Constructing optimized bosonic quantum codes via phase alteration

Bosonic quantum codes, such as cat codes and binomial codes, can address the key issue of correcting excitation loss in bosonic systems. Typically, there exists a basis for the codewords consisting of excitation probability amplitudes which are all real and positive. We argue that, by utilizing the phase degree of freedom in such probability amplitudes, improved orthogonality between logical states after the error process can be achieved. In particular, we show that a phase alteration of cat codes and binomial codes leads to much improved correction of photon losses, and can be experimentally implemented via a Kerr Hamiltonian. We discuss some extensions showing the general applicability of this principle in designing good quantum codes.