Coherent feedback control of a single qubit in diamond

Abstract

Engineering desired operations on qubits subjected to the deleterious effects of their environment is a critical task in quantum information processing, quantum simulation and sensing. The most common approach relies on open-loop quantum control techniques, including optimal-control algorithms based on analytical1 or numerical2 solutions, Lyapunov design3 and Hamiltonian engineering4. An alternative strategy, inspired by the success of classical control, is feedback control5. Because of the complications introduced by quantum measurement6, closed-loop control is less pervasive in the quantum setting and, with exceptions7,8, its experimental implementations have been mainly limited to quantum optics experiments. Here we implement a feedback-control algorithm using a solid-state spin qubit system associated with the nitrogen vacancy centre in diamond, using coherent feedback9 to overcome the limitations of measurement-based feedback, and show that it can protect the qubit against intrinsic dephasing noise for milliseconds. In coherent feedback, the quantum system is connected to an auxiliary quantum controller (ancilla) that acquires information about the output state of the system (by an entangling operation) and performs an appropriate feedback action (by a conditional gate). In contrast to open-loop dynamical decoupling techniques10, feedback control can protect the qubit even against Markovian noise and for an arbitrary period of time (limited only by the coherence time of the ancilla), while allowing gate operations. It is thus more closely related to quantum error-correction schemes11,12,13,14, although these require larger and increasing qubit overheads. Increasing the number of fresh ancillas enables protection beyond their coherence time. We further evaluate the robustness of the feedback protocol, which could be applied to quantum computation and sensing, by exploring a trade-off between information gain and decoherence protection, as measurement of the ancilla–qubit correlation after the feedback algorithm voids the protection, even if the rest of the dynamics is unchanged.

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Figure 1: Feedback algorithm and experimental implementation.
Figure 2: Experimental demonstration of the feedback-based protection algorithm.
Figure 3: Partial measurement of the ancilla.

References

  1. 1

    Boscain, U. & Mason, P. Time minimal trajectories for a spin 1/2 particle in a magnetic field. J. Math. Phys. 47, 062101 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2

    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, 296–305 (2005)

    CAS  ADS  Article  Google Scholar 

  3. 3

    Wang, X. & Schirmer, S. G. Analysis of Lyapunov method for control of quantum states. Automatic Control. IEEE Trans. Automatic Control 55, 2259–2270 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4

    Ajoy, A. & Cappellaro, P. Quantum simulation via filtered Hamiltonian engineering: application to perfect quantum transport in spin networks. Phys. Rev. Lett. 110, 220503 (2013)

    ADS  Article  Google Scholar 

  5. 5

    Wiseman, H. M. & Milburn, G. J. Quantum theory of optical feedback via homodyne detection. Phys. Rev. Lett. 70, 548–551 (1993)

    CAS  ADS  Article  Google Scholar 

  6. 6

    Doherty, A. C., Jacobs, K. & Jungman, G. Information, disturbance, and Hamiltonian quantum feedback control. Phys. Rev. A 63, 062306 (2001)

    ADS  Article  Google Scholar 

  7. 7

    Vijay, R. et al. Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback. Nature 490, 77–80 (2012)

    CAS  ADS  Article  Google Scholar 

  8. 8

    Sayrin, C. et al. Real-time quantum feedback prepares and stabilizes photon number states. Nature 477, 73–77 (2011)

    CAS  ADS  Article  Google Scholar 

  9. 9

    Lloyd, S. Coherent quantum feedback. Phys. Rev. A 62, 022108 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10

    Viola, L., Knill, E. & Lloyd, S. Dynamical decoupling of open quantum systems. Phys. Rev. Lett. 82, 2417–2421 (1999)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  11. 11

    Shor, P. W. Fault-tolerant quantum computation. In 37th Annual Symposium on Foundations of Computer Science 56–65 (IEEE Comput. Soc., 1996)

  12. 12

    Taminiau, H. T., Cramer, J., van der Sar, T., Dobrovitski, V. V. & Hanson, R. Universal control and error correction in multi-qubit spin registers in diamond. Nature Nanotechnol . 9, 171–176 (2014)

    CAS  ADS  Article  Google Scholar 

  13. 13

    Waldherr, G. et al. Quantum error correction in a solid-state hybrid spin register. Nature 506, 204–207 (2014)

    CAS  ADS  Article  Google Scholar 

  14. 14

    Cramer, J. et al. Repeated quantum error correction on a continuously encoded qubit by real-time feedback. Preprint at http://arXiv.org/abs/1508.01388 (2015)

  15. 15

    Ticozzi, F. & Viola, L. Single-bit feedback and quantum-dynamical decoupling. Phys. Rev. A 74, 052328 (2006)

    ADS  Article  Google Scholar 

  16. 16

    Griffiths, R. & Niu, C.-S. Semiclassical Fourier transform for quantum computation. Phys. Rev. Lett. 76, 3228–3231 (1996)

    CAS  ADS  Article  Google Scholar 

  17. 17

    Kessler, E. M., Lovchinsky, I., Sushkov, A. O. & Lukin, M. D. Quantum error correction for metrology. Phys. Rev. Lett. 112, 150802 (2014)

    CAS  ADS  Article  Google Scholar 

  18. 18

    Arrad, G., Vinkler, Y., Aharonov, D. & Retzker, A. Increasing sensing resolution with error correction. Phys. Rev. Lett. 112, 150801 (2014)

    CAS  ADS  Article  Google Scholar 

  19. 19

    Unden, T. et al. Quantum metrology enhanced by repetitive quantum error correction. Preprint at http://arXiv.org/abs/1602.07144 (2015)

  20. 20

    Chen, M., Hirose, M. & Cappellaro, P. Measurement of transverse hyperfine interaction by forbidden transitions. Phys. Rev. B 92, 020101 (2015)

    ADS  Article  Google Scholar 

  21. 21

    Jacques, V. et al. Dynamic polarization of single nuclear spins by optical pumping of nitrogen-vacancy color centers in diamond at room temperature. Phys. Rev. Lett. 102, 057403 (2009)

    CAS  ADS  Article  Google Scholar 

  22. 22

    Laraoui, A. et al. High-resolution correlation spectroscopy of 13C spins near a nitrogen-vacancy centre in diamond. Nature Commun . 4, 1651 (2013)

    ADS  Article  Google Scholar 

  23. 23

    Ryan, C. A., Hodges, J. S. & Cory, D. G. Robust decoupling techniques to extend quantum coherence in diamond. Phys. Rev. Lett. 105, 200402 (2010)

    CAS  ADS  Article  Google Scholar 

  24. 24

    Morton, J. J. L. et al. Solid-state quantum memory using the 31P nuclear spin. Nature 455, 1085–1088 (2008)

    CAS  ADS  Article  Google Scholar 

  25. 25

    Kim, Y.-H., Yu, R., Kulik, S. P., Shih, Y. & Scully, M. O. Delayed “choice” quantum eraser. Phys. Rev. Lett. 84, 1–5 (2000)

    CAS  ADS  Article  Google Scholar 

  26. 26

    Steiner, M., Neumann, P., Beck, J., Jelezko, F. & Wrachtrup, J. Universal enhancement of the optical readout fidelity of single electron spins at nitrogen-vacancy centers in diamond. Phys. Rev. B 81, 035205 (2010)

    ADS  Article  Google Scholar 

  27. 27

    Pla, J. J. et al. High-fidelity readout and control of a nuclear spin qubit in silicon. Nature 496, 334–338 (2013)

    CAS  ADS  Article  Google Scholar 

  28. 28

    Wolfowicz, G. et al. Conditional control of donor nuclear spins in silicon using stark shifts. Phys. Rev. Lett. 113, 157601 (2014)

    ADS  Article  Google Scholar 

  29. 29

    Widmann, M. et al. Coherent control of single spins in silicon carbide at room temperature. Nature Mater . 14, 164–168 (2015)

    CAS  ADS  Article  Google Scholar 

  30. 30

    Chekhovich, E. A. et al. Nuclear spin effects in semiconductor quantum dots. Nature Mater . 12, 494–504 (2013)

    CAS  ADS  Article  Google Scholar 

  31. 31

    Lindblad, G. On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976)

    ADS  MathSciNet  Article  Google Scholar 

  32. 32

    Bloch, F. Generalized theory of relaxation. Phys. Rev. 105, 1206–1222 (1957)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  33. 33

    Kälin, M. & Schweiger, A. Radio-frequency-driven electron spin echo envelope modulation spectroscopy on spin systems with isotropic hyperfine interactions. J. Chem. Phys . 115, 10863–10875 (2001)

    ADS  Article  Google Scholar 

  34. 34

    Mims, W. B. Envelope modulation in spin-echo experiments. Phys. Rev. B 5, 2409–2419 (1972)

    ADS  Article  Google Scholar 

  35. 35

    Neumann, P. et al. Excited-state spectroscopy of single NV defects in diamond using optically detected magnetic resonance. New J. Phys. 11, 013017 (2009)

    ADS  Article  Google Scholar 

  36. 36

    Meriles, C. A. et al. Imaging mesoscopic nuclear spin noise with a diamond magnetometer. J. Chem. Phys . 133, 124105 (2010)

    ADS  Article  Google Scholar 

  37. 37

    Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information Ch. 4 (Cambridge Univ. Press, 2003)

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Acknowledgements

We thank L. Viola, F. Ticozzi, M. Lukin and F. Jelezko for discussions. This work was supported in part by the US Air Force Office of Scientific Research grant number FA9550-12-1-0292 and by the US Office of Naval Research grant number N00014-14-1-0804.

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M.H. and P.C. contributed to all aspects of this work.

Corresponding author

Correspondence to Paola Cappellaro.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Feedback circuit.

See, for example, ref. 37 for an explanation of the notation. a, b, Measurement-based (a) and coherent (b) feedback algorithms. In the shaded regions we highlight the differences between the two strategies. The measurement-based protocol requires a measurement of the ancilla and subsequent classically controlled operation (double lines indicate a classical wire). The coherent feedback protocol does not perform a measurement, but requires a coherent controlled operation. c, Re-initializing the ancilla or using multiple fresh ancillas can extend the feedback protection beyond the coherence time of the ancilla. d, A concatenated feedback algorithm with two ancillas can protect the qubit from general noise (applied along any axis).

Extended Data Figure 2 Protected gate

. The feedback algorithm is compatible with the application of NOT gates at any point during the protection time. a, Rabi oscillations embedded in the feedback-based protection algorithm. b, When the NOT gate is applied in the middle of the protection time, it halves the period of the oscillations due to the hyperfine coupling; see equation (1). c, More complex evolution is obtained when inserting the NOT gate at other times. Here we show the behaviour for τπ = τ/4. Black circles are experimental data with error bars representing their standard deviation (see Fig. 2); the solid lines are fits using equation (1). PL, photoluminescence.

Extended Data Figure 3 Weak measurement circuit.

a, After the feedback algorithm is completed, a controlled-X rotation entangles the qubit with the ancilla, allowing us to perform a partial measurement of the ancilla: . The strength of the measurement can be adjusted by the angle ϕ. b, The control sequence in a is simplified by combining the partial measurement gates with the preceding gates of the feedback algorithm. As a result, to implement the weak correlated measurement in the experiment, we simply perform a controlled-phase rotation gate with θ = π/2 − ϕ, instead of the controlled-Z (θ = π) rotation that is required for the feedback algorithm.

Extended Data Figure 4 Control and coherence of the nitrogen nuclear spin.

a, Typical control sequence. The first laser pulse initializes the spins into the |0, 1〉 state. After a nuclear-spin gate (radio-frequency (RF) driving), to detect the nuclear spin state we use a microwave (MW)-selective π pulse that maps the nuclear spin state onto that of the NV spin, which is subsequently detected by the second laser pulse. b, Nuclear magnetic resonance at about 390 G. We sweep the radio frequency at fixed pulse length. The dip in the photoluminescence (PL) spectrum indicates the resonant frequency of the |0, 1〉 ↔ |0, 0〉 transition. c, Nuclear Rabi oscillations. We drive resonantly the |0, 1〉 ↔ |0, 0〉 transition. The measured nutation rate (Rabi frequency) is Ωn ≈ 26.3 kHz (ref. 20). d, Nuclear Ramsey fringes under the sequence π/2–τ–π/2, where τ is the free evolution time. We measure a dephasing time of the nuclear spin , which is limited by the NV electronic spin lattice relaxation process (T1 = 4.5 ms; red circles and dashed line). Error bars represent the standard deviation of the signal, calculated by error propagation from the photoluminescence intensity of the signal and reference photoluminescence curves acquired for each data point for mS = {0, −1}. Lines are fits to the data.

Extended Data Figure 5 Electronic- and nuclear-spin-dependent fluorescence at different magnetic field strengths.

At the lower magnetic field (left; B = 390 G), the fluorescence intensity shows only a weak dependence on the nuclear spin state in the mS = −1 manifold (indicated by the overlapping data), whereas at B = 514 G, which is very close to the ESLAC, a strong dependence on the nuclear spin state is observed in both manifolds (right; indicated by the non-overlapping data). From these fluorescence measurements, we obtained the parameters (ε, η) used to model the measurement operator. In the experiments, the detection time delay and window were optimized to obtain the maximum contrast of the state at each magnetic field.

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Hirose, M., Cappellaro, P. Coherent feedback control of a single qubit in diamond. Nature 532, 77–80 (2016). https://doi.org/10.1038/nature17404

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