Single-spin quantum sensors, for example based on nitrogen-vacancy centres in diamond, provide nanoscale mapping of magnetic fields. In applications where the magnetic field may be changing rapidly, total sensing time is crucial and must be minimised. Bayesian estimation and adaptive experiment optimisation can speed up the sensing process by reducing the number of measurements required. These protocols consist of computing and updating the probability distribution of the magnetic field based on measurement outcomes and of determining optimized acquisition settings for the next measurement. However, the computational steps feeding into the measurement settings of the next iteration must be performed quickly enough to allow real-time updates. This article addresses the issue of computational speed by implementing an approximate Bayesian estimation technique, where probability distributions are approximated by a finite sum of Gaussian functions. Given that only three parameters are required to fully describe a Gaussian density, we find that in many cases, the magnetic field probability distribution can be described by fewer than ten parameters, achieving a reduction in computation time by factor 10 compared to existing approaches. ForT2*=1μs, only a small decrease in computation time is achieved. However, in these regimes, the proposed Gaussian protocol outperforms the existing one in tracking accuracy.
Single-spin quantum sensors, for example based on nitrogen-vacancy centres in diamond, provide nanoscale mapping of magnetic fields. In applications where the magnetic field may be changing rapidly, total sensing time is crucial and must be minimised. Bayesian estimation and adaptive experiment optimisation can speed up the sensing process by reducing the number of measurements required. These protocols consist of computing and updating the probability distribution of the magnetic field based on measurement outcomes and of determining optimized acquisition settings for the next measurement. However, the computational steps feeding into the measurement settings of the next iteration must be performed quickly enough to allow real-time updates. This article addresses the issue of computational speed by implementing an approximate Bayesian estimation technique, where probability distributions are approximated by a finite sum of Gaussian functions. Given that only three parameters are required to fully describe a Gaussian density, we find that in many cases, the magnetic field probability distribution can be described by fewer than ten parameters, achieving a reduction in computation time by factor 10 compared to existing approaches. ForT2*=1μs, only a small decrease in computation time is achieved. However, in these regimes, the proposed Gaussian protocol outperforms the existing one in tracking accuracy.
Sensors based on single spins can enable magnetic-field detection with very high sensitivity and spatial resolution. Previous work has concentrated on sensing of a constant magnetic field or a periodic signal. Here, we instead investigate the problem of estimating a field with nonperiodic variation described by a Wiener process. We propose and study, by numerical simulations, an adaptive tracking protocol based on Bayesian estimation. The tracking protocol updates the probability distribution for the magnetic field based on measurement outcomes and adapts the choice of sensing time and phase in real time. By taking the statistical properties of the signal into account, our protocol strongly reduces the required measurement time. This leads to a reduction of the error in the estimation of a time-varying signal by up to a factor of four compare with protocols that do not take this information into account.
Quantum information protocols, such as quantum error correction and quantum phase estimation, have been widely used to enhance the performance of quantum sensors. While these protocols have relied on single-shot detection, in most practical applications only an averaged readout is available, as in the case of room-temperature sensing with the electron spin associated with a nitrogen-vacancy center in diamond. Here, we theoretically investigate the application of the quantum phase estimation algorithm for high dynamic-range magnetometry, when single-shot readout is not available. We show that, even in this case, Bayesian estimation provides a natural way to efficiently use the available information. We apply Bayesian analysis to achieve an optimized sensing protocol for estimating a time-independent magnetic field with a single electron spin associated to a nitrogen-vacancy center at room temperature and show that this protocol improves the sensitivity over previous protocols by more than a factor of 3. Moreover, we show that an extra enhancement can be achieved by considering the timing information in the detector clicks.
A large class of experiments consists of measuring the parameters of physical models. In these experiments, the goal is to learn about these parameters as accurately and, often, quickly as possible. Adaptive experiment design works by yielding instrument control to Bayesian-based algorithms that alter instrument settings based on potential information gain about the parameters. By actively learning from data in real-time where to measure instead of determining instrument settings a priori, striking improvements in experiment efficiency are possible. Here, two new algorithms that improve upon previous implementations of adaptive experiment design are introduced. The first algorithm focuses on learning the model parameters that matter the most. The second algorithm considers the expense of a measurement and prioritizes information that can be gained at a lower cost. We demonstrate the remarkable improvement in efficiency and sensitivity that these algorithms provide for quantum sensing, specifically magnetometry, with nitrogen-vacancy centers in diamond. Most notably, we find an almost fivefold improvement in magnetic field sensitivity.
The advancement and scaling of quantum technology has made the learning and identification of quantum systems and devices in highly multidimensional parameter spaces a pressing task for a variety of applications. In many cases, the integration of real-time feedback control and adaptive choice of measurement settings places strict demands on the speed of this task. Here we present a joint model selection and parameter estimation algorithm that is fast and operable on a large number of model parameters. The algorithm is based on variational Bayesian inference, which approximates the target posterior distribution by optimizing a tractable family of distributions, making it more scalable than exact inference methods relying on sampling and that generally suffer from high variance and computational cost in high-dimensional spaces. We show how a regularizing prior can be used to select between competing models, each comprising a different number of parameters, identifying the simplest model that explains the experimental data. The regularization can further separate the degrees of freedom, e.g., quantum systems in the environment or processes, which contribute to major features in the observed dynamics, with respect to others featuring small coupling, which only contribute to a background. As an application of the introduced framework, we consider the problem of the identification of multiple individual nuclear spins with a single electron spin quantum sensor, relevant for nanoscale nuclear magnetic resonance and for the implementation of multiqubit quantum networking nodes. With the number of environmental spins unknown a priori, our Bayesian approach is able to correctly identify the model, i.e., the number of spins and their couplings. We benchmark the algorithm on both simulated and experimental data, using standard figures of merit, and demonstrate that we can estimate dozens of parameters within minutes. Our methodology is compatible with the implementation of real-time adaptive choice of experimental settings in multiparameter estimation, which has the potential to greatly decrease the estimation time in a variety of metrology, tomography, and characterization tasks by prioritizing the measurements with the highest content of information.
This article introduces a real-time Bayesian estimation algorithm aimed at enhancing the spin readout efficiency of the nitrogen vacancy (NV) center ensemble in the diamond. By integrating both the prior probability distribution and the fluorescence likelihood function, stemming from the implementation of the NV center dynamics model, we offered an innovative solution to surpass the Cramer–Rao lower bound of the readout variance and improve the readout efficiency. The proposed real-time Bayesian estimation readout algorithm was experimentally realized by combining high-performance acquisition and processing hardware, and the Rabi oscillation experiments divulged that the signal-to-noise ratio of our approach was improved by 28.6%. This research illustrates the potential of the Bayesian estimation readout to enhance the NV ensemble's sensing capabilities and support the development of compact and scalable quantum sensors devices on a monolithic platform.
Quantum sensors offer significant advantages over classical devices in spatial resolution and sensitivity, enabling transformative applications across materials science, healthcare, and beyond. Their practical performance, however, is often constrained by unmodelled effects, including noise, imperfect state preparation, and non-ideal control fields. In this work, we report the first experimental implementation of a graybox modelling strategy for a solid-state open quantum system. The graybox framework integrates a physics-based system model with a data-driven description of experimental imperfections, achieving higher fidelity than purely analytical (whitebox) approaches while requiring fewer training resources than fully deep-learning blackbox models. We experimentally validate the method on the task of estimating a static magnetic field using a single-spin quantum sensor, performing Bayesian inference with a graybox model trained on prior experimental data. Using roughly 10,000 training datapoints, the graybox model yields several orders of magnitude improvement in mean squared error over the corresponding physics-only model, as well as significant improvement over a blackbox model with comparable size. These results are broadly applicable to a wide range of quantum sensing platforms, not limited to single-spin systems, and are particularly valuable for real-time adaptive protocols, where model inaccuracies can otherwise lead to suboptimal control and degraded performance.