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

We present, to our knowledge, 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. 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 $n_k$ simplex indices into $O(n_k^{1/\kappa})$ qubits with shallow, ancilla-free circuits. Because the resulting loss is rational rather than bilinear in the correlators, the barren-plateau bound of~\cite{Sciorilli25} 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~\cite{bauer2021ripser} on all 190 sliding windows (2007-2009). On the real market Laplacians ($\beta_1=1$--$22$), warm-starting from a classical null-space surrogate allows PCE-VQE to recover $\beta_1$ 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 out-of-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.