Implementing quantum stochastic differential equations on a quantum computer

  • Gé VissersEmail author
  • Luc Bouten


We study how to solve quantum stochastic differential equations (QSDEs) using a quantum computer. This is illustrated by an implementation of the QSDE that models the interaction of a laser driven two-level atom with the electromagnetic field in the vacuum state, on the IBMqx4 Tenerife quantum computer (IBM in The IBM Q experience. Accessed 23 Nov 2018, 2018). We compare the resulting master equation and quantum filtering equations to existing theory. In this way we characterize the performance of the computer.


Quantum computing Quantum optics Quantum stochastic differential equations Quantum filtering 



  1. 1.
    Aleksandrowicz, G., Alexander, T., Barkoutsos, P., Bello, L., Ben-Haim, Y., Bucher, D., et al.: Qiskit: An Open-source Framework for Quantum Computing. (2019).
  2. 2.
    Attal, S., Pautrat, Y.: From repeated to continuous quantum interactions. Ann. Henri Poincaré 7, 59–104 (2006)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Belavkin, V.P.: Quantum stochastic calculus and quantum nonlinear filtering. J. Multivar. Anal. 42, 171–201 (1992)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bouten, L., van Handel, R., James, M.: An introduction to quantum filtering. SIAM J. Control Optim. 46, 2199–2241 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bouten, L.M., van Handel, R.: Discrete approximation of quantum stochastic models. J. Math. Phys. 49, 102109 (2008)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bouten, L.M., van Handel, R., James, M.: A discrete invitation to quantum filtering and feedback control. SIAM Rev. 51, 239–316 (2009)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Brun, T.A.: A simple model of quantum trajectories. Am. J. Phys. 70, 719–737 (2002)ADSCrossRefGoogle Scholar
  8. 8.
    Carmichael, H.J.: An Open Systems Approach to Quantum Optics. Springer, Berlin (1993)zbMATHGoogle Scholar
  9. 9.
    Cross, A.W., Bishop, L.S., Smolin, J.A., Gambetta J.M.: Open quantum assembly language. ArXiv e-prints arXiv:1707.03429, July (2017)
  10. 10.
    Davies, E.B.: Quantum stochastic processes. Commun. Math. Phys. 15, 277–304 (1969)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Gough, J., James, M.: Quantum feedback networks: Hamiltonian formulation. Commun. Math. Phys. 287, 1109–1132 (2009)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gough, J., James, M.: The series product and its application to quantum feedforward and feedback networks. IEEE Trans. Autom. Control 54, 2530–2544 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gough, J., Sobolev, A.: Stochastic Schrödinger equations as limit of discrete filtering. Open Syst. Inf. Dyn. 11, 235–255 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gross, J.A., Caves, C.M., Milburn, G.J., Combes, J.: Qubit models of weak continuous measurements: Markovian conditional and open-system dynamics. Quantum Sci. Technol. 3(2), 024005 (2018)ADSCrossRefGoogle Scholar
  15. 15.
    Hudson, R.L., Parthasarathy, K.R.: Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984)ADSCrossRefGoogle Scholar
  16. 16.
    IBM.: The IBM Q experience. Accessed 23 Nov 2018 (2018)
  17. 17.
    Kümmerer, B.: Markov dilations on \(W^*\)-algebras. J. Funct. Anal. 63, 139–177 (1985)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lindsay, J.M., Parthasarathy, K.R.: The passage from random walk to diffusion in quantum probability II. Sankhya Indian J. Stat. 50, 151–170 (1988)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Maassen, H.: Quantum Markov processes on Fock space described by integral kernels. In: Accardi, L., von Waldenfels, W. (eds.) QP and Applications II, vol. 1136, pp. 361–374. Springer, Berlin (1985). Lecture Notes in MathematicsGoogle Scholar
  20. 20.
    Meyer, P.-A.: Quantum Probability for Probabilists. Springer, Berlin (1993)CrossRefGoogle Scholar
  21. 21.
    Nielsen, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  22. 22.
    Parthasarathy, K.R.: The passage from random walk to diffusion in quantum probability. J. Appl. Probab. 25A, 151–166 (1988)MathSciNetCrossRefGoogle Scholar
  23. 23.
    von Waldenfels, W.: A Measure Theoretical Approach to Quantum Stochastic Processes, vol. 878. Springer, Berlin (2014). Lecture notes in PhysicsCrossRefGoogle Scholar
  24. 24.
    Yip, K.W., Albash, T., Lidar, D.: Quantum trajectories for time-dependent adiabatic master equations. Phys. Rev. A 97, 022116 (2018)ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Q1t BVMolenhoekThe Netherlands

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