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We show that heat production in slowly driven quantum systems is linked tothe topological structure of the driving protocol through the Fubini-Studytensor. Analyzing a minimal model of a spin weakly coupled to a heat bath, wefind that dissipation is controlled by the quantum metric and a "qualityfactor" characterizing the spin's precession. Utilizing these findings, weestablish lower bounds on the heating rate in two-tone protocols, such as thoseemployed in topological frequency converters. Notably, these bounds aredetermined by the topology of the protocol, independent of its microscopicdetails. Our results bridge topological phenomena and energy dissipation inslowly driven quantum systems, providing a design principle for optimal drivingprotocols.

Quantum geometry and bounds on dissipation in slowly driven quantum systems