  Quantum annealing has shown promise for finding solutions to difficult
optimization problems, including protein folding. Recently, we used the D-Wave
Advantage quantum annealer to explore the folding problem in a coarse-grained
lattice model, the HP model, in which amino acids are classified into two broad
groups: hydrophobic (H) and polar (P). Using a set of 22 HP sequences with up
to 64 amino acids, we demonstrated the fast and consistent identification of
the correct HP model ground states using the D-Wave hybrid quantum-classical
solver. An equally relevant biophysical challenge, called the protein design
problem, is the inverse of the above, where the task is to predict protein
sequences that fold to a given structure. Here, we approach the design problem
by a two-step procedure, implemented and executed on a D-Wave machine. In the
first step, we perform a pure sequence-space search by varying the type of
amino acid at each sequence position, and seek sequences which minimize the
HP-model energy of the target structure. After mapping this task onto an Ising
spin glass representation, we employ a hybrid quantum-classical solver to
deliver energy-optimal sequences for structures with 30-64 amino acids, with a
100% success rate. In the second step, we filter the optimized sequences from
the first step according to their ability to fold to the intended structure. In
addition, we try solving the sequence optimization problem using only the QPU,
which confines us to sizes $\le$20, due to exponentially decreasing success
rates. To shed light on the pure QPU results, we investigate the effects of
control errors caused by an imperfect implementation of the intended
Hamiltonian on the QPU, by numerically analyzing the Schr\"odinger equation. We
find that the simulated success rates in the presence of control noise
semi-quantitatively reproduce the modest pure QPU results for larger chains.
