High-Rate Extended Binomial Codes for Multi-Qubit Encoding

Abstract: We introduce a new class of bosonic quantum error-correcting codes, termed \emph{extended binomial codes}, which generalize the structure of one-mode binomial codes by incorporating ideas from high-rate qubit stabilizer codes. These codes are constructed in close analogy to $[[n,k,d]]$ qubit codes, where the parameter $n$ corresponds to the total excitation budget rather than the number of physical qubits. Our construction achieves a significant reduction in average excitation per mode while preserving error-correcting capabilities, offering improved compatibility with hardware constraints in the strong-dispersive regime. We demonstrate that extended binomial codes not only reduce the mean excitation required for encoding but also simplify syndrome extraction and logical gate implementation, particularly the logical $\bar{X}$ operation. These advantages suggest that extended binomial codes offer a scalable and resource-efficient approach for bosonic quantum error correction.