  We study energetic exchanges and fluctuations in two-stroke quantum
thermodynamic engines where the working fluid is represented by two multilevel
quantum systems, i.e. qudits, the heat flow is allowed by relaxation with two
thermal reservoirs at different temperatures, and the work exchange is operated
by a partial-swap unitary interaction. We identify three regimes of operation
(heat engine, refrigerator, and thermal accelerator), present the thermodynamic
uncertainty relations between the entropy production and the signal-to-noise
ratio of work and heat, and derive the full joint probability of the stochastic
work and heat. Our results bridge the gap between two-qubit and two-mode
bosonic swap engines, and show which properties are maintained (e.g., a non
fluctuating Otto efficiency) and which are lost for increasing dimension (e.g.,
small violations of the standard thermodynamic uncertainty relations or the
possibility of beating the Curzon-Ahlborn efficiency).
