Noise-adapted qudit codes for amplitude-damping noise

Quantum error correction (QEC) plays a critical role in preventing information loss in quantum systems and provides a framework for reliable quantum computation. Identifying quantum codes with nice code parameters for physically motivated noise models remains an interesting challenge. While past work has primarily focused on qubit codes, here we identify a $[4,1]$ qudit error correcting code tailored to protect against amplitude-damping noise. We show that this four-qudit code satisfies the error correction conditions for all single-qudit and a few two-qudit damping errors up to the leading order in the damping parameter $\gamma$. We devise a protocol to extract syndromes that unambiguously identify this set of errors, leading to a noise-adapted recovery scheme that achieves a fidelity loss of $\mathcal{O}(\gamma^{2})$. For the $d=2$ case, our QEC scheme is identical to the known example of the $4$-qubit code and the associated syndrome-based recovery. We also assess the performance of this code using the Petz recovery map and note some interesting deviations from the qubit case. Finally, we generalize this construction to a family of $[2M+2, M]$ qudit codes that can approximately correct all the single-qudit and a few two-qudit amplitude-damping errors.