Universal SU(d) holonomic quantum computing with cat-qudits

We present a holonomic computation scheme with engineered dissipation of a multi-photon process, a generalization of the driven dissipative 2-photon process studied in [1]. The engineered $d$-photon process can stabilize a $d$-dimensional steady state manifold spanned by $d$ coherent states. Universal control is achieved with two types of non-Abelian holonomic gates [2]. The first type consists of adiabatically moving a coherent state around a closed path in phase space, resulting in a relative Berry phase between that state and the other states. The second type consists of adiabatically colliding two coherent states, resulting in a unitary evolution with coherent population transfer between those two components. We outline a way to realize the $d=2$ case using circuit QED. \newline\newline [1] M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, New J. Phys. {\bf 16}, 045014 (2014). \newline [2] A. Carollo, M. Santos, and V. Vedral, Phys. Rev. Lett. {\bf 96}, 020403 (2006).