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arXiv:2307.10175

Bayesian estimation for collisional thermometry and time-optimal holonomic quantum computation

Gabriel O. Alves

Published Jul 16, 2023• 0 citations
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Abstract

In this thesis we deal with two different topics. In the first half we investigate how the Bayesian formalism can be introduced into the problem of quantum thermometry -- a field which exploits the high level of control in coherent devices to offer enhanced precision for temperature estimation. In particular, we investigate concrete estimation strategies, with focus on collisional thermometry, a protocol where a series of ancillae are sent sequentially to probe the system's temperature. We put forth a complete framework for analyzing collisional thermometry using Bayesian inference. The approach is easily implementable and experimentally friendly. Moreover, it is guaranteed to always saturate the Cramér-Rao bound in the long-time limit. Subtleties concerning the prior information about the system's temperature are also discussed and analyzed in terms of a modified Cramér-Rao bound associated with Van Trees and Schützenberger. Meanwhile, in the last part of the thesis we approach the problem of non-adiabatic holonomic computation. Namely, we investigate the implementation based on $Λ$-systems. It is known that a three-level system can be used in a $Λ$-type configuration in order to construct a universal set of quantum gates through the use of non-Abelian nonadiabatic geometrical phases. Such construction allows for high-speed operation times which diminish the effects of decoherence. This might be, however, accompanied by a breakdown of the validity of the rotating-wave approximation (RWA) due to the comparable timescale between counter-rotating terms and the pulse length, which greatly affects the dynamics. Here, we investigate the trade-off between dissipative effects and the RWA validity, obtaining the optimal regime for the operation of the holonomic quantum gates.

DOI
10.48550/arxiv.2307.10175

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In this thesis we deal with two different topics. In the first half we investigate how the Bayesian formalism can be introduced into the problem of quantum thermometry -- a field which exploits the high level of control in coherent devices to offer enhanced precision for temperature estimation. In particular, we investigate concrete estimation strategies, with focus on collisional thermometry, a protocol where a series of ancillae are sent sequentially to probe the system's temperature. We put forth a complete framework for analyzing collisional thermometry using Bayesian inference. The approach is easily implementable and experimentally friendly. Moreover, it is guaranteed to always saturate the Cramér-Rao bound in the long-time limit. Subtleties concerning the prior information about the system's temperature are also discussed and analyzed in terms of a modified Cramér-Rao bound associated with Van Trees and Schützenberger. Meanwhile, in the last part of the thesis we approach the problem of non-adiabatic holonomic computation. Namely, we investigate the implementation based on ΛΛΛ-systems. It is known that a three-level system can be used in a ΛΛΛ-type configuration in order to construct a universal set of quantum gates through the use of non-Abelian nonadiabatic geometrical phases. Such construction allows for high-speed operation times which diminish the effects of decoherence. This might be, however, accompanied by a breakdown of the validity of the rotating-wave approximation (RWA) due to the comparable timescale between counter-rotating terms and the pulse length, which greatly affects the dynamics. Here, we investigate the trade-off between dissipative effects and the RWA validity, obtaining the optimal regime for the operation of the holonomic quantum gates.
Quantum thermometry exploits the high level of control in coherent devices to offer enhanced precision for temperature estimation. This highlights the need for constructing concrete estimation strategies. Of particular importance is collisional thermometry, where a series of ancillae are sent sequentially to probe the system's temperature. In this paper we put forth a complete framework for analyzing collisional thermometry using Bayesian inference. The approach is easily implementable and experimentally friendly. Moreover, it is guaranteed to always saturate the Cram\'er-Rao bound in the long-time limit. Subtleties concerning the prior information about the system's temperature are also discussed and analyzed in terms of a modified Cram\'er-Rao bound associated with Van Trees and Sch\"utzenberger.
We study the Bayesian approach to thermometry with no prior knowledge about the expected temperature scale, through the example of energy measurements on fully or partially thermalized qubit probes. We show that the most common Bayesian estimators, namely the mean and the median, lead to high-temperature divergences when used for uninformed thermometry. To circumvent this and achieve better overall accuracy, we propose two new estimators based on an optimization of relative deviations. Their global temperature-averaged behavior matches a modified van Trees bound, which complements the Cramer-Rao bound for smaller probe numbers and unrestricted temperature ranges. Furthermore, we show that, using partially thermalized probes, one can increase the range of temperatures to which the thermometer is sensitive at the cost of the local accuracy.
We investigate the dephasing dynamics of a qubit as an effective mechanism for estimating the temperature of its surrounding environment for different symmetrizes. Our approach is fundamentally quantum, leveraging the qubit's susceptibility to decoherence without necessitating thermal equilibrium with the system under study. We also examine how symmetry properties affect the accuracy of information retrieval and the robustness of quantum information storage in such systems, highlighting their potential advantages in mitigating decoherence effects. The optimization of quantum Fisher information is performed with respect to both the interaction duration and the environmental temperature, focusing on Ohmic-like spectral density environments. Furthermore, we explicitly identify the optimal qubit measurement that attains the quantum Cramer-Rao bound for precision. Our findings reveal that optimal estimation arises from a complex interplay between the qubit's dephasing dynamics and the Ohmic characteristics of the environment with a particular focus on non-Hermitian systems that exhibit enhanced resilience to decoherence. Notably, optimal estimation does not occur when the qubit reaches a stationary state nor under conditions of complete dephasing.
In this work we propose a theory of temperature estimation of quantum systems, which is applicable in the regime of non-negligible prior temperature uncertainty and limited measurement data. In this regime the problem of establishing a well-defined measure of estimation precision becomes nontrivial. Furthermore, the construction of a suitable criterion for optimal measurement design must be reexamined to account for the prior uncertainty. We propose a fully Bayesian approach to temperature estimation based on the concept of thermodynamic length, which solves both these problems. As an illustration of this framework, we consider thermal spin-1/21/21/2 particles and investigate the fundamental difference between two cases: on the one hand, when the spins are probing the temperature of a heat reservoir and, on the other, when the spins themselves constitute the sample.
Using quantum systems as sensors or probes has been shown to greatly improve the precision of parameter estimation by exploiting unique quantum features such as entanglement. A major task in quantum sensing is to design the optimal protocol, i.e., the most precise one. It has been solved for some specific instances of the problem, but in general even numerical methods are not known. Here, we focus on the single-shot Bayesian setting, where the goal is to find the optimal initial state of the probe (which can be entangled with an auxiliary system), the optimal measurement, and the optimal estimator function. We leverage the formalism of higher-order operations to develop a method based on semidefinite programming that finds a protocol that is close to the optimal one with arbitrary precision. Crucially, our method is not restricted to any specific quantum evolution, cost function, or prior distribution, and thus can be applied to any estimation problem. Moreover, it can be applied to both single-parameter or multiparameter estimation tasks. We demonstrate our method with three examples, consisting of unitary phase estimation, thermometry in a bosonic bath, and multiparameter estimation of an SU(2) transformation. Exploiting our methods, we extend several results from the literature. For example, in the thermometry case, we find the optimal protocol at any finite time and quantify the usefulness of entanglement. Published by the American Physical Society 2024
Thermometry is a fundamental parameter estimation problem that is crucial for the advancement of natural sciences. One widely adopted approach to address this issue is the local thermometry theory, which employs classical and quantum Cramér-Rao bounds as benchmarks for thermometric precision. However, this theory is constrained to decrease temperature fluctuations around a known temperature value, hardly tackling the precision thermometry problem over a wide temperature range. To overcome this limitation, we derive two basic bounds on thermometry precision within a global framework, i.e., classical and quantum optimal biased bounds. By implementing energy measurements on a thermal equilibrium system, the quantum optimal biased bound can be saturated. Furthermore, we demonstrate their thermometry performance through two specific applications: a noninteracting spin- 1 / 2 gas and a thermalized quantum harmonic oscillator. Our results indicate that, compared to local thermometry, global thermometry provides superior temperature estimation performance. Notably, global error bound approaches its local approximation under asymptotic cases. Published by the American Physical Society 2024