We propose an adaptive quantum state estimation method for nonstationary quantum states and verify the method by numerical simulations and experimental investigations. Adaptive quantum state estimation provides an asymptotically optimal scheme for estimating an unknown input quantum state by updating the measurement configuration upon the detection of each quantum. However, previous methods are only valid for quanta with the same quantum state (a stationary state). By adopting the likelihood function for a fixed number of recent detection results, our sequential adaptive quantum state estimation allows quantum states changing in time to be estimated. The numerical simulation results and experimental demonstration using photons agree well with the theoretical predictions. This method will find applications in various fields where dynamically changing quantum states need to be estimated.
We propose an adaptive quantum state estimation method for nonstationary quantum states and verify the method by numerical simulations and experimental investigations. Adaptive quantum state estimation provides an asymptotically optimal scheme for estimating an unknown input quantum state by updating the measurement configuration upon the detection of each quantum. However, previous methods are only valid for quanta with the same quantum state (a stationary state). By adopting the likelihood function for a fixed number of recent detection results, our sequential adaptive quantum state estimation allows quantum states changing in time to be estimated. The numerical simulation results and experimental demonstration using photons agree well with the theoretical predictions. This method will find applications in various fields where dynamically changing quantum states need to be estimated.
Accurate estimation of unknown quantum states using limited number of samples is very important but difficult task using fixed measurement basis states since the best local unbiased estimator (LUE) for estimating the parameter depends, in general, on the unknown parameter itself. In order to solve this problem, Nagaoka advocated an adaptive quantum state estimation (AQSE) procedure [1], where the measurement basis states are adaptively selected for each measurement result. Fujiwara proved the strong consistency and asymptotic efficiency for AQSE [2]. Recently, AQSE has been demonstrated by us for the linear polarizations of photons [3]. However, AQSE was designed for a static quantum state, and was not applicable to dynamic states where quantum states change in time. Here, we propose a new protocol, sequential adaptive quantum state estimation (SAQSE) for dynamic quantum states. Our numerical simulation (shown below) clearly show that the estimated value given by SAQSE can follow the true value changing in time whereas AQSE fails. We believe this new protocol will open up the use of quantum state estimation for various applications including biology, e.g. monitoring the motion of fluorescent molecules.
We herein report the experimental demonstration of adaptive quantum state estimation for totally unknown photonic qubits. Similar to our previous study [R. Okamoto et al., Phys. Rev. Lett. 109, 130404 (2012)], the measurement configuration is updated using the results of each photon detection event so that our method does not require prior knowledge of the total number of samples. The experimental results obtained herein demonstrate both strong consistency and asymptotic efficiency through several rigorous statistical tests. Furthermore, we demonstrate that the experimentally obtained distribution of the states estimated using an adaptive quantum state estimation is significantly different from that obtained by conventional state tomography and agrees well with theoretical predictions.
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.
Quantum phase estimation (QPE) is a fundamental technique for determining the phase of a quantum state, crucial for various quantum algorithms and quantum measurements. This paper introduces a novel adaptive optimization algorithm, dubbed the "Adaptive Quantum Phase Estimation" (AQPE) algorithm, designed to enhance the efficiency of QPE by dynamically adjusting parameters during the state estimation process. We demonstrate the effectiveness of AQPE through rigorous theoretical analysis and extensive numerical simulations, achieving significantly improved convergence rates compared to traditional optimization techniques. The algorithm leverages a feedback mechanism based on the observed phase error during the estimation, allowing it to iteratively refine the parameter set and ultimately locate the optimal phase estimation. We present preliminary results demonstrating the algorithm's capability to mitigate the challenges associated with conventional QPE, particularly in the presence of noisy data. The implications of this research extend to broader advancements in quantum algorithm design and computational power.
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.
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.