  Landauer's principle laid the main foundation for the development of modern
thermodynamics of information. However, in its original inception the principle
relies on semiformal arguments and dissipative dynamics. Hence, if and how
Landauer's principle applies to unitary quantum computing is less than obvious.
Here, we prove an inequality bounding the change of Shannon information encoded
in the logical quantum states by quantifying the energetic cost of Hamiltonian
gate operations. The utility of this bound is demonstrated by outlining how it
can be applied to identify energetically optimal quantum gates in theory and
experiment. The analysis is concluded by discussing the energetic cost of
quantum error correcting codes with non-interacting qubits, such as Shor's
code.
