  Algorithmic cooling is a method that employs thermalization to increase qubit
purification level, namely it reduces the qubit-system's entropy. We utilized
gradient ascent pulse engineering (GRAPE), an optimal control algorithm, to
implement algorithmic cooling in liquid state nuclear magnetic resonance.
Various cooling algorithms were applied onto the three qubits of
$^{13}$C$_2$-trichloroethylene, cooling the system beyond Shannon's entropy
bound in several different ways. In particular, in one experiment a carbon
qubit was cooled by a factor of 4.61. This work is a step towards potentially
integrating tools of NMR quantum computing into in vivo magnetic resonance
spectroscopy.
