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Physical Review A

Adaptive quantum tomography in high dimensions

Luciano Pereira, Leonardo Zambrano, J. Cortés-Vega, Sebastián Niklitschek, A. Delgado

Published Jul 31, 2018• 29 citations
Publisher

Abstract

Standard quantum tomography of a single qudit achieves an infidelity that scales in the worst case as $O(1/\sqrt{N})$ for a sample of size $N$. 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 ${N}_{0}$ 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 ${N}_{0}=N/2$ leads to an infidelity that scales approximately as $O(1/N)$ 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.

DOI
10.1103/physreva.98.012339

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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/N0(N\ensuremath−N0)]O[1/\sqrt{{N}_{0}(N\ensuremath{-}{N}_{0})}]O[1/N0​(N\ensuremath−N0​)​] for all quantum states, where N0{N}_{0}N0​ and N\ensuremath−N0N\ensuremath{-}{N}_{0}N\ensuremath−N0​ 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.
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 ($N\ensuremath{\approx}3\ifmmode\times\else\texttimes\fi{}{10}^{4}$).
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 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.
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.
Quantum tomography is a standard technique for characterizing, benchmarking and verifying quantum systems/devices and plays a vital role in advancing quantum technology and understanding the foundations of quantum mechanics. Achieving the highest possible tomography accuracy remains a central challenge. Here we unify the infidelity metrics for quantum state, detector and process tomography in a single index 1−F(S^,S)1-F(\hat S,S)1−F(S^,S), where SSS represents the true density matrix, POVM element, or process matrix, and S^\hat SS^ is its estimator. We establish a sufficient and necessary condition for any tomography protocol to attain the optimal scaling $1-F= O(1/N) $ where NNN is the number of state copies consumed, in contrast to the O(1/N)O(1/\sqrt{N})O(1/N​) worst-case scaling of static methods. Guided by this result, we propose adaptive algorithms with provably optimal infidelity scalings for state, detector, and process tomography. Numerical simulations and quantum optical experiments validate the proposed methods, with our experiments reaching, for the first time, the optimal infidelity scaling in ancilla-assisted process tomography.