  Silicon technology offers the enticing opportunity for monolithic integration
of quantum and classical electronic circuits. However, the power consumption
levels of classical electronics may compromise the local chip temperature and
hence the fidelity of qubit operations. Here, we utilize a quantum-dot-based
thermometer embedded in an industry-standard silicon field-effect transistor
(FET), to assess the local temperature increase produced by an active FET
placed in close proximity. We study the impact of both static and dynamic
operation regimes. When the FET is operated statically, we find a power budget
of 45 nW at 100 nm separation whereas at 216 $\mu$m the power budget raises to
150 $\mu$W. When operated dynamically, we observe negligible temperature
increase for the switch frequencies tested up to 10 MHz. Our work describes a
method to accurately map out the available power budget at a distance from a
solid-state quantum processor and indicate under which conditions
cryoelectronics circuits may allow the operation of hybrid quantum-classical
systems.
