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arXiv:1303.0436 · Physical Review Letters

Adaptive Quantum State Tomography Improves Accuracy Quadratically

Dylan H. Mahler, Lee A. Rozema, Ardavan Darabi, Christopher Ferrie, Robin Blume-Kohout, Aephraim M. Steinberg

Published Oct 29, 2013• 141 citations
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Abstract

We introduce a simple protocol for adaptive quantum state tomography, which reduces the worst-case infidelity [$1\ensuremath{-}F(\stackrel{^}{\ensuremath{\rho}},\ensuremath{\rho})$] between the estimate and the true state from $O(1/\sqrt{N})$ to $O(1/N)$. It uses a single adaptation step and just one extra measurement setting. In a linear optical qubit experiment, we demonstrate a full order of magnitude reduction in infidelity (from 0.1% to 0.01%) for a modest number of samples ($N\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}{10}^{4}$).

DOI
10.1103/physrevlett.111.183601

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We introduce a simple protocol for adaptive quantum state tomography, which reduces the worst-case infidelity [1\ensuremath{-}F(\stackrel{^}{\ensuremath{\rho}},\ensuremath{\rho})] between the estimate and the true state from O(1/N)O(1/\sqrt{N})O(1/N​) to O(1/N)O(1/N)O(1/N). It uses a single adaptation step and just one extra measurement setting. In a linear optical qubit experiment, we demonstrate a full order of magnitude reduction in infidelity (from 0.1% to 0.01%) for a modest number of samples (N\ensuremath≈3\ifmmode×\else\texttimes\fi104N\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}{10}^{4}N\ensuremath≈3\ifmmode×\else\texttimes\fi104).
Standard quantum tomography of a single qudit achieves an infidelity that scales in the worst case as O(1/N)O(1/\sqrt{N})O(1/N​) for a sample of size NNN. Here, we propose a suitable generalization of the two-stage adaptive quantum tomography for a qubit to the case of a single qudit. This achieves an infidelity of the order of $O[1/\sqrt{{N}_{0}(N\ensuremath{-}{N}_{0})}]$ for all quantum states, where N0{N}_{0}N0​ and $N\ensuremath{-}{N}_{0}$ are the ensemble sizes employed in the two stages of the method. This result is based on a second-order Taylor series expansion of the infidelity that is obtained by means of the Fr\'echet derivative and measurement outcomes modeled by a multinomial distribution. Numerical simulations indicate that the choice N0=N/2{N}_{0}=N/2N0​=N/2 leads to an infidelity that scales approximately as O(1/N)O(1/N)O(1/N) for all quantum states in a wide range of dimensions, that is, a quadratic improvement of the infidelity when compared to standard quantum tomography in the case of low-rank states.
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.
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.
Adaptive tomography has been widely investigated to achieve faster state tomography processing of quantum systems. Infidelity of the nearly pure states in a quantum information process generally scales as O(1/sqrt(N) ), which requires a large number of statistical ensembles in comparison to the infidelity scaling of 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). However, this dramatic improvement cannot be applied to weak-value-based measurement systems in which two quantum states cannot be distinguished with perfect measurement fidelity. We introduce in this work a new optimal measurement basis to achieve fast adaptive quantum state tomography and a minimum magnitude of infidelity in a weak 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 simulation without changing the O(1/sqrt(N)) scaling. Experimentally, we find a 14.81% measurement number reduction in a superconducting circuit system.
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.