  We propose a measurement setup reaching Heisenberg scaling precision for the
estimation of any distributed parameter $\varphi$ (not necessarily a phase)
encoded into a generic $M$-port linear network composed only of passive
elements. The scheme proposed can be easily implemented from an experimental
point of view since it employs only Gaussian states and Gaussian measurements.
Due to the complete generality of the estimation problem considered, it was
predicted that one would need to carry out an adaptive procedure which involves
both the input states employed and the measurement performed at the output; we
show that this is not necessary: Heisenberg scaling precision is still
achievable by only adapting a single stage. The non-adapted stage only affects
the value of a pre-factor multiplying the Heisenberg scaling precision: we show
that, for large values of $M$ and a random (unbiased) choice of the non-adapted
stage, this pre-factor takes a typical value which can be controlled through
the encoding of the parameter $\varphi$ into the linear network.
