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

ArXiv ID: 2507.03963 “View on arXiv”

Authors: Yen Jui Chang, Wei-Ting Wang, Yun-Yuan Wang, Chen-Yu Liu, Kuan-Cheng Chen, Ching-Ray Chang

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 represent securities while edges encode the return-covariance kernel. Portfolio weights are derived from the walk’s stationary distribution. Three empirical studies support the approach. (i) For the top 100 S&P 500 constituents over 2016-2024, six scenario portfolios calibrated on 1- and 2-year windows lift the out-of-sample Sharpe ratio by up to 27% while cutting annual turnover from 480% (mean-variance) to 2-90%. (ii) A $5^{“4”}=625$-point grid search identifies a robust sweet spot, $α,λ\lesssim0.5$ and $ω\in[“0.2,0.4”]$, that delivers Sharpe $\approx0.97$ at $\le 5%$ turnover and Herfindahl-Hirschman index $\sim0.01$. (iii) Repeating the full grid on 50 random 100-stock subsets of the S&P 500 adds 31,350 back-tests: the best-per-draw QSW beats re-optimised mean-variance on Sharpe in 54% of cases and always wins on trading efficiency, with median turnover 36% versus 351%. Overall, QSW raises the annualized Sharpe ratio by 15% and cuts turnover by 90% relative to classical optimisation, all while respecting the UCITS 5/10/40 rule. These results show that hybrid quantum-classical dynamics can uncover non-linear dependencies overlooked by quadratic models and offer a practical, low-cost weighting engine for themed ETFs and other systematic mandates.

Keywords: quantum stochastic walk, graph theory, Sharpe ratio, turnover reduction, stationary distribution, equities

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail
• Why: The paper introduces advanced mathematical constructs from quantum mechanics (Hamiltonian generators, density matrices, stochastic walks) and complex network theory. It provides extensive empirical validation with a large grid search (625 points), 31,350 back-tests across random universes, and robust performance metrics (Sharpe ratio, turnover, HHI) over a long historical period.

  flowchart TD
    A["Research Goal: Validate QSW optimizer for portfolio<br>weighting vs. classical mean-variance"] --> B["Data Input:<br>S&P 500 constituents (2016-2024)<br>50 random 100-stock subsets"]
    B --> C["Methodology: Graph Embedding<br>Nodes = Securities<br>Edges = Return-Covariance Kernel"]
    C --> D["Computational Process:<br>Quantum Stochastic Walk<br>Calibration via Grid Search (α, λ, ω)<br>Derive Stationary Distribution"]
    D --> E["Key Finding 1: Performance<br>Up to 27% higher Sharpe Ratio<br>vs. Mean-Variance"]
    D --> F["Key Finding 2: Efficiency<br>90% turnover reduction<br>(Median 36% vs 351%)"]
    D --> G["Key Finding 3: Robustness<br>Works under UCITS constraints<br>Optimal α,λ ≤ 0.5, ω ∈ [0.2,0.4"]]