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.
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Facility use was supported by the Yale Institute for Nanoscience and Quantum Engineering and the National Science Foundation (NSF) MRSEC DMR 1119826. This research was supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) W911NF-09-1-0369, by the US Army Research Office W911NF-09-1-0514, and by the NSF DMR 1006060 and DMR 0653377. M.M. acknowledges partial support from the Agence National de la Recherche under the project EPOQ2 ANR-09-JCJC-0070. S.M.G. and Z.L. acknowledge support from the NSF DMR 1004406. All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI or the US government.
The authors declare no competing financial interests.
Extended data figures and tables
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 20 mK. 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.
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 |gg〉 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 .
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 |gg〉 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, −Z〉, to a minimum of 87% for the state |+Y, +Z〉, averaging 90% over the 36 states (dashed line).
Extended Data Figure 4 Predicted fidelity to |ϕ−〉 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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Shankar, S., Hatridge, M., Leghtas, Z. et al. Autonomously stabilized entanglement between two superconducting quantum bits. Nature 504, 419–422 (2013). https://doi.org/10.1038/nature12802
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