  Phase estimation algorithms are key protocols in quantum information
processing. Besides applications in quantum computing, they can also be
employed in metrology as they allow for fast extraction of information stored
in the quantum state of a system. Here, we implement two suitably modified
phase estimation procedures, the Kitaev- and the semiclassical
Fourier-transform algorithms, using an artificial atom realized with a
superconducting transmon circuit. We demonstrate that both algorithms yield a
flux sensitivity exceeding the classical shot-noise limit of the device,
allowing one to approach the Heisenberg limit. Our experiment paves the way for
the use of superconducting qubits as metrological devices which are potentially
able to outperform the best existing flux sensors with a sensitivity enhanced
by few orders of magnitude.
