  The use of quantum resources can provide measurement precision beyond the
shot-noise limit (SNL). The task of ab initio optical phase measurement---the
estimation of a completely unknown phase---has been experimentally demonstrated
with precision beyond the SNL, and even scaling like the ultimate bound, the
Heisenberg limit (HL), but with an overhead factor. However, existing
approaches have not been able---even in principle---to achieve the best
possible precision, saturating the HL exactly. Here we demonstrate a scheme to
achieve true HL phase measurement, using a combination of three techniques:
entanglement, multiple samplings of the phase shift, and adaptive measurement.
Our experimental demonstration of the scheme uses two photonic qubits, one
double passed, so that, for a successful coincidence detection, the number of
photon-passes is $N=3$. We achieve a precision that is within $4\%$ of the HL,
surpassing the best precision theoretically achievable with simpler techniques
with $N=3$. This work represents a fundamental achievement of the ultimate
limits of metrology, and the scheme can be extended to higher $N$ and other
physical systems.
