Frequentist and Bayesian Quantum Phase Estimation

A Vision for the Future of Multiscale Modeling

The rise of modern computer science enabled physical chemistry to make enormous progresses in understanding and harnessing natural and artificial phenomena. Nevertheless, despite the advances achieved over past decades, computational resources are still insufficient to thoroughly simulate extended systems from first principles. Indeed, countless biological, catalytic and photophysical processes require ab initio treatments to be properly described, but the breadth of length and time scales involved makes it practically unfeasible. A way to address these issues is to couple theories and algorithms working at different scales by dividing the system into domains treated at different levels of approximation, ranging from quantum mechanics to classical molecular dynamics, even including continuum electrodynamics. This approach is known as multiscale modeling and its use over the past 60 years has led to remarkable results. Considering the rapid advances in theory, algorithm design, and computing power, we believe multiscale modeling will massively grow into a dominant research methodology in the forthcoming years. Hereby we describe the main approaches developed within its realm, highlighting their achievements and current drawbacks, eventually proposing a plausible direction for future developments considering also the emergence of new computational techniques such as machine learning and quantum computing. We then discuss how advanced multiscale modeling methods could be exploited to address critical scientific challenges, focusing on the simulation of complex light-harvesting processes, such as natural photosynthesis. While doing so, we suggest a cutting-edge computational paradigm consisting in performing simultaneous multiscale calculations on a system allowing the various domains, treated with appropriate accuracy, to move and extend while they properly interact with each other. Although this vision is very ambitious, we believe the quick development of computer science will lead to both massive improvements and widespread use of these techniques, resulting in enormous progresses in physical chemistry and, eventually, in our society.

Essay: Quantum Sensing with Atomic, Molecular, and Optical Platforms for Fundamental Physics

Atomic, molecular, and optical (AMO) physics has been at the forefront of the development of quantum science while laying the foundation for modern technology. With the growing capabilities of quantum control of many atoms for engineered many-body states and quantum entanglement, a key question emerges: what critical impact will the second quantum revolution with ubiquitous applications of entanglement bring to bear on fundamental physics? In this Essay, we argue that a compelling long-term vision for fundamental physics and novel applications is to harness the rapid development of quantum information science to define and advance the frontiers of measurement physics, with strong potential for fundamental discoveries. As quantum technologies, such as fault-tolerant quantum computing and entangled quantum sensor networks, become much more advanced than today's realization, we wonder what doors of basic science can these tools unlock. We anticipate that some of the most intriguing and challenging problems, such as quantum aspects of gravity, fundamental symmetries, or new physics beyond the minimal standard model, will be tackled at the emerging quantum measurement frontier. Part of a series of Essays which concisely present author visions for the future of their field.

Phase estimation of definite photon number states by using quantum circuits

We propose a method to map the conventional optical interferometry setup into quantum circuits. The unknown phase shift inside a Mach-Zehnder interferometer in the presence of photon loss is estimated by simulating the quantum circuits. For this aim, we use the Bayesian approach in which the likelihood functions are needed, and they are obtained by simulating the appropriate quantum circuits. The precision of four different definite photon-number states of light, which all possess six photons, is compared. The measurement scheme that we have considered is counting the number of photons detected after the final beam splitter of the interferometer, and photon loss is modeled by using fictitious beam splitters in the arms of the interferometer. Our results indicate that three of the four definite photon-number states considered can have better precision than the standard interferometry limit whenever the photon loss rate is in a specific range. In addition, the Fisher information for the four definite photon-number states in the setup is also estimated to check the optimality of the chosen measurement scheme.

Critical Quantum Metrology Assisted by Real-Time Feedback Control

We investigate critical quantum metrology, that is, the estimation of parameters in many-body systems close to a quantum critical point, through the lens of Bayesian inference theory. We first derive a no-go result stating that any nonadaptive strategy will fail to exploit quantum critical enhancement (i.e., precision beyond the shot-noise limit) for a sufficiently large number of particles N whenever our prior knowledge is limited. We then consider different adaptive strategies that can overcome this no-go result and illustrate their performance in the estimation of (i) a magnetic field using a probe of 1D spin Ising chain and (ii) the coupling strength in a Bose-Hubbard square lattice. Our results show that adaptive strategies with real-time feedback control can achieve sub-shot-noise scaling even with few measurements and substantial prior uncertainty.

Fundamental Limits in Bayesian Thermometry and Attainability via Adaptive Strategies

We investigate the limits of thermometry using quantum probes at thermal equilibrium within the Bayesian approach. We consider the possibility of engineering interactions between the probes in order to enhance their sensitivity, as well as feedback during the measurement process, i.e., adaptive protocols. On the one hand, we obtain an ultimate bound on thermometry precision in the Bayesian setting, valid for arbitrary interactions and measurement schemes, which lower bounds the error with a quadratic (Heisenberg-like) scaling with the number of probes. We develop a simple adaptive strategy that can saturate this limit. On the other hand, we derive a no-go theorem for nonadaptive protocols that does not allow for better than linear (shot-noise-like) scaling even if one has unlimited control over the probes, namely, access to arbitrary many-body interactions.

Adaptive Bayesian algorithm for achieving a desired magneto-sensitive transition

Bayesian methods that utilize Bayes' theorem to update the knowledge of desired parameters after each measurement are used in a wide range of quantum science. For various applications in quantum science, efficiently and accurately achieving a quantum transition frequency is essential. However, the exact relation between a desired transition frequency and the controllable experimental parameters is usually absent. Here, we propose an efficient scheme to search the suitable conditions for a desired magneto-sensitive transition via an adaptive Bayesian algorithm and experimentally demonstrate it by using coherent population trapping in an ensemble of laser-cooled ⁸⁷Rb atoms. The transition frequency is controlled by an external magnetic field, which can be tuned in realtime by applying a d.c. voltage. Through an adaptive Bayesian algorithm, the voltage can automatically converge to the desired one from a random initial value only after few iterations (N ≥ 10). The response time is limited by the time of obtaining the spectrum signal, which is about 50 s for 10 iterations in our experiment. In particular, when the relation between the target frequency and the applied voltage is nonlinear (e.g., quadratic), our algorithm shows significant advantages over traditional methods. This work provides a simple and efficient way to determine a transition frequency, which can be widely applied in the fields of precision spectroscopy, such as atomic clocks, magnetometers, and nuclear magnetic resonance.

Calibration of Quantum Sensors by Neural Networks

Introducing quantum sensors as a solution to real world problems demands reliability and controllability outside of laboratory conditions. Producers and operators ought to be assumed to have limited resources readily available for calibration, and yet, they should be able to trust the devices. Neural networks are almost ubiquitous for similar tasks for classical sensors: here we show the applications of this technique to calibrating a quantum photonic sensor. This is based on a set of training data, collected only relying on the available probe states, hence reducing overhead. We found that covering finely the parameter space is key to achieving uncertainties close to their ultimate level. This technique has the potential to become the standard approach to calibrate quantum sensors.

Phase noise estimation using Bayesian inference for continuous-variable quantum key distribution

Excess noise induced by the phase drifts is a serious impairment for the continuous-variable quantum key distribution with locally generated local oscillator scheme, which is recently proposed to avoid the side channel attacks due to the transmitted local oscillator. Theoretical and experimental studies on the phase estimation have been widely reported, while two frequency-locked laser sources are indispensable to achieve quantum coherent detection. Moreover, the self-referenced phase estimation scheme requires to propagate the strong reference pulse through optical fiber, which opens a security loophole through the manipulation of the reference pulse amplitude. Based on the theoretical security and Bayes' theorem, we propose a phase estimation protocol, which does not require propagating the strong reference pulse for performing phase estimation. Compared to the other related work, the protocol can avoid the security problem caused by strong reference pulse. Moreover, this algorithm is an iterative progress for each of experiment to obtain the phase estimation and its uncertainty. We hope the proposed scheme could further promote the performance of continuous-variable quantum key distribution.