Using Bayesian experimental design techniques, we have shown that for a single twolevel quantum mechanical system under strong (projective) measurement, the dynamical parameters of a model Hamiltonian can be estimated with exponentially improved accuracy over offline estimation strategies. To achieve this, we derive an adaptive protocol which finds the optimal experiments based on previous observations. We show that the risk associated with this algorithm is close to the global optimum, given a uniform prior. Additionally, we show that sampling at the Nyquist rate is not optimal.
As the size of quantum devices continues to grow, the development of scalable methods to characterise and diagnose noise is becoming an increasingly important problem. Recent methods have shown how to efficiently estimate Hamiltonians in principle, but they are poorly conditioned and can only characterize the system up to a scalar factor, making them difficult to use in practice. In this work we present a Bayesian methodology, called Bayesian Hamiltonian Learning (BHL), that addresses both of these issues by making use of any or all, of the following: well-characterised experimental control of Hamiltonian couplings, the preparation of multiple states, and the availability of any prior information for the Hamiltonian. Importantly, BHL can be used online as an adaptive measurement protocol, updating estimates and their corresponding uncertainties as experimental data become available. In addition, we show that multiple input states and control fields enable BHL to reconstruct Hamiltonians that are neither generic nor spatially local. We demonstrate the scalability and accuracy of our method with numerical simulations on up to 100 qubits. These practical results are complemented by several theoretical contributions. We prove that a
k-body Hamiltonian
H whose correlation matrix has a spectral gap
Δ can be estimated to precision
ε with only
O~(n3k/(εΔ)3/2) measurements. We use two subroutines that may be of independent interest: First, an algorithm to approximate a steady state of
H starting from an arbitrary input that converges factorially in the number of samples; and second, an algorithm to estimate the expectation values of
m Pauli operators with weight
≤k to precision
ε using only
O(ε−23klogm) measurements, which quadratically improves a recent result by Cotler and Wilczek.
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.
We investigate schemes for Hamiltonian parameter estimation of a two-level system using repeated measurements in a fixed basis. The simplest (Fourier based) schemes yield an estimate with a mean-square error (MSE) that decreases at best as a power law $\ensuremath{\sim}{N}^{\ensuremath{-}2}$ in the number of measurements
N. By contrast, we present numerical simulations indicating that an adaptive Bayesian algorithm, where the time between measurements can be adjusted based on prior measurement results, yields a MSE which appears to scale close to $\mathrm{exp}(\ensuremath{-}0.3N)$. That is, measurements in a single fixed basis are sufficient to achieve exponential scaling in
N.
This paper introduces an adaptive quantum sampling method designed to enhance the efficiency of Bayesian inference in complex, high-dimensional spaces. Traditional Bayesian inference methods often struggle with the computational complexity of exploring the vast parameter space, leading to slow convergence and limited accuracy. We propose a novel quantum sampling algorithm that dynamically adjusts the sampling process based on uncertainty estimates generated by quantum computations. This adaptive approach mitigates the inherent challenges of sampling in high-dimensional spaces, resulting in significantly faster convergence and improved inference accuracy. The core mechanism leverages quantum algorithms to identify promising regions of the parameter space, thereby reducing the computational burden and enhancing the effectiveness of Bayesian modeling. The paper details the theoretical framework underpinning the algorithm, outlines the implementation of the quantum sampling process, and provides a comprehensive analysis of its performance compared to traditional sampling techniques. Finally, we present preliminary experimental results demonstrating the effectiveness of the proposed method across a range of complex scenarios.
Bayesian quantum estimation provides a robust formalism for quantum technologies, particularly in scenarios with limited data and minimal prior information. Yet, its application to continuous-variable systems has remained limited and largely numerical due to the difficulty of the underlying parameter integrals. Here, we introduce a variational framework that restricts the optimisation over all measurements and estimators to a finite-dimensional subspace, reducing the problem to a linear system of equations with closed-form solutions. These solutions have a geometric interpretation as orthogonal projections of the global optimum onto the chosen subspace. For Gaussian states, a natural subspace is spanned by operators polynomial in the canonical quadratures, whose moments are computable from the first and second moments of the state. We further derive a necessary and sufficient condition for global optimality, and show that post-processing with the posterior mean, rather than using the estimator obtained from the subspace, improves performance towards the global optimum. Through single-shot examples, we show that the framework yields experimentally feasible strategies based on Gaussian operations and quadrature measurements that are either optimal or near-optimal.