The ability to efficiently infer system parameters is essential in any signal-processing task that requires fast operation. In quantum systems, a serious challenge arises due to substantial growth of the underlying Hilbert space with the system size. As the statistics of the measurement data observed—i.e., the —can no longer be easily computed, common approaches such as maximum-likelihood estimators or particle filters become impractical. To address this issue, we propose the use of the approximate Bayesian computation (ABC) algorithm, which evades likelihood computation by sampling from a library of measurement data— prepared for a given quantum device. We apply ABC to interpret photodetection click patterns arising from real-time probing of a two-level atom and an optomechanical system. For the latter, we consider both linear and nonlinear regimes to show how to tailor the ABC algorithm by understanding the quantum measurement statistics. Our work demonstrates that fast parameter inference may be possible regardless of the complexity of the quantum device and the measurement scheme involved.
The ability to efficiently infer system parameters is essential in any signal-processing task that requires fast operation. In quantum systems, a serious challenge arises due to substantial growth of the underlying Hilbert space with the system size. As the statistics of the measurement data observed—i.e., the —can no longer be easily computed, common approaches such as maximum-likelihood estimators or particle filters become impractical. To address this issue, we propose the use of the approximate Bayesian computation (ABC) algorithm, which evades likelihood computation by sampling from a library of measurement data— prepared for a given quantum device. We apply ABC to interpret photodetection click patterns arising from real-time probing of a two-level atom and an optomechanical system. For the latter, we consider both linear and nonlinear regimes to show how to tailor the ABC algorithm by understanding the quantum measurement statistics. Our work demonstrates that fast parameter inference may be possible regardless of the complexity of the quantum device and the measurement scheme involved.
To operate quantum sensors at their quantum limit in real time, it is crucial to identify efficient data inference tools for rapid parameter estimation. In photodetection, the key challenge is the fast interpretation of click-patterns that exhibit non-classical statistics-the very features responsible for the quantum enhancement of precision. We achieve this goal by comparing Bayesian likelihood-free methods with ones based on deep learning (DL). While the former are more conceptually intuitive, the latter, once trained, provide significantly faster estimates with comparable precision and yield similar predictions of the associated errors, challenging a common misconception that DL lacks such capabilities. We first verify both approaches for an analytically tractable, yet multiparameter, scenario of a two-level system emitting uncorrelated photons. Our main result, however, is the application to a driven non-linear optomechanical device emitting non-classical light with complex multiclick correlations. Although both methods facilitate inference, only DL provides the execution speed required for real-time performance while naturally learning complex multi-photon click-patterns. Our results pave the way for real-time sensing and dynamical control of quantum devices that leverage non-classical effects in photodetection.
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.
Models of quantum systems faithfully map system parameters to observations, but the inverse problem of parameter inference from measurement data presents a fundamental challenge: computationally intractable likelihoods due to an exponentially large Hilbert space. Here, we introduce simulation-based quantum system inference, a unified, likelihood-free framework that learns parameter posteriors directly from classical simulation data. The central idea is to pair polynomial-cost classical simulators, such as Pauli propagation and tensor networks, with normalizing flows or other neural density estimators for accurate, reusable inference. A single model, trained once, maps any new measurement record to its posterior in one forward pass---turning per-experiment inference into a fixed, up-front cost. We numerically demonstrate the framework's versatility across Pauli noise learning, quantum error mitigation, quantum state tomography, and Hamiltonian learning, with examples involving 81-qubit shallow circuits and 735-parameter inference. In each case, the approach yields accurate estimates of identifiable parameters, while posterior uncertainty provides additional diagnostics of non-identifiability and indicates where further characterization is needed. Our framework reduces data-acquisition requirements in quantum experiments and accelerates parameter inference, providing a practical route to characterizing and improving large-scale quantum systems.
Quantum-enhanced Bayesian Network Inference presents a novel approach to analyzing complex systems by leveraging the unique properties of quantum computation. This paper explores a quantum algorithm designed to significantly accelerate inference processes within Bayesian networks, addressing the limitations of classical methods when confronted with high-dimensional and complex dependencies. We detail the core mechanism, outlining how quantum algorithms will be employed to enhance the exploration of the network's state space and the efficient inference of its parameters. The proposed method offers a potential pathway for overcoming the computational bottlenecks inherent in traditional Bayesian network inference, particularly in scenarios with substantial data complexity. This research emphasizes the promise of quantum acceleration and improved accuracy in complex systems modeling.
Identifying and calibrating quantitative dynamical models for physical quantum systems is important for a variety of applications. Here we present a closed-loop Bayesian learning algorithm for estimating multiple unknown parameters in a dynamical model, using optimized experimental ``probe'' controls and measurement. The estimation algorithm is based on a Bayesian particle filter, and is designed to autonomously choose informationally optimized probe experiments with which to compare to model predictions. We demonstrate the performance of the algorithm in both simulated calibration tasks and in an experimental single-qubit ion-trap system. Experimentally, we find that, with 60 times fewer samples, we exceed the precision of conventional calibration methods, delivering an approximately 93 times improvement in efficiency (as quantified by the reduction of measurements and resets required to achieve a target residual uncertainty and multiplied by the increase in accuracy). In simulated and experimental demonstrations, we see that successively longer pulses are selected as the posterior uncertainty iteratively decreases, leading to an exponential improvement in the accuracy of model parameters with the number of experimental queries, and a commensurate increase in the per-query time.
Identifying and calibrating quantitative dynamical models for physical quantum systems is important for a variety of applications. Here we present a closed-loop Bayesian learning algorithm for estimating multiple unknown parameters in a dynamical model, using optimised experimental "probe" controls and measurement. The estimation algorithm is based on a Bayesian particle filter, and is designed to autonomously choose informationally-optimised probe experiments with which to compare to model predictions. We demonstrate the performance of the algorithm in both simulated calibration tasks and in an experimental single-qubit ion-trap system. Experimentally, we find that with 60x fewer samples, we exceed the precision of conventional calibration methods, delivering an approximately 93x improvement in efficiency (as quantified by the reduction of measurements required to achieve a target residual uncertainty and multiplied by the increase in accuracy). In simulated and experimental demonstrations, we see that successively longer pulses are selected as the posterior uncertainty iteratively decreases, leading to an exponential improvement in the accuracy of model parameters with the number of experimental queries.