  Superconducting single-flux-quantum (SFQ) circuits have so far been developed
and optimized for operation at or above helium temperatures. The SFQ approach,
however, should also provide potentially viable and scalable control and
read-out circuits for Josephson-junction qubits and other applications with
much lower, milli-kelvin, operating temperatures. This paper analyzes the
overheating problem which becomes important in this new temperature range. We
suggest a thermal model of the SFQ circuits at sub-kelvin temperatures and
present experimental results on overheating of electrons and silicon substrate
which support this model. The model establishes quantitative limitations on the
dissipated power both for "local" electron overheating in resistors and
"global" overheating due to ballistic phonon propagation along the substrate.
Possible changes in the thermal design of SFQ circuits in view of the
overheating problem are also discussed.
