  We report a proof-of-concept demonstration of a quantum order-finding
algorithm for factoring the integer 21. Our demonstration involves the use of a
compiled version of the quantum phase estimation routine, and builds upon a
previous demonstration by Mart\'in-L\'{o}pez et al. in Nature Photonics 6, 773
(2012). We go beyond this work by using a configuration of approximate Toffoli
gates with residual phase shifts, which preserves the functional correctness
and allows us to achieve a complete factoring of N=21. We implemented the
algorithm on IBM quantum processors using only 5 qubits and successfully
verified the presence of entanglement between the control and work register
qubits, which is a necessary condition for the algorithm's speedup in general.
The techniques we employ may be useful in carrying out Shor's algorithm for
larger integers, or other algorithms in systems with a limited number of noisy
qubits.
