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arXiv:2410.11955

Parameters estimation by fitting correlation functions of continuous quantum measurement

Pierre Guilmin, Pierre Rouchon, Antoine Tilloy

Published Oct 15, 2024• 0 citations
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Abstract

We propose a simple method to estimate the parameters of a continuously measured quantum system, by fitting correlation functions of the measured signal. We demonstrate the approach in simulation, both on toy examples and on a recent superconducting circuits experiment which proved particularly difficult to characterise using conventional methods. The idea is applicable to any system whose evolution is described by a jump or diffusive stochastic master equation. It allows the simultaneous estimation of many parameters, is practical for everyday use, is suitable for large Hilbert space dimensions, and takes into account experimental constraints such as detector imperfections and signal filtering and digitisation. Unlike existing methods, it also provides a direct way to understand how each parameter is estimated from the measured signal. This makes the approach interpretable, facilitates debugging, and enables validating the adequacy of a model with the observed data.

DOI
10.48550/arxiv.2410.11955

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This Rapid Communication provides an exact formula for the signal nnn-point correlation functions of detectors continuously measuring an arbitrary quantum system, in the presence of detection imperfections. The derivation uses only continuous stochastic calculus techniques, but the final result is easily understood from a discrete picture of repeated interactions with qubits or from a parallel with continuous matrix product states. This result provides a crude yet efficient a way to estimate system parameters directly from experimental data, without requiring nonlinear state reconstruction.
Continuous measurement of quantum systems gives rise to stochastic dynamics of the conditional quantum state, described by diffusive stochastic master equations. In this paper, we study parameter estimation for such equations when the Hamiltonian and measurement operators depend on unknown parameters. Based on multiple independent observed trajectories with a known initial state, we construct a contrast function using the deterministic averaged state and define a maximum contrast estimator for the unknown parameter. We prove strong consistency and asymptotic normality of the parameter estimator in a fixed-time, many-trajectory asymptotic regime. A key point is that the covariance matrix appearing in the asymptotic normality is given in a form that naturally leads to a consistent covariance estimator. This covariance estimator is computable from the observed data together with the deterministic averaged dynamics, so the asymptotic normality result can be used to construct standard errors and assess uncertainty for the parameter estimator.
Abstract We present an inference method utilizing artificial neural networks for parameter estimation of a quantum probe monitored through a single continuous measurement. Unlike existing approaches focusing on the diffusive signals generated by continuous weak measurements, our method harnesses quantum correlations in discrete photon-counting data characterized by quantum jumps. We benchmark the precision of this method against Bayesian inference, which is optimal in the sense of information retrieval. By using numerical experiments on a two-level quantum system, we demonstrate that our approach can achieve a similar optimal performance as Bayesian inference, while drastically reducing computational costs. Additionally, the method exhibits robustness against the presence of imperfections in both measurement and training data. This approach offers a promising and computationally efficient tool for quantum parameter estimation with photon-counting data, relevant for applications such as quantum sensing or quantum imaging, as well as robust calibration tasks in laboratory-based settings.
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