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So-called Thermal Operations seem to describe the most fundamental, andreasonable, set of operations allowable for state transformations at an ambientinverse temperature $\beta$. However, a priori, they require experimentaliststo manipulate very complex environments and have control over their internaldegrees of freedom. For this reason, the community has been working on creatingmore experimentally-friendly operations. In [Perry et al., Phys. Rev. X 8,041049] it was shown that for states diagonal in the energy basis, that ThermalOperations can be performed by so-called Coarse Operations, which need just oneauxiliary qubit, but are otherwise Markovian and classical in spirit. In thiswork, by providing an explicit counterexample, we show that this one qubit ofmemory is necessary. We also fully characterize the possible transitions thatdo not require memory for the system being a qubit. We do this by analyzingarbitrary control sequences comprising level energy changes and partialthermalizations in each step.

Thermal Operations in general are not memoryless