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arXiv:0907.5034 · Physical Review A

Frequency tracking and parameter estimation for robust quantum state estimation

Jason F. Ralph, Kurt A. Jacobs, Charles D. Hill

Published Nov 28, 2011• 42 citations
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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.

DOI
10.1103/physreva.84.052119

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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.
Quantum state estimation, based on the numerical integration of stochastic master equations (SMEs), provides estimates for the evolution of quantum systems subject to continuous weak measurements. The approach is similar to classical state estimation methods in that the ‘quantum trajectories’ produced by solving the SME are conditioned on continuous classical measurement signals. In this paper, we explore the use of classical state estimation for a candidate quantum system, one based on an experimentally realisable system: a material object undergoing continuous feedback cooling in an optical trap. In particular, we demonstrate that classical tracking methods based on particle filters can be used to track quantum states, and are particularly useful for higher temperature regimes where quantum state estimation becomes computationally demanding.
We discuss how continuous probing of a quantum system allows estimation of unknown classical parameters embodied in the Hamiltonian of the system. We generalize the stochastic master equation associated with continuous observation processes to a Bayesian filter equation for the probability distribution of the desired parameters, and we illustrate its application by estimating the direction of a magnetic field. In our example, the field causes a ground state spin precession in a two-level atom which is detected by the polarization rotation of off-resonant optical probes, interacting with the atomic spin components.
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.
We investigate the determination of a Hamiltonian parameter in a quantum system undergoing continuous measurement. We demonstrate a computationally rapid method to estimate an unknown and possibly time-dependent parameter, where we maximize the likelihood of the observed stochastic readout. By dealing directly with the raw measurement record rather than the quantum-state trajectories, the estimation can be performed while the data are being acquired, permitting continuous tracking of the parameter during slow drifts in real time. Furthermore, we incorporate realistic nonidealities, such as decoherence processes and measurement inefficiency. As an example, we focus on estimating the value of the Rabi frequency of a continuously measured qubit and compare maximum likelihood estimation to a simpler fast Fourier transform. Using this example, we discuss how the quality of the estimation depends on both the strength and the duration of the measurement; we also discuss the trade-off between the accuracy of the estimate and the sensitivity to drift as the estimation duration is varied.
Fast measurement, signal processing, and accurate estimation of Hamiltonian parameters are essential for feedback control in quantum-classical interface circuitry. However, existing frequentist and Bayesian inference methods typically require a large number of measurements to achieve the accuracy needed to mitigate qubit decoherence. Consequently, feedback control of semiconductor qubits has largely been limited to single-qubit frequency stabilization, whereas two-qubit parameter stabilization remains experimentally unexplored. Here, we demonstrate a real-time feedback framework based on sequential Monte Carlo estimation using one bit of data from a single-shot measurement. Using a four-qubit semiconductor quantum dot device, we rapidly estimate individual qubit frequencies, yielding an approximately twofold increase in coherence time compared with a conventional Bayesian strategy. Moreover, sequential two-qubit parameter estimation using two bits of data enables stabilization of qubit-qubit coupling, allowing both quasi-static frequency drift and exchange-interaction noise to be estimated and suppressed. By shortening the time required for precise parameter estimation, these results demonstrate the importance of the synergistic development of classical and quantum electronics for building robust and scalable quantum technologies in fluctuating environments.
We review the introduction of likelihood functions and Fisher information in classical estimation theory, and we show how they can be defined in a very similar manner within quantum measurement theory. We show that the stochastic master equations describing the dynamics of a quantum system subject to a definite set of measurements provides likelihood functions for unknown parameters in the system dynamics, and we show that the estimation error, given by the Fisher information, can be identified by stochastic master equation simulations. For large parameter spaces we describe and illustrate the efficient use of Markov chain Monte Carlo sampling of the likelihood function.