  The DQC1 complexity class, or power of one qubit model, is examined as an
open quantum system. We study the dynamics of a register of qubits carrying out
a DQC1 algorithm and show that, for any algorithm in the complexity class, the
evolution of the logical qubit can be described as an open quantum system
undergoing a dynamics which is unital. Unital quantum channels respect the
Tasaki-Crooks fluctuation theorem and we demonstrate how this is captured by
the thermodynamics of the logical qubit. As an application, we investigate the
equilibrium and non-equilibrium thermodynamics of the DQC1 trace estimation
algorithm. We show that different computational inputs, i.e. different traces
being estimated, lead to different energetic exchanges across the register of
qubits and that the temperature of the logical qubit impacts the magnitude of
fluctuations experienced and quality of the algorithm.
