Estimating parameters of quantum systems is usually done by performing a sequence of predetermined experiments and post-processing the resulting data. It is known that online design, where the choice of the next experiment is based on the most up-to-date knowledge about the system, can offer speedups to parameter estimation. We apply online Bayesian experiment design to a Nitrogen Vacancy (NV) in diamond to learn the values of a five-parameter model describing its Hamiltonian and decoherence process. Comparing this to standard pre-determined experiment sweeps, we find that we can achieve median posterior variances on some parameters that are between 10 and 100 times better given the same amount of data. This has applications to NV magnetometry where one of the Hamiltonian coefficients is the parameter of interest. Furthermore, the methods that we use are generic and can be adapted to any quantum device.
Estimating parameters of quantum systems is usually done by performing a sequence of predetermined experiments and post-processing the resulting data. It is known that online design, where the choice of the next experiment is based on the most up-to-date knowledge about the system, can offer speedups to parameter estimation. We apply online Bayesian experiment design to a Nitrogen Vacancy (NV) in diamond to learn the values of a five-parameter model describing its Hamiltonian and decoherence process. Comparing this to standard pre-determined experiment sweeps, we find that we can achieve median posterior variances on some parameters that are between 10 and 100 times better given the same amount of data. This has applications to NV magnetometry where one of the Hamiltonian coefficients is the parameter of interest. Furthermore, the methods that we use are generic and can be adapted to any quantum device.
In this work we combine two distinct machine learning methodologies, sequential Monte Carlo and Bayesian experimental design, and apply them to the problem of inferring the dynamical parameters of a quantum system. We design the algorithm with practicality in mind by including parameters that control trade-offs between the requirements on computational and experimental resources. The algorithm can be implemented online (during experimental data collection), avoiding the need for storage and post-processing. Most importantly, our algorithm is capable of learning Hamiltonian parameters even when the parameters change from experiment-to-experiment, and also when additional noise processes are present and unknown. The algorithm also numerically estimates the Cramer–Rao lower bound, certifying its own performance.
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.
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 analysis of photon count data from the standard nitrogen vacancy (NV) measurement process is treated as a statistical inference problem. This has applications toward gaining better and more rigorous error bars for tasks such as parameter estimation (e.g. magnetometry), tomography, and randomized benchmarking. We start by providing a summary of the standard phenomenological model of the NV optical process in terms of Lindblad jump operators. This model is used to derive random variables describing emitted photons during measurement, to which finite visibility, dark counts, and imperfect state preparation are added. NV spin-state measurement is then stated as an abstract statistical inference problem consisting of an underlying biased coin obstructed by three Poisson rates. Relevant frequentist and Bayesian estimators are provided, discussed, and quantitatively compared. We show numerically that the risk of the maximum likelihood estimator is well approximated by the Cramér–Rao bound, for which we provide a simple formula. Of the estimators, we in particular promote the Bayes estimator, owing to its slightly better risk performance, and straightforward error propagation into more complex experiments. This is illustrated on experimental data, where quantum Hamiltonian learning is performed and cross-validated in a fully Bayesian setting, and compared to a more traditional weighted least squares fit.
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.
We explore methodologies for characterizing and controlling small quantum systems. We are interested in starting with a description of a quantum system, designing estimators for parameters of the system, developing robust and high-fidelity gates for the system using knowledge of these parameters, and experimentally verifying the performance of these gates. A strong emphasis is placed on using rigorous statistical methods, especially Bayesian ones, to analyze quantum system data. Throughout this thesis, the Nitrogen Vacancy system is used as an experimental testbed. Characterization of system parameters is done using quantum Hamiltonian learning, where we explore the use of adaptive experiment design to speed up learning rates. Gates for the full three-level system are designed with numerical optimal control methods that take into account imperfections of the control hardware. Gate quality is assessed using randomized benchmarking protocols, including standard randomized benchmarking, unitarity benchmarking, and leakage/loss benchmarking.