  The precision of quantum metrology is widely believed to be restricted by the
Heisenberg limit, corresponding to a root mean square error that is inversely
proportional to the number of independent processes probed in an experiment, N.
In the past, some proposals have challenged this belief, for example using
non-linear interactions among the probes. However, these proposals turned out
to still obey the Heisenberg limit with respect to other relevant resources,
such as the total energy of the probes. Here, we present a photonic
implementation of a quantum metrology protocol surpassing the Heisenberg limit
by probing two groups of independent processes in a superposition of two
alternative causal orders. Each process creates a phase space displacement, and
our setup is able to estimate a geometric phase associated to two sets of N
displacements with an error that falls quadratically with N. Our results only
require a single-photon probe with an initial energy that is independent of N.
Using a superposition of causal orders outperforms every setup where the
displacements are probed in a definite order. Our experiment features the
demonstration of indefinite causal order in a continuous-variable system, and
opens up the experimental investigation of quantum metrology setups boosted by
indefinite causal order.
