Machine Learning approach to the Floquet--Lindbladian problem

Abstract: Similar to its classical version, quantum Markovian evolution can be either time-discrete or time-continuous. Discrete quantum Markovian evolution is usually modeled with completely-positive trace-preserving maps while time-continuous evolution is often specified with superoperators referred to as "Lindbladians". Here we address the following question: Being given a quantum map, can we find a Lindbladian which generates an evolution identical -- when monitored at discrete instances of time -- to the one induced by the map? It was demonstrated that the problem of getting the answer to this question can be reduced to an NP-complete (in the dimension $N$ of the Hilbert space the evolution takes place in) problem. We approach this question from a different perspective by considering a variety of Machine Learning (ML) methods and trying to estimate their potential ability to give the correct answer. Complimentary, we use the performance of different ML methods as a tool to check the hypothesis that the answer to the question is encoded in spectral properties of the so-called Choi matrix, which can be constructed from the given quantum map. As a test bed, we use two single-qubit models for which the answer can be obtained by using the reduction procedure. The outcome of our experiment is that, for a given map, the property of being generated by a time-independent Lindbladian is encoded both in the eigenvalues and the eigenstates of the corresponding Choi matrix.