Multi-parameter Estimation Problem Research Articles

Achieving the Multiparameter Quantum Cramér-Rao Bound with Antiunitary Symmetry.

The estimation of multiple parameters is a ubiquitous requirement in many quantum metrology applications. However, achieving the ultimate precision limit, i.e., the quantum Cramér-Rao bound, becomes challenging in these scenarios compared to single parameter estimation. To address this issue, optimizing the parameters encoding strategies with the aid of antiunitary symmetry is a novel and comprehensive approach. For demonstration, we propose two types of quantum statistical models exhibiting antiunitary symmetry in experiments. The results showcase the simultaneous achievement of ultimate precision for multiple parameters without any trade-off and the precision is improved at least twice compared to conventional encoding strategies. Our work emphasizes the significant potential of antiunitary symmetry in addressing multiparameter estimation problems.

Characterization of partially accessible anisotropic spin chains in the presence of anti-symmetric exchange

We address quantum characterization of anisotropic spin chains in the presence of anti-symmetric exchange, and investigate whether the Hamiltonian parameters of the chain may be estimated with precision approaching the ultimate limit imposed by quantum mechanics. At variance with previous approaches, we focus on the information that may be extracted by measuring only two neighboring spins rather than a global observable on the entire chain. We evaluate the Fisher information (FI) of a two-spin magnetization measure, and the corresponding quantum Fisher information (QFI), for all the relevant parameters, i.e. the spin coupling, the anisotropy, and the Dzyaloshinskii–Moriya (DM) parameter. Our results show that the reduced system made of two neighboring spins may be indeed exploited as a probe to characterize global properties of the entire system. In particular, we find that the ratio between the FI and the QFI is close to unit for a large range of the coupling values. The DM coupling is beneficial for coupling estimation, since it leads to the presence of additional bumps and peaks in the FI and QFI, which are not present in a model that neglects exchange interaction and may be exploited to increase the robustness of the overall estimation procedure. Finally, we address the multiparameter estimation problem, and show that the model is compatible but sloppy, i.e. both the Uhlmann curvature and the determinant of the QFI matrix vanish. Physically, this means that the state of the system actually depends only on a reduced numbers of combinations of parameters, and not on all of them separately.

Variational quantum algorithm for experimental photonic multiparameter estimation

Variational quantum metrology represents a powerful tool to optimize estimation strategies, resulting particularly beneficial for multiparameter estimation problems that often suffer from limitations due to the curse of dimensionality and computational complexity. To overcome these challenges, we develop a variational approach able to efficiently optimize a quantum multiphase sensor. Leveraging the reconfigurability of an integrated photonic device, we implement a hybrid quantum-classical feedback loop able to enhance the estimation performances. The quantum circuit evaluations are used to compute the system partial derivatives by applying the parameter-shift rule, and thus reconstruct experimentally the Fisher information matrix. This in turn is adopted as the cost function of a classical learning algorithm run to optimize the measurement settings. Our experimental results showcase significant improvements in estimation accuracy and noise robustness, highlighting the potential of variational techniques for practical applications in quantum sensing and more generally in quantum information processing using photonic circuits.

Multiband Delay Estimation for Localization Using a Two-Stage Global Estimation Scheme

The time of arrival (TOA)-based localization techniques, which need to estimate the delay of the line-of-sight (LoS) path, have been widely employed in location-aware networks. To achieve a high-accuracy delay estimation, a number of multiband-based algorithms have been proposed recently, which exploit the channel state information (CSI) measurements over multiple non-contiguous frequency bands. However, to the best of our knowledge, there still lacks an efficient scheme that fully exploits the multiband gains when the phase distortion factors caused by hardware imperfections are considered, due to that the associated multi-parameter estimation problem contains many local optimums and the existing algorithms can easily get stuck in a “bad” local optimum. To address these issues, we propose a novel two-stage global estimation (TSGE) scheme for multiband delay estimation. In the coarse stage, we exploit the group sparsity structure of the multiband channel and propose a Turbo Bayesian inference (Turbo-BI) algorithm to achieve a good initial delay estimation based on a coarse signal model, which is transformed from the original multiband signal model by absorbing the carrier frequency terms. The estimation problem derived from the coarse signal model contains fewer local optimums and thus a more stable estimation can be achieved than directly using the original signal model. Then in the refined stage, with the help of coarse estimation results to narrow down the search range, we perform a global delay estimation using a particle swarm optimization-least square (PSO-LS) algorithm based on a refined multiband signal model to exploit the multiband gains to further improve the estimation accuracy. Simulation results show that the proposed TSGE significantly outperforms the benchmarks with comparative computational complexity.

Parameter estimation for unsteady MHD oscillatory free convective flow of generalized second grade fluid with Hall effects and thermal radiation effects

This work first establishes an unsteady magnetohydrodynamic (MHD) oscillatory free convection flow model of the generalized second grade fluid with Hall heat and mass transfer effect in a straight rectangular duct. A fast second-order spectral method with fractional backward difference formula (FBDF) is proposed to obtain the numerical solution of the established model. The distribution of the velocity and the temperature fields is discussed, and the effect of each parameter on the velocity and temperature fields is shown through graphical experiments. Moreover, in order to avoid the problems of slow computational speed and low accuracy of traditional numerical methods, new fractional physics-informed neural networks (PINNs) are proposed for the multi-parameter estimation of this model. At the same time, a comparative study of our method with the Bayesian method is presented. Experimental results show that fractional PINNs is more effective for multi-parameter estimation problems and get better accuracy at the same number of parameters.

Joint optimal measurement for locating two incoherent optical point sources near the Rayleigh distance

Simultaneously optimizing the estimation of the centroid and separation of two incoherent optical point sources is constrained by a tradeoff relation determined by an incompatibility coefficient. At the Rayleigh distance, the incompatibility coefficient vanishes and thus the tradeoff relation no longer restricts the simultaneous optimization of measurement for a joint estimation. We construct such a joint optimal measurement by an elaborated analysis on the operator algebra of the symmetric logarithmic derivative. Our work not only confirms the existence of a joint optimal measurement for this specific imaging model, but also gives a promising method to characterize the condition on measurement compatibility for general multiparameter estimation problems.

Quantum-enhanced differential atom interferometers and clocks with spin-squeezing swapping

Thanks to common-mode noise rejection, differential configurations are crucial for realistic applications of phase and frequency estimation with atom interferometers. Currently, differential protocols with uncorrelated particles and mode-separable settings reach a sensitivity bounded by the standard quantum limit (SQL). Here we show that differential interferometry can be understood as a distributed multiparameter estimation problem and can benefit from both mode and particle entanglement. Our protocol uses a single spin-squeezed state that is mode-swapped among common interferometric modes. The mode swapping is optimized to estimate the differential phase shift with sub-SQL sensitivity. Numerical calculations are supported by analytical approximations that guide the optimization of the protocol. The scheme is also tested with simulation of noise in atomic clocks and interferometers.

Three-Parameter Estimation Method of Multiple Hybrid Weibull Distribution Based on the EM Optimization Algorithm

The hybrid Weibull distribution model can describe the failure rules of electromechanical products more accurately than the single Weibull distribution model, and it can improve the accuracy of reliability analysis. However, the hybrid Weibull distribution model is also more complex, and the multi-parameter estimation is more difficult. In this paper, a reliability mathematical model based on the two-fold three-parameter hybrid Weibull distribution model was established, an EM optimization algorithm was derived for its solution, and a practical initial parameter selection scheme was designed. The validity of the model and the algorithm were verified, and goodness-of-fit tests were conducted through an arithmetic example. The results showed that the initial value selection scheme proposed in this paper and the corresponding solution algorithm could solve all the parameters and weight coefficients to be estimated for each sub distribution, and the obtained failure probability fitting curve had a high fit with the actual sample data, which effectively solved the multi-parameter estimation problem of the multiple mixed Weibull distribution model.

Joint range and angle estimation of low-elevation target with bistatic meter-wave FDA-MIMO radar

Frequency diverse array multiple-input-multiple-output (FDA-MIMO) radar, which has gained much attention in recent years, can be used for the estimation of multidimensional parameters due to the range-dependent property of the beampattern. The localization of low-elevation targets is a key problem to be solved in meter-wave radar. Combining the advantages of the bistatic system, this paper investigates the joint direction of departure (DOD), direction of arrival (DOA), and range multiparameter estimation problem of the bistatic meter-wave FDA-MIMO radar. Firstly, the signal model of the bistatic meter-wave FDA-MIMO radar is derived. Then the signal model is deformed and then applied to the generalized multiple signal classification (MUSIC) algorithm which does not need to consider the signal coherence. A better performance steering vector matching (SVM) reduced dimension MUSIC algorithm is proposed by exploring the signal model characteristics. Then the computational complexity of the two algorithms and the derivation of Cramer Rao bound (CRB) for estimating low-elevation target parameters is given. The innovation of this paper is to provide a joint localization model for bistatic meter-wave FDA-MIMO, which extends the study of bistatic FDA-MIMO radar to the field of coherent signals. The simulation results verify the effectiveness of two localization algorithms without decoherence processing.

Estimation of Gaussian random displacement using non-Gaussian states

In continuous-variable quantum information processing, quantum error correction of Gaussian errors requires simultaneous estimation of both quadrature components of displacements on phase space. However, quadrature operators $x$ and $p$ are non-commutative conjugate observables, whose simultaneous measurement is prohibited by the uncertainty principle. Gottesman-Kitaev-Preskill (GKP) error correction deals with this problem using complex non-Gaussian states called GKP states. On the other hand, simultaneous estimation of displacement using experimentally feasible non-Gaussian states has not been well studied. In this paper, we consider a multi-parameter estimation problem of displacements assuming an isotropic Gaussian prior distribution and allowing post-selection of measurement outcomes. We derive a lower bound for the estimation error when only Gaussian operations are used, and show that even simple non-Gaussian states such as single-photon states can beat this bound. Based on Ghosh's bound, we also obtain a lower bound for the estimation error when the maximum photon number of the input state is given. Our results reveal the role of non-Gaussianity in the estimation of displacements, and pave the way toward the error correction of Gaussian errors using experimentally feasible non-Gaussian states.

A variational toolbox for quantum multi-parameter estimation

With an ever-expanding ecosystem of noisy and intermediate-scale quantum devices, exploring their possible applications is a rapidly growing field of quantum information science. In this work, we demonstrate that variational quantum algorithms feasible on such devices address a challenge central to the field of quantum metrology: The identification of near-optimal probes and measurement operators for noisy multi-parameter estimation problems. We first introduce a general framework that allows for sequential updates of variational parameters to improve probe states and measurements and is widely applicable to both discrete and continuous-variable settings. We then demonstrate the practical functioning of the approach through numerical simulations, showcasing how tailored probes and measurements improve over standard methods in the noisy regime. Along the way, we prove the validity of a general parameter-shift rule for noisy evolutions, expected to be of general interest in variational quantum algorithms. In our approach, we advocate the mindset of quantum-aided design, exploiting quantum technology to learn close to optimal, experimentally feasible quantum metrology protocols.

Confocal super-resolution microscopy based on a spatial mode sorter

Spatial resolution is one of the most important specifications of an imaging system. Recent results in the quantum parameter estimation theory reveal that an arbitrarily small distance between two incoherent point sources can always be efficiently determined through the use of a spatial mode sorter. However, extending this procedure to a general object consisting of many incoherent point sources remains challenging, due to the intrinsic complexity of multi-parameter estimation problems. Here, we generalize the Richardson-Lucy (RL) deconvolution algorithm to address this challenge. We simulate its application to an incoherent confocal microscope, with a Zernike spatial mode sorter replacing the pinhole used in a conventional confocal microscope. We test different spatially incoherent objects of arbitrary geometry, and we find that the resolution enhancement of sorter-based microscopy is on average over 30% higher than that of a conventional confocal microscope using the standard RL deconvolution algorithm. Our method could potentially be used in diverse applications such as fluorescence microscopy and astronomical imaging.

Tight Bounds on the Simultaneous Estimation of Incompatible Parameters

The estimation of multiple parameters in quantum metrology is important for a vast array of applications in quantum information processing. However, the unattainability of fundamental precision bounds for incompatible observables has greatly diminished the applicability of estimation theory in many practical implementations. The Holevo Cramer-Rao bound (HCRB) provides the most fundamental, simultaneously attainable bound for multi-parameter estimation problems. A general closed form for the HCRB is not known given that it requires a complex optimisation over multiple variables. In this work, we develop an analytic approach to solving the HCRB for two parameters. Our analysis reveals the role of the HCRB and its interplay with alternative bounds in estimation theory. For more parameters, we generate a lower bound to the HCRB. Our work greatly reduces the complexity of determining the HCRB to solving a set of linear equations that even numerically permits a quadratic speedup over previous state-of-the-art approaches. We apply our results to compare the performance of different probe states in magnetic field sensing, and characterise the performance of state tomography on the codespace of noisy bosonic error-correcting codes. The sensitivity of state tomography on noisy binomial codestates can be improved by tuning two coding parameters that relate to the number of correctable phase and amplitude damping errors. Our work provides fundamental insights and makes significant progress towards the estimation of multiple incompatible observables.

Acid hydrolysis-based sugarcane bagasse biorefining for levulinic acid production: Dynamic mechanistic modeling under varying operating conditions

The study on the lignocellulosic material conversion into bio-based platform chemicals, such as levulinic acid (LA), is one of the most promising routes to promote the development of advanced biorefineries. In this work, a dynamic mechanistic model is developed to simulate the LA production from lignocellulosic material. A wide operating range is used to estimate the parameters of the reaction kinetics. Because multi-parameter estimation problem is complex, a genetic algorithm-based optimization procedure is used to determine the optimum parameters values. Measurements are obtained for various reaction times (0 - 45 min) temperatures (150 – 200 °C) and acid concentration of 7.0 % w/v H2SO4. The calculated reaction rates for the state variables, concentrations of LA, glucose, 5-hydroxymethylfurfural and humins are used to construct the dynamic mechanistic model. The prediction of measured state variables was particularly accurate, as determined by the root mean square error (RMSE) and correlation coefficient (R2). Therefore, a satisfactory agreement between experimental LA yield of 57.2 mol% and computed LA yield of 56.4 mol% was achieved (at 200 °C, 7.0 % w/v H2SO4, 45 min). The proposed methodology drives the systematic development of an industrially reliable dynamic mechanistic model for LA production from sugarcane bagasse as a means to increase the LA yields in the biorefinery.

On the Quantumness of Multiparameter Estimation Problems for Qubit Systems.

The estimation of more than one parameter in quantum mechanics is a fundamental problem with relevant practical applications. In fact, the ultimate limits in the achievable estimation precision are ultimately linked with the non-commutativity of different observables, a peculiar property of quantum mechanics. We here consider several estimation problems for qubit systems and evaluate the corresponding quantumness , a measure that has been recently introduced in order to quantify how incompatible the parameters to be estimated are. In particular, is an upper bound for the renormalized difference between the (asymptotically achievable) Holevo bound and the SLD Cramér-Rao bound (i.e., the matrix generalization of the single-parameter quantum Cramér-Rao bound). For all the estimation problems considered, we evaluate the quantumness and, in order to better understand its usefulness in characterizing a multiparameter quantum statistical model, we compare it with the renormalized difference between the Holevo and the SLD-bound. Our results give evidence that is a useful quantity to characterize multiparameter estimation problems, as for several quantum statistical model, it is equal to the difference between the bounds and, in general, their behavior qualitatively coincide. On the other hand, we also find evidence that, for certain quantum statistical models, the bound is not in tight, and thus may overestimate the degree of quantum incompatibility between parameters.

On quantumness in multi-parameter quantum estimation

In this article we derive a measure of quantumness in quantum multi-parameter estimation problems. We can show that the ratio between the mean Uhlmann curvature and the Fisher information provides a figure of merit which estimates the amount of incompatibility arising from the quantum nature of the underlying physical system. This ratio accounts for the discrepancy between the attainable precision in the simultaneous estimation of multiple parameters and the precision predicted by the Cramér–Rao bound. As a testbed for this concept, we consider a quantum many-body system in thermal equilibrium, and explore the quantum compatibility of the model across its phase diagram.

On super-resolution imaging as a multiparameter estimation problem

We consider the problem of characterizing the spatial extent of a composite light source using the super-resolution imaging technique based on mode demultiplexing when the centroid of the source is not known precisely. We show that the essential features of this problem can be mapped onto a simple qubit model for joint estimation of a phase shift and a dephasing strength.

Joint Estimation of Carrier and Sampling Frequency Offsets in the Uplink of Multiuser OFDM Systems

This paper considers the joint estimation of the carrier frequency offset (CFO) and sampling frequency offset (SFO) in the uplink of multiuser orthogonal frequency division multiplexing systems, where the received signal at the base station is a mixture of all users’ signals and each user is affected by different frequency offsets. Usually, users’ separation is needed to cope with this challenging multi-parameter estimation problem. Conventional methods adopt the zero-forcing criterion, which will lead to noise enhancement. To overcome this problem, a novel joint CFO and SFO estimation algorithm is proposed. We use the QR decomposition to achieve users’ separation, and the successive interference cancellation technique is employed to estimate the frequency offsets of different users. To further improve the estimation accuracy, the minimum variance unbiased estimator is exploited. Numerical results show that the proposed algorithm offers better performance than existing methods in terms of mean squared error and bit error rate .

Quantum-enhanced multi-parameter estimation for unitary photonic systems

Precise device characterisation is a fundamental requirement for a large range of applications using photonic hardware, and constitutes a multi-parameter estimation problem. Estimates based on measurements using single photons or classical light have precision which is limited by shot noise, while quantum resources can be used to achieve sub-shot-noise precision. However, there are many open questions with regard to the best quantum protocols for multi-parameter estimation, including the ultimate limits to achievable precision, as well as optimal choices for probe states and measurements. In this paper, we develop a formalism based on Fisher information to tackle these questions for setups based on linear-optical components and photon-counting measurements. A key ingredient of our analysis is a mapping for equivalent protocols defined for photonic and spin systems, which allows us to draw upon results in the literature for general finite-dimensional systems. Motivated by the protocol in Zhou, et al Optica 2, 510 (2015), we present new results for quantum-enhanced tomography of unitary processes, including a comparison of Holland-Burnett and NOON probe states.

Quantum multiparameter estimation with generalized balanced multimode NOON-like states

The simultaneous multi-parameter estimation problem using a class of multi-mode entangled states is investigated in this paper. Specifically, the problem of optical phase imaging is considered and the quantum probe is taken to be a balanced coherent superposition of components with an arbitrary quantum state in one mode and vacuum states in all the other modes, which is a generalization of the multi-mode NOON state. The analytical form for the quantum Cramer-Rao bound (QCRB) is presented, which shows the performance by providing a lower bound of the estimation uncertainty. It is shown that the NOON state has the worst performance among those in the class of the entangled states considered. We also analyze in detail four different scenarios, which are the NOON state, the entangled coherent state, the entangled squeezed coherent state, and the entangled squeezed vacuum state. From the comparison among these four states, we find that when the mean photon number is fixed, the squeezed vacuum state has the smallest QCRB, followed by the squeezed coherent state, entangled coherent state, and NOON state. We also illustrate that the balanced entangled state can perform better than a more generalized unbalanced form studied in previous works for certain scenarios. Finally, we give an experimental setup for producing a two-mode entangled state that can beat the NOON state in quantum metrology.