Exploiting Translational Symmetry for Quantum Computing with Squeezed Cat Qubits

Squeezed cat quantum error correction (QEC) codes have garnered attention because of their robustness against photon-loss and excitation errors while maintaining the biased-noise property of cat codes. In this work, we reveal the utility of the unexplored translational symmetry of the squeezed cat codes, with applications to autonomous QEC, reliable logical operations, and readout in a non-orthogonal basis. Using the basis under subsystem decomposition spanned by squeezed displaced Fock states, we analytically show that our autonomous QEC protocol allows for correcting logical errors due to photon loss, although the translational symmetry in one direction does not uniquely specify the code space. We also introduce the implementation methods of reliable logical operations by repeated alternation of a small-step unitary operation with a subsequent step of QEC onto the code space. Finally, by appropriately treating the non-Hermitian nature of the logical $Z$ operator, we also propose a circuit for precisely reading out the squeezed cat code in a non-orthogonal basis.