  We develop a geometric formalism suited for describing the quantum
thermodynamics of a certain class of nanoscale systems (whose density matrix is
expressible in the McLennan--Zubarev form) at any arbitrary non-equilibrium
steady state. It is shown that the non-equilibrium steady states are points on
control parameter spaces which are in a sense generated by the steady state
Massieu--Planck function. By suitably altering the system's boundary
conditions, it is possible to take the system from one steady state to another.
We provide a contact Hamiltonian description of such transformations and show
that moving along the geodesics of the friction tensor results in a minimum
increase of the free entropy along the transformation. The control parameter
space is shown to be equipped with a natural Riemannian metric that is
compatible with the contact structure of the quantum thermodynamic phase space
which when expressed in a local coordinate chart, coincides with the
Schl\"{o}gl metric. Finally, we show that this metric is conformally related to
other thermodynamic Hessian metrics which might be written on control parameter
spaces. This provides various alternate ways of computing the Schl\"{o}gl
metric which is known to be equivalent to the Fisher information matrix.
