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We show how quantum metrology protocols that seek to estimate the parametersof a Hamiltonian that exhibits a quantum phase transition can be efficientlysimulated on an exponentially smaller quantum computer. Specifically, byexploiting the fact that the ground state of such a Hamiltonian changesdrastically around its phase transition point, we construct a suitableobservable from which one can estimate the relevant parameters of theHamiltonian with Heisenberg scaling precision. We then show how, for theone-dimensional Ising Hamiltonian with transverse magnetic field acting on Nspins, such a metrology protocol can be efficiently simulated on anexponentially smaller quantum computer while maintaining the same Heisenbergscaling, i.e., O(1/N^2) precision and derive the explicit circuit thataccomplishes the simulation.

Compressed quantum metrology for the Ising Hamiltonian