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We report a proof-of-concept demonstration of a quantum order-findingalgorithm for factoring the integer 21. Our demonstration involves the use of acompiled version of the quantum phase estimation routine, and builds upon aprevious 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 Toffoligates with residual phase shifts, which preserves the functional correctnessand allows us to achieve a complete factoring of N=21. We implemented thealgorithm on IBM quantum processors using only 5 qubits and successfullyverified the presence of entanglement between the control and work registerqubits, 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 forlarger integers, or other algorithms in systems with a limited number of noisyqubits.

Demonstration of Shor's factoring algorithm for N=21 on IBM quantum processors