Topological codes have emerged as one of the leading candidates for performing quantum computation. Twists are a form of lattice defects in topological codes which can be used to encode and process information. Twists have a rich structure which makes their study interesting. Also, twists have computational advantages over other forms of encoding. Although there is significant literature on the use of twists for quantum computation, many aspects are not fully explored. In this thesis, we undertake a study of twist-based topological quantum computation in two classes of topological codes namely, surface codes and color codes. The contributions of this thesis are threefold.

Our first contribution is developing the theory for twists in surface and color codes with emphasis on code construction Specifically, we present a systematic construction of qubit and qudit (in odd prime dimension) surface and color codes with twists. In qubit surface codes, the construction is generalized to arbitrary four-valent and two-face-colorable lattices and to qudits of odd prime dimension. In qubit color codes, we give systematic constructions for charge and color permuting twists, starting from a trivalent and three-colorable lattice. Further, we give the first constructions of twists in qudit color codes.

Secondly, making use of the constructions of topological codes with twists, we show how to implement encoded quantum gates. Specifically, we present protocols for implementing Clifford gates using twists. Non-Clifford gate implementation is done by using magic states. The techniques that are used for implementing Clifford gates are braiding twists, joint parity measurements and Pauli frame update. Braiding twists is an example of implementing gates by code deformation.

Thirdly, to track the evolution of logical operators during braiding, it is convenient to abstract out the lattice and express Pauli operators as strings. We develop string representation for the purpose of representing Pauli operators during encoded computations. The string representation of Pauli operators could be of independent interest.
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