Exploiting Translational Symmetry for Quantum Computing with Squeezed Cat Qubits

Translational symmetry plays an essential role in bosonic quantum error correction (QEC), most notably in the Gottesman-Kitaev-Preskill code. Squeezed cat (SC) codes provide a complementary platform, combining approximate protection against physical errors with the noise bias of cat codes, but a hardware-efficient route to exploit their translational symmetry for QEC has been lacking. Here we show that this symmetry provides a practical route to autonomous QEC and universal quantum computation with SC codes. We then propose a QEC protocol that autonomously restores states driven out of the code space by physical errors, even though translational symmetry along a single direction does not uniquely define the code space. Using a subsystem decomposition based on squeezed displaced Fock states, we analytically characterize the relaxation rate toward the code space induced by the protocol, thereby estimating the QEC-cycle rate required for effective error suppression. Within the same framework, we propose deterministic preparation of logical states, logical gates, and logical-$Z$ readout with improved error scaling. These results establish translational symmetry as a new perspective for approaching quantum computation with SC qubits.