  Quantum computers are operated by external driving fields, such as lasers,
microwaves or transmission lines, that execute logical operations on
multi-qubit registers, leaving the system in a pure state. However, the drive
and the logical system might become correlated in such a way that, after
tracing out the degrees of freedom of the driving field, the output state will
not be pure. Previous works have pointed out that the resulting error scales
inversely with the energy of the drive, thus imposing a limit on the
energy-efficiency of quantum computing. In this study, focusing on the
Jaynes-Cummings model, we show how the same scaling can be seen as a
consequence of two competing phenomena: the entanglement-induced error, which
grows with time, and a minimal time for computation imposed by quantum speed
limits. This evidence is made possible by quantifying, at any time, the
computation error via the spectral radius associated to the density operator of
the logical qubit. Moreover, we also prove that, in order to attain a given
target state at a chosen fidelity, it is energetically more efficient to
perform a single driven evolution of the logical qubits rather than to split
the computation in sub-routines, each operated by a dedicated pulse.
