Beyond Heisenberg Limit Quantum Metrology through Quantum Signal Processing

Leveraging quantum effects in metrology such as entanglement and coherence allows one to measure parameters with enhanced sensitivity. However, time-dependent noise can disrupt such Heisenberg-limited amplification. We propose a quantum-metrology method based on the quantum-signal-processing framework to overcome these realistic noise-induced limitations in practical quantum metrology. Our algorithm separates the gate parameter varphi (single-qubit Z phase) that is susceptible to time-dependent error from the target gate parameter θ (swap-angle between |10> and |01> states) that is largely free of time-dependent error. Our method achieves an accuracy of 10^-4 radians in standard deviation for learning θ in superconducting-qubit experiments, outperforming existing alternative schemes by two orders of magnitude. We also demonstrate the increased robustness in learning time-dependent gate parameters through fast Fourier transformation and sequential phase difference. We show both theoretically and numerically that there is an interesting transition of the optimal metrology variance scaling as a function of circuit depth d from the pre-asymptotic regime d ll 1/θ to Heisenberg limit d → infty. Remarkably, in the pre-asymptotic regime our method's estimation variance on time-sensitive parameter varphi scales faster than the asymptotic Heisenberg limit as a function of depth, Var\(hat{varphi}\)approx 1/d⁴. Our work is the first quantum-signal-processing algorithm that demonstrates practical application in laboratory quantum computers.