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Optimally band-limited controls for quantum multi-axis spectral estimation

Virginia Frey, Leigh Norris, Lorenza Viola, Michael J. Biercuk

Published Nov 1, 2019• 1 citation
OpenAlex

Abstract

The fragility of quantum systems makes them ideally suited for sensing applications at the nanoscale. However, interpreting the output signal of a qubit-based sensor is generally complicated by background clutter due to out-of-band spectral leakage, as well as ambiguity in signal origin when the sensor is operated with noisy hardware. We present a sensing protocol based on optimally band-limited Slepian functions that can overcome these challenges, by providing narrowband sensing of ambient dephasing noise, coupling additively to the sensor along the ${z}$-axis, while permitting isolation of the target noise spectrum from other contributions coupling along a different axis. This is achieved by introducing a finite-difference control modulation, which linearizes the sensor's response and affords tunable band-limited windowing in frequency. Building on these techniques, we experimentally demonstrate two spectral estimation capabilities using a trapped-ion qubit sensor. We first perform efficient experimental reconstruction of a mixed dephasing spectrum, composed of a broadband $1/f$-type spectrum with discrete spurs. We then demonstrate the simultaneous reconstruction of overlapping dephasing and control noise spectra from a single set of measurements, in a setting where the two noise sources contribute equally to the sensor's response. Our approach provides a direct means to augment quantum-sensor performance in the presence of both complex broadband noise environments and imperfect control signals, by optimally complying with realistic time-bandwidth constraints.

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The fragility of quantum systems makes them ideally suited for sensing applications at the nanoscale. However, interpreting the output signal of a qubit-based sensor is generally complicated by background clutter due to out-of-band spectral leakage, as well as ambiguity in signal origin when the sensor is operated with noisy hardware. We present a sensing protocol based on optimally band-limited Slepian functions that can overcome these challenges, by providing narrowband sensing of ambient dephasing noise, coupling additively to the sensor along the z{z}z-axis, while permitting isolation of the target noise spectrum from other contributions coupling along a different axis. This is achieved by introducing a finite-difference control modulation, which linearizes the sensor's response and affords tunable band-limited windowing in frequency. Building on these techniques, we experimentally demonstrate two spectral estimation capabilities using a trapped-ion qubit sensor. We first perform efficient experimental reconstruction of a mixed dephasing spectrum, composed of a broadband 1/f1/f1/f-type spectrum with discrete spurs. We then demonstrate the simultaneous reconstruction of overlapping dephasing and control noise spectra from a single set of measurements, in a setting where the two noise sources contribute equally to the sensor's response. Our approach provides a direct means to augment quantum-sensor performance in the presence of both complex broadband noise environments and imperfect control signals, by optimally complying with realistic time-bandwidth constraints.
The fragility of quantum systems makes them ideally suited for sensing applications at the nanoscale. However, interpreting the output signal of a qubit-based sensor is generally complicated by background clutter due to out-of-band spectral leakage, as well as ambiguity in signal origin when the sensor is operated with noisy hardware. We present a sensing protocol based on optimally band-limited ``Slepian functions'' that can overcome these challenges, by providing narrowband sensing of ambient dephasing noise, coupling additively to the sensor along the zzz axis, while permitting isolation of the target noise spectrum from other contributions coupling along a different axis. This is achieved by introducing a finite-difference control modulation, which linearizes the sensor's response and affords tunable band-limited ``windowing'' in frequency. Building on these techniques, we experimentally demonstrate two spectral estimation capabilities using a trapped-ion qubit sensor. We first perform efficient experimental reconstruction of a ``mixed'' dephasing spectrum, composed of a broadband 1/f1/f1/f-type spectrum with discrete spurs. We then demonstrate the simultaneous reconstruction of overlapping dephasing and control noise spectra from a single set of measurements, in a setting where the two noise sources contribute equally to the sensor's response. Our approach provides a direct means to augment quantum-sensor performance in the presence of both complex broadband noise environments and imperfect control signals, by optimally complying with realistic time-bandwidth constraints.
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.
Essential to the functionality of qubit-based sensors are control protocols, which shape their response in frequency space. However, in common control routines out-of-band spectral leakage complicates interpretation of the sensor's signal. In this work, we leverage discrete prolate spheroidal sequences (a.k.a. Slepian sequences) to synthesize provably optimal narrowband controls ideally suited to spectral estimation of a qubit's noisy environment. Experiments with trapped ions demonstrate how spectral leakage may be reduced by orders of magnitude over conventional controls when a near resonant driving field is modulated by Slepians, and how the desired narrowband sensitivity may be tuned using concepts from RF engineering. We demonstrate that classical multitaper techniques for spectral analysis can be ported to the quantum domain and combined with Bayesian estimation tools to experimentally reconstruct complex noise spectra. We then deploy these techniques to identify previously immeasurable frequency-resolved amplitude noise in our qubit's microwave synthesis chain.
Quantum-enabled technologies promise advancements across a huge range of industrial, metrological and medical applications and are already demonstrating significant impacts, especially in the realm of sensing and metrology. While the extreme sensitivity of quantum systems to their environment is fueling those applications, it also represents a major hurdle to technologies which require long-term stability such as quantum computing and quantum simulations. In addition to the inherent decoherence phenomenon, the challenge associated with controlling a quantum system accurately and precisely, does in fact impede all of the aforementioned applications. Such techniques have therefore been subject to extensive research in both the academic and industrial sector and as a result, sophisticated quantum control techniques are emerging in order to understand, characterize and mitigate errors in quantum systems. This thesis presents how quantum control techniques can be employed to characterize and suppress both control imperfections and environmental noise in quantum systems, using experiments with trapped ions as a model quantum platform. We demonstrate two distinct but interrelated approaches that leverage either time-domain or frequency-domain information about the noise. In the first approach, we show how supervised learning algorithms can efficiently extract time-domain correlations from time-stamped sequences of projective measurements on a qubit. This information can then be used to perform real-time predictive control in which we autonomously pre-compensate anticipated qubit noise in order to stabilize the system. The second approach deploys provably optimal narrowband controls in order to characterize the specific spectral components of noise experienced by a qubit. Here, frequency-shifted Slepian functions permit reconstruction of system noise with maximum out-of-band rejection, and full spectrum reconstruction is enabled using techniques based on multitaper and Bayesian methods.
We introduce a quantum control protocol that produces smooth, experimentally implementable control sequences optimized to combat temporally correlated noise for single-qubit systems. The control ansatz is specifically chosen to be a functional expansion of discrete prolate spheroidal sequences, a discrete time basis known to be optimally concentrated in time and frequency, and quite attractive when faced with experimental control hardware constraints. We leverage the filter function formalism to transform the control problem into a filter design problem, and show that the frequency response of a quantum system can be carefully tailored to avoid the most relevant dynamical contributions of noise processes. Using gradient ascent, we obtain optimized filter functions and exploit them to elucidate important details about the relationship between filter function design, control bandwidth, and noise characteristics. In particular, we identify regimes of optimal noise suppression and, in turn, optimal control bandwidth directly proportional to the size of the frequency bands where the noise power is large. In addition to providing guiding principles for filter design, our approach enables the development of controls that simultaneously yield robust noise filtering and high-fidelity single-qubit logic operations in a wide variety of complex noise environments.
We extend quantum noise spectroscopy (QNS) of amplitude control noise to settings where dephasing noise or detuning errors make significant contributions to qubit dynamics. Previous approaches to characterize amplitude noise are limited by their vulnerability to low-frequency dephasing noise and static detuning errors, which can overwhelm the target control noise signal and introduce bias into estimates of the amplitude noise spectrum. To overcome this problem, we leverage optimal control to identify a family of amplitude control waveforms that optimally suppress low-frequency dephasing noise and detuning errors, while maintaining the spectral concentration in the amplitude filter essential for spectral estimation. The waveforms found via numerical optimization have surprisingly simple analytic forms, consisting of oscillating sine waves obeying particular amplitude and frequency constraints. In numerically simulated QNS experiments, these waveforms demonstrate superior robustness, enabling accurate estimation of the amplitude noise spectrum in regimes where existing approaches are biased by low-frequency dephasing noise and detuning errors.