  We study the problem of dynamic cooling whereby a target qubit is cooled at
the expense of heating up $N-1$ further identical qubits, by means of a global
unitary operation. A standard back-of-the-envelope high temperature estimate
establishes that the target qubit temperature can only be dynamically cooled by
at most a factor of $1/\sqrt{N}$. Here, we provide the exact expression for the
minimum temperature to which the target qubit can be cooled and reveal that
there is a crossover from the high initial temperature regime where the scaling
is in fact $1/\sqrt{N}$ to a low initial temperature regime where a much faster
scaling of $1/N$ occurs. This slow $1/\sqrt{N}$ scaling, which was relevant for
early high-temperature NMR quantum computers, is the reason dynamic cooling was
dismissed as ineffectual around 20 years ago; the fact that current
low-temperature quantum computers fall in the fast $1/N$ scaling regime,
reinstates the appeal of dynamic cooling today. We further show that the
associated work cost of cooling is exponentially more advantageous in the low
temperature regime. We discuss the implementation of dynamic cooling in terms
of quantum circuits and examine the effects of hardware noise. We successfully
demonstrate dynamic cooling in a 3-qubit system on a real quantum processor.
Since the circuit size grows quickly with $N$, scaling dynamic cooling to
larger systems on noisy devices poses a challenge. We therefore propose a
suboptimal cooling algorithm, whereby relinquishing a small amount of cooling
capability results in a drastically reduced circuit complexity, greatly
facilitating the implementation of dynamic cooling on near-future quantum
computers.
