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Optics Express

Computing the information rate of discrete-time Wiener phase noise channels by parametric Bayesian tracking

Luca Barletta

Published Nov 9, 2016• 1 citation
Publisher

Abstract

A new upper bound (UB) on the information rate (IR) transferred through the additive white Gaussian noise channel affected by Wiener's laser phase noise is proposed in the paper. The bound is based on Bayesian tracking of the noisy phase. Specifically, the predictive and posterior densities involved in the tracking are expressed in parametric form, therefore tracking is made on parameters. This make the method less computationally demanding than known non-parametric methods, e.g. methods based on phase quantization and trellis representation of phase memory. Simulation results show that the UB is so close to the lower bound that we can claim of having virtually computed the actual IR.

DOI
10.1364/oe.24.026612

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A new upper bound (UB) on the information rate (IR) transferred through the additive white Gaussian noise channel affected by Wiener's laser phase noise is proposed in the paper. The bound is based on Bayesian tracking of the noisy phase. Specifically, the predictive and posterior densities involved in the tracking are expressed in parametric form, therefore tracking is made on parameters. This make the method less computationally demanding than known non-parametric methods, e.g. methods based on phase quantization and trellis representation of phase memory. Simulation results show that the UB is so close to the lower bound that we can claim of having virtually computed the actual IR.
Non-conventional receivers for phase-coherent states based on non-Gaussian measurements such as photon counting surpass the sensitivity limits of shot-noise-limited coherent receivers, the quantum noise limit (QNL). These non-Gaussian receivers can have a significant impact in future coherent communication technologies. However, random phase changes in realistic communication channels, such as optical fibers, present serious challenges for extracting the information encoded in coherent states. While there are methods for correcting random phase noise with conventional heterodyne detection, phase-tracking for non-Gaussian receivers surpassing the QNL is still an open problem. Here we demonstrate phase tracking for non-Gaussian receivers to correct for time-varying phase noise while allowing for decoding beyond the QNL. The phase-tracking method performs real-time parameter estimation and correction of phase drifts using the data from the non-Gaussian discrimination measurement, without relying on phase reference pilot fields. This method enables non-Gaussian receivers to achieve higher sensitivities and rates of information transfer than ideal coherent receivers in realistic channels with time-varying phase noise. This demonstration makes sub-QNL receivers a more robust, feasible, and practical quantum technology for classical and quantum communications.
Quantum frequency estimation under general noise is investigated in the Bayesian parameter estimation approach. To evaluate the accuracy of estimation, the Bayes cost is obtained analytically which can be applied to the common noisy channels, such as the phase-damping channel, the amplitude-damping channel, and the depolarizing channel. The Bayes cost formula clearly shows that the prior information imposes a restriction upon the effect of noise in the estimation process. Three examples of frequency estimations are provided to illustrate the roles of prior probability in the estimation process. It is found that, due to the restriction of prior information on the noise, the estimation accuracy is less sensitive to the noise in the Bayesian approach than that in the Cram\'er-Rao bound approach. More prior information can help us to use the non-Gaussianity of the noise channel to improve the estimation accuracy in the phase-damping channel.
Employing quantum receivers to discriminate laser coherent states makes it feasible to achieve error probabilities below the standard quantum limit (SQL), particularly for photon-starved optical communication. However, the quantum receiver performance for phase shift keyed coherent states is severely degraded by random phase and intensity noise in realistic communication channels. This paper proposes a phase and intensity noise-tracking method based on the cubature Kalman filter (CKF) to improve the performance of time-resolving quantum receivers. The filter leverages the data collected during time-resolving state discrimination to perform real-time estimation and correction of time-varying phase and intensity noise. By conducting numerical simulations, we demonstrate that the proposed noise-tracking method enables the time-resolving quantum receiver to retain its superiority over the SQL while requiring significantly less computational effort than a Bayesian estimator. Additionally, the robustness against practical imperfections, including dead time and dark count rate of single photon detector (SPD), is shown through the simulations.
Phase estimation is known to be a robust method for single-qubit gate calibration in quantum computers, while Bayesian estimation is widely used in devising optimal methods for learning in quantum systems. We present a Bayesian method that adaptively chooses a control phase and the time of coherent evolution. The uniqueness of the approach is to use analytics to implement an efficient numerical algorithm that offers exact calculation of the Bayesian probability and associated utility functions, even in the presence of noise. We determine the utility of control parameter values using functions of the prior probability of the phase that quantify expected knowledge gain in terms of either expected narrowing of the posterior or expected information gain. We find that by maximizing the of expected gain we obtain phase estimates having standard deviation a factor of 1.43 larger than the Heisenberg limit using a classical sequential strategy for noise-free experiments. In the presence of dephasing, we find near-optimal performance with respect to the Cramér-Rao bound. We also demonstrate some robustness of the estimates to noise that is not accounted for in the model of the estimator, making the methods suitable for calibrating operations in quantum computers. The rate of gain can be adjusted to account for times in the experiment other than the coherent evolution of the unknown phase, such as times required for state preparation or readout. The combined features of the method make it well suited to single-qubit gate calibration in trapped-ion or neutral atom experiments and promises a wide range of other applications by offering near-optimal phase estimation that straightforwardly accounts for experimental imperfections.
Background Quantum phase estimation (QPE) is a foundational subroutine in quantum algorithms that ranges from Shor’s factoring scheme to variational quantum eigensolvers and quantum metrology protocols. Classical post-processing replacements for the inverse quantum Fourier transform (iQFT) —specifically Kitaev-style iterative estimation and Bayesian update schemes—substantially reduce circuit depth, making them practical for noisy intermediate-scale quantum (NISQ) hardware. Despite this promise, the statistical performance of these methods under the conditions that actually arise in practice—non-eigenstate inputs characterised by eigenstate overlap p ≡ O = | ψ | ϕ | 2 < 1 and mixed-state purity γ ≡ P = Tr ( ρ 2 ) < 1 —remains uncharacterised in the literature. Methods We develop a rigorous information-theoretic framework for Bayesian QPE (BQPE) beyond the eigenstate assumption. Starting from the binary measurement likelihood attenuated by the joint factor O P , we derive exact classical Fisher information expressions and apply the Cramér–Rao inequality to obtain tight variance lower bounds. We then construct an adaptive measurement protocol based on von Mises posterior updates, prove its asymptotic efficiency in the frequentist sense, and establish matching upper and lower bounds on sample complexity. All theoretical results are validated against 2,000 -trial Monte Carlo simulations and cross-checked against published nitrogen-vacancy (NV) centre and photonic experimental data. Results We prove that the fundamental variance floor for any unbiased frequentist estimator is ( 1 − p γ ) / ( N p γ n ̄ 2 ) 1 / ( N ( p γ ) 2 n ̄ 2 ) , where N
Non-Gaussian receivers for optical communication with coherent states can achieve measurement sensitivities beyond the limits of conventional detection, given by the quantum-noise limit (QNL).However, the amount of information that can be reliably transmitted substantially degrades if there is noise in the communication channel, unless the receiver is able to efficiently compensate for such noise.Here, we investigate the use of a deep neural network as a computationally efficient estimator of phase and amplitude channel noise to enable a reliable method for noise tracking for non-Gaussian receivers.The neural network uses the data collected by the non-Gaussian receiver to estimate and correct for dynamic channel noise in real time.Using numerical simulations, we find that this noise tracking method allows the non-Gaussian receiver to maintain its benefit over the QNL across a broad range of strengths and bandwidths of phase and intensity noise.The noise tracking method based on neural networks can further include other types of noise to ensure sub-QNL performance in channels with many sources of noise.