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 \emph{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 \textbf{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 \textbf{+27\,\%} while slashing annual turnover from $480\%$ (mean–variance) to $2$–$90\%$. (ii)~A \mbox{$625$} grid search isolates a robust sweet-spot—$\alpha,\lambda\!\lesssim\!0.5$, $\omega\!\in[0.2,0.4]$—that delivers Sharpe $\approx0.97$ at $\le5\%$ turnover and Herfindahl–Hirschman index(HHI) ${\sim}0.01$. (iii)~Repeating the full grid on \textbf{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 \emph{always} wins on trading efficiency, with median turnover \textbf{36\,\%} versus \textbf{351\,\%}. Overall, QSW raises the annualized Sharpe ratio by \textbf{15\,\%} and cuts average turnover by \textbf{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.