Embedding a Biased-noise and Error-detectable Logical Qubit in a GKP Qudit*

Bosonic codes offer a path toward hardware-efficient quantum error correction (QEC) by redundantly encoding quantum information in the large Hilbert space of a harmonic oscillator. The Gottesman-Kitaev-Preskill (GKP) code is particularly promising, yielding in a circuit-QED architecture a QEC gain of 2.3, well beyond the break-even point [1]. The limiting factor in this architecture is the sensitivity of the QEC protocol to ancilla bit-flips, which propagate to the GKP qubit as random Pauli errors. Here we take a new approach to mitigate the effect of ancilla bit-flips, by embedding a logical qubit in two levels of a d=4 GKP qudit. With this encoding, single ancilla bit-flips propagate as either detectable leakage or logical phase-flips, but never logical bit-flips. This type of highly structured noise is a resource for logical qubits, enabling their concatenation into surface codes with improved thresholds and improved scaling with distance [2,3]. We present our experimental realization of this embedded logical qubit and demonstrate the first-order protection of its logical Z axis against ancilla bit-flip errors.