  Measurement underpins all quantitative science. A key example is the
measurement of optical phase, used in length metrology and many other
applications. Advances in precision measurement have consistently led to
important scientific discoveries. At the fundamental level, measurement
precision is limited by the number N of quantum resources (such as photons)
that are used. Standard measurement schemes, using each resource independently,
lead to a phase uncertainty that scales as 1/sqrt(N) - known as the standard
quantum limit. However, it has long been conjectured that it should be possible
to achieve a precision limited only by the Heisenberg uncertainty principle,
dramatically improving the scaling to 1/N. It is commonly thought that
achieving this improvement requires the use of exotic quantum entangled states,
such as the NOON state. These states are extremely difficult to generate.
Measurement schemes with counted photons or ions have been performed with N <=
6, but few have surpassed the standard quantum limit and none have shown
Heisenberg-limited scaling. Here we demonstrate experimentally a
Heisenberg-limited phase estimation procedure. We replace entangled input
states with multiple applications of the phase shift on unentangled
single-photon states. We generalize Kitaev's phase estimation algorithm using
adaptive measurement theory to achieve a standard deviation scaling at the
Heisenberg limit. For the largest number of resources used (N = 378), we
estimate an unknown phase with a variance more than 10 dB below the standard
quantum limit; achieving this variance would require more than 4,000 resources
using standard interferometry. Our results represent a drastic reduction in the
complexity of achieving quantum-enhanced measurement precision.
