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Practical Quantum Topological Data Analysis with Applications to High-Dimensional Feature Extraction and Time Series Analysis

This paper establishes the practical utility of quantum topological data analysis by reframing it as a feature-extraction method that utilizes low-order spectral moments of the combinatorial Laplacian to improve predictive performance in time-series applications, supported by a novel moment-based quantum algorithm and experimental validation on Barium-based quantum hardware.

Original authors: Jason Iaconis, Sayonee Ray, Samwel Sekwao, Claudio Girotto, Martin Roetteler

Published 2026-07-30

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Jason Iaconis, Sayonee Ray, Samwel Sekwao, Claudio Girotto, Martin Roetteler

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 understand a complex system, like a bustling city, a human brain, or the stock market. Usually, we look at these systems by checking individual parts or how two parts talk to each other. But what if the real secret lies in how groups of parts interact all at once? This is the realm of Topological Data Analysis (TDA). Think of TDA as a way to look at the "shape" of data. Instead of just counting dots, it asks: "Are there loops?" "Are there hollow bubbles?" "Is there a giant tunnel running through the middle?" These shapes, called topological features, can reveal hidden patterns that standard math misses.

However, there's a catch. Calculating these shapes for high-dimensional data is incredibly hard for classical computers. It's like trying to count every possible way a group of friends can form a secret club in a city of millions; the number of possibilities explodes so fast that even the fastest supercomputers get stuck. This is where Quantum Computing enters the story. Quantum computers use the weird rules of physics to handle massive amounts of possibilities simultaneously. The big question has been: Can quantum computers actually do this useful work for real-world problems, or is it just a theoretical dream?

This paper, written by researchers at IonQ, says "Yes, but with a twist." They argue that we shouldn't try to use quantum computers to calculate the exact number of these shapes (which is still very hard). Instead, we should use them to extract specific "fingerprints" or features of the shape that are good enough to help us make predictions. The authors tested this idea on two very different real-world problems: diagnosing brain diseases using MRI scans and predicting stock market crashes. They found that these higher-level "shape" features contain valuable information that helps computers predict disease and market instability better than looking at simple connections alone.

To prove this works, the team didn't just run simulations; they built a quantum circuit and ran it on a real, working quantum computer made of trapped barium ions. They successfully extracted these "shape fingerprints" from small graphs and showed that the results matched what classical math predicted. While the quantum computer they used is still small, their work suggests a clear path forward: by focusing on these specific, useful features rather than perfect exactness, quantum computers could soon help us solve data problems that are currently impossible for classical machines.

The Shape of Things to Come

Let's dive into the details. The researchers started by looking at two very different worlds: the human brain and the financial market.

1. The Brain: Finding Disease in the Shape of Thoughts
The team looked at data from functional MRI (fMRI) scans, which track blood flow in the brain to see which parts are active. Usually, scientists look at how two brain regions connect. But the researchers asked: What if we look at how groups of regions connect? They turned the brain activity data into a "point cloud" (a bunch of dots in space) and then used TDA to find loops and voids in the shape formed by these dots.

They tested this on data from patients with Alzheimer's disease and healthy individuals. They found that looking at higher-dimensional shapes (like 3D voids or 4D tunnels) gave them better clues about who had the disease than just looking at simple connections. For example, when they used a neural network to classify patients, adding these complex shape features boosted the accuracy. While the accuracy wasn't perfect yet (around 74% for the best model), the key takeaway is that the "higher-order" shapes held information that simpler methods missed. This suggests that the way brain regions interact in groups might be a crucial clue for diagnosing neurodegenerative diseases.

2. The Market: Seeing the Crash Before It Happens
Next, they turned to the stock market. Financial data is messy and noisy, but the researchers believed that the "shape" of how different stocks move together could predict a crash. They took the daily closing prices of 17 major stock indexes and turned them into a point cloud using a technique called "time-delay embedding." This is like taking a snapshot of the market's shape every day.

They found that as the market approached a major crash (like in 2008, 2020, or 2022), the "shape" of the data changed dramatically. Specifically, the higher-dimensional features (like 3D and 4D voids) showed huge spikes right before the market dipped. These spikes were much clearer and more predictive than the simple loops (1D features) that other analysts had looked at before. In fact, their "shape" signal acted as a leading indicator, warning of a downturn about 100 days before it happened, outperforming traditional financial tools like the MACD indicator.

The Quantum Leap: From Theory to Hardware

So, we know these "shape features" are useful. But can a quantum computer actually find them? The traditional way to use quantum computers for this was to try and calculate the exact number of holes (Betti numbers) in the data. The authors argue this is too hard and unnecessary. Instead, they proposed a new method: Moment-Based Quantum TDA.

Think of it like this: Instead of counting every single hole in a Swiss cheese to know if it's a good cheese, you just measure the average density of the holes. If the density is right, you know it's a good cheese. The researchers developed a quantum algorithm that measures these "moments" (specifically, the relative trace of a mathematical object called the Combinatorial Laplacian). They showed through simulations that these moments are strongly correlated with the actual number of holes, even when the number of holes is tiny.

The Hardware Test
To prove this wasn't just a computer simulation, they ran their algorithm on a real quantum computer at IonQ. They used a system with 40 qubits (quantum bits) based on trapped barium ions. They tested it on two types of graphs:

A small graph with 8 nodes.
A slightly larger graph with 16 nodes.

The results were promising. The quantum computer successfully measured the "shape fingerprints" (the relative trace) and the values matched the exact classical calculations very closely. Even with the noise inherent in current quantum hardware, the device could distinguish between graphs with different topological features. This is a big deal because it's one of the first times a quantum computer has been used to extract topological features from real data and compare them directly to the "ground truth."

Why This Matters

The paper concludes that we don't need to wait for a perfect, error-free quantum computer to start doing useful work. By shifting the goal from "exact calculation" to "feature extraction," the quantum advantage becomes reachable much sooner.

The researchers suggest that for complex, high-dimensional data like brain scans or financial markets, the "sweet spot" for quantum advantage might be right around the corner. They estimate that for graphs with high connectivity (dense networks), a quantum computer could solve these problems faster than a classical supercomputer once the graph size reaches somewhere between 70 and 1,200 nodes, depending on the complexity.

In short, this paper bridges the gap between abstract math and real-world utility. It shows that quantum computers can already help us see the hidden shapes in our data, offering a new lens to understand everything from the human mind to the global economy. While the technology is still in its early stages, the path forward looks clear: use quantum computers to find the "shape" of the problem, and let classical computers do the rest.

Technical Summary: Practical Quantum Topological Data Analysis with Applications to High-Dimensional Feature Extraction and Time Series Analysis

Problem Statement

Topological Data Analysis (TDA) offers a robust framework for characterizing the shape of complex, unstructured data through topological invariants such as Betti numbers. However, the computational cost of calculating high-dimensional topological features (specifically, Betti numbers for homological degrees $d > 1$ ) on classical hardware presents a significant bottleneck. The number of potential $d$ -simplices in a complex grows combinatorially, making the calculation of high-order features infeasible for large, dense datasets. While quantum algorithms for TDA have been proposed to bypass this bottleneck, existing approaches typically focus on exact or high-precision estimation of Betti numbers. This narrow focus limits the regime for practical quantum advantage, as it often requires the relative Betti number (the ratio of Betti numbers to the total number of cliques) to be large, a condition not always met in real-world data.

The authors argue that for practical data science applications, the goal is often not to compute exact Betti numbers in the most general setting, but rather to extract topological information that improves downstream predictive tasks (e.g., classification or anomaly detection). They posit that a quantum approach should be reframed as a feature-extraction method, utilizing low-order spectral information from the combinatorial Laplacian as a proxy for high-dimensional topology.

Methodology

The paper adopts a two-pronged approach: validating the utility of high-order TDA features in classical applications and developing a resource-efficient quantum algorithm to extract these features.

1. Classical Validation of High-Order Features

The authors first demonstrate that higher-order homological features ( $H_2$ through $H_4$ ) contain predictive signal in two commercially relevant time-series domains:

Neurodegenerative Disease Classification: Using functional MRI (fMRI) data from the OASIS dataset, the authors construct point clouds from Region of Interest (ROI) time series using Takens embedding. They compute persistent homology up to dimension 4. Machine learning classifiers (Neural Networks and SVMs) trained on these features show that including higher-order homologies ( $H_2, H_3, H_4$ ) improves classification accuracy for Alzheimer's disease compared to models using only $H_0$ and $H_1$ .
Financial Time Series Analysis: Using stock market data (17 indices from 2005–2024), the authors construct point clouds via time-delay embedding. They analyze persistent homology to detect structural changes preceding market crashes. Results indicate that higher-order homologies ( $H_3, H_4$ ) produce stronger and more predictive signals (peaks in persistence) during market downturns (e.g., 2008 crisis, 2020 pandemic) than lower-order features. Furthermore, time-delayed Pearson correlation analysis shows that higher-order TDA signals act as leading indicators, preceding market price changes, whereas traditional indicators like MACD do not exhibit this leading behavior.

2. Quantum Algorithm: Moment-Based Feature Extraction

To address the computational bottleneck, the authors propose a moment-based quantum algorithm that estimates low-order spectral moments of the combinatorial Laplacian ( $\Delta_k^\Gamma$ ) rather than computing exact Betti numbers.

Core Concept: Instead of counting zero eigenvalues (which requires high precision when the relative Betti number is small), the algorithm estimates the relative trace of the Laplacian, $\text{tr}[\Delta_k^\Gamma]$ . The authors demonstrate numerically that the relative trace is strongly correlated with the Betti number $\beta_{k-1}$ , even when the relative Betti number is small.
Circuit Construction: The algorithm utilizes three main components:
1. Data Loading: Preparation of a Dicke state $|D_N^k\rangle$ representing a superposition of all $k$ -cliques with random phases.
2. Projection: Projection onto the specific simplicial complex $\Gamma$ using mid-circuit measurements (MCM) and ancilla qubits to filter out non-cliques.
3. Trace Estimation: Application of the boundary operator $B = \partial + \partial^\dagger$ (implemented via Trotterization of a fermionic Hamiltonian) and measurement of the resulting state's norm to estimate the trace.
Resource Efficiency: The authors utilize mid-circuit measurements and reset capabilities to reduce qubit requirements. The circuit width scales as $O(N)$ , and the shot count depends on the desired precision and the density of $k$ -cliques ( $\zeta_k$ ), but notably, the complexity does not scale strongly with the relative Betti number.

Key Results

Classical Performance

fMRI: In a binary classification task (Alzheimer's vs. Healthy) on a reduced OASIS subset, adding higher-order homology features ( $H_2$ to $H_4$ ) to a Neural Network increased balanced accuracy from ~69% ( $H_0$ ) to ~74% (all features). SVM models showed similar trends, with performance peaking at ~66% with specific feature compression.
Finance: Higher-order homology curves ( $H_3, H_4$ ) exhibited larger peaks and clearer signals near market crashes compared to $H_1$ . Time-delayed correlation analysis revealed that $H_2$ through $H_4$ signals act as leading indicators (negative correlation with future price drops), outperforming the MACD indicator in predictive power.

Quantum Algorithm Performance

Correlation Validation: Numerical simulations on Erdős-Rényi graphs and fMRI point clouds confirmed a strong linear correlation between the relative trace $\text{tr}[\Delta_k^\Gamma]$ and the Betti number $\beta_{k-1}$ , even for small relative Betti numbers. This validates the use of the trace as a proxy feature.
Resource Estimates: The authors provide crossover estimates comparing the Time-to-Solution (TTS) of their quantum algorithm against classical TDA software (giotto-ph). For high-dimensional Betti numbers ( $d=4$ to $10$) on graphs with high edge density ( $\zeta_2 \approx 0.9$ ), the quantum algorithm is projected to outperform classical methods for graphs with 70 to 1,200 nodes, with crossover times ranging from hours to days.
Hardware Demonstration: The algorithm was executed on a Barium-based trapped-ion quantum processor (40 qubits, similar to the IonQ Tempo line).
- Experiments: Successfully ran on 8-node graphs ( $N=8$ , measuring $\beta_1$ ) and 16-node graphs ( $N=16$ , measuring $\beta_3$ ).
- Technique: Utilized 4 rounds of mid-circuit measurements to project states, reducing the required qubit count from a theoretical 56 to 32 for the 16-node case.
- Outcome: The hardware successfully distinguished between graph instances with different Betti numbers based on the measured relative trace observable. The results closely matched ideal simulations, though increased circuit depth for the 16-node case introduced observable noise.

Significance and Claims

The paper claims to establish a "clear pathway to potential quantum utility" for TDA by shifting the paradigm from exact Betti number estimation to topological feature extraction.

Practical Utility: The authors demonstrate that high-order topological features are not just theoretical constructs but contain actionable information for real-world problems like medical diagnosis and financial forecasting, often outperforming lower-dimensional features.
Algorithmic Feasibility: By reframing the task as estimating low-order moments (relative trace), the authors show that quantum advantage is achievable in regimes where the relative Betti number is small, a condition that previously seemed to preclude quantum advantage.
Experimental Validation: The work provides the first quantum-hardware demonstration of moment-based TDA where a measured Laplacian observable is quantitatively compared against exact Betti information to distinguish graph instances.
Roadmap: The combination of application-driven feature validation and resource-efficient algorithm design outlines a realistic roadmap for achieving quantum advantage in data science before the advent of fully fault-tolerant quantum computers.

The authors remain modest in their claims, acknowledging that their quantum algorithm estimates an observable correlated with Betti numbers rather than computing exact values, and that the current hardware demonstration is limited to small graph sizes. However, they argue that the scaling properties and the demonstrated correlation suggest that quantum TDA can provide practical value for classically challenging, high-dimensional data analysis tasks.