Frequency tracking and parameter estimation for robust quantum state estimation

Abstract

In this paper we consider the problem of tracking the state of a quantum system via a continuous weak measurement. If the system Hamiltonian is known precisely, this merely requires integrating the appropriate stochastic master equation. However, even a small error in the assumed Hamiltonian can render this approach useless. The natural answer to this problem is to include the parameters of the Hamiltonian as part of the estimation problem, and the full Bayesian solution to this task provides a state estimate that is robust against uncertainties. However, this approach requires considerable computational overhead. Here we consider a single qubit in which the Hamiltonian contains a single unknown parameter. We show that classical frequency estimation techniques greatly reduce the computational overhead associated with Bayesian estimation and provide accurate estimates for the qubit frequency.

Confidence and backaction in the quantum filter equation

“…[22], even a small error in the Hamiltonian of the above equation can induce errors in the estimate of the state provided by the quantum filter equation. We expect the same to be true of the estimates of the noise spectrum of the QPC.…”

Multiparameter estimation along quantum trajectories with sequential Monte Carlo methods

“…The secret to working with SMC methods is to pick a suitable proposal distribution to solve the problem in a robust manner using limited computational resources. In particular, a good choice of proposal distribution allows classical parameters to be estimated significantly more efficiently than when using an enumerative or grid based method [38], as was used for a one parameter Hybrid SME problem in [15,27].…”

“…A full description of the problem involves starting with a prior probability density for the parameters one wishes to determine and then using Bayes' theorem to continually update this probability density from the stream of measurement results as they are obtained. A number of authors have considered this problem [15,[18][19][20][21][22][23][24][25][26][27][28]. This is of particular interest when the parameters of a system change slowly with time, and one wishes to be able to track the variations in the parameters.…”

“…[15], where an approach was presented based on a set of parallel SMEs, each using a different set of parameter values contained in a vector λ, which have an associated probability. The final mixed state is then constructed by averaging over the probabilities for the classical parameters.…”

“…We begin our presentation by first reviewing other approaches to Hamiltonian parameter estimation, and the development of a set of Hybrid SMEs for the quantum evolution and the Kushner-Stratonovich equation for the classical parameters [15] in sections II and III, respectively. In section IV, we then discuss how efficient classical parameter estimation techniques [16] can be applied to the solution of the classical aspects of the Hybrid SME.…”