Phase estimation algorithms are key protocols in quantum informationprocessing. Besides applications in quantum computing, they can also beemployed in metrology as they allow for fast extraction of information storedin the quantum state of a system. Here, we implement two suitably modifiedphase estimation procedures, the Kitaev- and the semiclassicalFourier-transform algorithms, using an artificial atom realized with asuperconducting transmon circuit. We demonstrate that both algorithms yield aflux sensitivity exceeding the classical shot-noise limit of the device,allowing one to approach the Heisenberg limit. Our experiment paves the way forthe use of superconducting qubits as metrological devices which are potentiallyable to outperform the best existing flux sensors with a sensitivity enhancedby few orders of magnitude.
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