Systematic construction of digital autonomous quantum error correction for state preparation and error suppression via conditional Gaussian operations

Abstract

In continuous-variable quantum computing, autonomous quantum error correction (QEC) can dissipatively steer a noisy quantum state into a target state or manifold, enabling robust quantum information processing without explicit syndrome measurements and feedback. Here, we propose a nullifier-based digital autonomous QEC enabled by conditional Gaussian operations. By designing jump operators for target nullifiers and compiling the resulting Lindbladian into a Trotterized sequence of elementary conditional Gaussian operations, we demonstrate two use cases: (i) deterministic preparation of non-Gaussian resource states for universal computation, including finitely squeezed cubic phase states and approximate trisqueezed states, and (ii) autonomous suppression of dephasing error for cat and squeezed cat states. We provide explicit gate decompositions for the required conditional Gaussian operations and numerically evaluate the performance under realistic imperfections, including photon loss in the bosonic mode and ancillary-qubit decoherence. Our results clarify the resource requirements and trade-offs, such as circuit depth, time-step choices, and the required set of conditional Gaussian operations, for scalable, gate-level implementations of autonomous state preparation and error suppression.

Figures (9)

• Figure 1: Concept of autonomous QEC. Given an operator $\hat{\delta}$, our goal is to produce an output state $\hat{\rho}_{\mathrm{out}}$ satisfying $\hat{\delta}\hat{\rho}_{\mathrm{out}}\hat{\delta}^\dagger=0$. The autonomous QEC mimics the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) dynamics with jump operator $\hat{\delta}$ so that, in the long-time limit, the system relaxes to $\hat{\rho}_{\mathrm{out}}$. With this autonomous QEC, we consider two use cases in this paper: (1) preparing a target quantum state and (2) suppressing an error of a target state. For the state preparation, we consider the preparation of (a) the cubic phase state with the dissipator $\hat{\delta}_{\mathrm{CPS}}$ and the preparation of (b) the trisqueezed state with the dissipator $\hat{\delta}_{\mathrm{TSS}}$. For the error suppression, we consider the suppression of the dephasing error of (c) the cat state with the dissipator $\hat{\delta}_{\mathrm{CAT}}$ and (d) the squeezed cat state with the dissipator $\hat{\delta}_{\mathrm{SqCAT}}$. $\hat{G}$ indicates the Gaussian operations, which are implemented easily in the experiment.
• Figure 2: Gate-level circuits for the autonomous QEC. Define $\hat{A}(\Delta t)=\exp\!\left[-i\Gamma\Delta t\,\{\hat{X}\otimes(\hat{\delta}+\hat{\delta}^\dagger)/2\}\right]$ and $\hat{B}(\Delta t)=\exp\!\left[-i\Gamma\Delta t\,\{-\hat{Y}\otimes(\hat{\delta}-\hat{\delta}^\dagger)/(2i)\}\right]$. Repeated application of Trotterized products of $\hat{A}$ and $\hat{B}$ stabilizes the desired oscillator subspace.
• Figure 3: Numerical simulations of preparing quantum states. We prepare various states from the vacuum state with our autonomous QEC. The horizontal axis represents the circuit depth $N$, the vertical axis represents the time increment $\Delta t$, and the color scale indicates the fidelity with respect to the corresponding target state. (a) Cubic phase state with squeezing level $5\ \mathrm{dB}$ and $\eta=0.3$. (b) Trisqueezed state with trisqueezing level $2\ \mathrm{dB}$.
• Figure 4: Numerical simulations of preparing quantum states with different squeezing or trisqueezing levels. The vertical axis is the squeezing or trisqueezing level instead of the time increment $\Delta t$. Here, $\Delta t$ is fixed at 50 ns; all other conditions are the same as those in Fig. \ref{['fig:QEC_generating']}.
• Figure 5: Numerical simulations of suppressing the error of quantum states. Time evolution of the fidelity between the ideal target state and the noisy state under three strategies: no error correction ("no QEC"), a single application of the autonomous QEC immediately before readout ("single QEC"), and repeated applications during storage ("interleaved QEC"). Here, we consider the dephasing error as the target error during the storage. Lines show the idealized numerical results without imperfections during the QEC operations, whereas points show the results including such imperfections. (a) Cat state with the coherent-state amplitude $\alpha = 3$. (b) Squeezed cat state with the amplitude $\alpha = 3$ and squeezing level $5~\mathrm{dB}$.