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arXiv:1507.07682 · Scientific Reports

Exact quantum Bayesian rule for qubit measurements in circuit QED

Wei Feng, Pengfei Liang, Lupei Qin, Xin-Qi Li

Published Feb 4, 2016• 11 citations
arXivHTMLPDF

Abstract

Developing efficient framework for quantum measurements is of essential importance to quantum science and technology. In this work, for the important superconducting circuit-QED setup, we present a rigorous and analytic solution for the effective quantum trajectory equation (QTE) after polaron transformation and converted to the form of Stratonovich calculus. We find that the solution is a generalization of the elegant quantum Bayesian approach developed in arXiv:1111.4016 by Korotokov and currently applied to circuit-QED measurements. The new result improves both the diagonal and off-diagonal elements of the qubit density matrix, via amending the distribution probabilities of the output currents and several important phase factors. Compared to numerical integration of the QTE, the resultant quantum Bayesian rule promises higher efficiency to update the measured state, and allows more efficient and analytical studies for some interesting problems such as quantum weak values, past quantum state, and quantum state smoothing. The method of this work opens also a new way to obtain quantum Bayesian formulas for other systems and in more complicated cases.

DOI
10.1038/srep20492

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Developing efficient framework for quantum measurements is of essential importance to quantum science and technology. In this work, for the important superconducting circuit-QED setup, we present a rigorous and analytic solution for the effective quantum trajectory equation (QTE) after polaron transformation and converted to the form of Stratonovich calculus. We find that the solution is a generalization of the elegant quantum Bayesian approach developed in arXiv:1111.4016 by Korotokov and currently applied to circuit-QED measurements. The new result improves both the diagonal and off-diagonal elements of the qubit density matrix, via amending the distribution probabilities of the output currents and several important phase factors. Compared to numerical integration of the QTE, the resultant quantum Bayesian rule promises higher efficiency to update the measured state, and allows more efficient and analytical studies for some interesting problems such as quantum weak values, past quantum state, and quantum state smoothing. The method of this work opens also a new way to obtain quantum Bayesian formulas for other systems and in more complicated cases.
Compared with the quantum trajectory equation (QTE), the quantum Bayesian approach has the advantage of being more efficient to infer a quantum state under monitoring, based on the integrated output of measurements. For weak measurement of qubits in circuit quantum electrodynamics (cQED), properly accounting for the measurement backaction effects within the Bayesian framework is an important problem of current interest. Elegant work towards this task was carried out by Korotkov in 'bad-cavity' and weak-response limits (Korotkov 2011 Quantum Bayesian approach to circuit QED measurement (arXiv: 1111.4016 )). In the present work, based on insights from the cavity-field states (dynamics) and the help of an effective QTE, we generalize the results of Korotkov to more general system parameters. The obtained Bayesian rule is in full agreement with Korotkovʼs result in limiting cases and as well holds satisfactory accuracy in non-limiting cases in comparison with the QTE simulations. We expect the proposed Bayesian rule to be useful for future cQED measurement and control experiments.
Designing high-precision and efficient protocols is of crucial importance for quantum parameter estimation in practice. Estimation based on continuous quantum measurement is one possible type of this, which also appears to be the most natural choice for continuous dynamical processes. In this work we consider the state-of-the-art superconducting circuit quantum-electrodynamics (QED) systems, where high-quality continuous measurements have been extensively performed in the past decade. Within the framework of Bayesian estimation and particularly using the quantum Bayesian rule in circuit QED, we numerically simulate the likelihood function as an estimator for the Rabi frequency of qubit oscillations. We find that, by proper design of the interaction strength of measurement, the estimate precision can scale with the measurement time beyond the standard quantum limit, which is usually assumed for this type of continuous measurement. This unexpected result is supported by the simulated Fisher information and can be understood as a consequence of the quantum correlation between the output signals by simulating the effect of quantum efficiency of measurement.
Using a superconducting circuit QED device, we present a theoretical study of real-time quantum state estimation via the quantum Bayesian approach. Suitable conditions under which the quantum Bayesian approach can accurately update the density matrix of the qubit are analyzed. We also consider the correlation between some basic and physically meaningful parameters of the circuit QED and the performance of the Bayesian approach. Our results advance the understanding of the quantum Bayesian approach.
Solid-state superconducting circuit-quantum electrodynamics (QED) system is a promising candidate for quantum information processing and an ideal platform for quantum measurement and quantum control studies. As an extension to our previous simulation for single qubit circuit-QED, in this work we simulate the quantum measurement and control of multi-qubit system. Particularly, we consider the deterministic generation of a two-qubit Bell state. In this context we examine the validity conditions of two cavity-photon-elimination scheme. On the level of quantum trajectory simulation, we find that, owing to the qubit flip caused by feedback, the advanced polaron-transformation scheme is no longer applicable if the measurement is not weak, which also makes meaningless the elegant effective measurement operator.
We present a theoretical study of a superconducting charge qubit dispersively coupled to a transmission line resonator. Starting from a master equation description of this coupled system and using a polaron transformation, we obtain an exact effective master equation for the qubit. We then use quantum trajectory theory to investigate the measurement of the qubit by continuous homodyne measurement of the resonator out field. Using the same polaron transformation, a stochastic master equation for the conditional state of the qubit is obtained. From this result, various definitions of the measurement time are studied. Furthermore, we find that in the limit of strong homodyne measurement, typical quantum trajectories for the qubit exhibit a crossover from diffusive to jumplike behavior. Finally, in the presence of Rabi drive on the qubit, the qubit dynamics is shown to exhibit quantum Zeno behavior.
We consider continuous quantum measurement of a superconducting qubit in the circuit QED setup with a moderate bandwidth of the measurement resonator, i.e., when the ``bad-cavity'' limit is not applicable. The goal is a simple description of the quantum evolution due to measurement, i.e., the measurement backaction. Extending the quantum Bayesian approach previously developed for the bad-cavity regime, we show that the evolution equations remain the same, but now they should be applied to the entangled qubit-resonator state, instead of the qubit state alone. The derivation uses only elementary quantum mechanics and basic properties of coherent states, thus being accessible to nonexperts.