Superresolution in Quantum Noise Spectroscopy via Filter Design

TL;DR

This work develops a filter-function–based framework to achieve spectral superresolution in quantum sensing by shaping the control of a qubit under a two-tone spectrum with known centroid $ω_c$ and small separation $Δω$. It derives necessary and sufficient SR conditions in terms of the filter function, establishes a tight Fisher information bound, and demonstrates SR via two principal protocols, FE-SR and CPMG-SR, with extensions to continuous and multiaxis controls, plus entanglement-enhanced sensing. The authors analyze performance under realistic noise (white and Lorentzian) and provide a numerical optimization approach to design improved continuous controls, yielding practical SR advantages in short-coherence regimes. Comparisons to quantum noise spectroscopy and classical strategies highlight regimes where control-based SR outperforms conventional methods and where entanglement can reduce resource requirements. The results offer a systematic path to surpass conventional spectral-resolution limits across quantum-sensing platforms.

Abstract

Resolving signals with closely spaced frequencies is central to applications in communications, spectroscopy and sensing. Recent results have shown that quantum sensing protocols can exhibit superresolution, the ability to discriminate between spectral lines with arbitrarily small frequency separation. Here, we revisit this problem from the perspective of quantum control theory, utilizing the filter function formalism to derive general, analytic conditions on quantum control protocols for achieving superresolution. Building on these conditions, we develop an optimal control framework, the utility of which is demonstrated through numerical identification of superresolution control protocols in the presence of realistic, experimentally-relevant constraints. We further extend our results to entangled initial states and assess their potential advantage. Our approach is broadly applicable to a wide variety of quantum sensing platforms, and it provides a systematic path to discover novel protocols that surpass conventional resolution limits in these systems.

Figures (7)

• Figure 1: Illustration of the filter functions corresponding to the free evolution and CPMG superresolution protocols (with $\kappa=2$), and the (approximate) signal $S(\omega)$ and noise spectrum $S_\lambda(\omega)$.
• Figure 2: Given the parameters $g = 0.1$ MHz, $\omega_c = 1$ MHz, and desired relative error $\delta=0.1$, we numerically simulate the free evolution protocol with $\kappa = 5/2$ which is not a superresolution protocol (FE Non-SR), the free evolution superresolution protocol with $\kappa = 2$ (FE SR), and the CPMG superresolution protocol with $\kappa = 2$ (CPMG SR). For each of the protocols, we utilize $N_{\rm free}$ samples as given in \ref{['eq:number-measurements']}, which sets the number of samples for the FE-SR protocol, with $\delta=0.1$ . (Left): Noiseless case; the Lorentzian noise parameters are set to $g_\lambda = W = 0$. As $\Delta\omega\to 0$, the SR protocols achieve the desired relative accuracy while the non-SR protocol fails. (Right): Noisy case; the Lorentzian noise parameters are set to $g_\lambda = 0.001$ MHz and FWHM $W = 0.1$ Hz. The CPMG superresolution protocol performs significantly better due to the fact that the corresponding filter function overlaps with the noise spectrum significantly less. In both figures, the analytic error bounds are from \ref{['ap:error-analysis']}.
• Figure 3: The filter functions for the $c_1(t)$ protocol (continuous controls) and for the CPMG and free superresolution protocols (instantaneous controls). Note that we are plotting the lowest order filter function. For the instantaneous, dephasing-preserving control protocols, the lowest order filter function is the only one that matters; however, for continuous controls protocols, higher order filter functions become important. (Left) Filter functions for $\kappa = 2$. (Right) Filter functions for $\kappa =4$.
• Figure 4: For $\kappa=4$. (Left): $c_1$, $c_1^{\rm (opt)}$, and CPMG control sequences. (Right): Corresponding filter functions. The green and red filter functions are the same as in the right of \ref{['fig:filterfunctions']}.
• Figure 5: This plot is analogous to \ref{['fig:MSE-numerics']}, except we now also consider continuous control protocols. We numerically simulate the CPMG protocol, the $c_1$ protocol, and the optimized $c_1^{\rm (opt)}$ protocol (where recall we optimized for a small and smooth control amplitude with small fourth filter function), all with $\kappa =4$. For each of the protocols, we utilize $N_{\rm CPMG}$ samples for a desired noiseless relative error of $\delta = 0.1$ given in \ref{['eq:number-measurements']}. Note that all the protocols fail at large enough $\Delta\omega$ because we are using a second order Taylor expansion approximation for $*{P}$. We could go to higher order to remove this effect, but in this work we are primarily concerned with the limit where $\Delta\omega$ is small. (Top left): Noiseless case and $g=0.06$. The Lorentzian noise parameters are set to $g_\lambda = W = 0$. As $\Delta\omega\to 0$, the instantaneous control protocols achieve the desired relative accuracy while the $c_1$ protocol fails due to the higher order filter functions. However, there is a range of $\Delta\omega$ values where the $c_1$ protocol outperforms the instantaneous control protocols due to their filter functions having a larger second derivative at the centroid. Meanwhile, due to its small fourth filter function, the $c_1^{\rm opt}$ protocol performs well for smaller values of $\Delta\omega$. (Top right): Noiseless case and $g=0.02$. The Lorentzian noise parameters are set to $g_\lambda = W = 0$. This is similar to the top left case, except that now because $g$ is smaller, the effect of the higher order filter functions is reduced, resulting in a larger range of values of $\Delta\omega$ for which the continuous control protocols outperform the instantaneous control protocols. (Middle left): Noisy case ($g_\lambda = g/30$) and $g = 0.06$. The Lorentzian noise parameters are set to $g_\lambda = g/30$ and FWHM $W = 0.1$. (Middle right): Noisy case ($g_\lambda = g/30$) and $g = 0.02$. This case is the same as the Middle left, except again because of the smaller $g$ value, the continuous control protocols perform better for longer. (Bottom left): Noisy case ($g_\lambda = g/15$) and $g = 0.06$. The Lorentzian noise parameters are set to $g_\lambda = g/15$ and FWHM $W = 0.1$. (Bottom right): Noisy case ($g_\lambda = g/15$) and $g = 0.02$. This case is the same as the bottom left, except again because of the smaller $g$ value, the continuous control protocols perform better for longer.