A Magnus-expansion analysis of product formula errors for quantum phase estimation yields custom product formulas with up to quartic energy-error scaling and a low-energy bound with up to quadratic speedup in the target error.
This is because the lowest contribution comes from the first term above 1 s R s 0 dσE(σ), where we recall E(σ) = O(σp)
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Better product formulas for quantum phase estimation
A Magnus-expansion analysis of product formula errors for quantum phase estimation yields custom product formulas with up to quartic energy-error scaling and a low-energy bound with up to quadratic speedup in the target error.