No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

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

### Abstract

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 [Quantum Inf. Comput. 13(7,8), 607–630 (2013)]. 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
. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of
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
. 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%.

© 2015 AIP Publishing LLC

Received 13 February 2015
Accepted 09 July 2015
Published online 05 August 2015

Acknowledgments:
We thank Jean-Francois Biasse and Michele Mosca for helpful discussions. D.G. and D.M. were supported in part by NSERC. D.G. was supported in part by ARO. IQC is supported in part by the Government of Canada and the province of Ontario.

Article outline:

I. INTRODUCTION
II. CLIFFORD-CYCLOTOMIC GATE SETS
A. The single-qubit Clifford group
B. Clifford-cyclotomic gate sets
C. Optimal decomposition
D. Bloch sphere representation
III. CANONICAL FORM
IV. OPTIMAL EXACT SYNTHESIS
V. FOR WHICH *n* IS EQUAL TO THE GROUP OF 2 × 2 UNITARY MATRICES WITH ENTRIES IN THE RING R_{
n
}?
VI. PROOF OF THEOREM 4.1
VII. PROOF OF THEOREM 5.1
A. Part 1
B. Part 2
C. Part 3
VIII. PROOF OF THEOREM 5.3

/content/aip/journal/jmp/56/8/10.1063/1.4927100

http://aip.metastore.ingenta.com/content/aip/journal/jmp/56/8/10.1063/1.4927100

Article metrics loading...

/content/aip/journal/jmp/56/8/10.1063/1.4927100

2015-08-05

2016-10-20

Full text loading...

###
Most read this month

Article

content/aip/journal/jmp

Journal

5

3

Commenting has been disabled for this content