245 research outputs found
Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates
In this paper, we show the equivalence of the set of unitaries computable by
the circuits over the Clifford and T library and the set of unitaries over the
ring , in the single-qubit case. We report an
efficient synthesis algorithm, with an exact optimality guarantee on the number
of Hadamard and T gates used. We conjecture that the equivalence of the sets of
unitaries implementable by circuits over the Clifford and T library and
unitaries over the ring holds in the
-qubit case.Comment: 23 pages, 3 figures, added the proof of T-optimality of the circuits
synthesized by Algorithm
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 we consider the "Clifford-cyclotomic" gate set consisting of
the Clifford group plus a z-rotation by . We present an
efficient exact synthesis algorithm which outputs a decomposition using the
minimum number of z-rotations. For the Clifford+T case
the group of exactly synthesizable unitaries was shown to be equal to the group
of unitaries with entries over the ring .
We prove that this characterization holds for a handful of other small values
of but the fraction of positive even integers for which it fails to hold is
100%.Comment: v2: published versio
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