We study decoupling, one of the most important primitives in quantum Shannon theory, with random diagonal-unitary matrices. A decoupling protocol aims at destroying all correlations in an initial state, shared by two parties, by applying an appropriate unitary followed by a quantum channel only on one party. In Ref. [1,2], decouipling has been shown to be achievable using uniformly distributed random unitaries and their approximate versions called unitary 2-designs. In this talk based on Ref. [3,4], we show that decoupling at the same rate is achievable by alternately applying random diagonal-unitaries in the Pauli-Z and -X bases although the resulting unitary is not a very precise unitary 2-design. This implies that, for achieving decoupling, precise unitary 2-designs may not be needed. We also provide a simple quantum circuit achieving decoupling, most of which consists of commuting gates and the non-commuting part is only constant depth. Since the commuting parts can be, in principle, applied simultaneously, this implmentation results in a vast reduction of the execution time and has a practical advantage.

[1] F. Dupuis, M. Berta, J. Wullschleger, and R. Renner. One-shot decoupling. Commun. Math. Phys. 328:251 (2014).
[2] O. Szehr, F. Dupuis, M. Tomamichel, and R. Renner. Decoupling with unitary approximate two-designs. New J. Phys., 15:053022 (2013).
[3] Y. Nakata, C. Hirche, C. Morgan, and A. Winter, Implementing unitary 2-designs using random diagonal-unitary matrices, arXiv: 1502.07514 (2015).
[4] Y. Nakata, C. Hirche, C. Morgan, and A. Winter, in preparation.