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Quantum Portfolio Optimization: An Extensive Benchmark

This paper presents an extensive benchmark comparing quantum optimization methods (quantum annealing and QAOA) against state-of-the-art classical algorithms on real-world portfolio optimization instances, concluding that classical mixed-integer programming and tailored heuristics significantly outperform quantum approaches in both solution quality and speed, thereby indicating very limited potential for quantum advantage in this specific domain.

Original authors: Eric Stopfer, Friedrich Wagner

Published 2026-07-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Eric Stopfer, Friedrich Wagner

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the captain of a massive treasure ship, and your job is to decide how to split your gold among 1,000 different islands. You want to maximize your loot (return) but also keep the ship from rocking too hard in the storm (volatility). This is the "Portfolio Optimization" problem, a classic puzzle in finance that researchers have been trying to solve with the help of the universe's newest, shiniest tool: quantum computers.

Recently, a team of scientists from the Fraunhofer Institute decided to put these quantum machines to the ultimate test. They didn't just guess; they ran a massive, real-world experiment with 250 different treasure maps, some involving as many as 1,000 islands. Their goal? To see if quantum computers could beat the best human-made strategies (classical methods) at finding the perfect gold distribution.

The Heavyweight Champion: The Old-School Solver
First, let's talk about the "classical" methods—the ones we've been using for decades. The researchers found that modern, super-smart classical solvers (like a program called Gurobi) are absolute beasts. When they tried to solve the problem for 1,000 assets, these solvers found the mathematically perfect answer in the order of seconds. It was like having a librarian who could find the one perfect book in a library of a million books before you could even finish saying "hello."

The Quantum Contenders: The New Kids on the Block
Then came the quantum challengers. The researchers tested two main types of quantum "magic":

Quantum Annealing: Think of this as a magical marble rolling down a complex, bumpy landscape to find the deepest valley (the best solution). They used a D-Wave machine with over 4,400 physical qubits.
QAOA (Quantum Approximate Optimization Algorithm): This is like a gate-based quantum computer (specifically an IBM machine with 127 qubits) that tries to "tune" a quantum state to find the answer, similar to tuning a radio to the perfect station.

The Big Reveal: The Quantum Struggle
Here is where the plot thickens. The researchers set a strict time limit of 60 seconds for the quantum computers to do their work. Why? Because in the real world, you can't wait forever for an answer.

The results were a bit of a bummer for the quantum hype train.

The Size Limit: The quantum machines hit a wall very quickly. Because the problem requires every asset to be connected to every other asset (a "dense" problem), the quantum computers could only handle instances with up to 30 assets. Anything bigger, and the machines simply couldn't fit the puzzle in their memory.
The Quality Gap: Even within that small range, the quantum methods struggled. When the researchers looked at the solutions found in 60 seconds, the quantum computers often couldn't even find a valid solution (one that followed all the rules). When they did find one, it was usually far from perfect.
The "Random" Surprise: In some cases, the quantum computers performed no better than just picking random answers. Imagine trying to find a needle in a haystack by throwing darts blindfolded; the quantum computers were sometimes just as likely to hit the needle as they were to hit the straw.

The Real Winner: A Custom-Built Tool
But wait, there's a twist! The researchers didn't just compare quantum to "standard" classical solvers; they also built a special, custom-made tool (a problem-specific heuristic) designed just for this treasure map.

This custom tool was a superstar. It consistently found better solutions than the quantum computers in the same 60 seconds.
It was so good that it made the quantum machines look like they were playing with their toys. The custom tool found valid solutions for almost every instance, while the quantum ones often failed to find even a single valid one for larger problems.

What Does This Mean?
The paper concludes that for this specific type of portfolio problem (minimizing volatility), there is currently very little room for a "quantum advantage."

Classical solvers can solve huge problems perfectly in seconds.
Custom classical heuristics beat quantum computers at finding good solutions quickly.
Quantum computers (both annealing and QAOA) are currently stuck. They struggle with the "dense" nature of the problem, which forces them to use too many resources just to map the problem onto the chip.

The authors are careful to say this doesn't mean quantum computing is useless forever. They suggest that for more complex versions of this problem (with extra rules and variables), quantum might have a chance. But for the version they tested? The old-school methods are still the kings of the hill, and the quantum challengers are still in the training camp, unable to even finish the race against a custom-built classical runner.

In short: If you need to optimize a portfolio of 1,000 assets today, you don't need a quantum computer. You need a good classical solver and maybe a custom script. The quantum revolution for this specific task is still waiting in the wings, not quite ready to take the stage.

Technical Summary: Quantum Portfolio Optimization: An Extensive Benchmark

Problem Definition
This study investigates the computational potential of quantum algorithms for portfolio optimization, specifically focusing on a volatility-minimizing variant. Unlike "choose-asset-or-not" formulations, this work addresses the more practical problem of determining optimal continuous asset weights ( $\omega_i \in [0, 1]$ ) to minimize portfolio volatility ( $\sigma^2$ ) subject to a minimum expected return ( $\epsilon$ ) and full investment constraints ( $\sum \omega_i = 1$ ). The authors identify this specific variant, termed MinVola, as the most computationally challenging for classical solvers compared to return-maximizing or multi-objective formulations, making it the most promising candidate for demonstrating a quantum advantage.

The study utilizes a dataset of 250 instances derived from real-world Nasdaq stock data (2020–2023), ranging from 3 to 1,000 assets. Due to current hardware limitations, quantum methods were tested on instances with up to 30 assets, while classical methods were evaluated across the full range.

Methodology
The authors conducted a comparative benchmark of quantum and classical approaches under a strict 60-second time limit for sampling and result analysis. The study adheres to principles of fairness, reproducibility, and verifiability.

Problem Transformation: To interface with quantum hardware, the continuous MinVola problem was transformed into a Quadratic Unconstrained Binary Optimization (QUBO) model. Continuous weights were discretized using a binary expansion with precision parameter $d=3$ , resulting in $4n$ binary variables for $n$ assets. Penalty factors ( $\phi, \psi = 1000$ ) were applied to enforce return and normalization constraints.
Quantum Methods:
- Quantum Annealing (QA): Executed on a D-Wave Advantage 2 processor (4,400+ physical qubits). The study tested various annealing times (1–50 $\mu$ s) and chain strengths. Due to the fully connected nature of the QUBO, a clique embedding strategy was employed, which significantly increased the qubit overhead.
- Quantum Approximate Optimization Algorithm (QAOA): Implemented on the IBM ibm_strasbourg (127 qubits) and ideal simulators. The study tested three parameter selection strategies: grid search (for $p=1$ ), linear ramp initialization, and COBYLA optimization. The study was limited to small instances ( $n \le 30$ ) due to the exponential memory requirements of classical simulation and hardware connectivity constraints.
Classical Methods:
- Exact Solvers: Gurobi and SCIP were used to establish proven optimality for the discretized problem.
- Heuristics: The benchmark included Simulated Annealing, Steepest Descent, Tabu Search, and a novel problem-specific heuristic designed to iteratively adjust asset weights to reduce variance without building a full mathematical model.
Metrics: Performance was evaluated based on the approximation ratio (objective value relative to the optimal solution), feasibility percentage (fraction of instances yielding at least one feasible solution), and the number of samples generated within the time limit.

Key Results

Classical Performance: Modern classical solvers, particularly Gurobi, solved instances with up to 1,000 assets to proven optimality in seconds. Gurobi was found to be over 1,000 times faster than the open-source SCIP solver for large instances, largely due to its ability to detect problem convexity and utilize interior-point methods.
Quantum vs. Classical Heuristics: The problem-specific heuristic consistently outperformed all quantum approaches (both QA and QAOA) in terms of solution quality (approximation ratio) and feasibility for fixed runtimes.
Quantum Performance:
- Feasibility: Both QA and QAOA struggled to find feasible solutions, with feasibility percentages dropping significantly as problem size increased.
- Quality: The approximation ratios for quantum methods were often suboptimal (e.g., $\ge 2$ for $n \ge 20$ in QAOA).
- Comparison to Randomness: In many configurations, quantum methods performed roughly on par with, or worse than, random sampling regarding feasibility and approximation ratio.
- Hardware Limitations: The dense, all-to-all connectivity required by the QUBO formulation led to significant embedding overhead for QA (limiting solvable instances to $n=25$ ) and massive gate overhead (swaps) for QAOA during transpilation, which degraded performance as circuit depth increased.
Quantum Annealing vs. QAOA: Quantum annealing slightly outperformed gate-based QAOA in feasibility and solution quality, but this advantage was contingent on fine-tuning chain strength parameters. Even with tuning, QA did not surpass classical heuristics.

Significance and Claims
The paper concludes that for the considered variant of portfolio optimization (volatility minimization with continuous weights), there is currently very limited room for a practical quantum advantage.

Classical Dominance: The study demonstrates that state-of-the-art classical solvers can solve large-scale instances to optimality in seconds, and specialized classical heuristics outperform quantum methods in solution quality within fixed timeframes.
Hardware Constraints: The poor performance of quantum algorithms is attributed primarily to the dense connectivity of the portfolio optimization problem, which exacerbates embedding and transpilation overheads on current hardware.
Benchmarking Standard: The authors assert that their work provides a comprehensive and fair benchmark by isolating quantum contributions from classical components (unlike hybrid black-box solvers) and using real-world data.
Modest Outlook: While the results set a high bar for future quantum advantage, the authors note that more complex problem variants with additional constraints might yield different results, though such scenarios require further investigation. The study does not claim that quantum computing is useless for finance, but rather that for this specific, practically relevant formulation, classical methods remain superior.

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