constraint (i.e., selection of only $M$ assets) would be to add a term $O_{\textrm{penalty}}({\bf q})$ to the objective function given by expression (\[eq:QUBO\]) such that the unsatisfying selections would be penalized by a large value $P \gg 1$, which would force the global minimum to be such that $\sum_{i=1}^N q_i=M$: $$\label{eq:Penalty}
O_{\textrm{penalty}}({\bf q}) = P \left( M - \sum_{i=1}^{N} q_i \right)^{2} \; .$$

Unfortunately, the introduction of such a large energy scale $P$ is typically associated with precision issues connected to the analog nature of the machine and the fact that there is a physical maximum to the energy that can be controllably programmed on local elements of the D-Wave chip [@johnson2011quantum]. However, several hybrid quantum-classical strategies can be put in place to overcome the limitation. For instance, we observe that shifting artificially the Sharpe ratio values by a constant amount $\pm \Delta$ (and adding buckets according to the prescription chosen, e.g., Table \[tab:tableBuckets\]) will essentially amount to forcing the ground state solution of the unconstrained problem to have more or less desired number of assets selected. Hence, while not solving the same problem, we could imagine a solver of a similarly constrained problem that will enclose the quantum annealing 