Coherent feedback control of a single qubit in diamond

Journal name:
Nature
Volume:
532,
Pages:
77–80
Date published:
DOI:
doi:10.1038/nature17404
Received
Accepted
Published online

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.

At a glance

Figures

  1. Feedback algorithm and experimental implementation.
    Figure 1: Feedback algorithm and experimental implementation.

    a, b, Nitrogen-vacancy (NV) centre in diamond (a) and relevant energy levels of the spin system (b), showing polarization processes under optical illumination21. c, Quantum circuit. Hadamard gates (H) prepare and read out a superposition state of the qubit, . Amid entangling gates between qubit (‘q’) and ancilla (‘a’), the qubit is subjected to noise (lightning bolt) and possibly unitary gates U. We assume the ancilla is not affected by the bath (‘b’), yielding , with Uqb(τ) the qubit–bath joint evolution. Given a dephasing bath, we set the entangling gate to Uc = σx (conditional-NOT gate). More generally, upon undoing the entangling operation, the system is left in the state , with . The entangling gate Uc is designed such that and , in which act on the bath only, and Uq acts on the qubit only. After measuring the ancilla, we could use a feedback operation Uq to restore the correct qubit state. The ancilla measurement is replaced by coherent feedback (shaded region) obtained by a controlled-correction gate (here Uq = σz (Z) for dephasing noise). The final state of the combined system is then , which reveals how the qubit is now decoupled from the bath. d, Experimental implementation. The laser excitation polarizes both spins. Black sinusoidal lines refer to selective microwave (MW) pulses acting only in the mI = 1 manifold (thus mimicking controlled-NOT gates); solid bars indicate non-selective pulses. The radio-frequency (RF) excitation describes selective pulses in the ms = 0 manifold. We use π/2 rotations about x to approximate Hadamard gates. To implement a non-selective radio-frequency π/2 gate on the nuclear spin we embed a non-selective microwave π pulse into two consecutive radio-frequency π/2 pulses. The controlled-correction gate is implemented by free evolution (tz) under hyperfine coupling.

  2. Experimental demonstration of the feedback-based protection algorithm.
    Figure 2: Experimental demonstration of the feedback-based protection algorithm.

    a, The signal—normalized photoluminescence intensity—oscillates at the hyperfine coupling frequency, A = −2.16 MHz. The initial coherent superposition state of the qubit is preserved for a time τ > 1 ms at B = 390 G (red circles), while we observe a sharp decrease in the signal amplitude at B = 514 G (black squares), where correlations between the qubit and ancilla states are partly measured. b, c, This behaviour is evident when we compare the fidelity at short (b) and long (c) times, for B = 390 G (red circles) and B = 514 G (black squares). To highlight the differences while taking into account different photoluminescence intensities at the two fields, we normalized all the data so that at short times the signal has the same (maximum) contrast. d, Protected NOT gate (red circles). The coherence of the qubit is protected for a time longer than the dephasing time ( ) even when a NOT gate is applied. We compare the dynamics to the NOOP dynamics (black squares), demonstrating that the NOT gate inverts the state of the qubit, as indicated by the out-of-phase oscillations. e, Weak measurement of the ancilla. Normalized photoluminescence signal after a protection time τ = 8 μs, as a function of the strength of the ancilla measurement. In the experiment, we vary the strength of the ancilla measurement by changing the angle of the last controlled phase rotation gate by varying its time. All error bars represent the standard deviation of the signal, calculated by error propagation from the photoluminescence intensity of the signal and of reference photoluminescence curves acquired for each data point for mS = {0, −1}. The lines are fits to the model presented in Methods.

  3. Partial measurement of the ancilla.
    Figure 3: Partial measurement of the ancilla.

    ad, Comparison of the fidelity signal with (grey) and without (black) a π pulse on the qubit, revealing the amount of information acquired about the ancilla qubit state. Signals were measured at short protection times, (0−2 μs; a, b), and at longer times, (20−22 μs; c, d). Error bars represent the standard deviation of the signal (see Fig. 2). The data are fitted (lines) using the model described in Methods. At the lower magnetic field (B = 390 G; a, c), the average signals (red lines, obtained from the fits to the data) exhibit only weak oscillations, indicating that at this field the measurement carries very little information about the ancilla state. For magnetic fields close to the level anti-crossing (B = 514 G; b, d), the oscillation of the average signal is more pronounced and is observed until , whereas it disappears at longer times. This signal behavior is an indication that the ancillary spin effectively decoheres on the timescale as a result of the feedback algorithm.

  4. Feedback circuit.
    Extended Data Fig. 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).

  5. Protected gate
    Extended Data Fig. 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.

  6. Weak measurement circuit.
    Extended Data Fig. 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.

  7. Control and coherence of the nitrogen nuclear spin.
    Extended Data Fig. 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.

  8. Electronic- and nuclear-spin-dependent fluorescence at different magnetic field strengths.
    Extended Data Fig. 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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Author information

Affiliations

  1. Research Laboratory of Electronics and Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • Masashi Hirose &
    • Paola Cappellaro

Contributions

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

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

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Extended data figures and tables

Extended Data Figures

  1. Extended Data Figure 1: Feedback circuit. (112 KB)

    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).

  2. Extended Data Figure 2: Protected gate (155 KB)

    . 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.

  3. Extended Data Figure 3: Weak measurement circuit. (83 KB)

    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.

  4. Extended Data Figure 4: Control and coherence of the nitrogen nuclear spin. (200 KB)

    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.

  5. Extended Data Figure 5: Electronic- and nuclear-spin-dependent fluorescence at different magnetic field strengths. (161 KB)

    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.

Additional data