  Quantum effects like entanglement and coherent amplification can be used to
drastically enhance the accuracy of quantum parameter estimation beyond
classical limits. However, challenges such as decoherence and time-dependent
errors hinder Heisenberg-limited amplification. We introduce Quantum
Signal-Processing Phase Estimation algorithms that are robust against these
challenges and achieve optimal performance as dictated by the Cram\'{e}r-Rao
bound. These algorithms use quantum signal transformation to decouple
interdependent phase parameters into largely orthogonal ones, ensuring that
time-dependent errors in one do not compromise the accuracy of learning the
other. Combining provably optimal classical estimation with near-optimal
quantum circuit design, our approach achieves an unprecedented standard
deviation accuracy of $10^{-4}$ radians for estimating unwanted swap angles in
superconducting two-qubit experiments, using low-depth ($<10$) circuits. This
represents up to two orders of magnitude improvement over existing methods.
Theoretically and numerically, we demonstrate the optimality of our algorithm
against time-dependent phase errors, observing that the variance of the
time-sensitive parameter $\varphi$ scales faster than the asymptotic Heisenberg
scaling in the small-depth regime. Our results are rigorously validated against
the quantum Fisher information, confirming our protocol's ability to achieve
unmatched precision for two-qubit gate learning.
