In Bayesian estimation theory, the estimator
θ^=E[θ∣l] attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest
θ from the observation of
l through a noisy channel
Pl∣θ, given a prior
Pθ on
θ. In quantum sensing tasks, the user gets
ρθ, the quantum state that encodes
θ. They choose a measurement, a positive-operator valued measure (POVM)
Πl, which induces the channel
Pl∣θ=Tr(ρθΠl) to the measurement outcome
l, on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM
Πl that minimizes the MMSE over all possible measurements, and that MMSE. This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements, when
Pθ is unavailable. For multi-parameter estimation, i.e., when
θ is a vector, in Fisher quantum estimation theory, the inverse of the QFI matrix provides an operator lower bound to the covariance of an unbiased estimator. However, there has been little work on quantifying quantum limits and measurement designs, for multi-parameter quantum estimation in the {\em Bayesian} setting. In this paper, we build upon Personick's result to construct a Bayesian adaptive measurement scheme for multi-parameter estimation when
N copies of
ρθ are available. We illustrate an application to localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs far superior to quantum-noise-limited focal-plane direct imaging.
Quantum sensing harnesses the unique properties of quantum systems to enable precision measurements of physical quantities such as time, magnetic and electric fields, acceleration, and gravitational gradients well beyond the limits of classical sensors. However, identifying suitable sensing probes and measurement schemes can be a classically intractable task, as it requires optimizing over Hilbert spaces of high dimension. In variational quantum sensing, a probe quantum system is generated via a parameterized quantum circuit (PQC), exposed to an unknown physical parameter through a quantum channel, and measured to collect classical data. PQCs and measurements are typically optimized using offline strategies based on frequentist learning criteria. This paper introduces an adaptive protocol that uses Bayesian inference to optimize the sensing policy via the maximization of the active information gain. The proposed variational methodology is tailored for non-asymptotic regimes where a single probe can be deployed in each time step, and is extended to support the fusion of estimates from multiple quantum sensing agents.
It is well known in Bayesian estimation theory that the conditional estimator attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest. In quantum, e.g., optical and atomic, imaging and sensing tasks the user has access to the quantum state that encodes the parameter. The choice of a measurement operator, i.e. a positive-operator valued measure (POVM), leads to a measurement outcome on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM that attains the MMSE over all possible physically allowable measurements and the resulting MMSE (Personick, 1997). This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements without considering any prior probability. For multi-parameter estimation, in quantum Fisher estimation theory the inverse of the QFI matrix provides an operator lower bound on the covariance of an unbiased estimator, and this bound is understood in the positive semidefinite sense. However, there has been little work on quantifying the quantum limits and measurement designs, for multi-parameter quantum estimation in aBayesiansetting. In this work, we build upon Personick's result to construct a Bayesian adaptive (greedy) measurement scheme for multi-parameter estimation. We illustrate our proposed measurement scheme with the application of localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs superior to quantum-noise-limited focal-plane direct imaging.
Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary N tot times, the variance of the estimates can scale as O ( 1 / N tot 2 ) , compared to the best “classical” strategy with O ( 1 / N tot ) . The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O ( 1 / N tot 2 ) . Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps. Published by the American Physical Society 2024
Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary
Ntot times, the variance of the estimates can scale as
O(1/Ntot2), compared to the best ``classical'' strategy with
O(1/Ntot). The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without inter-probe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling
O(1/Ntot2). Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analyses the results of previous steps.