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Physical Review A

Experimental demonstration of real-time adaptive one-qubit quantum-state tomography

Qi Yin, Li Li, Xiao Xiang, Guo‐Yong Xiang, Chuang-Feng Li, Guang‐Can Guo

Published Jan 20, 2017• 7 citations
Publisher

Abstract

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 $1/\sqrt{N}$ of nonadaptive schemes.

DOI
10.1103/physreva.95.012129

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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.
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 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 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.
Quantum state tomography (QST) estimates an unknown quantum state from a measurement process typically performed on identical copies of the target state and plays a core role in characterizing quantum information systems. However, because quantum measurement outcomes are probabilistic, QST methods often require many shots per basis to achieve high-fidelity estimates, which hinders scalability to many-qubit applications and slows the transmission of quantum information. A more efficient procedure is therefore required for modern-day quantum operations in quantum computing, communications, and sensing. We address this need with an adaptive QST approach that measures a single-qubit state in a wide set of bases. We include bases beyond the Pauli triad {X,Y,Z} by leveraging the infinite-dimensional Hilbert space of quantum mechanics, increasing the likelihood of producing highly polarized measurement outcomes and finding an optimally informative basis. With this basis, the state can be identified with fewer measurements by focusing on directions that most reduce uncertainty of the state. In addition to theoretical innovation in [9], we experimentally verify in a simplified photonic realization using 13 bases that having many more available measurement bases than the Pauli triad benefits QST. This engineering implementation demonstrates that adaptive QST is both efficient in performance and practical to implement.
Adaptive tomography has been widely investigated to obtain faster state tomography process for quantum computation. Infidelity [1 − F(ρ,σ)] on nearly pure states in quantum information process scales generally as O(1/N)O\left( {1/\sqrt N } \right)O(1/N​), which requires a large number of statistical ensembles in comparison to the infidelity scaling on mixed states of O(1/N). Recently, optimization of a measurement basis in a photonic qubit system, whose state tomography uses projective measurements, reported improved infidelity scaled as O(1/N). However, this dramatic improvement cannot be applied to a weak-value based measurement system, which can be attributed to the fact that one cannot distinguish two quantum states with perfect measurement reliability. We introduce new optimal measurement basis to achieve a fast adaptive quantum state tomography and minimum magnitude of infidelity in this weak measurement system. This novel protocol allows us to accomplish approximately 80% error reduction without changing the scaling of O(1/N)O\left( {1/\sqrt N } \right)O(1/N​) in numerical simulations and realize approximately 90% error reduction in superconducting circuit QED experiments.
Adaptive tomography has been widely investigated with the aim of achieving faster state tomography processing of quantum systems. Infidelity of the nearly pure states in a quantum information process generally scales as O(1/N)O(1/\sqrt{N})O(1/N​), which requires a large number of statistical ensembles in comparison to the infidelity scaling of O(1/N)O(1/N)O(1/N) for mixed states. One previous report optimized the measurement basis in a photonic qubit system, whose state tomography uses projective measurements, to obtain an infidelity scaling of O(1/N)O(1/N)O(1/N). However, this dramatic improvement cannot be applied to indistinct measurement systems in which two quantum states cannot be distinguished with perfect measurement fidelity. We introduce in this work an optimal measurement basis to achieve fast adaptive quantum state tomography and a minimum magnitude of infidelity in an indistinct measurement system. We expect that the adaptive quantum state tomography protocol can lead to a reduction in the number of required measurements of approximately 33.74% via analytical expectation without changing the O(1/N)O(1/\sqrt{N})O(1/N​) scaling. Experimentally, we find a 15.06% measurement number reduction in a superconducting circuit system.