  The Portfolio Optimization task has long been studied in the Financial
Services literature as a procedure to identify the basket of assets that
satisfy desired conditions on the expected return and the associated risk. A
well-known approach to tackle this task is the maximization of the Sharpe
Ratio, achievable with a problem reformulation as Quadratic Programming. While
the sole Sharpe Ratio could be efficiently optimized via classical solvers, in
business scenarios it is common that multiple additional needs arise, which
have to be integrated in the optimization model as either new constraints or
objective function terms. Then, in general, the problem may become non-convex
and hence could potentially be not efficiently solvable via classical
techniques anymore. One example of such additional objective function term
consists of maximizing a diversification measure penalizing portfolios holding
significant portions of investments on assets belonging to the same sector,
while favouring solutions that diversify over multiple sectors. The problem of
optimizing both the Sharpe Ratio and a diversification term can be mapped to a
QUBO and be solved via quantum annealing devices or Hybrid Computing
approaches, which are expected to find high quality solutions. We propose a new
QUBO formulation for the task described and provide the mathematical details
and required assumptions, showing the ease of modeling the optimization as QUBO
against the effort that would be required by classical strategies. We derive
results via the available QUBO solvers, as well as discussing the behaviour of
Hybrid approaches to tackle large scale problems in the near term. We finally
elaborate on the results showing the trade-off between the observed values of
the portfolio's Sharpe Ratio and diversification, as a natural consequence of
solving a multi-objective optimization problem.
