RIGA is a General Algorithm for Quantum Gate Generation

RIGA (Reference Input Generation Algorithm) is a monotonic numerical method for generating quantum gates for closed systems that are described by Schrödinger equations. In a previous paper, the authors have presented a monotonic quantum gate generation algorithm, called here by L-RIGA (Lindblad-RIGA), that is able to consider open quantum systems described by Lindblad master equations. The authors have claimed in that paper, without proof, that L-RIGA was originally obtained from a version of RIGA. In this paper we present this version of RIGA, called here F-RIGA (Fock-RIGA) that can consider open quantum systems after converting them to the Fock-Liouville descripton. This conversion is based on the Fock-Map, that is, the map F that sends an × n Hermitian matrix into an 2-vector of a real Euclidean space. The contribution of this paper is to show that that L-RIGA and F-RIGA are equivalent in the sense that, for each step ℓ, the data that is obtained by F-RIGA is transformed by the inverse of the Fock-Map into the data that is obtained in the same step of L-RIGA, letting the corresponding Lyapunov functions invariant. Furthermore, thanks to the great similarity between L-RIGA and a version of the Krotov method, a subproduct of this work is also establish a strong connection between that version of the Krotov method and the family of algorithms that are called by RIGA.