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In quantum error correction, information is encoded in a high-dimensional system to protect it from the environment. A crucial step is to use natural, low-weight operations with an ancilla to extract information about errors without causing backaction on the encoded system. Essentially, ancilla errors must not propagate to the encoded system and induce errors beyond those which can be corrected. The current schemes for achieving this fault-tolerance to ancilla errors come at the cost of increased overhead requirements. An efficient way to extract error syndromes in a fault-tolerant manner is by using a single ancilla with strongly biased noise channel. Typically, however, required elementary operations can become challenging when the noise is extremely biased. We propose to overcome this shortcoming by using a bosonic-cat ancilla in a parametrically driven nonlinear cavity. Such a cat-qubit experiences only bit-flip noise and is stabilized against phase-flips. To highlight the flexibility of this approach, we illustrate the syndrome extraction process in a variety of codes such as qubit-based toric codes, bosonic cat-and Gottesman-Kitaev-Preskill (GKP) codes. Our results open a path for realizing hardware-efficient, fault-tolerant error syndrome extraction.

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CONTENTS S1. Experimental design S2. System Hamiltonian and Parameters S3. System Characterization and Calibration A. Semiclassical phase space trajectories 1. Simulating in the displaced frame 2. Semiclassical trajectories B. Measurement of Hamiltonian parameters using out-and-back C. Oscillator drive strength calibration using geometric phase S4. The Echoed Conditional Displacement Gate A. Derivation of the ECD gate B. Optimization of the ECD gate S5. Characteristic function tomography A. Measurement and Post-Processing B. Density matrix reconstruction C. Effective squeezing measurement D. Binomial code analysis E. GKP code analysis S6. Sources of Infidelity A. Decoherence-Free Error Budget B. Impact of decoherence C. Discussion S7. Optimization of ECD circuit parameters S8. Optimization of logical gates on a finite energy GKP code S9. Optimization of GRAPE and SNAP pulses A. Optimization of GRAPE B. Optimization of SNAP S10. Code Availability References