Autonomously stabilized entanglement between two superconducting quantum bits

Journal name:
Nature
Volume:
504,
Pages:
419–422
Date published:
DOI:
doi:10.1038/nature12802
Received
Accepted
Published online

Quantum error correction codes are designed to protect an arbitrary state of a multi-qubit register from decoherence-induced errors1, but their implementation is an outstanding challenge in the development of large-scale quantum computers. The first step is to stabilize a non-equilibrium state of a simple quantum system, such as a quantum bit (qubit) or a cavity mode, in the presence of decoherence. This has recently been accomplished using measurement-based feedback schemes2, 3, 4, 5. The next step is to prepare and stabilize a state of a composite system6, 7, 8. Here we demonstrate the stabilization of an entangled Bell state of a quantum register of two superconducting qubits for an arbitrary time. Our result is achieved using an autonomous feedback scheme that combines continuous drives along with a specifically engineered coupling between the two-qubit register and a dissipative reservoir. Similar autonomous feedback techniques have been used for qubit reset9, single-qubit state stabilization10, and the creation11 and stabilization6 of states of multipartite quantum systems. Unlike conventional, measurement-based schemes, the autonomous approach uses engineered dissipation to counteract decoherence12, 13, 14, 15, obviating the need for a complicated external feedback loop to correct errors. Instead, the feedback loop is built into the Hamiltonian such that the steady state of the system in the presence of drives and dissipation is a Bell state, an essential building block for quantum information processing. Such autonomous schemes, which are broadly applicable to a variety of physical systems, as demonstrated by the accompanying paper on trapped ion qubits16, will be an essential tool for the implementation of quantum error correction.

At a glance

Figures

  1. Bell state stabilization set-up schematic and frequency landscape of autonomous feedback loop.
    Figure 1: Bell state stabilization set-up schematic and frequency landscape of autonomous feedback loop.

    a, The qubits (Alice and Bob) are coupled to the fundamental mode of a 3D cavity. Six continuous drives applied to the cavity input stabilize the Bell state |ϕ,0right fence. The cavity output jumps between low and high amplitude depending on whether the qubits are in the desired Bell state or not. The output is monitored by a quantum-limited amplifier (JPC). b, Spectra of the qubits and cavity coupled with nearly equal dispersive shifts (χA and χB); κ is the cavity linewidth. Colours denote transitions that are driven to establish the autonomous feedback loop. c, Effective states of the system involved in the feedback loop. Qubit states consist of the odd-parity states in the Bell basis {|ϕright fence,|ϕ+right fence} and the even-parity computational states {|ggright fence,|eeright fence}. Cavity states, arrayed horizontally, are the photon-number basis kets |nright fence. Sinusoidal double lines represent the two cavity tones whose amplitudes create on average photons in the cavity when the qubits are in even-parity states. The cavity level populations are Poisson distributed with mean and we show only |nright fence such that . Straight double lines represent four tones on qubit transitions. Collectively, the six tones and the cavity decay (decaying sinusoidal lines) drive the system towards the ‘dark’ state, |ϕ,0right fence, which builds up a steady-state population.

  2. Convergence of two qubit-state to the target Bell state.
    Figure 2: Convergence of two qubit-state to the target Bell state.

    a, The six drives are turned on during the stabilization period for time TS. Next, the drives are turned off and the system is left idle for 500ns, allowing any remaining cavity photons to decay away. Finally, two-qubit tomography is performed using single-qubit rotations followed by single-shot joint readout. The system is then allowed to reach thermal equilibrium by waiting at least 5T1 before repetition. b, Time variation of the relevant Pauli operator averages, showing the system’s evolution from thermal equilibrium (nearly |ggright fence) towards |ϕright fence. The system remains in this steady state for an arbitrarily long time, as demonstrated by data acquired at TS = 50, 100 and 500μs. c, Fidelity to the target state |ϕright fence as a function of stabilization time, TS. The dashed line at 50% is the entanglement threshold. The fidelity converges to 67% with a time constant, τ, of about ten cavity lifetimes, in good agreement with the theoretical prediction; inset, F for TS = 0–10μs (red circles), with exponential fit (blue line).

  3. Fidelity improved by monitoring the feedback loop.
    Figure 3: Fidelity improved by monitoring the feedback loop.

    a, Pulse sequence consisting of a TS = 10-μs period followed by two-qubit tomography. Here, the cavity output at is recorded during the last 240ns of the stabilization period (M1). The outcomes obtained during M1 are used to condition the tomography in post-processing. After waiting 100ns for any cavity photons to decay away, two-qubit state tomography is performed using a second 240-ns-long measurement (M2) similar to that used in the unconditioned tomography (Fig. 2). b, Reference histogram for M1, with qubits prepared in thermal equilibrium (denoted GG) and after a π-pulse on Alice (denoted ). The standard deviation, σ, of the Gaussian distributions scales the horizontal axis of measurement outcomes, Im. c, Complete set of Pauli operator averages measured by tomography without conditioning, as in Fig. 2, showing a fidelity of 67% to |ϕright fence. d, Tomography conditioned on M1 being , that is, outcomes Im/σ during , resulting in an increased fidelity of 77%.

  4. Experiment schematic.
    Extended Data Fig. 1: Experiment schematic.

    The qubit–cavity set-up as well as the JPC amplifier is mounted on the base stage of a dilution refrigerator (bottom of diagram) which is operated at less than 20mK. The room-temperature set-up consists of electronics used for qubit control (top left) and for qubit measurement (top right). The experiment is controlled by an arbitrary waveform generator (AWG), which produces analogue waveforms and also supplies digital markers (not shown) to the pulsed microwave sources. The drives for stabilization and qubit control are generated from four microwave sources in the present experiment, although the two cavity drives, and , could be produced in principle from the same source. These drives were combined with a measurement drive and sent through filtered and attenuated lines to the cavity input at the base of the fridge. The cavity output is directed to the signal port of a JPC, whose idler is terminated in a 50-Ω load. The JPC is powered by a drive applied to its pump port. The fridge input for JPC tuning is used solely for initial tune up and is terminated during the stabilization experiment. The cavity output signal is amplified in reflection by the JPC and then output from the fridge after further amplification. The output signal is demodulated at room temperature and then digitized by an analogue-to-digital converter along with a reference copy of the measurement drive.

  5. Single-shot readout of the observable |gg[rang][lang]gg|.
    Extended Data Fig. 2: Single-shot readout of the observable |ggright fenceleft fencegg|.

    a, Histogram of measurement outcomes recorded by the projective readout used for tomography. Outcome Im = 0 implies that no microwave field was received in the I quadrature for that measurement. The GG histogram (blue dots) was recorded with the qubits initially prepared in |ggright fence with a fidelity of 99.5%. The histogram (red dots) was recorded after identical preparation followed by a π-pulse on Alice. Solid lines are Gaussian fits. The horizontal axis of measurement outcomes Im is scaled by the average of the standard deviations of the two Gaussians, showing 5.5 standard deviations between the centres of the two distributions. Dashed line indicates the threshold that distinguishes GG from : an outcome of is associated with GG, whereas is associated with . b. Summary of the fidelity of a single projective readout of the state of the two qubits assuming the separatrix .

  6. Calibration of systematic errors in tomography.
    Extended Data Fig. 3: Calibration of systematic errors in tomography.

    Fidelity of two-qubit Clifford states measured by tomography identical to that used in the Bell state stabilization protocol. Clifford states are prepared by starting in |ggright fence with a fidelity of 99.5% and then performing individual single-qubit rotations. The fidelity varies from a maximum of 94% for the state |−Z,Zright fence, to a minimum of 87% for the state |+Y,+Zright fence, averaging 90% over the 36 states (dashed line).

  7. Predicted fidelity to |[phiv]-[rang] as a function of drive parameters  and [OHgr]n under the conditions of the present experiment.
    Extended Data Fig. 4: Predicted fidelity to |ϕright fence as a function of drive parameters and Ωn under the conditions of the present experiment.

    Ω0 is taken to be κ/2 in this simulation. A broad distribution of parameter values resulting in a fidelity of about 70% indicates the robustness of the autonomous feedback protocol to variations in the drives.

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Author information

Affiliations

  1. Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA

    • S. Shankar,
    • M. Hatridge,
    • Z. Leghtas,
    • K. M. Sliwa,
    • A. Narla,
    • U. Vool,
    • S. M. Girvin,
    • L. Frunzio,
    • M. Mirrahimi &
    • M. H. Devoret
  2. INRIA Paris-Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France

    • M. Mirrahimi

Contributions

S.S. performed the experiment and analysed the data with assistance from M.H. and Z.L. Z.L. proposed the autonomous feedback protocol and performed numerical simulations under the guidance of M.M. K.M.S., A.N. and L.F. contributed to the experimental apparatus, and U.V. contributed to the theoretical modelling under the guidance of S.M.G. M.H.D. supervised the project. S.S., M.H. and M.H.D. wrote the manuscript. All authors provided suggestions for the experiment, discussed the results and contributed to the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

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Author details

Extended data figures and tables

Extended Data Figures

  1. Extended Data Figure 1: Experiment schematic. (212 KB)

    The qubit–cavity set-up as well as the JPC amplifier is mounted on the base stage of a dilution refrigerator (bottom of diagram) which is operated at less than 20mK. The room-temperature set-up consists of electronics used for qubit control (top left) and for qubit measurement (top right). The experiment is controlled by an arbitrary waveform generator (AWG), which produces analogue waveforms and also supplies digital markers (not shown) to the pulsed microwave sources. The drives for stabilization and qubit control are generated from four microwave sources in the present experiment, although the two cavity drives, and , could be produced in principle from the same source. These drives were combined with a measurement drive and sent through filtered and attenuated lines to the cavity input at the base of the fridge. The cavity output is directed to the signal port of a JPC, whose idler is terminated in a 50-Ω load. The JPC is powered by a drive applied to its pump port. The fridge input for JPC tuning is used solely for initial tune up and is terminated during the stabilization experiment. The cavity output signal is amplified in reflection by the JPC and then output from the fridge after further amplification. The output signal is demodulated at room temperature and then digitized by an analogue-to-digital converter along with a reference copy of the measurement drive.

  2. Extended Data Figure 2: Single-shot readout of the observable |ggright fenceleft fencegg|. (168 KB)

    a, Histogram of measurement outcomes recorded by the projective readout used for tomography. Outcome Im = 0 implies that no microwave field was received in the I quadrature for that measurement. The GG histogram (blue dots) was recorded with the qubits initially prepared in |ggright fence with a fidelity of 99.5%. The histogram (red dots) was recorded after identical preparation followed by a π-pulse on Alice. Solid lines are Gaussian fits. The horizontal axis of measurement outcomes Im is scaled by the average of the standard deviations of the two Gaussians, showing 5.5 standard deviations between the centres of the two distributions. Dashed line indicates the threshold that distinguishes GG from : an outcome of is associated with GG, whereas is associated with . b. Summary of the fidelity of a single projective readout of the state of the two qubits assuming the separatrix .

  3. Extended Data Figure 3: Calibration of systematic errors in tomography. (163 KB)

    Fidelity of two-qubit Clifford states measured by tomography identical to that used in the Bell state stabilization protocol. Clifford states are prepared by starting in |ggright fence with a fidelity of 99.5% and then performing individual single-qubit rotations. The fidelity varies from a maximum of 94% for the state |−Z,Zright fence, to a minimum of 87% for the state |+Y,+Zright fence, averaging 90% over the 36 states (dashed line).

  4. Extended Data Figure 4: Predicted fidelity to |ϕright fence as a function of drive parameters and Ωn under the conditions of the present experiment. (249 KB)

    Ω0 is taken to be κ/2 in this simulation. A broad distribution of parameter values resulting in a fidelity of about 70% indicates the robustness of the autonomous feedback protocol to variations in the drives.

Additional data