High-fidelity qubit measurements play a crucial role in quantum computation, communication, and metrology. In recent experiments, it has been shown that readout fidelity may be improved by performing repeated quantum non-demolition (QND) readouts of a qubits state through an ancilla. For a qubit encoded in a two-level system, the fidelity of such schemes is limited by the fact that a single error can destroy the information in the qubit. On the other hand, if a bosonic system is used, this fundamental limit could be overcome by utilizing higher levels such that a single error still leaves states distinguishable. In this work, we present a robust readout scheme, applicable to bosonic systems dispersively coupled to an ancilla, which leverages both repeated QND readouts and higher-level encodings to asymptotically suppress the effects of qubit/cavity relaxation and individual measurement infidelity. We calculate the measurement fidelity in terms of general experimental parameters, provide an information-theoretic description of the scheme, and describe its application to several encodings, including cat and binomial codes.
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