Heisenberg-limited metrology in the presence of non-Markovian noise with finite control rates

Quantum metrology aims to estimate physical parameters, such as frequency, with the highest possible precision. While the quantum Cramér-Rao bound allows for Heisenberg-limited (HL) scaling—where precision improves quadratically with interrogation time—this is typically destroyed by environmental noise. This paper investigates whether one can restore HL scaling in the presence of general non-Markovian noise by using fast control operations to suppress decoherence.

The authors model the system-environment interaction using a Stinespring dilation and leverage the quantum Zeno effect to freeze the evolution of the system. They define a codespace where the signal generator acts non-trivially while the noise generators act trivially. They analyze three distinct control strategies: (a) approximate quantum error detection (AQED), which relies on projective measurements to reset the state; (b) approximate quantum error correction (AQEC), which uses active recovery operations; and (c) dynamical decoupling (DD), which uses unitary control pulses.

The study establishes that HL scaling can be achieved in non-Markovian settings provided the signal generator satisfies a specific 'not-in-span' condition relative to the noise generators. The authors prove that while all three protocols achieve HL scaling in the limit of infinitely fast controls, they diverge at finite control rates. Specifically, AQEC and DD protocols exhibit a quadratic convergence of the quantum Fisher information (QFI) with respect to the control interval, whereas AQED exhibits only linear convergence. This makes active recovery or dynamical decoupling significantly more robust than simple error detection in realistic, finite-rate scenarios.