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EPJ Web of Conferences

Experimental Adaptive Bayesian Tomography

G. I. Struchalin, Ivan Pogorelov, Konstantin Kravtsov, I. Radchenko, S. Straup, S. P. Kulik

Published Jan 1, 2015• 0 citations
Publisher

Abstract

We discuss an experimental realization of an adaptive quantum state tomography protocol. The method we suggested and tested takes advantage of a Bayesian approach to statistical inference and is naturally tailored for adaptive strategies. For pure states we observe close to 1/N scaling of infidelity with overall number of registered events N, while best non-adaptive protocols allow for 1 N scaling only. This is the theoretical limit for any tomographic protocol, and further improvement may only affect pre-factors in this power law. Also we consider particular strategies of the state reconstruction based on adequate and inadequate models and compare their scaling. Experiments have been performed for polarization qubits and ququarts, but the approach is readily adapted to any dimension. Our method does not take into account systematic errors caused, for example, by inaccuracies in retardant plates rotation. However for the reached values of infidelities of on the order of 10 – 10 we did not observe any deviations from expected behavior and were not able to identify the influence of systematic errors.

DOI
10.1051/epjconf/201510301008

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We discuss an experimental realization of an adaptive quantum state tomography protocol. The method we suggested and tested takes advantage of a Bayesian approach to statistical inference and is naturally tailored for adaptive strategies. For pure states we observe close to 1/N scaling of infidelity with overall number of registered events N, while best non-adaptive protocols allow for 1 N scaling only. This is the theoretical limit for any tomographic protocol, and further improvement may only affect pre-factors in this power law. Also we consider particular strategies of the state reconstruction based on adequate and inadequate models and compare their scaling. Experiments have been performed for polarization qubits and ququarts, but the approach is readily adapted to any dimension. Our method does not take into account systematic errors caused, for example, by inaccuracies in retardant plates rotation. However for the reached values of infidelities of on the order of 10 – 10 we did not observe any deviations from expected behavior and were not able to identify the influence of systematic errors.
We report an experimental realization of an adaptive quantum state tomography protocol. Our method takes advantage of a Bayesian approach to statistical inference and is naturally tailored for adaptive strategies. For pure states, we observe close to ${N}^{\ensuremath{-}1}$ scaling of infidelity with overall number of registered events, while the best nonadaptive protocols allow for ${N}^{\ensuremath{-}1/2}$ scaling only. Experiments are performed for polarization qubits, but the approach is readily adapted to any dimension.
We report an experimental realization of adaptive Bayesian quantum state tomography for two-qubit states. Our implementation is based on the adaptive experimental design strategy proposed in the work by Husz\'ar and Houlsby [F. Husz\'ar and N. M. T. Houlsby, Phys. Rev. A 85, 052120 (2012).] and provides an optimal measurement approach in terms of the information gain. We address the practical questions which one faces in any experimental application: the influence of technical noise and the behavior of the tomographic algorithm for an easy-to-implement class of factorized measurements. In an experiment with polarization states of entangled photon pairs, we observe a lower instrumental noise floor and superior reconstruction accuracy for nearly pure states of the adaptive protocol compared to a nonadaptive protocol. At the same time, we show that for the mixed states, the restriction to factorized measurements results in no advantage for adaptive measurements, so general measurements have to be used.
Adaptive measurements have recently been shown to significantly improve the performance of quantum state and process tomography. Adaptive tomography is especially useful in complicated experiments with low outcome generation rate, since it allows one to extract maximal information from a limited amount of data. However, the existing methods either cannot be simply applied to high-dimensional systems or are prohibitively computationally expensive. Here we propose and experimentally implement a tomographic protocol specially designed for the reconstruction of high-dimensional quantum states. The protocol shows qualitative improvement in infidelity scaling with the number of measurements and is fast enough to allow for complete state tomography of states with dimensionality up to 36.
Abstract Adaptive techniques have great potential for wide application in enhancing the precision of quantum parameter estimation. We present an adaptive quantum state tomography protocol for finite dimensional quantum systems and experimentally implement the adaptive tomography protocol on two-qubit systems. In this adaptive quantum state tomography protocol, an adaptive measurement strategy and a recursive linear regression estimation algorithm are performed. Numerical results show that our adaptive quantum state tomography protocol can outperform tomography protocols using mutually unbiased bases and the two-stage mutually unbiased bases adaptive strategy, even with the simplest product measurements. When nonlocal measurements are available, our adaptive quantum state tomography can beat the Gill–Massar bound for a wide range of quantum states with a modest number of copies. We use only the simplest product measurements to implement two-qubit tomography experiments. In the experiments, we use error-compensation techniques to tackle systematic error due to misalignments and imperfection of wave plates, and achieve about a 100-fold reduction of the systematic error. The experimental results demonstrate that the improvement of adaptive quantum state tomography over nonadaptive tomography is significant for states with a high level of purity. Our results also show that this adaptive tomography method is particularly effective for the reconstruction of maximally entangled states, which are important resources in quantum information.
Assumption-free reconstruction of quantum states from measurements is essential for benchmarking and certifying quantum devices, but it remains difficult due to the extensive measurement statistics and experimental resources it demands. An approach to alleviating these demands is provided by adaptive measurement strategies, which can yield up to a quadratic improvement in reconstruction accuracy for pure states by dynamically optimizing measurement settings during data acquisition. A key open question is whether these asymptotic advantages remain in realistic experiments, where readout is inevitably noisy. In this work, we analyze the impact of readout noise on adaptive quantum state tomography with readout-error mitigation, focusing on the challenging regime of reconstructing pure states using mixed-state estimators. Using analytical arguments based on Fisher information optimization and extensive numerical simulations using Bayesian inference, we show that any nonzero readout noise eliminates the asymptotic quadratic scaling advantage of adaptive strategies. We numerically investigate the behavior for finite measurement statistics for single- and two-qubit systems with exact readout-error mitigation and find a gradual transition from ideal to sub-optimal scaling. We furthermore investigate realistic scenarios where detector tomography is performed with a limited number of state copies for calibration, showing that insufficient detector characterization leads to estimator bias and limited reconstruction accuracy. Although our result imposes an upper bound on the reconstruction accuracy that can be achieved with adaptive strategies, we nevertheless observe numerically a constant-factor gain in reconstruction accuracy, which becomes larger as the readout noise decreases. This indicates potential practical benefits in using adaptive measurement strategies in well-calibrated experiments.
Quantum-state tomography plays a pivotal role in quantum computation and information processing. To improve the accuracy in estimating an unknown state, carefully designed measurement schemes, such as adopting an adaptive strategy, are necessarily needed, which have gained great interest recently. In this work, based on the proposal of Sugiyama et al. [Phys. Rev. A 85, 052107 (2012)], we experimentally realize an adaptive quantum-state tomography for one qubit in an optical system. Since this scheme gives an analytical solution to the optimal measurement basis problem, our experiment is updated in real time and the infidelity between the real state and the estimated state is tracked with the detected photons. We observe an almost 1/N1/N1/N scaling rule of averaged infidelity against the overall number of photons, NNN, in our experiment, which outperforms 1/N1/\sqrt{N}1/N​ of nonadaptive schemes.