Aspects of Topological Quantum Computation

Author

Elias KokkasFollow

Date of Award

8-2023

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Physics

Major Professor

George Siopsis

Committee Members

Steven Johnston, Joon Sue Lee, Ahmedullah Aziz

Abstract

Topological quantum computation is a promising scheme leading towards fault-tolerant quantum computation. This can be achieved by harnessing systems in topological phases of matter and, specifically, the non-Abelian anyons’ exotic exchange (braiding) statistics. By doing so, quantum information gets encoded and processed in a robust way against small local perturbations. This is guaranteed by the existence of an energy gap and the topology depended nature of the “braiding” gates. However, to this day, experimental observation of non-Abelian anyons remains elusive and a major bottleneck. The most prominent candidate for realizing Ising (non-Abelian) anyons is the fractional quantum Hall effect at ν=5/2. Α wavefunction to describe this state, exhibiting Ising statistics, was proposed by Moore and Read via the Ising Minimal Model M(4, 3) = SU(2)1^⊗2/SU(2)₂. In this thesis, we build a “family” of such wavefunctions based on the coset SU(2)₁^⊗k/SU(2)_k CFT. Unlike, Minimal Models with k > 2, this gives gapped wavefunctions with generalized quasi-hole or quasi-particle statistics. Specifically, for k = 3, we obtain the Fibonacci braid group, which, together with the k ≥ 5 theories, offer universal fault-tolerant quantum computation. We find the four and six-anyon wavefunctions and their braiding matrices and discuss the generalization to an arbitrary number of anyons. The coset wavefunction construction offers new directions for experimental observation of non-Abelian anyons in fractional quantum Hall effect and fast-rotating cold atoms. An alternative approach in searching for non- Abelian anyons is by working with lattice models (e.g., toric code, honeycomb model). Recently, experimental evidence for realizing Abelian anyons with Z₂ topological order was discovered by employing Rydberg atoms in a Ruby lattice. However, Abelian braiding statistics cannot lead to non-trivial quantum computation schemes. Here, we provide numerical results that support the emergence of non-Abelian statistics in these systems by adding mixed-boundary punctures and ancillary qubits. Specifically, we realize the Ising braiding and fusion matrices which can be used to construct several quantum gates such as Hadamard, Pauli, and CNOT gates.

Recommended Citation

Kokkas, Elias, "Aspects of Topological Quantum Computation. " PhD diss., University of Tennessee, 2023.
https://trace.tennessee.edu/utk_graddiss/8644