Optimal control with flag qubits

Abstract

High-fidelity quantum operations are the cornerstone of fault-tolerant quantum computation. In open quantum systems, traditional optimal control only passively resists decoherence, leaving environment-induced uncertainty as a fundamental performance bottleneck. To overcome this, we propose a new optimal control framework with flag ancillas and the Flag-GRAPE algorithm, which can actively tailor the system's noise structure. Through embedding post-selection directly into the objective function, Flag-GRAPE correlates decoherence errors with the ancilla's unexpected state. Subsequent measurement and post-selection effectively expel this uncertainty, circumventing the fidelity bounds of traditional control. Numerical simulations in a superconducting quantum circuit demonstrate a $51\%$ reduction in infidelity compared to traditional closed-system pulses and also show that such enhancement is robust across broad noise regimes. Furthermore, by actively converting unstructured decoherence into heralded erasure errors, Flag-GRAPE is inherently compatible with quantum error correction. We demonstrate this by initializing a logical cat-code state, showing that the combination between Flag-GRAPE and QEC yields immediate state preparation enhancements. This new framework can reduce hardware overhead for fault-tolerant architectures and open up a practical path toward logical state preparation gain in near-term experiments.

Figures (4)

• Figure 1: Schematic of the suppression of noise-induced uncertainty via a flag ancilla. The target system inevitably suffers from environmental noise during gate operations. By introducing an auxiliary flag qubit, a post-selection measurement can be performed immediately after the target operation to evaluate the process. (a) If the ancilla is measured in the expected state, no error is flagged, and the resulting state of the target system is accepted. (b) Conversely, if the ancilla measurement yields an unexpected outcome and the flag is raised, it deterministically indicates that an error has occurred during the evolution, and the corresponding corrupted state is rejected.
• Figure 2: Simulation results of the Flag-GRAPE algorithm (a) Schematic of the superconducting platform and the optimization workflows. Both Closed-GRAPE and Flag-GRAPE can suppress decoherence-induced uncertainty via measurement of the ancilla and post-selection. The flag-based scheme actively correlates decoherence errors with the ancilla's excited state, allowing noisy evolutions to be effectively flagged and discarded. The inset Wigner functions schematic illustratively compare the target state $\left|0\right\rangle+e^{-i\pi/4}\left|1\right\rangle$ prepared under these distinct strategies. (b) Infidelity distribution of 500 independently optimized Closed-GRAPE pulses and corresponding Flag-GRAPE pulses. Blue and red histograms denote the infidelities $f_\text{post}$ of Flag-GRAPE and Closed-GRAPE pulses with post-selection, respectively. The yellow histogram represents infidelities $f_\text{pre}$ from Closed-GRAPE pulses without post-selection. For visual clarity, the results for Flag-GRAPE pulses without post-selection, which distribute much further to the right, are shown in the Appendix. (c) Scatter plot illustrating the correlation between the final infidelity and the post-selection success probability for the post-selected Flag-GRAPE and Closed-GRAPE pulses shown in (b).
• Figure 3: Performance of the algorithms with varying error rate.(a) Comparison of average infidelity of $10$ optimized pulses with Flag-GRAPE (blue) and Closed-GRAPE (red) where all error rates are multiplied equally by an error scaling factor $\Gamma$. The curves are fitted linearly with the respective prefactor $\beta$. (b) Average post-selection probability $p_0$ for the same set. (c) Reduction in the best infidelity by Flag-GRAPE pulses compared with Closed-GRAPE pulse for different error scalings in the cavity (decay $\hat{a}$) and qubit (decay $\sigma_-$ and dephasing $\sigma_z$, varied together).
• Figure 4: Logical state preparation of the cat-code within the flag-based scheme. Probability distributions of infidelities for logical state $\left|\bar{\psi}_t\right\rangle= \left|0_L\right\rangle+e^{-i\pi/4}\left|1_L\right\rangle$ initialization with post-selection. The data is derived from 500 independently optimized Closed-GRAPE pulses (red histogram) and their corresponding Flag-GRAPE pulses (green histogram). For comparison, the blue histogram shows the result of unencoded state from Flag-GRAPE presented in Fig. \ref{['fig:Fig2']}(b). The pink shaded region highlights the critical regime where the encoded infidelity surpasses the absolute best value ($\Phi<0.036\%$) achieved by the unencoded state. The gate duration for the encoded state is set to $0.2$$\mu s$, and this value is determined via preliminary time optimizations as the near-shortest feasible time (see Appendix for details). All other system parameters remain the same as those from Fig. \ref{['fig:Fig2']}(b).