Asymptotically optimal topological quantum compiling.

We address the problem of compiling quantum operations into braid representations for non-Abelian quasiparticles described by the Fibonacci anyon model. We classify the single-qubit unitaries that can be represented exactly by Fibonacci anyon braids and use the classification to develop a probabilistically polynomial algorithm that approximates any given single-qubit unitary to a desired precision by an asymptotically depth-optimal braid pattern. We extend our algorithm in two directions: to produce braids that allow only single-strand movement, called weaves, and to produce depth-optimal approximations of two-qubit gates.

Asymptotically optimal approximation of single qubit unitaries by Clifford and T circuits using a constant number of ancillary qubits.

Decomposing unitaries into a sequence of elementary operations is at the core of quantum computing. Information theoretic arguments show that approximating a random unitary with precision ε requires Ω(log(1/ε)) gates. Prior to our work, the state of the art in approximating a single qubit unitary included the Solovay-Kitaev algorithm that requires O(log(3+δ)(1/ε)) gates and does not use ancillae and the phase kickback approach that requires O(log(2)(1/ε)loglog(1/ε)) gates but uses O(log(2)(1/ε)) ancillae.