Quantum-Assisted Optimal Rebalancing with Uncorrelated Asset Selection for Algorithmic Trading Walk-Forward QUBO Scheduling via QAOA

1. Problem Statement

The paper addresses two fundamental challenges in portfolio management that are traditionally treated as separate, sequential problems:

1. Asset Selection: Choosing a subset of assets that are maximally uncorrelated to ensure diversification while avoiding estimation errors and survivorship bias.
2. Rebalancing Scheduling: Determining the optimal timing for rebalancing a portfolio to target weights. Classical approaches often rely on fixed schedules (e.g., calendar-based) or simple thresholds, which fail to account for the complex interaction between weight drift, transaction costs, and marginal risk-adjusted returns.

The authors propose a unified framework that solves these simultaneously, specifically formulating the rebalancing schedule as a combinatorial optimization problem solvable via quantum algorithms.

2. Methodology

The proposed framework consists of four main stages:

A. Uncorrelated Asset Selection

• Universe Construction: A survivorship-bias-free S&P 500 universe was constructed by reversing index changes after the training cutoff (Dec 31, 2024), resulting in 422 assets.
• Covariance Estimation: The Ledoit-Wolf shrinkage estimator is applied to the sample covariance matrix to reduce estimation error in high-dimensional, finite-sample regimes.
• Clustering & Selection:
  • Hierarchical Clustering: Assets are clustered based on angular distance derived from the shrinkage correlation matrix using Ward's linkage.
  • Representative Selection: From $n=10$ clusters, the single stock with the highest annualized Sharpe ratio in the training set is selected. This ensures the final portfolio consists of 10 stocks that are both high-performing and mutually uncorrelated.
  • Selected Assets: EXR, NI, CHD, LLY, COST, AVGO, TPL, CTAS, AMP, AJG.

B. Weight Optimization

Three weight optimization strategies are compared:

1. Entropy-Regularized Genetic Algorithm (GA): Maximizes a regularized Sharpe ratio objective: $F(w) = \text{Sharpe} + \lambda_{ent} \cdot H(w)$. The entropy term prevents degenerate solutions (concentration in 1-2 assets). This is accelerated via GPU.
2. Minimum Variance (MinVar): Closed-form solution using the inverse of the Ledoit-Wolf covariance matrix.
3. Ensemble: An equal-weight blend of GA, MinVar, and Equal-Weight portfolios.

C. QUBO Formulation of Rebalancing

The core novelty is casting the rebalancing schedule as a Quadratic Unconstrained Binary Optimization (QUBO) problem.

• Variables: A binary vector $x \in \{0, 1\}^W$ represents whether to rebalance on $W$ candidate dates.
• Objective Function: Minimize $x^T Q x$, where $Q$ encodes:
  • Marginal Sharpe Gain ($g_k$): The improvement in Sharpe ratio if rebalancing occurs at time $t_k$ versus holding drifted weights.
  • Transaction Cost Penalty: A linear cost proportional to the $L_1$ norm of weight drift.
  • Over-Frequency Penalty: An exponentially decaying interaction term ($e^{-|t_k - t_l|/\Delta t}$) penalizing consecutive rebalances to avoid unnecessary trading.

D. Quantum Solution (QAOA)

• Algorithm: The Quantum Approximate Optimization Algorithm (QAOA) is used to solve the QUBO.
• Circuit: Depth $p=2$ with 8 qubits (for 8 candidate dates).
• Optimization: Variational parameters ($\gamma, \beta$) are optimized using the COBYLA classical optimizer.
• Walk-Forward Framework: To eliminate lookahead bias, the test period (2025) is split into 3 windows. The QUBO is constructed and solved only on data available up to that window, and the resulting schedule is applied strictly within that window.
• Multi-Restart: 5 random initializations are used to avoid local minima, with the best solution selected based on measurement shots (4,096 shots for final evaluation).

3. Key Contributions

1. Novel QUBO Formulation: First application of QAOA to the rebalancing timing problem (distinct from asset selection), incorporating marginal Sharpe gains and transaction costs.
2. Robust Asset Selection: A principled pipeline combining Ledoit-Wolf shrinkage and hierarchical clustering to select uncorrelated assets without survivorship bias.
3. Entropy-Regularized GA: A GPU-accelerated weight optimizer that prevents degenerate portfolios while maximizing risk-adjusted returns.
4. Walk-Forward Quantum Backtesting: A rigorous experimental design that eliminates lookahead bias, a common flaw in quantum finance simulations.
5. Empirical Evidence of Advantage: Demonstrates that a quantum-assisted strategy can outperform classical baselines in both risk-adjusted returns and transaction cost efficiency.

4. Experimental Results

The framework was backtested on S&P 500 data (Training: 2010–2024; Test: Jan–Dec 2025).

• Performance Metrics (GA + QAOA vs. Best Classical Baseline):

  • Sharpe Ratio: 0.588 (GA + QAOA) vs. 0.575 (GA Rebal/10d).
  • Total Return: 10.1% vs. 9.77%.
  • Transaction Costs: 6.1 basis points (8 rebalances) vs. 11.0 basis points (24 rebalances).
  • Cost Reduction: 44.5% reduction in trading costs while achieving superior returns.
  • Drawdown: Comparable maximum drawdown to classical strategies, with tighter control in some metrics.

• Quantum Convergence:

  • The QAOA measurement distribution showed significant concentration of probability mass on the optimal bitstring (top bitstring >7% of 4,096 shots vs. uniform baseline of ~0.4%), indicating successful convergence to a low-energy solution.

• Key Findings:

  • Scheduling Matters: The QAOA schedule significantly outperformed fixed periodic schedules and threshold-based rules.
  • Weight Method Dependency: The GA weights were critical; MinVar weights underperformed in the 2025 momentum-driven market regardless of the scheduling method.
  • Threshold Failure: A standard 5% drift threshold triggered too few trades (2 trades) and underperformed buy-and-hold, highlighting the need for dynamic, data-driven scheduling.

5. Significance and Conclusion

This paper provides preliminary empirical evidence of quantum advantage in a practical financial application. While the specific combinatorial problem (8 dates) is small enough to be solved classically, the methodology demonstrates that:

1. Quantum algorithms (QAOA) can effectively solve combinatorial scheduling problems in finance.
2. The approach yields economically meaningful improvements (higher Sharpe, lower costs) over state-of-the-art classical heuristics.
3. The framework is compatible with NISQ (Noisy Intermediate-Scale Quantum) hardware, as the circuit depth and qubit count are within current coherence limits.

The authors conclude that variational quantum optimization offers a viable path for enhancing algorithmic trading strategies, particularly in the complex, non-convex domain of rebalancing scheduling, and propose future work on scaling to larger windows and joint weight-schedule optimization.