⚛️ quantum physics

Evaluating Quantum Amplitude Estimation for Pricing Multi-Asset Basket Options

This paper evaluates the performance of quantum amplitude estimation for pricing multi-asset basket options by analyzing the trade-offs between accuracy and computational resources through a hybrid quantum-classical benchmarking framework.

Original authors: Muhammad Kashif, Shaf Khalid, Nouhaila Innan, Alberto Marchisio, Muhammad Shafique

Published 2026-02-12

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Muhammad Kashif, Shaf Khalid, Nouhaila Innan, Alberto Marchisio, Muhammad Shafique

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 trying to predict the final price of a "fruit basket" at a massive supermarket. This basket isn't just one thing; it’s a mix of apples, bananas, cherries, and dragon fruit. Because the prices of these fruits change every second based on weather, shipping, and demand, predicting the total cost of the basket is incredibly hard.

In the financial world, this is called a "Multi-Asset Basket Option." Big banks use complex math to guess these prices so they don't lose money. Currently, they use "Classical Computers" (the ones we use today), but as the basket gets bigger (more types of fruit), these computers start to struggle and get very slow.

This paper explores using Quantum Computers to solve this problem faster and more accurately. Here is the breakdown of how they did it:

1. The Problem: The "Blurry Photo" Effect

To predict the price, you have to turn a continuous, moving world into something a computer can understand. You do this by "discretizing" it—basically, instead of saying a price could be any number, you create a grid of possible prices (like pixels in a photo).

The Qubit Dilemma: In a quantum computer, the "pixels" are controlled by things called qubits.
- If you use only 1 or 2 qubits, it’s like looking at a photo made of only 4 giant, blurry blocks. You can tell there is a fruit in the picture, but you can't tell if it's a grape or a blueberry. The prediction is way off.
- If you use more qubits, the photo becomes high-definition. The "pixels" get smaller, and the prediction becomes much sharper.

2. The Experiment: Finding the "Sweet Spot"

The researchers didn't use fake, "perfect" math numbers. They used real-world data from actual stocks (like Apple, Google, and Microsoft) to see if this quantum method actually works in the messy real world.

They tested two main things:

Adding more "Pixels" (Qubits): They found that if you use too few qubits, the computer "overshoots" or "undershoots" the price wildly. But once you hit about 3 or 4 qubits, the accuracy jumps up significantly. It’s like moving from a 144p video to a 1080p video—suddenly, you can actually see what's happening.
Adding more "Fruit" (Assets): They found that as you add more stocks to the basket (moving from 3 stocks to 9), the job gets much harder. If you keep the number of qubits the same, the computer starts to get "confused" again because it doesn't have enough "pixels" to describe such a complex basket.

3. The "Diminishing Returns" Rule

You might think, "Why not just use 1,000 qubits to make it perfect?"

The researchers discovered a catch: The Law of Diminishing Returns. Adding more qubits makes the "photo" clearer, but it also makes the "camera" (the quantum circuit) incredibly heavy and difficult to operate. After a certain point (around 4 qubits), adding more doesn't actually make the price much more accurate, but it makes the computer work exponentially harder.

The Bottom Line (The "TL;DR")

The researchers found the "Goldilocks Zone" for quantum finance.

If you want to price a basket of stocks using a quantum computer today, you shouldn't use too few qubits (too blurry) or too many (too heavy/slow). Instead, 3 to 4 qubits per asset is the "just right" amount. It provides a high-definition enough picture to compete with the best classical computers without breaking the machine.

In short: Quantum computers are getting ready to take over the stock market, but we need to know exactly how much "resolution" we need to get the job done efficiently.

Technical Summary: Evaluating Quantum Amplitude Estimation for Pricing Multi-Asset Basket Options

1. Problem Statement

The pricing of multi-asset basket options—derivatives whose payoff depends on the weighted average of several underlying assets—is a computationally intensive task in quantitative finance.

Classical Limitations: Traditional analytical models like Black-Scholes struggle with multi-asset correlations and rely on simplifying assumptions (e.g., constant volatility). Monte Carlo (MC) simulations, while flexible, suffer from slow convergence rates ( $O(\epsilon^{-2})$ ) and become prohibitively expensive as the number of assets (dimensionality) increases.
Quantum Gap: While Quantum Amplitude Estimation (QAE) offers a theoretical quadratic speedup ( $O(\epsilon^{-1})$ ), most existing research relies on synthetic, idealized data. There is a lack of empirical evidence regarding how QAE performs when applied to real-world market data and how it scales with both asset dimensionality and the number of qubits used for discretization.

2. Methodology

The authors propose a hybrid quantum-classical workflow designed to bridge the gap between theory and practice.

Data Integration: Instead of synthetic models, the study uses historical market data (e.g., AAPL, GOOG, MSFT) retrieved via the yfinance library. They calculate logarithmic returns, estimate mean ( $\mu$ ) and volatility ( $\sigma$ ), and scale these parameters to the option's maturity ( $T$ ).
Discretization & State Preparation:
- The continuous log-normal distribution of each asset is discretized into a grid of $2^n$ points, where $n$ is the number of uncertainty qubits per asset.
- A quantum circuit is constructed to load these probability distributions into a quantum register.
Payoff Encoding: The basket payoff— $\max(0, \sum w_i S_i - K)$ —is encoded using reversible arithmetic (a weighted adder) and a piecewise-linear approximation function implemented via controlled rotations on an objective qubit.
Parametric Study Design: The researchers systematically varied two dimensions:
1. Qubit Count ( $n$ ): Increasing the number of uncertainty qubits per asset to refine the price grid.
2. Asset Count ( $d$ ): Increasing the number of assets in the basket to test dimensionality scaling.
Benchmarking: Quantum results are compared against Classical Monte Carlo and Black-Scholes models.

3. Key Contributions

Real-World Empirical Validation: Moving beyond synthetic datasets to use actual historical stock behaviors, including realistic volatility and drift.
Resource-Accuracy Trade-off Analysis: Providing a granular look at how increasing qubit counts improves precision but exponentially increases circuit complexity.
Hybrid Framework: Establishing a scalable pipeline that allows for the systematic comparison of quantum estimates against established classical financial benchmarks.

4. Results and Discussion

The study reveals a critical relationship between discretization resolution and estimation accuracy:

Impact of Uncertainty Qubits:
- Low Qubit Counts (1–2): Result in "coarse" discretization, leading to severe errors (significant underestimation or drastic overshooting).
- The "Sweet Spot" (3–4 qubits): At this level, QAE estimates converge closely with Monte Carlo results. For small baskets (3–4 assets), the error drops to $<1\%$ .
- Diminishing Returns ( $>4$ qubits): While accuracy continues to improve slightly, the computational cost (circuit depth) increases exponentially, making it less practical for near-term (NISQ) hardware.
Impact of Asset Dimensionality:
- As the number of assets increases (from 3 to 9), the accuracy of the QAE estimate tends to deteriorate if the qubit count remains fixed.
- For larger baskets (6–9 assets), the model showed a tendency to "overshoot" classical benchmarks, likely due to the current implementation's assumption of asset independence and the challenges of capturing complex correlations in a discretized space.

5. Significance

This paper provides actionable insights for the deployment of quantum algorithms in finance. It identifies that 3–4 uncertainty qubits per asset is the optimal allocation for near-term quantum devices to balance precision and circuit depth. By highlighting the limitations of current discretization and correlation modeling, the work sets a roadmap for future research, emphasizing the need for advanced state-preparation techniques and improved error mitigation to handle high-dimensional, highly correlated financial portfolios.