Quantum TDA for Financial Bubble Detection

Overview

Financial markets can look stable until they hit a tipping point: a bubble or crash.
While traditional indicators (volatility, moving averages, etc.) often lag, Topological Data Analysis (TDA) captures deeper geometric changes in the market’s structure.

This repository applies classical TDA and other quantum methods to S&P 500 data to identify when the “shape” of the market changes in ways consistent with bubbles.

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Pipeline

• Log transformation of prices for numerical stability.
• Takens embedding to reconstruct the market’s phase space.
• Sliding windows to analyze evolving structure over time.
• Betti₀ curves across multiple ε-scales to measure fragmentation.
• Pairwise L² deltas to quantify sudden topological shifts.

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Files

• src/takens.py – Builds Takens embeddings and sliding windows from time series
• src/betti_curves.py – Computes Betti curves, Lᵖ deltas, and statistical spike detection
• src/rips_laplacian.py – Constructs Vietoris–Rips complex and Laplacian for quantum demo
• src/qpe.py – Demonstrates Quantum Phase Estimation (QPE) using Qiskit
• scripts/run_qpe.py – QPE demo on one window/ε (bar chart comparing classical vs QPE Betti)
• scripts/run_qtda.py – Main script to run the full analysis
• data/sp500.csv – Daily S&P 500 closing prices (date, value)
• plots/betti_curves.png – Betti₀ curves over time
• plots/crash_spikes.png – Pairwise L² deltas with spikes flagged
• plots/qpe_betti_k*_eps*_*.png – QPE bar chart (filename includes window’s center date)

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Results & Interpretation

1. Betti Curves over Windows

Tracks the number of connected components (Betti₀) for each sliding window at several neighborhood radii (ε).
Axes:

• X: Calendar date (center of each window)
• Y: Betti₀ (connectivity)

How to read:

• Values near 1 show cohesive, stable market behavior.
• Spikes show fragmentation in the geometry, consistent with instability or bubble formation.
  These spikes often precede major events (e.g., 2008, 2020).

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2. Pairwise L² Deltas (Spikes Flagged)

Measures how abruptly the topology changes between consecutive windows via the L² distance between their Betti curves.
Axes:

• X: Date (of the later window)
• Y: Δ (L² distance between consecutive Betti curves)

How to read:

• Taller bars indicate larger step-changes in topology.
• “×” markers denote statistically significant spikes (z-score > 2.0).
  When Betti₀ spikes and Δ spikes occur together, they strongly align with bubble bursts.

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3. Quantum Betti Estimation (QPE)

Builds a Vietoris–Rips Laplacian for one window and uses Quantum Phase Estimation (QPE) on a simulator to estimate the Betti number from the Laplacian’s spectrum (zero eigenvalues ↔ connected components).

What the QPE bar chart shows

• Classical (left): Betti₀ from persistent homology (ripser) — the true connectivity for that window.
• QPE est. (right): Betti₀ estimated from the QPE zero-phase probability.

How to interpret:

• The classical Betti₀ reflects the actual connectivity in that window.
• The QPE estimate is an approximation; with few phase qubits and limited shots it can overestimate, but it illustrates how topological information can be extracted from the Laplacian using a quantum routine.

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Accuracy & Insight

• Detects early fragmentation during the 2007–2008 financial crisis.
• Shows a sharp topological shift around March 2020 (COVID-19 crash).
  These findings demonstrate that TDA-based indicators can act as early signals of market regime change, often preceding traditional metrics.

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Credit

This work was inspired by the Moody’s Quantum Challenge at iQuHACK 2025, which encouraged exploration of quantum and topological methods for financial risk analysis.

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License

MIT License © 2025