# Classical simulation of quantum-coherent thermal machines. (arXiv:1810.04174v3 [quant-ph] UPDATED)

The performance enhancements observed in various models of continuous quantum
thermal machines have been linked to the buildup of coherences in a preferred
basis. But, is this connection always an evidence of quantum-thermodynamic
supremacy'? By force of example, we show that this is not the case. In
particular, we compare a power-driven three-level continuous quantum
refrigerator with a four-level combined cycle, partly driven by power and
partly by heat. We focus on the weak driving regime and find the four-level
model to be superior since it can operate in parameter regimes in which the
three-level model cannot, it may exhibit a larger cooling rate, and,
simultaneously, a better coefficient of performance. Furthermore, we find that
the improvement in the cooling rate matches the increase in the stationary
quantum coherences exactly. Crucially, though, we also show that the
thermodynamic variables for both models follow from a classical representation
based on graph theory. This implies that we can build incoherent
stochastic-thermodynamic models with the same steady-state operation or,
equivalently, that both coherent refrigerators can be emulated classically.
More generally, we prove this for any N-level weakly driven device with a
cyclic' pattern of transitions. Therefore, even if coherence is present in a
specific quantum thermal machine, it is often not essential to replicate the
underlying energy conversion process.