Machine Learning Versus Semidefinite Programming Approach to a Particular Problem of the Theory of Open Quantum Systems

• Published in: Lobachevskii journal of mathematics Vol. 42; no. 7; pp. 1622 - 1629
• Main Authors: Yusipov, I. I., Volokitin, V. D., Liniov, A. V., Ivanchenko, M. V., Meyerov, I. B., Denisov, S. V.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.07.2021; Springer Nature B.V
• Subjects: Algebra; Analysis; Complexity; Evolutionary algorithms; Geometry; Hilbert space; Image classification; Machine learning; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Neural networks; Probability Theory and Stochastic Processes; Quantum theory; Qubits (quantum computing); Semidefinite programming; integer programming; machine learning; open quantum systems
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S199508022107026X

Most of the problems of theoretical quantum physics are characterized by a high complexity. Practically, this means that solution of such a problem demands computational effort and resources that often scale exponentially with the size of the quantum model and the solution can be obtained—even for relatively small models—only by resorting to high performance computing (HPC) technologies. Here we discuss a particular problem of the theory of open quantum systems, the so-called ‘‘Markovianity problem’’. To get the answer to the question whether a given quantum map can be obtained as the result of a time-continuous open quantum evolution, one needs to implement an algorithm whose complexity grows exponentially with the dimension of the model’s Hilbert space. We discuss how machine learning (ML) methods can be used to get the answer and provide some evidence for potential efficiency of the ML-based approach. We demonstrate that neural networks that are used to classify images can be used to determine the boundary between answers ‘‘yes’’ and ‘‘no’’ in the parameter space of a particular open quantum model. For an open model consisting of two qubits, our ML algorithm is able to give correct answer with accuracy. The computational experiments were performed on the Lomonosov-2 supercomputer.