⚛️ quantum physics

Option Pricing on Noisy Intermediate-Scale Quantum Computers: A Quantum Neural Network Approach

This paper demonstrates the viability of using a compact 2-qubit Quantum Neural Network to accurately approximate Black-Scholes-Merton option pricing across multiple Noisy Intermediate-Scale Quantum hardware platforms, establishing a foundational framework for applying quantum machine learning to more complex financial models.

Original authors: Sebastian Zając, Rafał Pracht

Published 2026-04-23

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sebastian Zając, Rafał Pracht

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

The Big Picture: A Quantum Chef in a Noisy Kitchen

Imagine the global financial market as a massive, chaotic kitchen where chefs (banks and traders) are constantly trying to calculate the price of complex dishes (financial options). These calculations are vital for managing risk, but they are incredibly hard to do perfectly, especially when the dishes get complicated.

For decades, chefs have used a standard recipe book called the Black-Scholes-Merton (BSM) model. It's like a classic cookbook that works perfectly for simple cakes (European options) but gets messy and slow when you try to bake a multi-layered, custom-shaped wedding cake (complex derivatives).

The Problem: The current computers we use (classical computers) are like super-fast calculators, but they struggle when the recipe gets too complex. They have to simulate millions of scenarios to get a good guess, which takes time and energy.

The Solution: The authors of this paper asked, "What if we used a Quantum Computer?" But there's a catch: today's quantum computers are like newborn babies with shaky hands. They are powerful but very "noisy" (prone to errors) and can't hold a complex thought for long. This era is called NISQ (Noisy Intermediate-Scale Quantum).

The team, from a company called finQbit, built a special "Quantum Neural Network" (QNN)—think of it as a tiny, super-smart quantum brain—and tested if it could learn to price these options accurately, even with its shaky hands.

The Experiment: Training the Quantum Brain

1. The Training Ground (The "Toy" Problem)

To test their new quantum brain, they didn't start with a real, chaotic market. They started with a controlled playground: the Black-Scholes model.

The Analogy: Imagine you are teaching a child to ride a bike. You don't start them on a mountain trail with rocks and traffic. You start them on a flat, empty parking lot.
The Goal: If the quantum brain can learn the "parking lot" (the simple math of the Black-Scholes model) perfectly, maybe it can eventually learn the "mountain trail" (complex, real-world markets).

They fed the quantum brain data about four things that change an option's price:

How much the stock is worth compared to the strike price (Moneyness).
How much time is left before the option expires (Time).
The interest rate (Interest).
How wild the stock market is (Volatility).

2. The Quantum Architecture (The "finQbit" Design)

Most quantum computers today are very small and fragile. They can only hold a few "qubits" (quantum bits) at once.

The Challenge: Usually, you need one qubit for every piece of data. With 4 data points, you'd need 4 qubits. But the team wanted to be efficient.
The Innovation: They built a 2-qubit model (the finQbit architecture).
- Analogy: Imagine you have two buckets (qubits) but you need to carry four different liquids (data). Instead of getting four buckets, they figured out how to mix the liquids cleverly into the two buckets so nothing is lost. They also "shuffled" the liquids around inside the buckets as the calculation progressed to make sure every liquid touched every other liquid.
The Result: This tiny, efficient model learned the pricing rules surprisingly well, matching the accuracy of the best classical computer models (like XGBoost) but using far fewer resources.

The Real-World Test: The "Noisy" Hardware

This is the most exciting part. They didn't just run the code on a perfect computer simulation. They ran it on real, physical quantum computers from four different companies: IBM, IQM, IonQ, and Rigetti.

Think of these as four different brands of "shaky hands":

IBM & Rigetti (Superconducting): These are like fast, metallic robots. They are quick but tend to get "cold" (lose signal) quickly, causing prices to drop slightly too low.
IonQ (Trapped Ions): These use lasers to hold atoms. They are very stable and steady, but they tend to "overestimate" slightly, making prices a bit too high.
IQM: Another fast robot, similar to IBM.

The Findings:

It Worked! Even with the noise and the "shaky hands," the quantum brain could still predict the prices with high accuracy.
The "Sweet Spot": They found that taking more measurements (called "shots") helps smooth out the noise, but only up to a point. After a certain number, the machine gets tired (drifts), and taking more measurements doesn't help.
The Winner: The IBM Fez processor performed the best, getting an accuracy score (R²) of over 93%, which is incredibly close to the perfect theoretical answer.

Why Does This Matter?

You might ask, "Why bother with a quantum computer if the math is already solved for simple options?"

The Answer: The Future.

Proof of Concept: This paper proves that quantum computers aren't just science fiction. They can actually do useful financial work today, even while they are still "noisy" and imperfect.
The "Curse of Dimensionality": Classical computers get stuck when you add too many variables (like 10 different stocks in one option). They have to check every single possibility, which takes forever. Quantum computers, by using the weird geometry of "Hilbert Space" (a fancy math term for a multi-dimensional universe), can navigate these complex landscapes much faster.
Efficiency: The team showed that a tiny 2-qubit model could do the job of a much larger classical model. This suggests that in the future, we might not need massive supercomputers to price complex risks; a small quantum chip might do it in seconds.

The Bottom Line

The authors successfully taught a tiny, noisy quantum brain to price a financial option. They did this on real hardware from four different companies and showed that it works surprisingly well.

The Metaphor:
Imagine trying to paint a perfect portrait of a person using a brush that drips paint and shakes. Most people would say, "Impossible, use a computer." But this team said, "Let's try anyway." They found that even with the dripping and shaking, they could still paint a recognizable, accurate portrait.

This doesn't mean we will replace our banks' computers tomorrow. But it proves that the foundation is solid. As the quantum "brushes" get steadier and the "paint" gets cleaner, these quantum brains will be able to solve the most complex, expensive financial puzzles that classical computers simply cannot handle.

1. Problem Statement

The global derivatives market, valued in the hundreds of trillions of dollars, relies heavily on accurate and efficient option pricing for risk management and capital allocation. While classical models like Black-Scholes-Merton (BSM) offer closed-form solutions for European options, they fail for complex, high-dimensional, or path-dependent derivatives (e.g., American options, local volatility models). Classical numerical methods (Monte Carlo, Finite Difference) suffer from the "curse of dimensionality" and slow convergence.

Although Quantum Machine Learning (QML) promises to overcome these bottlenecks, most existing research remains theoretical or relies on hybrid approaches where quantum components are isolated subroutines. There is a lack of empirical evidence demonstrating that fully quantum approaches can accurately approximate pricing functions on currently available Noisy Intermediate-Scale Quantum (NISQ) hardware.

2. Methodology

The authors propose a Quantum Neural Network (QNN) framework to approximate the BSM option pricing function. The study uses the BSM model not as a replacement for classical solvers, but as a controlled benchmark to rigorously test QML capabilities.

Data Framework

Inputs: Four features: Moneyness ( $m=S/K$ ), Time to Maturity ( $T$ ), Risk-free Rate ( $r$ ), and Volatility ( $\sigma$ ).
Target: Normalized European call price ( $\hat{C} = C/K$ ).
Dataset: 500 training samples and 1,001 test samples generated via analytical BSM solutions, covering diverse market regimes (OTM, ATM, ITM).

Architecture Evolution

The authors developed and tested three distinct architectures:

Baseline 4-Qubit QNN: Maps each input feature to a dedicated qubit. Uses a post-processing max(0, ...) function to ensure non-negative prices.
Fourier-Enhanced 4-Qubit QNN: Utilizes "Data Re-uploading" (encoding data multiple times through the circuit) to expand the frequency spectrum of the model, theoretically allowing it to capture high-frequency components of the pricing surface.
finQbit Architecture (2-Qubit): A highly compressed, hardware-efficient model designed for NISQ constraints.
- High-Density Encoding: Encodes two features per qubit using orthogonal rotation axes ( $R_x, R_y$ ).
- Dynamic Feature Permutation: The mapping of features to qubits changes across the 3 re-uploading layers to capture diverse inter-feature correlations.
- Compression: The circuit was further optimized by decomposing the multi-gate structure into a single 2-qubit unitary block ( $U(4)$ ) to minimize gate count and decoherence.

Experimental Setup

Classical Benchmarks: Ordinary Least Squares (OLS) and XGBoost (Gradient Boosted Trees) were used to establish performance baselines.
Hardware Execution: The final finQbit model was executed on four distinct physical Quantum Processing Units (QPUs):
- IBM Fez (Superconducting Transmon)
- IQM Garnet (Superconducting Transmon)
- Rigetti Ankaa-3 (Superconducting Transmon)
- IonQ Forte (Trapped Ion)
Error Mitigation: Techniques included readout error mitigation (calibration matrices) and shot noise analysis (varying shot counts from 500 to 5,000).

3. Key Contributions

First End-to-End Implementation: One of the first demonstrations of a fully quantum option pricing workflow executed on real, diverse quantum hardware, moving beyond idealized simulations.
The finQbit Architecture: Introduction of a compact 2-qubit QNN that achieves high precision with only 36 trainable parameters (an order of magnitude fewer than the ~400 parameters in the optimized XGBoost benchmark).
Cross-Platform Validation: A rigorous comparative study across superconducting and trapped-ion technologies, revealing distinct hardware-specific noise signatures (e.g., downward bias in superconducting vs. upward bias in trapped ions).
ATM Convexity Challenge: Demonstration that QNNs, leveraging the continuous geometry of Hilbert space, outperform tree-based models (XGBoost) specifically in the At-The-Money (ATM) region where pricing surface curvature (Gamma) is highest.

4. Key Results

Accuracy vs. Classical Models:
- The finQbit model achieved an $R^2$ of 0.97958 on the test set, slightly outperforming XGBoost ( $R^2 = 0.97854$ ).
- Crucially, in the ATM region (where classical tree models struggle due to piecewise-constant approximations), finQbit reduced the Mean Squared Error (MSE) by approximately 50% compared to XGBoost (0.000163 vs. 0.000328).
Hardware Performance:
- IBM Fez emerged as the most accurate physical platform ( $R^2 \approx 0.93$ ), closely approximating ideal state-vector simulations.
- IonQ Forte showed high temporal stability but a persistent upward price bias.
- Rigetti and IQM exhibited systematic downward biases, particularly in deep In-The-Money (ITM) scenarios.
Noise Resilience: Despite NISQ constraints, the compressed 2-qubit architecture maintained predictive power ( $R^2 > 0.64$ ) across all tested devices.
Shot Noise Analysis: Increasing shot counts improved accuracy up to a point (diminishing returns observed beyond 2,000–3,000 shots), after which systematic hardware drift became the dominant error source.

5. Significance and Outlook

Proof of Concept: The study validates that QNNs are not merely speculative tools for future fault-tolerant machines but are viable approximation frameworks for financial functions on current hardware.
Inductive Bias: The results suggest that the continuous geometry of Hilbert space provides a superior inductive bias for modeling smooth financial manifolds (like the BSM surface) compared to the discrete boundaries of classical decision trees.
Future Directions: The authors propose extending this framework to more complex, non-analytical models (Heston, SABR, Local Volatility) and path-dependent derivatives (Asian, Barrier, American options) where classical methods face severe computational bottlenecks.
Operational Feasibility: The work highlights the critical trade-off between statistical precision (shot count) and operational cost, suggesting that for financial QML, optimizing for hardware-specific noise characteristics is as important as algorithmic design.

In conclusion, this paper establishes a foundational step toward quantum-enhanced derivative pricing, demonstrating that compact, noise-resilient QNNs can effectively learn complex financial mappings on real-world quantum hardware.