General Transfer-Function Approach to Noise Filtering in Open-Loop Quantum Control

We present a general transfer-function approach to noise filtering in open-loop Hamiltonian engineering protocols for open quantum systems. We show how to identify a computationally tractable set of fundamental filter functions, out of which arbitrary transfer filter functions may be assembled up to arbitrary high order in principle. Besides avoiding the infinite recursive hierarchy of filter functions that arises in general control scenarios, this fundamental filter-function set suffices to characterize the error suppression capabilities of the control protocol in both the time and the frequency domain. We prove that the resulting notion of filtering order reveals conceptually distinct, albeit complementary, features of the controlled dynamics as compared to the order of error cancellation, traditionally defined in the Magnus sense. Examples and implications are discussed.

Figure 1

Performance of ${\mathrm{CDD}}_3$ and ${\mathrm{UDD}}_4$ for “commuting” (dashed blue lines) vs “noncommuting” (solid red lines) single-qubit dephasing models (see text). A relative metric of performance $\mathbf{(}1-{\mathcal{F}}_{{\mathrm{UDD}}_4}^{(\text{av})}(T)\mathbf{)}/\mathbf{(}1-{\mathcal{F}}_{{\mathrm{CDD}}_3}^{(\text{av})}(T)\mathbf{)}<1$ (black dotted line) implies that ${\mathrm{UDD}}_4$ outperforms ${\mathrm{CDD}}_3$ (e.g., below $\omega \lesssim 10^{-3}$ for the largest noise strength, grey vertical line). Only model 1 is sensitive to the difference in FO, which manifests in the change in slope of the solid red lines in the low-frequency regime. The parameters are $T=1$ and $g=9/40$, $9/400$, $9/4000$ in appropriate units, such that $\parallel T{\tilde{H}}_{e,1(2)}(t)\parallel <1$ for all $t\in [0,T]$ and convergence is ensured.