  Quantum thermodynamics often deals with the dynamics of small quantum
machines interfacing with a large and complex environment. Virtual qubits,
collisional models and reset master equations have become highly useful tools
for predicting the qualitative behaviour of two-dimensional target systems
coupled to few-qubit machines and a thermal environment. While few successes in
matching the simplified model parameters for all possible physical systems are
known, the qualitative predictions still allow for a general design of quantum
machines irrespective of the implementation. We generalise these tools by
introducing multiple competing virtual qubits for modelling multi-dimensional
systems coupled to larger and more complex machines. By simulating the full
physical dynamics for targets with three dimensions, we uncover general
properties of reset models that can be used as `dials' to correctly predict the
qualitative features of physical changes in a realistic setup and thus design
autonomous quantum machines beyond a few qubits. We then present a general
analytic solution of the reset model for arbitrary-dimensional systems coupled
to multi-qubit machines. Finally, we showcase an improved three-level laser as
an exemplary application of our results.
