Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets

We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov and Mosca. Their algorithm takes as input an exactly synthesizable single-qubit unitary--one which can be expressed without error as a product of Clifford and T gates--and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer $n$ we consider the "Clifford-cyclotomic" gate set consisting of the Clifford group plus a z-rotation by $\frac{\pi}{n}$. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of $\frac{\pi}{n}$ z-rotations. For the Clifford+T case $n=4$ the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring $\mathbb{Z}[e^{i\frac{\pi}{n}},1/2]$. We prove that this characterization holds for a handful of other small values of $n$ but the fraction of positive even integers for which it fails to hold is 100%.Comment: v2: published versio

QFAST: Conflating Search and Numerical Optimization for Scalable Quantum Circuit Synthesis

We present a quantum synthesis algorithm designed to produce short circuits and to scale well in practice. The main contribution is a novel representation of circuits able to encode placement and topology using generic "gates", which allows the QFAST algorithm to replace expensive searches over circuit structures with few steps of numerical optimization. When compared against optimal depth, search based state-of-the-art techniques, QFAST produces comparable results: 1.19x longer circuits up to four qubits, with an increase in compilation speed of 3.6x. In addition, QFAST scales up to seven qubits. When compared with the state-of-the-art "rule" based decomposition techniques in Qiskit, QFAST produces circuits shorter by up to two orders of magnitude (331x), albeit 5.6x slower. We also demonstrate the composability with other techniques and the tunability of our formulation in terms of circuit depth and running time

Density and unitarity of the Burau representation from a non-semisimple TQFT

We study the density of the Burau representation from the perspective of a non-semisimple TQFT at a fourth root of unity. This gives a TQFT construction of Squier's Hermitian form on the Burau representation with possibly mixed signature. We prove that the image of the braid group in the space of possibly indefinite unitary representations is dense. We also argue for the potential applications of non-semisimple TQFTs toward topological quantum computation.Comment: 27 pages, 4 fgure