Quantum Stochastic Walks for Portfolio Optimization: Theory and Implementation on Financial Networks

Abstract Financial markets are noisy yet contain a latent graph–theoretic structure that can be exploited for superior risk-adjusted returns. We propose a quantum-stochastic-walk (QSW) optimizer that embeds assets in a weighted graph—nodes are securities, edges encode the return–covariance kernel—and derives portfolio weights from the stationary distribution of the walk. Three empirical studies support the method. (i)~On the Top-100 S&P constituents (2016–2024) six scenario portfolios fitted on 1- and 2-year windows lift the out-of-sample Sharpe ratio by up to +27\,% while slashing annual turnover from $480%$ (mean–variance) to $2$–$90%$. (ii)~A $625$ grid search isolates a robust sweet-spot—α,λ\!\!0.5, ω\!∈[0.2,0.4]—that delivers Sharpe ≈0.97 at ≤5% turnover and Herfindahl–Hirschman index(HHI) ~0.01. (iii)~Repeating the full grid on 50 random 100-stock subsets of the S&P\,500 generates 31,350 additional back-tests: the best-per-draw QSW beats re-optimized mean–variance on Sharpe in 54\,% of samples and always wins on trading efficiency, with median turnover 36\,% versus 351\,%. Overall, QSW raises the annualized Sharpe ratio by 15\,% and cuts average turnover by 90\,% relative to classical optimization, all while remaining comfortably within UCITS 5/10/40 rule. The findings demonstrate that hybrid quantum–classical dynamics can uncover non-linear dependencies overlooked by quadratic models, offering a practical low-cost weighting engine for themed ETFs and other systematic mandates.