  Quantum simulation advantage over classical memory limitations would allow
compact quantum circuits to yield insight into intractable quantum many-body
problems, but the interrelated obstacles of large circuit depth in quantum time
evolution and noise seem to rule out unbiased quantum simulation in the near
term. We prove that classical post-processing, i.e., long-time filtering of an
offline time series, exponentially improves the circuit depth needed for
quantum time evolution. We apply the filtering method to the construction of a
hybrid quantum-classical algorithm to estimate energy gap, an important
observable not governed by the variational theorem. We demonstrate, within an
operating range of filtering, the success of the algorithm in proof-of-concept
simulation for finite-size scaling of a minimal spin model. Our findings set
the stage for unbiased quantum simulation to offer memory advantage in the near
term.
