Comparative Analysis of Quantum State Estimation Algorithm Based on Compressive Sensing
CONG Shuang1 , ZHANG Hui1, LI Kezhi2
1.Department of Automation, University of Science and Technology of China, Hefei 230027 2.Magnetic Resonance Research Center, University of Cambridge, Cambridge, CB2 3RA, UK
Abstract The alternating direction method of multipliers (ADMM) is used to estimate quantum density matrix with 6 qubits based on the completed research on 5 qubits estimation. In addition, the comparison with least squares and Dantzig optimization method is studied under the situations with and without external interference. The optimization schemes are implemented in Matlab environment to realize the fast estimation of quantum pure state. The experimental results show that ADMM is superior to two other algorithms in estimation accuracy and resistance to external disturbances.
[1] DONOHO D L. Compressed Sensing. IEEE Trans on Information Theory, 2006, 52(4): 1289-1306. [2] HEINOSAARI T, MAZZARELLA L, WOLF M M. Quantum Tomography under Prior Information. Communications in Mathematical Physics, 2013, 318(2): 355-374. [3] GROSS D, LIU Y K, FLAMMIA S T, et al. Quantum State Tomography via Compressed Sensing. Physical Review Letters, 2010, 105(15). DOI: 10.1103/PhysRevLett.105.150401. [4] LIU W T, ZHANG T, LIU J Y, et al. Experimental Quantum State Tomography via Compressed Sampling. Physical Review Letters, 2012, 108(17). DOI: 10.1103/PhysRevLett.108.170403. [5] BALDWIN C, KALEV A, DEUTSCH I. Quantum Process Estimation via Compressed Sensing with Convex Optimization[EB/OL].[2014-6-30].http://absimage.aps.org/image/MAR14/MWS_MAR14-2013-005989.pdf. [6] BANASZEK K, CRAMER M, GROSS D. Focus on Quantum Tomography. New Journal of Physics, 2013, 15(12). DOI: 10.1088/1367-2630/15/12/125020. [7] JELONEK J, KRAWIEC K, SLOWIN′ SKI R. Rough Set Reduction of Attributes and Their Domains for Neural Networks. Computational Intelligence, 1995, 11(2): 339-347. [8] 丛 爽,匡 森.量子系统中状态估计方法的综述.控制与决策, 2008, 23(2): 121-126, 132. (CONG S, KUANG S. Review of State Estimation Methods in Quantum Systems. Control and Decision, 2008, 23(2): 121-126, 132.) [9] 丛 爽.基于李雅普诺夫量子系统控制方法的状态调控.控制理论与应用, 2012, 29(3): 273-281. (CONG S. State Manipulation in Lyapunov-Based Quantum System Control Methods. Control Theory & Applications, 2012, 29(3): 273-281.) [10] FLAMMIA S T, GROSS D, LIU Y K, et al. Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators. New Journal of Physics, 2012, 14(9). DOI: 10.1088/1367-2630/14/9/095022. [11] MIOSSO C J, VON BORRIES R, ARGAEZ M, et al. Compressive Sensing Reconstruction with Prior Information by Iteratively Reweighted Least-Squares. IEEE Trans on Signal Processing, 2009, 57(6): 2424-2431. [12] CANDS E J. The Restricted Isometry Property and Its Implications for Compressed Sensing. Comptes Rendus Mathematique, 2008, 346(9/10): 589-592. [13] DANTZIG G B, INFANGER G. Multi-stage Stochastic Linear Programs for Portfolio Optimization. Annals of Operations Research, 1993, 45(1): 59-76. [14] BOYD S, VANDENBERGHE L. Convex Optimization. Cambridge, UK: Cambridge University Press, 2004. [15] SMITH A, RIOFRIO C A, ANDERSON B E, et al. Quantum State Tomography by Continuous Measurement and Compressed Sensing. Physical Review A, 2013, 87(3). DOI: 10.1103/PhysRevA.87.030102. [16] LIU Y K. Universal Low-Rank Matrix Recovery from Pauli Mea-surements // SHAWE-TAYLOR J, ZEMEL R S, BARTLETT P L, et al., eds. Advances in Neural Information Processing Systems 24. Cambridge, USA: MIT Press, 2011: 1638-1646. [17] BOYD S, PARIKH N, CHU E, et al. Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Foundations and Trends in Machine Learning, 2011, 3(1): 1-122. [18] CANDES E J, PLAN Y. Tight Oracle Inequalities for Low-Rank Matrix Recovery from a Minimal Number of Noisy Random Measurements. IEEE Trans on Information Theory, 2011, 57(4): 2342-2359. [19] LI K, CONG S. A Robust Compressive Quantum State Tomography Algorithm Using ADMM // Proc of the 19th World Congress of the International Federation of Automation Control. Cape Town, South Africa, 2014: 6878-6883.
2016, Vol. 29 
