Resource-efficient adaptive Bayesian tracking of magnetic fields with a quantum sensor
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2021-02-05 |
| Journal | Journal of Physics Condensed Matter |
| Authors | K Craigie, E M Gauger, Y. Altmann, C Bonato, K Craigie |
| Institutions | Heriot-Watt University |
| Citations | 12 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled âExecutive SummaryâThis research introduces a resource-efficient adaptive Bayesian protocol designed to significantly accelerate the tracking of fluctuating magnetic fields using quantum sensors, specifically Nitrogen-Vacancy (NV) centers in diamond.
- Computational Efficiency: The protocol achieves a substantial reduction in computational time, yielding speed increases ranging from 1.3x up to 13.5x (typically an order of magnitude) compared to existing non-approximate Bayesian methods.
- Methodology: The speedup is realized by approximating the complex likelihood and posterior probability distributions using Gaussian Mixture Models (GMMs), rather than computationally intensive grid discretization or particle filtering.
- Parameter Reduction: GMMs drastically reduce the computational load by requiring the propagation of only a small set of parameters (typically 8 to 9 parameters, corresponding to 1-3 Gaussian peaks) instead of thousands of data points.
- Real-Time Compatibility: This computational efficiency allows the adaptive optimization steps (Bayesian update and prediction) to be performed quickly enough (on the microsecond scale) to enable real-time updates, compatible with fast digital electronic systems like FPGAs.
- Performance Trade-offs: For long coherence times (T2 = 100 ”s), the Gaussian protocol provides maximum speed gains while maintaining tracking accuracy comparable to the original method.
- Low T2 Advantage: Crucially, for short coherence times (T2 = 1 ”s), where the original method fails to track, the proposed Gaussian protocol demonstrates superior tracking accuracy, compensating for the smaller speed increase observed in this regime.
Technical Specifications
Section titled âTechnical Specificationsâ| Parameter | Value | Unit | Context |
|---|---|---|---|
| Maximum Speed Increase | 13.5 | Factor | Achieved when T2 = 100 ”s and overhead time (toh) = 2 ”s. |
| Typical Speed Increase | ~10 | Factor | Observed across most high T2 and toh regimes. |
| Minimum Speed Increase | 1.3 | Factor | Observed when T2 = 1 ”s and toh = 2 ”s. |
| Coherence Time (T2) Tested | 100, 10, 1 | ”s | Range of NV center coherence times used in simulations. |
| Measurement Overhead Time (toh) | 10, 6, 2 | ”s | Time allocated for computation and experimental setup between Ramsey measurements. |
| Diffusion Coefficient (Îș) | 10 to 50 | MHz Hz1/2 | Rate of magnetic field change tested for protocol robustness. |
| Average Parameters (Gaussian Protocol) | 8 to 9 | Parameters | Number of parameters (3x number of Gaussians) used during tracking phase. |
| Average Parameters (Original Protocol) | Thousands | Points | Number of frequency points used for grid discretization. |
| Tracking Fail Rate Threshold | >0.15 | MHz/ms | Defined Mean Squared Error (MSE) threshold for a âfailed run.â |
| Gaussian Pruning Threshold (Ath) | 0.04 | Amplitude | Threshold for discarding low-amplitude Gaussian terms in the mixture. |
| KL Divergence Threshold (KLth) | 0.001 | Unitless | Threshold used for merging highly similar Gaussian terms. |
Key Methodologies
Section titled âKey Methodologiesâ- Ramsey Measurement Basis: Magnetic field tracking is performed using Ramsey experiments, where the Larmor frequency (fB) is measured via the relative phase acquired by an electron spin superposition.
- Wiener Process Model: Magnetic field fluctuations (fB) are modeled as a Gaussian random walk (Wiener process) characterized by a diffusion coefficient (Îș), defining the prediction step distribution.
- Gaussian Mixture Approximation (GMM): The core technique involves approximating the oscillatory, cosine-shaped likelihood function P(”|fB, Ξ, Ï) using a finite sum of Gaussian functions derived from a Taylor expansion (assuming ideal conditions: no dephasing and perfect readout).
- Analytical Bayesian Update: The posterior distribution is calculated by multiplying the prior GMM (from the previous step) and the likelihood GMM. Since the product of two Gaussians is a new Gaussian, the update is performed analytically, avoiding numerical integration.
- Prediction (Convolution): The prior for the next time step is generated by convolving the current posterior GMM with the Gaussian random walk distribution, resulting in a new GMM with broadened variances.
- Adaptive Control Phase (Ξn): The measurement basis angle is optimized using the Fourier transform of the prior distribution (P2Ïn) to maximize the information gain from the subsequent measurement.
- Adaptive Sensing Time (Ïn): The sensing time is dynamically adjusted (doubled or halved) based on a figure of merit derived from the standard deviation of the posterior distribution, ensuring optimal accuracy per measurement.
- Pruning and Merging: To maintain computational efficiency, a pruning step removes Gaussian terms below a specified amplitude threshold (Ath), and a merging step combines similar Gaussian terms based on a Kullback-Leibler (KL) divergence threshold (KLth).
Commercial Applications
Section titled âCommercial Applicationsâ- Quantum Sensing and Metrology: Enabling real-time, adaptive operation of quantum sensors, particularly those based on single-qubit systems (like NV centers or silicon carbide defects), where fast feedback is essential.
- Nanoscale Current Detection: High-speed mapping and tracking of fluctuating currents in advanced electronic devices, such as integrated circuits and 2D materials, with nanoscale spatial resolution.
- Biomagnetism and Bio-Sensing: Monitoring dynamic processes in biological systems, including temperature drift and energy metabolism tracking inside living cells using nanodiamonds.
- Quantum Inertial Navigation: Fast tracking of the rotation and orientation of levitated nanodiamonds in traps (optical, magnetic, or ion traps), providing critical feedback for stabilization and realignment mechanisms.
- Coherence Engineering: Application in quantum computing and memory to track and suppress noise sources (like fluctuating nuclear spin baths) to extend the coherence time (T2) of central electron spins.
- Materials Characterization: Tracking magnetization dynamics and stray fields in novel solid-state systems and magnetic materials.
View Original Abstract
Abstract 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. For <mml:math xmlns:mml=âhttp://www.w3.org/1998/Math/MathMLâ display=âinlineâ overflow=âscrollâ> <mml:msubsup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>*</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mspace class=ânbspâ width=â0.3333emâ/> <mml:mi>ÎŒ</mml:mi> <mml:mi mathvariant=ânormalâ>s</mml:mi> </mml:math> , only a small decrease in computation time is achieved. However, in these regimes, the proposed Gaussian protocol outperforms the existing one in tracking accuracy.