# Strategic Portfolio Optimization Using Simulated, Digital, and Quantum Annealing
## 6 Portfolio Optimization Implementation
### Theoretical Remarks
The solutions to our QUBO formulation create portfolios in which a certain fraction of the overall budget is dedicated to certain strategies, while also respecting the amount of risk that should be taken and the magnitude of return that should be achieved. We evaluated our QUBO formulation via simulated annealing, digital annealing, and quantum annealing. The simulated annealing results show that our approach works as intended, meaning that portfolios were generated that respect the given preferences of the investor (expressed by the chosen values of \(\theta_{1},\theta_{2},\theta_{3}\)) while also creating diversified portfolios. The results of quantum annealing were also promising, yet not as good as the simulated annealing and digital annealing results. The probable cause for this is a combination of inherent noise, missing error correction, scaling of the parameters, and others. We note, however, that we only used default parameters for all of our evaluations. Hence, for future work, finding and using more suitable parameter configurations or adding techniques such as reverse annealing or certain types of low polynomial running-time post-processing might significantly increase solution quality in all cases. The combination of classical algorithms and reverse annealing might also provide benefits over purely classical or purely quantum methods. Conceptualization, S.F. and J.L. ; Methodology, J.L., S.Z. and S.F. ; Software, J.L. ; Validation, J.L., S.Z. and S.F. ; Writing--original draft preparation, J.L. ; Writing--review and editing, J.L., S.Z. and S.F. ; Supervision, S.F. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The authors declare no conflict of interest. ## References

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