Towards a universal set of topologically protected gates for quantum computation with Pfaffian qubits

TL;DR

This paper advances topological quantum computation by explicitly constructing universal quantum gates, including single- and multi-qubit gates, using braiding transformations in Pfaffian qubits, aiming for robust, topologically protected quantum operations.

Contribution

It extends the Pfaffian qubit scheme by explicitly realizing universal gates like Hadamard, phase, and CNOT through braid group representations, and discusses constructing a topologically protected Toffoli gate.

Findings

Explicit braid matrices for 4, 6, and 8 quasiholes

Implementation of single- and two-qubit gates via braiding

Proposal for a topologically protected Toffoli gate

Abstract

We review the topological quantum computation scheme of Das Sarma et al. from the perspective of the conformal field theory for the two-dimensional critical Ising model. This scheme originally used the monodromy properties of the non-Abelian excitations in the Pfaffian quantum Hall state to construct elementary qubits and execute logical NOT on them. We extend the scheme of Das Sarma et al. by exploiting the explicit braiding transformations for the Pfaffian wave functions containing 4 and 6 quasiholes to implement, for the first time in this context, the single-qubit Hadamard and phase gates and the two-qubit Controlled-NOT gate over Pfaffian qubits in a topologically protected way. In more detail, we explicitly construct the unitary representations of the braid groups B_4, B_6 and B_8 and use the elementary braid matrices to implement one-, two- and three-qubit gates. We also propose to construct a topologically protected Toffoli gate, in terms of a braid-group based Controlled-Controlled-Z gate precursor. Finally we discuss some difficulties arising in the embedding of the Clifford gates and address several important questions about topological quantum computation in general.