发表机构
Carnegie Mellon University; Cambridge Rindge and Latin School（卡内基梅隆大学; 剑桥拉丁学校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究首次将泡利相关编码用于量子拓扑数据分析，通过塔克嵌入和维托里斯 - 里普斯过滤提取数据，用浅电路编码，能在一定程度上恢复贝蒂数，但跨危机状态校准不能泛化。
AI 中文摘要
据我们所知，我们首次将泡利相关编码（PCE）应用于量子拓扑数据分析，将贝蒂数估计重新表述为对压缩量子比特寄存器的深度高效变分优化。从标准普尔500指数回报的塔克嵌入和维托里斯 - 里普斯过滤中，我们提取组合拉普拉斯算子，并将零空间计数重铸为具有变分收缩的连续PCE瑞利商最小化，用浅的、无辅助量子比特的电路将\(n_k\)个单纯形索引编码到\(O(n_k^{1/\kappa})\)个量子比特中。由于所得损失在相关器中是有理而非双线性的，文献[Sciorilli25]的贫瘠高原界限不适用；经验上，梯度方差仅多项式衰减，在\(n = 4\) - \(12\)个量子比特上没有指数贫瘠高原。经典阶段在所有190个滑动窗口（2007 - 2009）上与ripser[鲍尔2021ripser]匹配。在实际市场拉普拉斯算子（\(\beta_1 = 1\) - \(22\)）上，从经典零空间替代物进行热启动允许PCE - VQE在每个尺度上精确恢复\(\beta_1\)，将障碍置于优化景观而非编码中。按时间顺序拆分分类在 regime 内的ROC AUC为\(0.818\)，但在2020年新冠冲击和2022年利率周期上的分布外评估（AUC为\(0.009\)，\(0.515\)）表明校准不能跨危机状态泛化。
英文摘要
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.
Comments12 pages, 6 figures, 5 tables, Accepted to IEEE International Conference of Quantum Computing and Engineering - QCE 2026 in the Quantum Applications (QAPP) Technical Papers track
