Skip to main content

Advertisement

Springer Nature Link
Log in

Optimal control methods for quantum gate preparation: a comparative study

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this study, optimal control methods for quantum gate preparation are investigated. Quantum computation demands very high fidelity and requires controls to be easily tunable to achieve different computational tasks. Here, NOT and Controlled-NOT gates are prepared using four optimal control approaches on single- and two-qubit spin systems. Techniques we employed and compared are Krotov method, gradient ascent pulse engineering (GRAPE), chopped random basis optimization (CRAB) and gradient optimization of analytic controls (GOAT). For the preparation of NOT gate both unitary and Lindbladian dynamics are considered. From the numerical simulations, it is observed that GOAT achieves better results as compared to Krotov, GRAPE and CRAB, in terms of minimum infidelity, algorithmic simplicity and analyticity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 10th edn. Cambridge University Press, Cambridge (2010). https://doi.org/10.1017/CBO9780511976667

    Book  MATH  Google Scholar 

  2. Vandersypen, L.M., Chuang, I.L.: NMR techniques for quantum control and computation. Rev. Mod. Phys. 76(4), 1037 (2005)

    Article  ADS  Google Scholar 

  3. Rowland, B., Jones, J.A.: Implementing quantum logic gates with gradient ascent pulse engineering: principles and practicalities. Philos. Trans. R. Soc. A 370(1976), 4636 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Veldhorst, M., Yang, C., Hwang, J., Huang, W., Dehollain, J., Muhonen, J., Simmons, S., Laucht, A., Hudson, F., Itoh, K.M., et al.: A two-qubit logic gate in silicon. Nature 526(7573), 410 (2015)

    Article  ADS  Google Scholar 

  5. Zu, C., Wang, W.B., He, L., Zhang, W.G., Dai, C.Y., Wang, F., Duan, L.M.: Experimental realization of universal geometric quantum gates with solid-state spins. Nature 514(7520), 72 (2014)

    Article  ADS  Google Scholar 

  6. Shim, Y.P., Tahan, C.: Semiconductor-inspired design principles for superconducting quantum computing. Nat. Commun. 7, 11059 (2016)

    Article  ADS  Google Scholar 

  7. DAlessandro, D.: Small time controllability of systems on compact Lie groups and spin angular momentum. J. Math. Phys. 42(9), 4488 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Albertini, F., D’Alessandro, D.: Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Autom. Control 48(8), 1399 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dür, W., Vidal, G., Cirac, J., Linden, N., Popescu, S.: Entanglement capabilities of nonlocal Hamiltonians. Phys. Rev. Lett. 87(13), 137901 (2001)

    Article  ADS  Google Scholar 

  10. Kraus, B., Cirac, J.: Optimal creation of entanglement using a two-qubit gate. Phys. Rev. A 63(6), 062309 (2001)

    Article  ADS  Google Scholar 

  11. Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52(5), 3457 (1995)

    Article  ADS  Google Scholar 

  12. DiVincenzo, D.P., Bacon, D., Kempe, J., Burkard, G., Whaley, K.B.: Universal quantum computation with the exchange interaction. Nature 408(6810), 339 (2000)

    Article  ADS  Google Scholar 

  13. Makhlin, Y.: Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations. Quantum Inf. Process. 1(4), 243 (2002)

    Article  MathSciNet  Google Scholar 

  14. Zhang, J., Vala, J., Sastry, S., Whaley, K.B.: Geometric theory of nonlocal two-qubit operations. Phys. Rev. A 67(4), 042313 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  15. Zhang, J., Whaley, K.B.: Generation of quantum logic operations from physical Hamiltonians. Phys. Rev. A 71(5), 052317 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Galiautdinov, A.: Generation of high-fidelity controlled-NOT logic gates by coupled superconducting qubits. Phys. Rev. A 75(5), 052303 (2007)

    Article  ADS  Google Scholar 

  17. Geller, M.R., Pritchett, E.J., Galiautdinov, A., Martinis, J.M.: Quantum logic with weakly coupled qubits. Phys. Rev. A 81(1), 012320 (2010)

    Article  ADS  Google Scholar 

  18. Pal, A., Rashba, E.I., Halperin, B.I.: Exact CNOT gates with a single nonlocal rotation for quantum-dot qubits. Phys. Rev. B 92(12), 125409 (2015)

    Article  ADS  Google Scholar 

  19. Watts, P., Vala, J., Müller, M.M., Calarco, T., Whaley, K.B., Reich, D.M., Goerz, M.H., Koch, C.P.: Optimizing for an arbitrary perfect entangler. I. Functionals. Phys. Rev. A 91(6), 062306 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  20. Goerz, M.H., Gualdi, G., Reich, D.M., Koch, C.P., Motzoi, F., Whaley, K.B., Vala, J., Müller, M.M., Montangero, S., Calarco, T.: Optimizing for an arbitrary perfect entangler. II. Application. Phys. Rev. A 91(6), 062307 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  21. Palao, J.P., Kosloff, R.: Quantum computing by an optimal control algorithm for unitary transformations. Phys. Rev. Lett. 89(18), 188301 (2002). https://doi.org/10.1103/PhysRevLett.89.188301

    Article  ADS  Google Scholar 

  22. Palao, J.P., Kosloff, R.: Optimal control theory for unitary transformations. Phys. Rev. A 68(6), 062308 (2003). https://doi.org/10.1103/PhysRevA.68.062308

    Article  ADS  Google Scholar 

  23. Carlini, A., Hosoya, A., Koike, T., Okudaira, Y.: Time-optimal unitary operations. Phys. Rev. A 75(4), 042308 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  24. Schulte-Herbrüggen, T., Spörl, A., Khaneja, N., Glaser, S.: Optimal control-based efficient synthesis of building blocks of quantum algorithms: A perspective from network complexity towards time complexity. Phys. Rev. A 72(4), 042331 (2005)

    Article  ADS  Google Scholar 

  25. Nigmatullin, R., Schirmer, S.: Implementation of fault-tolerant quantum logic gates via optimal control. New J. Phys. 11(10), 105032 (2009)

    Article  ADS  Google Scholar 

  26. Schulte-Herbrüggen, T., Spörl, A., Khaneja, N., Glaser, S.: Optimal control for generating quantum gates in open dissipative systems. J. Phys. B Atom. Mol. Opt. Phys. 44(15), 154013 (2011)

    Article  ADS  Google Scholar 

  27. Zhou, W., Schirmer, S., Zhang, M., Dai, H.Y.: Bang-bang control design for quantum state transfer based on hyperspherical coordinates and optimal time-energy control. J. Phys. A Math. Theoret. 44(10), 105303 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Huang, S.Y., Goan, H.S.: Optimal control for fast and high-fidelity quantum gates in coupled superconducting flux qubits. Phys. Rev. A 90(1), 012318 (2014)

    Article  ADS  Google Scholar 

  29. Bhole, G., Anjusha, V., Mahesh, T.: Steering quantum dynamics via bang-bang control: implementing optimal fixed-point quantum search algorithm. Phys. Rev. A 93(4), 042339 (2016)

    Article  ADS  Google Scholar 

  30. Hirose, M., Cappellaro, P.: Time-optimal control with finite bandwidth. Quantum Inf. Process. 17(4), 88 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Wen, J., Cong, S.: Preparation of quantum gates for open quantum systems by Lyapunov control method. Open Syst. Inf. Dyn. 23(01), 1650005 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Silveira, H.B., da Silva, P.P., Rouchon, P.: Quantum gate generation for systems with drift in U (n) using Lyapunov–LaSalle techniques. Int. J. Control 89(12), 2466 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Khaneja, N., Reiss, T., Kehlet, C., Schulte-Herbrüggen, T., Glaser, S.J.: Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms. J. Magn. Reson. 172(2), 296 (2005)

    Article  ADS  Google Scholar 

  34. De Fouquieres, P., Schirmer, S., Glaser, S., Kuprov, I.: Second order gradient ascent pulse engineering. J. Magn. Reson. 212(2), 412 (2011)

    Article  ADS  Google Scholar 

  35. Leung, N., Abdelhafez, M., Koch, J., Schuster, D.: Speedup for quantum optimal control from automatic differentiation based on graphics processing units. Phys. Rev. A 95(4), 042318 (2017)

    Article  ADS  Google Scholar 

  36. Caneva, T., Calarco, T., Montangero, S.: Chopped random-basis quantum optimization. Phys. Rev. A 84(2), 022326 (2011)

    Article  ADS  Google Scholar 

  37. Rach, N., Müller, M.M., Calarco, T., Montangero, S.: Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape. Phys. Rev. A 92(6), 062343 (2015)

    Article  ADS  Google Scholar 

  38. Machnes, S., Assémat, E., Tannor, D., Wilhelm, F.K.: Tunable, flexible, and efficient optimization of control pulses for practical qubits. Phys. Rev. Lett. 120(15), 150401 (2018)

    Article  ADS  Google Scholar 

  39. Sørensen, J., Aranburu, M., Heinzel, T., Sherson, J.: Quantum optimal control in a chopped basis: applications in control of Bose–Einstein condensates. Phys. Rev. A 98(2), 022119 (2018)

    Article  ADS  Google Scholar 

  40. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48(2), 119 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17(5), 821 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  42. Lendi, K.: Evolution matrix in a coherence vector formulation for quantum Markovian master equations of N-level systems. J. Phys. A Math. Gen. 20(1), 15 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  43. d’Alessandro, D.: Introduction to Quantum Control and Dynamics. Chapman and Hall/CRC, New York (2007)

    Book  MATH  Google Scholar 

  44. Dirr, G., Helmke, U.: Lie theory for quantum control. GAMM-Mitteilungen 31(1), 59 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Koch, C.P.: Controlling open quantum systems: tools, achievements, and limitations. J. Phys. Condens. Matter 28(21), 213001 (2016)

    Article  ADS  Google Scholar 

  46. Schirmer, S.G., de Fouquieres, P.: Efficient algorithms for optimal control of quantum dynamics: the Krotov method unencumbered. New J. Phys. 13(7), 073029 (2011)

    Article  ADS  Google Scholar 

  47. Boutin, S., Andersen, C.K., Venkatraman, J., Ferris, A.J., Blais, A.: Resonator reset in circuit QED by optimal control for large open quantum systems. Phys. Rev. A 96(4), 042315 (2017)

    Article  ADS  Google Scholar 

  48. Machnes, S., Sander, U., Glaser, S., de Fouquieres, P., Gruslys, A., Schirmer, S., Schulte-Herbrüggen, T.: Comparing, optimizing, and benchmarking quantum-control algorithms in a unifying programming framework. Phys. Rev. A 84(2), 022305 (2011)

    Article  ADS  Google Scholar 

  49. Ben-Israel, A.: A Newton-Raphson method for the solution of systems of equations. J. Math. Anal. Appl. 15(2), 243 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  50. Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  51. Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  52. Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E.: Convergence properties of the Nelder–Mead simplex method in low dimensions. SIAM J. Optim. 9(1), 112 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  53. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2012). https://doi.org/10.1017/9781139020411

    Book  Google Scholar 

  54. de Fouquieres, P.: Implementing quantum gates by optimal control with doubly exponential convergence. Phys. Rev. Lett. 108(11), 110504 (2012)

    Article  Google Scholar 

  55. Zhu, W., Rabitz, H.: A rapid monotonically convergent iteration algorithm for quantum optimal control over the expectation value of a positive definite operator. J. Chem. Phys. 109(2), 385 (1998)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Automation, University of Science and Technology of China, Hefei, 230027, China

    Bilal Riaz, Cong Shuang & Shahid Qamar

Authors
  1. Bilal Riaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cong Shuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shahid Qamar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bilal Riaz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Riaz, B., Shuang, C. & Qamar, S. Optimal control methods for quantum gate preparation: a comparative study. Quantum Inf Process 18, 100 (2019). https://doi.org/10.1007/s11128-019-2190-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-019-2190-0

Keywords