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An improved robust ADMM algorithm for quantum state tomography

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Abstract

In this paper, an improved adaptive weights alternating direction method of multipliers algorithm is developed to implement the optimization scheme for recovering the quantum state in nearly pure states. The proposed approach is superior to many existing methods because it exploits the low-rank property of density matrices, and it can deal with unexpected sparse outliers as well. The numerical experiments are provided to verify our statements by comparing the results to three different optimization algorithms, using both adaptive and fixed weights in the algorithm, in the cases of with and without external noise, respectively. The results indicate that the improved algorithm has better performances in both estimation accuracy and robustness to external noise. The further simulation results show that the successful recovery rate increases when more qubits are estimated, which in fact satisfies the compressive sensing theory and makes the proposed approach more promising.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (61573330).

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Authors and Affiliations

  1. Department of Automation, University of Science and Technology of China, Hefei, 230027, People’s Republic of China

    Kezhi Li, Hui Zhang, Sen Kuang & Shuang Cong

  2. Imperial College London, London, UK

    Kezhi Li

  3. Hefei Uinversity, Hefei, 230022, People’s Republic of China

    Fangfang Meng

Authors
  1. Kezhi Li
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  2. Hui Zhang
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  3. Sen Kuang
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  4. Fangfang Meng
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  5. Shuang Cong
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Corresponding author

Correspondence to Shuang Cong.

Appendix

Appendix

Definition 1

(Rank RIP) [14, 23] The \(\mathcal {A}\) satisfies the rank-restricted isometry property (RIP) if for all \(d \times d\) \(\mathbf{X}\), we have

$$\begin{aligned} (1-\delta )||\mathbf{X}||_F \le ||\mathcal {A} (\mathbf{X})||_2 \le (1+\delta )||\mathbf{X}||_F \end{aligned}$$
(29)

where some constant \(0 < \delta <1\).

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Li, K., Zhang, H., Kuang, S. et al. An improved robust ADMM algorithm for quantum state tomography. Quantum Inf Process 15, 2343–2358 (2016). https://doi.org/10.1007/s11128-016-1288-x

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