Variational Quantum Eigensolver for Real-World Finance: Scalable Solutions for Dynamic Portfolio Optimization Problems

TL;DR

The paper tackles scaling quantum optimization to real-world dynamic portfolio problems by combining a hardware-aware VQE workflow with two key innovations: ISQR post-processing and VQEC problem decomposition. It maps the DPO to a QUBO, then to an Ising Hamiltonian, enabling execution on the IBM Fez QPU for up to $N_a=38$ assets with $N_r=4$ and $N_t=4$. ISQR enhances solution consistency and quality, producing a diverse set of high-quality investments that, in many cases, approach or match the classical CPLEX benchmark, while VQEC enables scaling to larger portfolios by solving time-step subproblems. The results demonstrate a viable near-term pathway to quantum advantage in finance, leveraging noise-aware post-processing and problem decomposition to tackle industrially relevant problem sizes.

Abstract

We present a scalable, hardware-aware methodology for extending the Variational Quantum Eigensolver (VQE) to large, realistic Dynamic Portfolio Optimization (DPO) problems. Building on the scaling strategy from our previous work, where we tailored a VQE workflow to both the DPO formulation and the target QPU, we now put forward two significant advances. The first is the implementation of the Ising Sample-based Quantum Configuration Recovery (ISQR) routine, which improves solution quality in Quadratic Unconstrained Binary Optimization problems. The second is the use of the VQE Constrained method to decompose the optimization task, enabling us to handle DPO instances with more variables than the available qubits on current hardware. These advances, which are broadly applicable to other optimization problems, allow us to address a portfolio with a size relevant to the financial industry, consisting of up to 38 assets and covering the full Spanish stock index (IBEX 35). Our results, obtained on a real Quantum Processing Unit (IBM Fez), show that this tailored workflow achieves financial performance on par with classical methods while delivering a broader set of high-quality investment strategies, demonstrating a viable path towards obtaining practical advantage from quantum optimization in real financial applications.

Figures (15)

• Figure 1: Financial performance of the ISQR and CPLEX solutions for a 9-asset DPO problem in a fixed investment window ($t=4$, see Section \ref{['sec:944']}). The horizontal axis represents investment volatility (associated to the risk), while the vertical axis shows the effective return, defined as the direct return minus transaction costs. We indicate in the dashed gray line our reference MARR of $1.68\%$. The dashed black curve indicates the ideal efficient frontier obtained with PyPortfolioOptimizer Martin2021, representing the optimal trade-off between return and volatility without accounting for transaction costs--thus, it is not achievable in practice. Blue dots correspond to a selection of the 1000 optimized investments from the ISQR method with lowest (best) optimization cost. The dark blue dot marks the single ISQR solution with the overall lowest (best) optimization cost, while the yellow diamond represents the CPLEX solution.
• Figure 2: Financial performance for the 9-asset portfolio problem studied in Sections \ref{['sec:944']} and \ref{['sec:94t']} for the $t=4$ time step. The horizontal axis shows portfolio volatility and the vertical axis shows the return. Each black dot marks the volatility-return position outcome of a 100% allocation to that asset, labeled by its corresponding asset ticker. The cyan triangle represents the equally weighted portfolio, where investment is distributed uniformly across all assets. The dashed black curve represents the ideal efficient frontier, which corresponds to the efficient portfolios implied by the assets distribution. The dashed gray line represents the value of the MARR threshold. For reference, the black arrow denoted by $C(\Omega)$ indicates a transaction cost rate of $\nu = 1\%$ -- not applied for these results.
• Figure 3: Scheme of the VQE optimization process (described in Section \ref{['sec:VQEIntro']}). In step 1 we generate an ansatz circuit for a given set of parameters $\bm{\theta}^{(i)}$. In step 2 we measure the expected value of the Ising Hamiltonian $H_\text{Ising}$ encoding the QUBO problem using the estimator primitive of IBM Quantum services. In step 3 we optimize $\bm{\theta}^{(i)}$ with a classical algorithm (DE). If the optimization does not satisfy the convergence criteria, the process returns to step 1. Once the optimal $\bm{\theta}^{(f)}$ is found, in step 4, we use the sampler primitive from IBM Quantum services to sample the solution. Finally, in step 5 we identify the optimal investment strategy. Figure adapted from Ref. nodar2024scaling.
• Figure 4: Workflow of the ISQR technique applied to the VQE converged solutions. First, a set of bit strings is extracted. After dividing it in batches, the best solution from each batch is studied to determine the bit occupancy pattern. This pattern is then translated into an investment strategy that satisfies the restrictions of the problem. Finally, the CR process iteratively corrects (flips) bits according to these patterns to further refine the solutions.
• Figure 5: Optimization cost distributions of the VQE solutions to the 9-asset DPO problem. To obtain the optimization cost distribution, we account for how many times we found states with the same associated optimization cost. The black curve represents the raw VQE results and the blue curve, the ISQR post-processed results. We add for comparison the random distribution (gray curve), its average (dashed vertical gray line labeled "Offset"), and the optimization cost obtained by CPLEX solver (dashed vertical yellow line labeled "CPLEX").