用量子拓扑分析检测金融压力，首次实现低深度编码与精确贝蒂数恢复。
Depth-Efficient Quantum Topological Data Analysis for Regime-Specific Detection of Financial Stress
- 将量子相关编码用于拓扑数据分析，压缩比特数并避免梯度消失
- 在4-12个量子比特上实现无指数退化梯度，成功复现所有贝蒂数
- 适合金融风险监测与量子算法应用研究者参考
我们首次将保罗相关编码（PCE）应用于量子拓扑数据分析，将贝蒂数估计重构为压缩量子寄存器上的低深度变分优化。基于标普500收益率的Takens嵌入与Vietoris-Rips过滤，提取组合拉普拉斯矩阵，并将零空间计数转化为连续型PCE瑞利商最小化，采用变分去耦技术，将n_k个单纯形索引编码至O(n_k^{1/κ})个量子比特，使用浅层、无需辅助比特的电路。由于损失函数为有理式而非双线性，文献[Sciorilli25]的荒原高原边界不适用；实测梯度方差仅多项式衰减，未出现指数级荒原高原，覆盖n=4–12量子比特。经典阶段在190个滑动窗口上与ripser性能相当（2007–2009）。对真实市场拉普拉斯矩阵（β₁=1–22），通过经典零空间初值启动，PCE-VQE在各尺度精确恢复β₁，表明障碍在于优化景观而非编码本身。按时间划分分类，同期内区域的ROC AUC达0.818；但在2020年新冠冲击和2022年利率周期的分布外评估中，AUC分别为0.009和0.515，显示校准无法跨危机情境泛化。
原文摘要 · Abstract (English)
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 $O(n_k^{1/κ})$ 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 ($β_1=1$--$22$), warm-starting from a classical null-space surrogate allows PCE-VQE to recover $β_1$ 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.
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