Depth Efficient Quantum Topological Data Analysis for Regime Specific Detection of Financial Stress

Jul 16, 2026

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Depth-efficient adaptation of Pauli Correlation Encoding (PCE) for quantum topological data analysis reformulates Betti number estimation as variational optimization over a compressed qubit register. Using Takens embedding and Vietoris–Rips filtration of S&P 500 returns, combinatorial Laplacians are extracted and null-space counting becomes a continuous-PCE Rayleigh-quotient minimization with variational deflation. nk simplex indices are encoded into O(n1k/κ) qubits with shallow, ancilla-free circuits; the loss is rational, avoiding exponential barren plateaus. Classical verification matches ripser on 190 sliding windows (2007–2009), while in-market evaluation shows regime-dependent generalization limits across crisis shocks.

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CategoryResearch & Report

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Depth-Efficient Quantum Topological Data Analysis for Regime-Specific Detection of Financial Stress Arul Rhik Mazumder Carnegie Mellon University Pittsburgh, PA, USA ORCID: 0000-0002-2395-4400 Shreyan Ronit Mazumder Cambridge Rindge and Latin School Cambridge, MA, USA arXiv :2607 .09906v1 [ quant-ph] 10 Jul 2026 Abstract—We present, to our knowledge, the first adaptation of Pauli Correlation Encoding (PCE) to quantum topological data analysis, reformulating Betti number estimation as a depthefficient variational optimization over a compressed qubit register. From a Takens embedding and Vietoris–Rips filtration of S&P 500 returns, we extract combinatorial Laplacians and recast null-space counting as a continuous-PCE Rayleigh-quotient minimization with variational deflation, encoding nk simplex indices into O (n1k/κ ) qubits with shallow, ancilla-free circuits. Because the resulting loss is rational rather than bilinear in the correlators, the barren-plateau bound of [1] does not transfer; empirically the gradient variance decays only polynomially, with no exponential barren plateau, over n = 4–12 qubits. The classical stage matches ripser [2] on all 190 sliding windows (2007–2009). On the real market Laplacians (β1 = 1–22), warmstarting from a classical null-space surrogate allows PCE-VQE to recover β1 exactly at every scale, placing the obstacle in the optimisation landscape rather than the encoding. Chronologically split classification gives in-regime ROC AUC 0.818, but outof-distribution evaluation on the 2020 COVID shock and 2022 rate cycle (AUC 0.009, 0.515) shows the calibration does not generalize across crisis regimes. Index Terms—quantum topological data analysis, Betti numbers, Pauli correlation encoding, variational quantum eigensolver, financial crash detection, combinatorial Laplacian I. INTRODUCTION Financial markets exhibit complex nonlinear dynamics that defy classical statistical characterization. The 2008 financial crisis—which erased over $10 trillion in US household wealth—emerged from correlations and structural instabilities invisible to conventional volatility measures such as variance or VaR. During stable regimes a time series {xt } exhibits small, roughly Gaussian fluctuations; approaching a crash, the underlying dynamical attractor undergoes topological deformation that can precede the dislocation. In particular, Gidea and Katz [3] showed that persistence-based topological features of a multi-index point cloud (S&P 500, DJIA, NASDAQ, and Russell 2000 viewed jointly as coordinates in R4 ) exhibit a sustained rise in the spectral density of Lp-norms of persistence landscapes for roughly 250 trading days prior to the Lehman bankruptcy. This motivates the search for early warning signals that operate on the geometry of market data rather than its pointwise statistics, and in particular on its topology, which is stable under perturbations and captures qualitative structural change that purely local statistics miss [4]–[6] . Our pipeline is inspired by this line of work but adopts a different (single-index Takens delay) embedding, described in Section IV. Topological Data Analysis (TDA) offers a robust framework for extracting such geometric features. The central invariants are Betti numbers βk , which count the number of kdimensional holes in the data manifold: β0 counts connected components, β1 counts loops, and β2 counts voids. However, computing Betti numbers classically requires constructing and diagonalizing combinatorial Laplacians whose dimensions grow combinatorially with the number of simplices—an O (n3k) cost per window that becomes prohibitive for real-time deployment at scale. Quantum algorithms for TDA (qTDA) have been proposed to address this computational bottleneck, beginning with the seminal LGZ algorithm [7] and its subsequent refinements [8]–[13] . These approaches use quantum phase estimation (QPE) to estimate the eigenspectrum of the combinatorial Laplacian, fro

FAQs

What key method does the document propose for quantum topological data analysis?

It proposes the first adaptation of Pauli Correlation Encoding (PCE) to quantum topological data analysis, reformulating Betti number estimation as a depth-efficient variational optimization over a compressed qubit register.

How are financial data and topology connected in the workflow?

S&P 500 returns are processed with a Takens embedding and a Vietoris–Rips filtration to derive combinatorial Laplacians, then null-space counting is recast as a continuous-PCE Rayleigh-quotient minimization.

Does the approach generalize across different market crisis regimes?

Chronologically split classification achieves in-regime ROC AUC of 0.818, but out-of-distribution tests on the 2020 COVID shock and the 2022 rate cycle show poor calibration and limited generalization (AUC values 0.009 and 0.515).

Jul 16, 2026

Upload by Maeve