  The third law of thermodynamics, also known as the Nernst unattainability
principle, puts a fundamental bound on how close a system, whether classical or
quantum, can be cooled to a temperature near to absolute zero. On the other
hand, a fundamental assumption of quantum computing is to start each
computation from a register of qubits initialized in a pure state, i.e., at
zero temperature. These conflicting aspects, at the interface between quantum
computing and thermodynamics, are often overlooked or, at best, addressed only
at a single-qubit level. In this work, we argue how the existence of a small,
but finite, effective temperature, which makes the initial state a mixed state,
poses a real challenge to the fidelity constraints required for the scaling of
quantum computers. Our theoretical results, carried out for a generic quantum
circuit with $N$-qubit input states, are validated by test runs performed on a
real quantum processor.
