Continuous-time measurements are instrumental for a multitude of tasks in quantum engineering and quantum control, including the estimation of dynamical parameters of open quantum systems monitored through the environment. However, such measurements do not extract the maximum amount of information available in the output state, so finding alternative optimal measurement strategies is a major open problem. In this paper we solve this problem in the setting of discrete-time input-output quantum Markov chains. We present an efficient algorithm for optimal estimation of one-dimensional dynamical parameters which consists of an iterative procedure for updating a `measurement filter' operator and determining successive measurement bases for the output units. A key ingredient of the scheme is the use of a coherent quantum absorber as a way to post-process the output after the interaction with the system. This is designed adaptively such that the joint system and absorber stationary state is pure at a reference parameter value. The scheme offers an exciting prospect for optimal continuous-time adaptive measurements, but more work is needed to find realistic practical implementations.
Continuous-time measurements are instrumental for a multitude of tasks in quantum engineering and quantum control, including the estimation of dynamical parameters of open quantum systems monitored through the environment. However, such measurements do not extract the maximum amount of information available in the output state, so finding alternative optimal measurement strategies is a major open problem. In this paper we solve this problem in the setting of discrete-time input-output quantum Markov chains. We present an efficient algorithm for optimal estimation of one-dimensional dynamical parameters which consists of an iterative procedure for updating a `measurement filter' operator and determining successive measurement bases for the output units. A key ingredient of the scheme is the use of a coherent quantum absorber as a way to post-process the output after the interaction with the system. This is designed adaptively such that the joint system and absorber stationary state is pure at a reference parameter value. The scheme offers an exciting prospect for optimal continuous-time adaptive measurements, but more work is needed to find realistic practical implementations.
We propose a two step strategy for estimating one-dimensional dynamical parameters of a quantum Markov chain, which involves quantum post-processing the output using a coherent quantum absorber and a "pattern counting'' estimator computed as a simple additive functional of the outcomes trajectory produced by sequential, identical measurements on the output units. We provide strong theoretical and numerical evidence that the estimator achieves the quantum Cramer-Rao bound in the limit of large output size. Our estimation method is underpinned by an asymptotic theory of translationally invariant modes (TIMs) built as averages of shifted tensor products of output operators, labelled by binary patterns. For large times, the TIMs form a bosonic algebra and the output state approaches a joint coherent state of the TIMs whose amplitude depends linearly on the mismatch between system and absorber parameters. Moreover, in the asymptotic regime the TIMs capture the full quantum Fisher information of the output state. While directly probing the TIMs' quadratures seems impractical, we show that the standard sequential measurement is an effective joint measurement of all the TIMs number operators; indeed, we show that counts of different binary patterns extracted from the measurement trajectory have the expected joint Poisson distribution. Together with the displaced-null methodology of \cite{GiGoGu} this provides a computationally efficient estimator which only depends on the total number of patterns. This opens the way for similar estimation strategies in continuous-time dynamics, expanding the results of \cite{DayouCounting}.
Abstract Estimation of physical observables for unknown quantum states is an important problem that underlies a wide range of fields, including quantum information processing, quantum physics, and quantum chemistry. In the context of quantum computation, in particular, existing studies have mainly focused on holistic state tomography or estimation on specific observables with known classical descriptions, while this lacks the important class of problems where the estimation target itself relies on the measurement outcome. In this work, we propose an adaptive measurement optimization method that is useful for the quantum subspace methods, namely the variational simulation methods that utilize classical postprocessing on measurement outcomes. The proposed method first determines the measurement protocol for classically simulatable states, and then adaptively updates the protocol of quantum subspace expansion (QSE) according to the quantum measurement result. As a numerical demonstration, we have shown for excited-state simulation of molecules that (i) we are able to reduce the number of measurements by an order of magnitude by constructing an appropriate measurement strategy (ii) the adaptive iteration converges successfully even for a strongly correlated molecule of H4. Our work reveals that the potential of the QSE method can be empowered by elaborated measurement protocols, and opens a path to further pursue efficient quantum measurement techniques in practical computations.
This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence--contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement--estimation--feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.
Estimating correctly the quantum phase of a physical system is a central problem in quantum parameter estimation theory due to its wide range of applications from quantum metrology to cryptography. Ideally, the optimal quantum estimator is given by the so-called quantum Cramér-Rao bound, so any measurement strategy aims to obtain estimations as close as possible to it. However, more often than not, the current state-of-the-art methods to estimate quantum phases fail to reach this bound as they rely on maximum likelihood estimators of non-identifiable likelihood functions. In this work we thoroughly review various schemes for estimating the phase of a qubit, identifying the underlying problem which prohibits these methods to reach the quantum Cramér-Rao bound, and propose a new adaptive scheme based on covariant measurements to circumvent this problem. Our findings are carefully checked by Monte Carlo simulations, showing that the method we propose is both mathematically and experimentally more realistic and more efficient than the methods currently available.
A stochastic model and an adaptive method for estimating the dominant states of a quantum system have been developed. The proposed approach demonstrates a stable advantage over the frequency-based method under the influence of noise and fluctuations in the measurement process. The dominant state is defined as the most probable basis state into which the wave function of the quantum system collapses after the act of measurement. Determining the dominant state is essential for improving the accuracy of quantum measurements and quantum state reconstruction. The measurement process is modeled on an ensemble of identical quantum systems prepared in the same superposition of basis states. This approach allows the continuation of a series of measurements after the collapse of individual ensemble elements, enabling the accumulation of statistical data and the formation of a reliable estimate of the observation probabilities for each state. The obtained data are used for adaptive updating of the algorithm’s parameters and for constructing a more accurate estimate of the system’s dominant state. The adaptive method is based on a Markovian mechanism of stochastic amplification of the probability of correctly predicting the outcome of measurements of discrete quantum states. The deviation between the predicted and actual results is treated as a penalty function, which is used for dynamic adjustment of the prior probabilities. This approach enables gradual refinement of the dominant state prediction through iterative learning. Based on the proposed method, an adaptive algorithm for predicting the dominant state of a quantum system has been developed and implemented as a computational procedure. The operating principle of the algorithm lies in the multi-stage updating of prediction probabilities for the dominant state, depending on the current value of the penalty function, with the goal of minimizing the expected loss. The convergence of the adaptive process to the actual dominant state of the system has been confirmed by computer simulations. Studies were conducted on the influence of algorithm parameters and quantum system characteristics on accuracy and convergence speed. It was found that the average convergence rate of the adaptive method depends on the intensity of noise affecting the measurement channel, the probabilistic gap between the quantum states, and the learning step parameters. Increasing the variance of white noise or decreasing the probabilistic gap between states leads to slower algorithm convergence, which is consistent with analytical estimates from stochastic optimization theory. The obtained results make it possible to define the necessary constraints on learning parameters to ensure stable convergence under uncertainty. Practical recommendations have been formulated for applying the developed method to problems of quantum computing and quantum information processing.
This paper introduces a novel approach to optimizing quantum algorithms for complex problems, specifically focusing on mitigating the detrimental effects of decoherence. The core idea centers around an adaptive strategy where the algorithm's parameters and execution are dynamically adjusted based on real-time monitoring of the algorithm's decoherence level. We propose a quantum decoherence monitoring system employing quantum sensors to measure coherence times of qubits. These measurements then inform adjustments to the algorithm's parameters, such as the angles and durations of quantum gates. The system aims to maximize computational efficiency by minimizing the impact of decoherence. This approach represents a significant departure from traditional quantum algorithm design, which typically lacks this dynamic suppression mechanism. The theoretical framework and the proposed methodology are presented, outlining a promising pathway for enhancing the performance of quantum algorithms across various optimization tasks. The key metric of success is a quantifiable increase in the accuracy and speed of the algorithm, directly correlated to the effectiveness of the decoherence suppression strategy. The presented system provides a framework for continuous optimization, potentially leading to algorithms capable of tackling increasingly complex problems.