  We show how quantum metrology protocols that seek to estimate the parameters
of a Hamiltonian that exhibits a quantum phase transition can be efficiently
simulated on an exponentially smaller quantum computer. Specifically, by
exploiting the fact that the ground state of such a Hamiltonian changes
drastically around its phase transition point, we construct a suitable
observable from which one can estimate the relevant parameters of the
Hamiltonian with Heisenberg scaling precision. We then show how, for the
one-dimensional Ising Hamiltonian with transverse magnetic field acting on N
spins, such a metrology protocol can be efficiently simulated on an
exponentially smaller quantum computer while maintaining the same Heisenberg
scaling, i.e., O(1/N^2) precision and derive the explicit circuit that
accomplishes the simulation.
