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Quantum parameter estimation via dispersive measurement in circuit QED

  • Beili Gong
  • Yang Yang
  • Wei CuiEmail author
Article
  • 108 Downloads

Abstract

We investigate the quantum parameter estimation in circuit quantum electrodynamics via dispersive measurement. Based on the Metropolis–Hastings algorithm and the Markov chain Monte Carlo (MCMC) integration, a new algorithm is proposed to calculate the Fisher information by the stochastic master equation. The Fisher information is expressed in the form of log-likelihood functions and further approximated by the MCMC integration. Numerical results show that the evolution of the Fisher information can approach the quantum Fisher information in a short time interval. These results demonstrate the effectiveness of the proposed algorithm. Finally, based on the proposed algorithm, we consider the effects of the measurement operator and the measurement efficiency on the Fisher information.

Keywords

Quantum Fisher information Quantum parameter estimation Stochastic master equation 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant 61873317 and by the Fundamental Research Funds for the Central Universities.

References

  1. 1.
    Helstrom, C.W.: Quantum Detection and Estimation Theory, Mathematics in Science and Engineering, vol. 123. Academic Press, New York (1976)zbMATHGoogle Scholar
  2. 2.
    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. Edizioni della Normale, Pisa (2011)CrossRefGoogle Scholar
  3. 3.
    Wiseman, H.M., Milburn, G.J.: Quantum Measurements and Control. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  4. 4.
    Escher, B.M., de Matos Filho, R.L., Davidovich, L.: General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phys. 7, 406–411 (2011)CrossRefGoogle Scholar
  5. 5.
    Giovannetti, V., Lloyd, S., Maccone, L.: Quantum-enhanced measurements: beating the standard quantum limit. Science 306, 1330–1336 (2004)ADSCrossRefGoogle Scholar
  6. 6.
    Chapeau-Blondeau, F.: Entanglement-assisted quantum parameter estimation from a noisy qubit pair: a Fisher information analysis. Phys. Lett. A 381, 1369–1378 (2017)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bradshaw, M., Assad, S.M., Lam, P.K.: A tight Cramér–Rao bound for joint parameter estimation with a pure two-mode squeezed probe. Phys. Lett. A 381, 2598–2607 (2017)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Li, X.L., Li, J.G., Wang, Y.M.: The influence of non-Gaussian noise on the accuracy of parameter estimation. Phys. Lett. A 381, 216–220 (2017)CrossRefGoogle Scholar
  9. 9.
    Seveso, L., Rossi, M.A.C., Paris, M.G.A.: Quantum metrology beyond the quantum Cramér–Rao theorem. Phys. Rev. A 95, 012111 (2017)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Caves, C.M., Thorne, K.S., Drever, R.W., Sandberg, V.D., Zimmermann, M.: On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle. Rev. Mod. Phys. 52, 341–392 (1980)ADSCrossRefGoogle Scholar
  11. 11.
    Caves, C.M.: Quantum-mechanical noise in an interferometer. Phys. Rev. D 23, 1693–1708 (1981)ADSCrossRefGoogle Scholar
  12. 12.
    Zwierz, M., Pérez-Delgado, C.A., Kok, P.: General optimality of the Heisenberg limit for quantum metrology. Phys. Rev. Lett. 105, 180402 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    Yuan, H., Fung, C.H.F.: Optimal feedback scheme and universal time scaling for hamiltonian parameter estimation. Phys. Rev. Lett. 115, 110401 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    Gong, B., Cui, W.: Multi-objective optimization in quantum parameter estimation. Sci. China-Phys. Mech. Astron. 61(4), 040312 (2018)ADSCrossRefGoogle Scholar
  15. 15.
    Cui, W.: Research opportunities arising from measurement and estimation of quantum systems. Control Theory Technol. 16(3), 241–243 (2018)CrossRefGoogle Scholar
  16. 16.
    Hodges, J.L., Lehmann, E.L.: Some Applications of the Cramér–Rao Inequality. Springer, Boston (2012)CrossRefGoogle Scholar
  17. 17.
    Zhong, W., Sun, Z., Ma, J., Wang, X.G., Nori, F.: Fisher information under decoherence in Bloch representation. Phys. Rev. A 87, 022337 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    Li, N., Luo, S.: Entanglement detection via quantum Fisher information. Phys. Rev. A 88, 014301 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    Zhang, Y., Li, X.W., Yang, W., Jin, G.R.: Quantum Fisher information of entangled coherent states in the presence of photon loss. Phys. Rev. A 88, 043832 (2013)ADSCrossRefGoogle Scholar
  20. 20.
    Smith, G.A., Silberfarb, A., Deutsch, I.H., Jessen, P.S.: Efficient quantum state estimation by continuous weak measurement and dynamical control. Phys. Rev. Lett. 97, 180403 (2006)ADSCrossRefGoogle Scholar
  21. 21.
    Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351–1354 (1988)ADSCrossRefGoogle Scholar
  22. 22.
    Zhang, L., Datta, A., Walmsley, I.A.: Precision metrology using weak measurements. Phys. Rev. Lett. 114, 210801 (2015)ADSCrossRefGoogle Scholar
  23. 23.
    Xu, X.Y., Kedem, Y., Sun, K., Vaidman, L., Li, C.F., Guo, G.C.: Phase estimation with weak measurement using a white light source. Phys. Rev. Lett. 111, 033604 (2013)ADSCrossRefGoogle Scholar
  24. 24.
    Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, New York (2002)zbMATHGoogle Scholar
  25. 25.
    Albarelli, F., Rossi, M.A.C., Paris, M.G.A., Genoni, M.G.: Ultimate limits for quantum magnetometry via time-continuous measurements. New J. Phys. 19, 123011 (2017)ADSCrossRefGoogle Scholar
  26. 26.
    Ralph, J.F., Jacobs, K., Hill, C.D.: Frequency tracking and parameter estimation for robust quantum state estimation. Phys. Rev. A 84, 052119 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    Gammelmark, S., Mølmer, K.: Bayesian parameter inference from continuously monitored quantum systems. Phys. Rev. A 87, 032115 (2013)ADSCrossRefGoogle Scholar
  28. 28.
    Gammelmark, S., Mølmer, K.: Fisher information and the quantum Cramér–Rao sensitivity limit of continuous measurements. Phys. Rev. Lett. 112, 170401 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    Genoni, M.G.: Cramér–Rao bound for time-continuous measurements in linear Gaussian quantum systems. Phys. Rev. A 95, 012116 (2017)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    You, J.Q., Nori, F.: Superconducting circuits and quantum information. Phys. Today 58(11), 42–47 (2005)CrossRefGoogle Scholar
  31. 31.
    You, J.Q., Nori, F.: Atomic physics and quantum optics using superconducting circuits. Nature 474, 589–597 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    Blais, A., Huang, R.S., Wallraff, A., Girvin, S.M., Schoelkopf, R.J.: Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004)ADSCrossRefGoogle Scholar
  33. 33.
    Vijay, R., Macklin, C., Slichter, D.H., Weber, S.J., Murch, K.W., Naik, R., Korotkov, A.N., Siddiqi, I.: Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback. Nature 490, 77–80 (2012)ADSCrossRefGoogle Scholar
  34. 34.
    Xiang, Z.L., Ashhab, S., You, J.Q., Nori, F.: Hybrid quantum circuit consisting of a superconducting flux qubit coupled to both a spin ensemble and a transmission-line resonator. Phys. Rev. B 87, 144516 (2013)ADSCrossRefGoogle Scholar
  35. 35.
    Cui, W., Nori, F.: Feedback control of Rabi oscillations in circuit QED. Phys. Rev. A 88, 063823 (2013)ADSCrossRefGoogle Scholar
  36. 36.
    Gilks, W.R., Richardson, S., Spiegelhalter, D.J.: Markov Chain Monte Carlo in Practice. Chapman & Hall, London (1996)zbMATHGoogle Scholar
  37. 37.
    Johnston, I.G.: Efficient parametric inference for stochastic biological systems with measured variability. Stat. Appl. Genet. Mol. Biol. 13, 379–390 (2014)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Efendiev, Y., Datta-Gupta, A., Ma, X., Mallick, B.: Modified Markov chain Monte Carlo method for dynamic data integration using streamline approach. Math. Geosci. 40, 213–232 (2008)CrossRefGoogle Scholar
  39. 39.
    Braunstein, S.L., Caves, C.M.: Statistical distance and the geometry of quantum states. Phys. Rev. Lett. 72(22), 3439–3443 (1994)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Ciampini, M.A., Spagnolo, N., Vitelli, C., Pezzè, L., Smerzi, A., Sciarrino, F.: Quantum-enhanced multiparameter estimation in multiarm interferometers. Sci. Rep. 6, 28881 (2016)ADSCrossRefGoogle Scholar
  41. 41.
    Kiilerich, A.H., Mølmer, K.: Bayesian parameter estimation by continuous homodyne detection. Phys. Rev. A 94, 032103 (2016)ADSCrossRefGoogle Scholar
  42. 42.
    Devoret, M.H., Schoelkopf, J.R.: Superconducting circuits and quantum information: an outlook. Science 339, 1169–1174 (2013)ADSCrossRefGoogle Scholar
  43. 43.
    Gambetta, J., Blais, A., Boissonneault, M., Houck, A.A., Schuster, D.I., Girvin, S.M.: Quantum trajectory approach to circuit QED: quantum jumps and the Zeno effect. Phys. Rev. A 77, 012112 (2008)ADSCrossRefGoogle Scholar
  44. 44.
    Yang, Y., Gong, B., Cui, W.: Real-time quantum state estimation in circuit QED via the Bayesian approach. Phys. Rev. A 97, 012119 (2018)ADSCrossRefGoogle Scholar
  45. 45.
    Qi, B., Guo, L.: Is measurement-based feedback still better for quantum control systems? Syst. Control Lett. 59, 333–339 (2010)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Feng, W., Liang, P.F., Qin, L.P., Li, X.Q.: Exact quantum Bayesian rule for qubit measurement in circuit QED. Sci. Rep. 6, 20492 (2016)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Automation Science and EngineeringSouth China University of TechnologyGuangzhouChina

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