  Quantum metrology allows for measuring properties of a quantum system at the
optimal Heisenberg limit. However, when the relevant quantum states are
prepared using digital Hamiltonian simulation, the accrued algorithmic errors
will cause deviations from this fundamental limit. In this work, we show how
algorithmic errors due to Trotterized time evolution can be mitigated through
the use of standard polynomial interpolation techniques. Our approach is to
extrapolate to zero Trotter step size, akin to zero-noise extrapolation
techniques for mitigating hardware errors. We perform a rigorous error analysis
of the interpolation approach for estimating eigenvalues and time-evolved
expectation values, and show that the Heisenberg limit is achieved up to
polylogarithmic factors in the error. Our work suggests that accuracies
approaching those of state-of-the-art simulation algorithms may be achieved
using Trotter and classical resources alone for a number of relevant
algorithmic tasks.
