2 research outputs found
A polynomial time algorithm for computing the HNF of a module over the integers of a number field
We present a variation of the modular algorithm for computing the Hermite
Normal Form of an \OK-module presented by Cohen, where \OK is the ring of
integers of a number field K. The modular strategy was conjectured to run in
polynomial time by Cohen, but so far, no such proof was available in the
literature. In this paper, we provide a new method to prevent the coefficient
explosion and we rigorously assess its complexity with respect to the size of
the input and the invariants of the field K.Comment: 11 page
A Framework for Approximating Qubit Unitaries
We present an algorithm for efficiently approximating of qubit unitaries over
gate sets derived from totally definite quaternion algebras. It achieves
-approximations using circuits of length ,
which is asymptotically optimal. The algorithm achieves the same quality of
approximation as previously-known algorithms for Clifford+T [arXiv:1212.6253],
V-basis [arXiv:1303.1411] and Clifford+ [arXiv:1409.3552], running on
average in time polynomial in (conditional on a
number-theoretic conjecture). Ours is the first such algorithm that works for a
wide range of gate sets and provides insight into what should constitute a
"good" gate set for a fault-tolerant quantum computer.Comment: 60 pages, 16 figure