  We can encode a qubit in the energy levels of a quantum system. Relaxation
and other dissipation processes lead to decay of the fidelity of this stored
information. Is it possible to preserve the quantum information for a longer
time by introducing additional drives and dissipation? The existence of
autonomous quantum error correcting codes answers this question in the
positive. Nonetheless, discovering these codes for a real physical system,
i.e., finding the encoding and the associated driving fields and bath
couplings, remains a challenge that has required intuition and inspiration to
overcome. In this work, we develop and demonstrate a computational approach
based on adjoint optimization for discovering autonomous quantum error
correcting codes given a description of a physical system. We implement an
optimizer that searches for a logical subspace and control parameters to better
preserve quantum information. We demonstrate our method on a system of a
harmonic oscillator coupled to a lossy qubit, and find that varying the
Hamiltonian distance in Fock space -- a proxy for the control hardware
complexity -- leads to discovery of different and new error correcting schemes.
We discover what we call the $\sqrt{3}$ code, realizable with a Hamiltonian
distance $d=2$, and propose a hardware-efficient implementation based on
superconducting circuits.
