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arXiv:1911.10598 · International Journal of Quantum Information

Role of the filter functions in noise spectroscopy

Nicola Dalla Pozza, Stefano Gherardini, Matthias M. Müller, Filippo Caruso

Published Dec 1, 2019• 6 citations
arXivHTMLPDF

Abstract

The success of quantum noise sensing methods depends on the optimal interplay between properly designed control pulses and statistically informative measurement data on a specific quantum-probe observable. To enhance the information content of the data and reduce as much as possible the number of measurements on the probe, the filter orthogonalization method has been recently introduced. The latter is able to transform the control filter functions on an orthogonal basis allowing for the optimal reconstruction of the noise power spectral density. In this paper, we formalize this method within the standard formalism of minimum mean squared error estimation and we show the equivalence between the solutions of the two approaches. Then, we introduce a nonnegative least squares formulation that ensures the nonnegativeness of the estimated noise spectral density. Moreover, we also propose a novel protocol for the design in the frequency domain of the set of filter functions. The frequency-designed filter functions and the nonnegative least squares reconstruction are numerically tested on noise spectra with multiple components and as a function of the estimation parameters.

DOI
10.1142/s0219749919410089

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The success of quantum noise sensing methods depends on the optimal interplay between properly designed control pulses and statistically informative measurement data on a specific quantum-probe observable. To enhance the information content of the data and reduce as much as possible the number of measurements on the probe, the filter orthogonalization method has been recently introduced. The latter is able to transform the control filter functions on an orthogonal basis allowing for the optimal reconstruction of the noise power spectral density. In this paper, we formalize this method within the standard formalism of minimum mean squared error estimation and we show the equivalence between the solutions of the two approaches. Then, we introduce a nonnegative least squares formulation that ensures the nonnegativeness of the estimated noise spectral density. Moreover, we also propose a novel protocol for the design in the frequency domain of the set of filter functions. The frequency-designed filter functions and the nonnegative least squares reconstruction are numerically tested on noise spectra with multiple components and as a function of the estimation parameters.
The filter function formalism from quantum control theory is typically used to determine the noise susceptibility of pulse sequences by looking at the overlap between the filter function of the sequence and the noise power spectral density. Importantly, the square modulus of the filter function is used for this method. In this work we show that by using the square modulus one neglects valuable information about the system dynamics. We take advantage of the full filter function by including information about the phase of the perturbation and the resulting rotation axis. By decomposing the filter function with phase preservation before taking the modulus, we are able to consider the contributions to x,yx, yx,y, and zzz rotations separately. Continuously driven systems provide noise protection in the form of dynamical decoupling by canceling low-frequency noise; however, generating control pulses synchronously with an arbitrary driving field is not trivial. Using the decomposed filter function we look at the controllability of a system under arbitrary driving fields, as well as the noise susceptibility, and also relate the filter function to the geometric formalism.
The dynamics of quantum systems are unavoidably influenced by their environment, but in turn observing a quantum system (probe) can allow one to measure its environment: Measurements and controlled manipulation of the probe such as dynamical decoupling sequences as an extension of the Ramsey interference measurement allow to spectrally resolve a noise field coupled to the probe. Here, we introduce fast and robust estimation strategies for the characterization of the spectral properties of classical and quantum dephasing environments. These strategies are based on filter function orthogonalization, optimal control filters maximizing the relevant Fisher Information and multi-qubit entanglement. We investigate and quantify the robustness of the schemes under different types of noise such as finite-precision measurements, dephasing of the probe, spectral leakage and slow temporal fluctuations of the spectrum.
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.
Classical control noise is ubiquitous in qubit devices, making its accurate spectral characterization essential for designing optimized error suppression strategies at the physical level. Here, we focus on multiplicative Gaussian amplitude control noise on a driven qubit sensor and show that sensing protocols using optimally band-limited Slepian modulation offer substantial benefit in realistic scenarios. Special emphasis is given to laying out the theoretical framework necessary for extending nonparametric multitaper spectral estimation to the quantum setting by highlighting key points of contact and differences with respect to the classical formulation. In particular, we introduce and analyze two approaches (adaptive vs single setting) to quantum multitaper estimation, and show how they provide a practical means to both identify fine spectral features not otherwise detectable by existing protocols and to obtain reliable prior estimates for use in subsequent parametric estimation, including high-resolution Bayesian techniques. We quantitatively characterize the performance of both single- and multitaper Slepian estimation protocols by numerically reconstructing representative spectral densities, and demonstrate their advantage over dynamical-decoupling noise spectroscopy approaches in reducing bias from spectral leakage as well as in compensating for aliasing effects while maintaining a desired sampling resolution.
Abstract We introduce and experimentally demonstrate a quantum sensing protocol to sample and reconstruct the autocorrelation of a noise process using a single-qubit sensor under digital control modulation. This Walsh noise spectroscopy method exploits simple sequences of spin-flip pulses to generate a complete basis of digital filters that directly sample the power spectrum of the target noise in the sequency domain, from which the autocorrelation function in the time domain, as well as the power spectrum in the frequency domain, can be reconstructed using linear transformations. Our method, which can also be seen as an implementation of frame-based noise spectroscopy, solves the fundamental difficulty in sampling continuous functions with digital filters by introducing a transformation that relates the arithmetic and logical time domains. In comparison to standard, frequency-based dynamical-decoupling noise spectroscopy protocols, the accuracy of our method is only limited by sampling and discretization in the time domain and can be easily improved, even under limited evolution time due to decoherence and hardware limitations. Finally, we experimentally reconstruct the autocorrelation function of the effective magnetic field produced by the nuclear-spin bath on the electronic spin of a single nitrogen-vacancy center in diamond, discuss practical limitations of the method, and avenues to further improve the reconstruction accuracy.
Abstract The field of quantum control and sensing encounters challenges due to various forms of time-dependent noise. We introduce a generalized filter-function framework that treats noise coupling strength as a tunable control parameter, enabling target noise suppression across user-defined frequency bands. By optimizing this generalized filter function, we design band-selective control pulses that achieve high fidelity of single- and two-qubit gates under strong noise with diverse spectral profiles. We further extend the method to selectively enhance the signal-to-noise ratio for quantum sensing of AC signals with an enhanced precision of up to 12.58 dB. The resulting control pulses are experimentally feasible, offering a practical pathway toward robust quantum operations and high-precision sensing under spectrally complex noises.