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Quantum Information Processing

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01-06-2016

An improved robust ADMM algorithm for quantum state tomography

Authors: Kezhi Li, Hui Zhang, Sen Kuang, Fangfang Meng, Shuang Cong

Published in: Quantum Information Processing | Issue 6/2016

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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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Smith, A., Riofro, C., Anderson, B., Martinez, H., Deutsch, I., Jessen, P.: Quantum state tomography by continuous measurement and compressed sensing. Phys. Rev. A 87, 030102 (2013)ADSCrossRef

Wu, X., Xu, K.: Partial standard quantum process tomography. Quantum Inf. Process. 12(2), 1379–1393 (2013). doi:10.1007/s11128-012-0473-9 ADSMathSciNetCrossRefMATH

Heinosaari, T., Mazzarella, L., Wolf, M.: Quantum tomography under prior information. Commun. Math. Phys. 318(2), 355–374 (2013). doi:10.1007/s00220-013-1671-8 ADSMathSciNetCrossRefMATH

Wu, L.A., Byrd, M.: Self-protected quantum algorithms based on quantum state tomography. Quantum Inf. Process. 8(1), 1–12 (2009). doi:10.1007/s11128-008-0090-9 MathSciNetCrossRefMATH

Baraniuk, R.: Compressive sensing. IEEE Signal Process. Mag. 24(4), 118–121 (2007). doi:10.1109/MSP.2007.4286571 ADSMathSciNetCrossRef

Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52, 1289–1306 (2006)MathSciNetCrossRefMATH

Gross, D., Liu, Y., Flammia, S.T., Becker, S., Eisert, J.: Quantum state tomography via compressed sensing. Phys. Rev. Lett. 105(15), 150401 (2010)ADSCrossRef

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), 0401503 (2014)

Lloyd, S., Mohseni, M., Rebentrost, P.: Quantum principal component analysis. Nat. Phys. 10(9), 631–633 (2014)

10.

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). doi:10.1103/PhysRevLett.106.100401 ADSCrossRef

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)ADSCrossRef

12.

Miosso, C., von Borries, R., Argaez, M., Velazquez, L., Quintero, C., Potes, C.: Compressive sensing reconstruction with prior information by iteratively reweighted least-squares. IEEE Trans. Signal Process. 57(6), 2424–2431 (2009). doi:10.1109/TSP.2009.2016889 ADSMathSciNetCrossRef

13.

Kosut, R., Lidar, D.: Quantum error correction via convex optimization. Quantum Inf. Process. 8(5), 443–459 (2009). doi:10.1007/s11128-009-0120-2 MathSciNetCrossRefMATH

14.

Liu, Y.: Universal low-rank matrix recovery from pauli measurements. In: Proceedings of Advances in Neural Information Processing Systems, pp. 1638–1646 (2011)

15.

Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–22 (2011)CrossRefMATH

16.

He, B., Yang, H., Wang, S.: Alternating direction method with self-adaptive penalty parameters for monotone variational inequalities. J. Optim. Theory Appl. 106(2), 337–356 (2000). doi:10.1023/A:1004603514434 MathSciNetCrossRefMATH

17.

Lin, Z., Liu, R., Su, Z.: Linearized alternating direction method with adaptive penalty for low rank representation. In: Proceedings of Advances in Neural Information Processing Systems, pp. 612–620 (2011)

18.

Gross, D.: Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory 57(3), 1548–1566 (2011). doi:10.1109/TIT.2011.2104999 MathSciNetCrossRef

19.

Wright, J., Ganesh, A., Min, K., Ma, Y.: Compressive principal component pursuit. J. IMA 2, 32–68 (2013)MathSciNetMATH

20.

Yuan, X.M., Yang, J.: Sparse and low-rank matrix decomposition via alternating direction methods. Pac. J. Optim. (2009)

21.

Li, K., Cong, S.: A robust compressive quantum state tomography algorithm using admm. In: The 19th World Congress of the International Federation of Automatic Control, pp. 6878–6883 (2014)

22.

Cong S., Z.H., K., L.: An improved quantum state estimation algorithm via compressive sensing. In: 2014 IEEE international conference on Robio and Biomimetics, 5–10, pp. 2238–2343 (2014)

23.

Recht, B., Fazel, M., Parillo, P.: Guaranteed minimum rank solution of matrix equations via nuclear norm minimization. SIAM Rev. 52, 471–501 (2007)CrossRefMATH

24.

Zyczkowski, K., Penson, K.A., Nechita, I., Collins, B.: Generating random density matrices. J. Math. Phys. 52(6), 062201 (2011)

25.

Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Univ. Press, Cambridge, U.K. (2004)CrossRefMATH

26.

Candés, E.J., Li, X., Ma, Y., Wright, J.: Robust principal component analysis? J. ACM 58(3), 1–37 (2011)MathSciNetCrossRefMATH

27.

Candès, E., Romberg, J.: Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6(8), 227–254 (2006)MathSciNetCrossRefMATH

28.

Candès, E., Romberg, J.: Sparsity and incoherence in compressive sampling. Inverse Probl. 23, 969–985 (2007)ADSMathSciNetCrossRefMATH

29.

Sturm, J.F.: Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)

Title: An improved robust ADMM algorithm for quantum state tomography
Authors: Kezhi Li
Hui Zhang
Sen Kuang
Fangfang Meng
Shuang Cong
Publication date: 01-06-2016
Publisher: Springer US
Published in: Quantum Information Processing / Issue 6/2016
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI: https://doi.org/10.1007/s11128-016-1288-x

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