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Efficient synthesis of quantum circuits by number-theoretic methods
by
Peter Selinger
Dalhousie University
Recall that a set of quantum gates is called universal if it generates a dense subgroup of the group of all unitary operators, or in other words, if every unitary operator can be approximated up to arbitrary epsilon by a quantum circuit built from the gate set. The problem of finding such approximations is sometimes called the approximate synthesis problem. Until recently, the state-of-the-art solution to this problem was the Solovay-Kitaev algorithm, which yields circuits of size O(log^c(1/epsilon)), where c > 3. I will talk about a new efficient class of synthesis algorithms that were developed in the last two years and that are based on algebraic number theory. These algorithms achieve circuit sizes of O(log(1/epsilon)), which is also a lower bound. For some gate sets, such as the Clifford+T set or the Pauli+V set, the algorithms are strictly optimal (in absolute terms, i.e., not just up to big-O notation). For other gate sets, such as the Fibonacci anyon gate set, the algorithms are only asymptotically optimal. To round off the presentation, I will also give a presentation of the group of single-qubit Fibonacci circuits in terms of generators and relations, also based on algebraic number theory. Based on joint work with Neil J. Ross and Travis Russell.
Date received: December 30, 2014
Copyright © 2014 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbjz-78.