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Explore the mathematical connection between quantum thermodynamics and semi-definite programming in this 21-minute conference talk from QTML 2025. Learn how quantum systems described by Hamiltonians and non-commuting conserved charges can be analyzed using the same mathematical framework as semi-definite optimization problems, despite originating from different scientific communities with distinct terminologies. Discover how adopting Jaynes' approach to minimize free energy instead of energy leads to an elegant dual chemical potential maximization problem that is concave in chemical potential parameters, enabling the use of standard gradient ascent methods with guaranteed quick convergence. Understand how this Jaynes-inspired gradient-ascent approach can be implemented in both classical and hybrid quantum-classical algorithms for energy minimization and SDP solving, with specific runtime guarantees. Examine the development of quantum Boltzmann machine learning algorithms for energy minimization and gain insight into why classical algorithms like matrix multiplicative weights update and matrix exponentiated gradient update methods, along with their quantum generalizations, perform effectively at solving semi-definite programs through the lens of quantum thermodynamics principles.
