  Multilevel autonomous quantum thermal machines are discussed. In particular,
we explore the relation between the size of the machine (captured by Hilbert
space dimension), and the performance of the machine. Using the concepts of
virtual qubits and virtual temperatures, we show that higher dimensional
machines can outperform smaller ones. For instance, by considering
refrigerators with more levels, lower temperatures can be achieved, as well as
higher power. We discuss the optimal design for refrigerators of a given
dimension. As a consequence we obtain a statement of the third law in terms of
Hilbert space dimension: reaching absolute zero temperature requires infinite
dimension. These results demonstrate that Hilbert space dimension should be
considered a thermodynamic resource.
