Comparing and Combining Measurement-Based and Driven-Dissipative Entanglement Stabilization^*

We demonstrate and contrast two approaches to the stabilization of qubit entanglement by feedback. Our demonstration is built on a feedback platform consisting of two superconducting qubits coupled to a cavity, which are measured by a nearly quantum-limited measurement chain and controlled by high-speed classical logic circuits. This platform is used to stabilize entanglement by two nominally distinct schemes: a “passive” reservoir engineering method and an “active” correction based on conditional parity measurements. In view of the instrumental roles that these two feedback paradigms play in quantum error correction and quantum control, we directly compare them on the same experimental setup. Furthermore, we show that a second layer of feedback can be added to each of these schemes, which heralds the presence of a high-fidelity entangled state in real time. This “nested” feedback brings about a marked entanglement fidelity improvement without sacrificing success probability.

Popular Summary

Errors caused by decoherence inevitably occur in quantum computation systems, and quantum error correction is accordingly essential for building fault-tolerant quantum computers. Error correction methods can be broadly classified into two groups, the first of which relies on projective measurements and low-latency classical control actuator drives. The second group of error correction methods makes use of the driven-dissipative approach, in which particular coupling dynamics are engineered between the quantum system and the environment in order to transfer the entropy caused by errors out of the quantum system. An important question that naturally arises is whether one approach is better than the other, particularly for large multiqubit systems. Another outstanding question that is rarely considered is whether these two approaches can be integrated to obtain the best of both worlds.

To investigate these questions, we build a feedback platform consisting of two superconducting qubits coupled to an aluminum cavity connected to a high-fidelity measurement chain and a very-low-latency classical controller. We then implement and compare the driven-dissipative and measurement-based schemes on this setup to stabilize an entangled Bell state between two superconducting qubits. Our performance metrics include the state fidelity and the success probability. With the goal of determining whether the two feedback methods can be integrated, we design a nested feedback protocol to monitor either entanglement stabilization scheme and herald a successful stabilization run in real time. Using the nested protocol, we find that the combination of both approaches delivers the best performance in terms of the fidelity to the target Bell state.

Our experiment demonstrates that measurement-based and driven-dissipative approaches, far from being antagonistic, can be integrated to perform better than either approach on its own. We expect that this strategy of integration can be extended to many other stabilization and error correction experiments, included those that require stabilizing multiple states.

Figure 1

(a) Schematic of the experimental setup. Two independently addressable transmon qubits, Alice and Bob, are dispersively coupled to a three-dimensional microwave cavity. The cavity output is directed to a nearly quantum-limited measurement chain consisting of a Josephson amplifier (JPC) followed by a semiconductor amplifier (HEMT). A pair of custom field-programmable gate array boards (FPGA a, b) monitor the amplified output and generate real-time modulated microwave drives to control the cavity-qubit system. (b) Transmission spectra of the cavity. Cavity outputs at $f_{gg}$ and $f_{ee}$ are fed to FPGA a and b, respectively.

Figure 2

Comparison between the DD and the MB entanglement stabilization, shown in the left (DD) and right (MB) panels. (a) Diagram of qubit- and cavity-state evolution in the DD feedback loop. Two-qubit state manifolds are laddered for different photon numbers in the cavity (labeled by $|n⟩$). The green and pink sinusoidal double arrows represent cavity drives, while the straight double red/blue and cyan/yellow arrows are Rabi drives on the qubits. These six drives and the cavity dissipation (black decaying arrow) couple the different states of the system such that the target state $|{\varphi }_{-}⟩$ is stabilized. (b) Functional pulse sequence for DD. The duration needed to empty the cavity of residual photons (see text) before tomography is indicated by $T_{\mathrm{w}}$. (c) Fidelity to the target as a function of stabilization duration ($T_{\mathrm{s}}$). The dashed line at 0.5 denotes the threshold for entanglement. The time given in the white box, $\tau$, is the characteristic time constant of the exponential rise of fidelity. (d) State machine representation of the MB feedback loop. The quasiparity measurement reports $|gg⟩$, $|ee⟩$ as $\tilde{p}=+1$ (even) and $|{\varphi }_{-}⟩$, $|{\varphi }_{+}⟩$ as $\tilde{p}=-1$ (odd). For odd (even) parity, two $\frac{\pi }{2}$ pulses, with identical (opposite) phases, are applied to Alice and Bob, respectively. (e) Sequence of correction steps conditioned by the quasiparity measurement and leading into tomography. Counter $k$ limits the number of steps to $N$. (f) Fidelity to the target Bell state as a function of stabilization duration ($T_S$) or number of correction steps ($N$).

Figure 3

Trade-off between success probability and fidelity for both DD and MB schemes obtained by experiment. (a) Pulse sequence for heralding DD stabilization by postselection. At the end of the stabilization period, the cavity outputs, $I_{gg}$ and $I_{ee}$, are measured at their respective frequencies. (b) Color plot of fidelity to the target state for DD as a function of thresholds chosen for $I_{gg}$ and $I_{ee}$ (see Appendix pp2 for details of the thresholds). Also plotted as white dashes are contour lines of the success probability associated with each choice. The solid black square indicates the thresholds chosen for the condition $C$ in the nested feedback protocol described in Fig. 4. (c,d) Same as above for MB, including the corresponding thresholds for $C$. The grey circle indicates the thresholds chosen for the quasiparity measurement $\tilde{p}$ used to condition MB correction steps.

Figure 4

Nested feedback protocol for boosting fidelity and heralding successful stabilization runs in real time. (a) State machine representation. The control variable $C$ (see main text and Fig. 3) determines the repetition of a boost cycle or the heralding of a successful run. The maximum number of boost attempts allowed is set to 11 in the experiment. (b,e) 200 single-shot sequence trajectories of nested feedback for DD and MB, respectively. The trajectories are colored yellow during boost attempts and black after $C$ is satisfied. Trajectories that are entirely black satisfy the success criterion without the need for boost. The inset in (e) shows an example of a MB trajectory consisting of both $\tilde{p}$ and $C$ outcomes. (c,f) Cumulative success probability (grey shade) of having completed at most a given number of boost attempts before tomography for DD and MB, respectively. Yellow bars indicate differential success probability. (d,g) Fidelity to target Bell state for DD and MB, respectively. Green squares show the corresponding fidelity as a function of the number of boost attempts. The blue dashed line denotes fidelity without any boosting (i.e., unconditioned by $C$). Blue arrows denote the improvement in fidelity due to nested feedback. While the absolute fidelity of MB is worse than DD because of latency, the fidelity boost is higher.

Figure 5

Experimental calibration of measurement strength and duration for MB. Fidelity to the state $|{\varphi }_{+}⟩=\frac{1}{\sqrt{2}}(|ge⟩+|eg⟩)$ after preparing the qubits in a maximally superposed state followed by a quasiparity measurement with postselection. The fidelity is plotted as a function of measurement duration and shown for a set of measurement strengths. The optimal values chosen for the experiment are 660 ns and $\overline{n}=4.5$.

Figure 6

(a) The cavity outputs at $f_{gg}$ and $f_{ee}$ are integrated to obtain measurement outcomes $I_{gg}$ and $I_{ee}$, respectively. (b)–(e) Histograms of the measurement outcomes $I_{gg}$ and $I_{ee}$ for DD and MB, respectively. The leftmost dashed line (labeled $I_{gg}^{\text{herald}}$ and $I_{ee}^{\text{herald}}$) indicates a particular choice of threshold for heralding the measurement outcomes. Counts to the left of both thresholds are declared successful. The right dashed line ($I_{gg}^{\text{parity}}$ and $I_{ee}^{\text{parity}}$) indicates the threshold for quasiparity measurement in MB.

Figure 7

Markov model of a correction step in MB. Transition between any two nodes is possible and is represented as a directional edge. The 16 possible transitions make up the 16 matrix elements in the stochastic transition matrix $\mathcal{T}$, corresponding to a correction step in MB. Numbers next to the edges represent the elements of $\mathcal{T}$ calculated for the current experiment parameters by a master-equation simulation.

Figure 8

Markov model of the NFP for real-time heralding. This model is an extension of the model introduced for MB. During a boost attempt, transitions occur inside the “boosting” node for further stabilization. The edge leading from the boosting to the “heralded” node represents the real-time heralding that selects some fraction of trajectories by the threshold-dependent matrix $\tilde{c}$ (see text) at the end of a boost attempt. Those that are not selected ($\mathbf{1}-\tilde{c}$) enter into another boost attempt.

Figure 9

Simulation of real-time heralding by NFP. The left (right) panel presents the results for DD (MB). (a,c) Cumulative success probability of having completed, at most, a given number of boost attempts before tomography for DD and MB, respectively. The green curve shows the experimental result as presented in the main text. The blue curve is the simulation result using the same experimental parameters. (b,d) Fidelity to $|{\varphi }_{-}⟩$ for DD and MB, respectively. The cyan dashed line denotes the unconditioned steady-state fidelity obtained in the experiment. Green squares (blue circles) show the corresponding fidelity as a function of the number of boost attempts during NFP obtained in the experiment (simulation).

Figure 10

Correcting for the deterministic measurement-induced ac Stark shift. (a) Deterministic phase shift induced by measurement as a function of measurement duration. (b) One example of a sequence trajectory illustrating the phase correction $R_z^b(\theta )$ at work. See the detailed explanation in accompanying text. (c) Fidelity of the steady state to the target Bell state as a function of the correction angle of the effective $Z$ gate on Bob.

Figure 11

Experimental measurement of $f$-state population plotted as a function of the boost attempt number in NFP for DD.

Figure 12

Setup of the experiment. The qubit-cavity system was placed at the base stage of a dilution refrigerator (Oxford Triton200) below 20 mK. On the input side, two FPGAs (Innovative Integration X6-1000M) generated the pulse envelopes in I and Q quadratures to modulate the qubit drives at $f_{\text{Alice}}$ and $f_{\mathrm{Bob}}$ frequencies (Vaunix Labbrick LMS-802) for Alice and Bob, respectively. The I/Q modulations were output by the FPGAs with one and two single-sideband modulations in MB and DD (so that both zero-photon and $n$-photon qubit frequencies were addressed), respectively. The microwave frequency drives and I/Q modulation were mixed by IQ mixers. Two Agilent N5183 microwave generators produced the cavity drives at $f_{ee}$ and $f_{gg}$, respectively. The cavity drives were also pulsed by the FPGAs. On the output side, the transmitted signal through the cavity was directed by two circulators to the JPC for nearly quantum-limited amplification. It was further amplified at the 3-K stage by a cryogenic HEMT amplifier. After additional room-temperature amplification, the signal was split into two interferometric setups for readouts at the $f_{gg}$ and $f_{ee}$ frequencies, respectively. In each of the interferometers, the signal arriving from the fridge was mixed with a local oscillator set $+50\mathrm{MHz}$ away to produce a down-converted signal at 50 MHz. A copy of the cavity drive that did not go through the fridge was also down-converted in the same manner to produce a reference. Finally, the two signals with their respective references were sent to the analog-to-digital converters on the FPGA boards for digitization and were further demodulated inside the FPGAs. The two FPGAs jointly estimate the qubit state by communicating their results with each other. Along the input and output lines, attenuators, low-pass filters, and homemade Eccosorb filters were placed at various stages to protect the qubits from thermal noise and undesired microwave and optical frequency radiation. Moreover, the cavity and JPC were shielded from stray magnetic fields by aluminum and cryogenic $\mu$-metal (Amumetal A4K) shields.