  In the field of fault-tolerant quantum computing, continuous-variable systems
can be utilized to protect quantum information from noise through the use of
bosonic codes. These codes map qubit-type quantum information onto the larger
bosonic Hilbert space, and can be divided into two main categories:
translational-symmetric codes, such as Gottesman-Kitaev-Preskill (GKP) codes,
and rotational-symmetric codes, including cat and binomial codes. The
relationship between these families of codes has not yet been fully understood.
We present an iterative protocol for converting between two instances of these
codes GKP qunaught states and four-foldsymmetric binomial states corresponding
to a zero-logical encoded qubit - using only Gaussian operations. This
conversion demonstrates the potential for universality of binomial states for
all-Gaussian quantum computation and provides a new method for the heraladed
preparation of GKP states. Through numerical simulation, we obtain GKP qunaught
states with a fidelity of over 98% and a probability of approximately 3.14%,
after only two steps of our iterative protocol, though higher fidelities can be
achieved with additional iterations at the cost of lower success probabilities.
