allows to approach a given unitary operation to any desired accuracy. In this paper, the latter task is fulfilled with an alternative method, in the SU(2) case, based on a...