Adaptive measurements were recently shown to significantly improve the performance of quantum state tomography. Utilizing information about the system for the online choice of optimal measurements allows one to reach the ultimate bounds of precision for state reconstruction. In this article we generalize an adaptive Bayesian approach to the case of process tomography and experimentally show its superiority in the task of learning unknown quantum operations. Our experiments with photonic polarization qubits cover all types of single-qubit channels. We also discuss instrumental errors and the criteria for evaluation of the ultimate achievable precision in an experiment. It turns out that adaptive tomography provides a lower noise floor in the presence of strong technical noise.
Adaptive measurements were recently shown to significantly improve the performance of quantum state tomography. Utilizing information about the system for the online choice of optimal measurements allows one to reach the ultimate bounds of precision for state reconstruction. In this article we generalize an adaptive Bayesian approach to the case of process tomography and experimentally show its superiority in the task of learning unknown quantum operations. Our experiments with photonic polarization qubits cover all types of single-qubit channels. We also discuss instrumental errors and the criteria for evaluation of the ultimate achievable precision in an experiment. It turns out that adaptive tomography provides a lower noise floor in the presence of strong technical noise.
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.
We introduce a fast and accurate heuristic for adaptive tomography that addresses many of the limitations of prior methods.Previous approaches were either too computationally intensive or tailored to handle special cases such as single qubits or pure states.By contrast, our approach combines the efficiency of online optimization with generally applicable and well-motivated data-processing techniques.We numerically demonstrate these advantages in several scenarios including mixed states, higher-dimensional systems, and restricted measurements.Quantum information processing (QIP) promises advantages in a wide range of different contexts, including machine learning [2-4], chemistry simulation [5][6][7], and number theory [8,9].As such, the experimental effort to build useful QIP devices has exploded in recent years.In the course of this effort, quantum tomography is a valuable tool for diagnosing and debugging small quantum devices, and has subsequently seen a variety of different advances.In particular, Bayesian approaches to tomography which are especially well suited to utilizing prior information and adapting to changing experimental conditions have developed significantly in recent years [10-13], presenting a useful experimental tool [14][15][16].In this paper we demonstrate the efficiency and accuracy of an adaptive tomography protocol that we call PAQT: practical adaptive quantum tomography.PAQT intelligently selects new measurements based on the outcomes of previous ones [10,[17][18][19][20][21].Adaptivity has been experimentally demonstrated [14,15,[22][23][24][25], but is not currently standard practice.Though adaptivity increases accuracy, the computational costs incurred outweigh that of simply repeating standard measurements many times.The PAQT approach employs a simple heuristic that can be efficiently computed between measurements, even with embedded hardware [26][27][28][29].The algorithm we propose is therefore compatible with modern experimental design and avoids an important limitation of previous approaches.We base our algorithm off of self-guided quantum tomography (SGQT), which treats adaptive tomography as a direct optimization problem rather than a new optimization problem between each measurement [30].Though this affords an efficient and easy to implement adaptive heuristic, SGQT is not without its limitations.It requires assuming that the target state is pure, and it does not return rich region estimates for a state.What PAQT achieves is to effectively combine SGQT with conventional and easily-implemented tomographic estimators, such as the Bayesian particle filter or least-squares fit (LSF) estimators.Under this approach, an experimentalist can collect data using SGQT (even if its assumptions are not met), and then post-process this data using particle filtering or LSF.The benefit of PAQT is two-fold.(1) From the point of view of traditional tomography, it gives an adaptive tomography protocol requiring only modest computational resources, as the bulk of the computational cost is offloaded to post-processing.(2) From the point of view of simulation-based optimization tomography (such as SGQT), it effectively augments the output with region estimation providing a statistically robust quantification of uncertainty.Thus, while we do not explicitly demonstrate that the improved scaling of Ferrie [30] remains in the more general case considered here, PAQT does provide a practical and efficient procedure for performing adaptive quantum tomography with rigorous statistical principles.
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.
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/N scaling rule of averaged infidelity against the overall number of photons, N, in our experiment, which outperforms 1/N of nonadaptive schemes.
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 process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be Θ(din3dout3/ε2), where din and dout are the channel input and output dimensions, respectively, and ε is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires Ω(din3dout3/ε2) queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound O(din3dout3/ε2). Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case din=1. By contrast, coherent protocols with quantum memory achieve query complexity Θ(din2dout2/ε2). Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.