Hybrid reconstruction of quantum density matrix: when low-rank meets sparsity

  • Kezhi Li
  • Kai Zheng
  • Jingbei Yang
  • Shuang CongEmail author
  • Xiaomei Liu
  • Zhaokai Li


Both the mathematical theory and experiments have verified that the quantum state tomography based on compressive sensing is an efficient framework for the reconstruction of quantum density states. In recent physical experiments, we found that many unknown density matrices in which people are interested in are low-rank as well as sparse. Bearing this information in mind, in this paper we propose a reconstruction algorithm that combines the low-rank and the sparsity property of density matrices and further theoretically prove that the solution of the optimization function can be, and only be, the true density matrix satisfying the model with overwhelming probability, as long as a necessary number of measurements are allowed. The solver leverages the fixed-point equation technique in which a step-by-step strategy is developed by utilizing an extended soft threshold operator that copes with complex values. Numerical experiments of the density matrix estimation for real nuclear magnetic resonance devices reveal that the proposed method achieves a better accuracy compared to some existing methods. We believe that the proposed method could be leveraged as a generalized approach and widely implemented in the quantum state estimation.


Quantum state estimation Compressive sensing Low-rank Sparse Reconstruction Fixed-point equation 



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


  1. 1.
    Heinosaari, T., Mazzarella, L., Wolf, M.M.: Quantum tomography under prior information. Commun. Math. Phys. 318(2), 355–374 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Wu, L.-A., Byrd, M.S.: Self-protected quantum algorithms based on quantum state tomography. Quantum Inf. Process. 8(1), 1–12 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Candès, E., Tao, T.: Decoding by linear programming. IEEE Trans. Inf. Theory 51, 4203–4215 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52, 1289–1306 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Li, K., Ling, C., Gan, L.: Deterministic compressed-sensing matrices: where Toeplitz meets Golay. In: IEEE Int. Conf. on Aco., Spe. and Sig. Proc. (ICASSP), pp. 3748–3751 (2011)Google Scholar
  6. 6.
    Liu, W., Zhang, T., Liu, J., Chen, P., Yuan, J.: Experimental quantum state tomography via compressed sampling. Phys. Rev. Lett. 108(17), 170403 (2012)ADSCrossRefGoogle Scholar
  7. 7.
    Li, K., Cong, S.: State of the art and prospects of structured sensing matrices in compressed sensing. Front. Comput. Sci. 9(5), 665–677 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    Gross, D., Liu, Y., Flammia, S.T., Becker, S., Eisert, J.: Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105(15), 150401–150404 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    Schwemmer, C., Tóth, G., Niggebaum, A., Moroder, T., Gross, D., Gühne, O., Weinfurter, H.: Experimental comparison of efficient tomography schemes for a six-qubit state. Phys. Rev. Lett. 113(5), 040503 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    Lloyd, S., Mohseni, M., Rebentrost, P.: Quantum principal component analysis. Nat. Phys. 10(9), 631–633 (2014)CrossRefGoogle Scholar
  11. 11.
    Flammia, S.T., Gross, D., Liu, Y.-K., Eisert, J.: Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators. New J. Phys. 14(9), 095022 (2012)ADSCrossRefGoogle Scholar
  12. 12.
    Kosut, R.L., Lidar, D.A.: Quantum error correction via convex optimization. Quantum Inf. Process. 8(5), 443–459 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Li, K., Cong, S.: A robust compressive quantum state tomography algorithm using ADMM. In: The 19th World Congress of the IFAC, vol. 47(3), pp. 6878–6883 (2014)Google Scholar
  14. 14.
    Liu, Y.-K.: Universal low-rank matrix recovery from Pauli measurements. In: Advances in Neural Information Processing Systems (NIPS), pp. 1638–1646 (2011)Google Scholar
  15. 15.
    Wu, X., Xu, K.: Partial standard quantum process tomography. Quantum Inf. Process. 12(2), 1379–1393 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shabani, A., Kosut, R.L., Mohseni, M., Rabitz, H., Broome, M.A., Almeida, M.P., Fedrizzi, A., White, A.G.: Efficient measurement of quantum dynamics via compressive sensing. Phys. Rev. Lett. 106(4), 100401 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    Li, K., Zhang, H., Kuang, S., Meng, F., Cong, S.: An improved robust admm algorithm for quantum state tomography. Quantum Inf. Process. 15(6), 2343–2358 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Candés, E.J., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM 58(3), 1–37 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, Z., Yung, M., Chen, H., Lu, D., Whitfield, J.D., Peng, X., Aspuru-Guzik, A., Du, J.: Solving quantum ground-state problems with nuclear magnetic resonance. Sci. Rep. 1, 88 (2011)CrossRefGoogle Scholar
  20. 20.
    Zheng, K., Li, K., Cong, S.: A reconstruction algorithm for compressive quantum tomography using various measurement sets. Sci. Rep. 6, 38497 (2016)ADSCrossRefGoogle Scholar
  21. 21.
    Li, K., Zhang, J., Cong, S.: Fast reconstruction of high-qubit-number quantum states via low-rate measurements. Phys. Rev. A 96(1), 012334 (2017)ADSCrossRefGoogle Scholar
  22. 22.
    Zyczkowski, K., Penson, K.A., Nechita, I., Collins, B.: Generating random density matrices. J. Math. Phys. 52, 062201 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Flammia, S.T., Liu, Y.: Direct fidelity estimation from few Pauli measurements. Phys. Rev. Lett. 106(23), 230501 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    Recht, B., Fazel, M., Parillo, P.: Guaranteed minimum rank solution of matrix equations via nuclear norm minimization. SIAM Rev. 52, 471–01 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Candès, E.J.: The restricted isometry property and its implications for compressed sensing. C.R. Math. 346(9), 589–592 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Chen, M., Lin, Z., Ma, Y.: The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices. UIUC Technical Report UILU-ENG-09-2215 (2009)Google Scholar
  27. 27.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Univ. Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of AutomationUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.Department of Electrical and Electronic EngineeringImperial College LondonLondonUK
  3. 3.Department of PhysicsUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

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