  We address the study of quantum metrology enhanced by indefinite causal
order, demonstrating a quadratic advantage in the estimation of the product of
two average displacements in a continuous variable system. We prove that no
setup where the displacements are probed in a fixed order can have
root-mean-square error vanishing faster than the Heisenberg limit 1/N, where N
is the number of displacements contributing to the average. In stark contrast,
we show that a setup that probes the displacements in a superposition of two
alternative orders yields a root-mean-square error vanishing with
super-Heisenberg scaling 1/N^2. This result opens up the study of new
measurement setups where quantum processes are probed in an indefinite order,
and suggests enhanced tests of the canonical commutation relations, with
potential applications to quantum gravity.
