通过Mayer同调实现拓扑数据分析中的量子优势

中文摘要 AI 辅助

先前的工作已探索了用于拓扑数据分析（TDA）的量子算法，揭示了在估计贝蒂数与组合拉普拉斯算子维度之比时存在指数级量子加速的可能性。然而，仅当贝蒂数呈指数级大时，该量才非零且可被高效地量子估计，而这种情况的具体实例鲜为人知。此外，某些随机经典算法在该机制下有时是高效的。因此，在传统TDA中实现量子优势的前景显得相当狭窄。在此，我们通过开发用于Mayer同调的量子算法来应对这些TDA中量子优势的挑战，Mayer同调将单纯同调推广到$N$-幂零边界算子（$\partial^N =0$），并且最近已成功应用于现实世界的TDA场景。我们引入了一种用于估计Mayer贝蒂数及其持久对应物的高效量子算法。随后，我们证明对于高阶单纯形，在密集机制下Mayer贝蒂数通常呈指数级大，这改善了传统量子TDA的归一化瓶颈。在同一机制下，我们认为现有的为传统TDA开发的去量子化算法，当应用于Mayer同调时，通常会失去理论保证，面临某些阻碍其实际效用的结构性障碍。我们还提供了逻辑资源估计，表明一台拥有约数百个量子比特和六千万个Toffoli门的量子计算机可以解决超出已知经典方法能力的Mayer同调问题。最后，我们讨论了Mayer同调在基因组学、超对称性、药物发现和神经科学中的实际应用，揭示了我们的量子算法通过Mayer同调产生现实世界影响的潜力。

英文摘要

Prior work has explored quantum algorithms for topological data analysis (TDA), revealing the possibility of exponential quantum speedups in estimating the ratios of Betti numbers to the dimension of the combinatorial Laplacian. However, this quantity is only non-vanishing and efficient-to-quantumly-estimate when Betti numbers are exponentially large, a case for which concrete examples are rarely known. Furthermore, certain randomized classical algorithms are sometimes efficient in this regime. Thus, the prospect of achieving quantum advantage in conventional TDA appears fairly narrow. Here, we address these challenges to the quantum advantage in TDA by developing quantum algorithms for Mayer homology, which generalize simplicial homology to $N$-nilpotent boundary operators ($\partial^N =0$) and have recently been successfully applied to real-world TDA contexts. We introduce an efficient quantum algorithm for estimating Mayer Betti numbers and their persistent counterparts. We then prove that for high-order simplices, Mayer Betti numbers are often exponentially large in the dense regime, which ameliorates the normalization bottleneck of conventional quantum TDA. In the same regime, we argue that existing dequantization algorithms developed for conventional TDA, when applied to Mayer homology, generally lose theoretical guaranties, facing certain structural barriers that prevent their practical utilities. We also provide logical resource estimates revealing that a quantum computer with roughly a few hundred qubits and sixty million Toffoli gates could solve Mayer homology problems beyond the capabilities of known classical approaches. Finally, we discuss real-world applications of Mayer homology in genomics, supersymmetry, drug discovery, and neuroscience, revealing the potential of our quantum algorithm to deliver real-world impacts via Mayer homology.

发表机构

• Google Quantum AI（谷歌量子人工智能）
• C. N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook（纽约州立大学石溪分校陈省身理论物理研究所）
• School of Mathematical and Physical Sciences, Macquarie University（麦考瑞大学数学与物理科学学院）
• Scripps and Pitzer Colleges, Claremont Colleges Consortium（克莱蒙特学院联盟斯克里普斯学院和皮策学院）
• University of Georgia（佐治亚大学）
• Yukawa Institute for Theoretical Physics & The Hakubi Center, Kyoto University（京都大学汤川理论物理研究所及博雅中心）

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