Fault-tolerant error syndrome detection in the GKP code*

The bosonic encoding proposed by Gottesman, Kitaev, and Preskill (GKP) is one of the
leading candidates to implement a robust logical qubit in a hardware-efficient manner. In
recent experiments, stabilization and error-correction of GKP states encoded in a harmonic
oscillator are achieved through repeated detection of error syndromes and applied feedback
displacements. This detection consists of sequential Rabi interactions between the target
oscillator and an ancillary two-level system, which is subsequently measured. However, in this scheme, ancilla noise propagates back to the target oscillator and induces logical errors. We present a novel error correction scheme in which logical errors induced by ancilla noise are suppressed, realizing a fault-tolerant error syndrome detection. To this end, a second harmonic oscillator, playing the role of a buffer, is inserted between the target and the ancilla. A two-mode quadrature coupling maps the error syndromes from the target to the buffer oscillator, prepared in a Schrödinger cat state. We utilize the toolbox developed in previous GKP stabilization experiments to retrieve the error information contained in the buffer oscillator while efficiently resetting its state.