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

Yen-Jui Chang, Weiting Wang, Yun-Yuan Wang, Chen-Yu Liu, Kuan‐Cheng Chen, Ching‐Ray Chang

Published Jul 21, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.21203/rs.3.rs-7045880/v1

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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%480\%480% (mean–variance) to 222–90%90\%90%. (ii)~A \mbox{625625625} grid search isolates a robust sweet-spot—α,λ ⁣≲ ⁣0.5\alpha,\lambda\!\lesssim\!0.5α,λ≲0.5, ω ⁣∈[0.2,0.4]\omega\!\in[0.2,0.4]ω∈[0.2,0.4]—that delivers Sharpe ≈0.97\approx0.97≈0.97 at ≤5%\le5\%≤5% turnover and Herfindahl–Hirschman index(HHI) ∼0.01{\sim}0.01∼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.
Classical mean-variance optimization is powerful in theory but fragile in practice, often producing highly concentrated, high-turnover portfolios. Naive equal-weight (1/N) portfolios are more robust but largely ignore cross-sectional information. We propose a quantum stochastic walk (QSW) framework that embeds assets in a weighted graph and derives portfolio weights from the stationary distribution of a hybrid quantum-classical walk. The resulting allocations behave as a “smart 1/N” portfolio: structurally close to equal-weight, but with small, data-driven tilts and a controllable level of trading. On recent S&P 500 universes, QSW portfolios match the diversification and stability of 1/N while delivering higher risk-adjusted returns than both mean-variance and naive benchmarks. A comprehensive hyper-parameter grid search shows that this behavior is structural rather than the result of fine-tuning and yields simple design rules for practitioners. A 34-year, multi-universe robustness study with rolling re-optimization further demonstrates that the QSW optimizer preserves these advantages across market regimes. Overall, the QSW framework improves risk-adjusted performance while maintaining strong diversification and moderate turnover.
This study explores the integration of quantum computing techniques, specifically the Quantum Approximate Optimization Algorithm (QAOA), with classical portfolio optimization frameworks through the Quadratic Unconstrained Binary Optimization (QUBO) approach. Using monthly turnover data from the National Stock Exchange of India (NSE), this paper demonstrates the construction of a QUBO matrix by encoding expected returns and covariance-based risk profiles. Binary asset inclusion constraints are introduced into portfolios and then optimised using both brute force enumeration and quantum inspired solvers. The analysis reveals those combinations including Index Futures and Index Options exhibit superior risk-adjusted performance (Sharpe Ratio≈0.286\approx 0.286≈0.286). The hybrid approach suggests the promise of quantum inspired models as means to improve financial decision making under uncertainty. The results also give a robust template for scalable asset selection models that operate within the constraints of quantum computing infrastructure.
The increasing complexity of financial markets has heightened the demand for advanced computational methods capable of optimizing portfolios under multi-dimensional constraints involving risk, return, diversification, and liquidity. Traditional optimization frameworks, such as mean-variance analysis and heuristic meta-optimization, are limited by their scalability and sensitivity to non-convex, high-dimensional data structures. This study explores the application of quantum algorithms to portfolio optimization, demonstrating how Quantum Approximate Optimization Algorithm (QAOA), Quantum Annealing, and Variational Quantum Eigensolver (VQE) can address computational bottlenecks inherent in classical approaches. Quantum computation offers the potential to encode complex financial systems as Hamiltonian functions, allowing parallel exploration of vast solution spaces through superposition and entanglement. This enables faster convergence toward optimal portfolio allocations while simultaneously accounting for multi-factor correlations and dynamic market conditions. The research further investigates the role of hybrid quantum–classical optimization pipelines, in which quantum solvers perform core optimization tasks while classical systems manage data preprocessing and constraint validation. Empirical modeling and simulation indicate that quantum-based approaches can outperform conventional methods in risk-adjusted return optimization, efficient frontier computation, and real-time portfolio rebalancing. By bridging the gap between quantum computing and financial engineering, this study contributes to the development of next-generation financial analytics systems, aligning with the broader goal of achieving more adaptive, transparent, and data-resilient financial decision-making.
We present a hybrid classical-quantum framework for portfolio construction and rebalancing. Asset selection is performed using Ledoit-Wolf shrinkage covariance estimation combined with hierarchical correlation clustering to extract n = 10 decorrelated stocks from the S&P 500 universe without survivorship bias. Portfolio weights are optimised via an entropy-regularised Genetic Algorithm (GA) accelerated on GPU, alongside closed-form minimum-variance and equal-weight benchmarks. Our primary contribution is the formulation of the portfolio rebalancing schedule as a Quadratic Unconstrained Binary Optimisation (QUBO) problem. The resulting combinatorial optimisation task is solved using the Quantum Approximate Optimisation Algorithm (QAOA) within a walk-forward framework designed to eliminate lookahead bias. This approach recasts dynamic rebalancing as a structured binary scheduling problem amenable to variational quantum methods. Backtests on S&P 500 data (training: 2010-2024; out-of-sample test: 2025, n = 249 trading days) show that the GA + QAOA strategy attains a Sharpe ratio of 0.588 and total return of 10.1%, modestly outperforming the strongest classical baseline (GA with 10-day periodic rebalancing, Sharpe 0.575) while executing 8 rebalances versus 24, corresponding to a 44.5% reduction in transaction costs. Multi-restart QAOA (4096 measurement shots per run) exhibits concentrated probability mass on high-quality schedules, indicating stable convergence of the variational procedure. These findings suggest that hybrid classical-quantum architectures can reduce turnover in portfolio rebalancing while preserving competitive risk-adjusted performance, providing a structured testbed for near-term quantum optimisation in financial applications.
We test whether quantum-inspired optimization, specifically Quadratic Unconstrained Binary Optimization (QUBO) solved via simulated annealing, can improve implementable portfolio performance compared to classical baselines when realistic frictions are included. Our experimental design uses daily returns from the Center for Research in Security Prices (CRSP) for S&P500 constituents spanning 2014–2025, with monthly rebalancing, explicit transaction costs, and turnover accounting. We compare QUBO portfolios against equal-weight and convex mean–variance portfolios across multiple universe sizes (20, 50, 67 assets), lookback windows, and transaction-cost levels. Statistical significance is assessed via paired moving-block bootstrap on out-of-sample net returns. A hyperparameter sweep over encoding depth, penalty coefficients, and turnover reveals 19 of 81 QUBO configurations beat the equal-weight baseline, with the best achieving a Sharpe of 1.59 versus 1.42. However, under a fair walk forward model-selection protocol that eliminates look-ahead bias, the QUBO strategy does not achieve statistically significant superiority over either classical benchmark (∆Sharpe vs. equal weight: −0.13, 95% CI [−0.46,0.16]; vs. mean–variance: −0.29, 95% CI [−0.63,0.06]). Excessive turnover (averaging ∼1.0 per rebalance) is identified as the primary bottleneck. We conclude that current QUBO formulations with simulated annealing do not yet deliver a robust, implementable edge over well-tuned convex optimizers, though the framework remains a promising research direction as quantum hardware and hybrid solvers mature.
Abstract Recent research on quantum walk-based optimization has shown theoretical potential as an alternative to variational algorithms, avoiding local minima entrapment or barren plateaus by utilizing physically motivated structure rather than unconstrained parameter search. Despite this, the Guided Quantum Walk (GQW) has never been used for portfolio optimization or tested on existing noisy quantum hardware. In this paper we propose a hybrid quantum-classical solver for portfolio optimization that integrates Guided Quantum Walks (GQW) with domain-specific knowledge elicited from real Indian equity data. The core technical contributions are threefold. First, we implement a parameterized GQW circuit in Qiskit (v2.2) with a B\'{e}zier-curve-parameterized hopping rate schedule, mapping the portfolio selection problem directly to the GQW Hilbert space via an Ising Hamiltonian encoding. Second, we introduce a problem-aware warm-start strategy that initializes the energy scale parameters from classical sampling of the feasible portfolio subspace, bypassing the high-energy exploration phase that characterizes naive initialization. Third, we incorporate Conditional Value-at-Risk (CVaR) aggregation into the quantum walk measurement to improve solution space sampling, as well as a hardware-adaptive pipeline that adjusts concentration to per-backend circuit fidelity for IBM Heron devices.\\Our simulation results demonstrate near-optimal approximation ratios exceeding 0.99 for problem sizes up to N=20 on a noiseless setting with CVaR concentration ((α∈{0.1,0.25}))((\alpha \in \{0.1,0.25\}))((α∈{0.1,0.25})), significantly outperforming standard energy expectation values ((α=1.0))((\alpha=1.0))((α=1.0)). Hardware validation on IBM Heron R2 processors, confirms that parameters optimized on noiseless simulation transfer effectively to real devices without re-optimization, with median approximation ratios above 0.85 across all backends for N=12 drawn from live NIFTY 50 market data. This constitutes the first end-to-end GQW implementation for financial portfolio optimization using IBM quantum hardware, and its application as a scalable quantum solver for real-world financial decision support systems.