Quantum Algorithm for Estimating Betti Numbers Using Cohomology Approach

TL;DR

This work introduces a dual quantum approach to estimating Betti numbers by leveraging discrete Hodge theory and cohomology on triangulated manifolds, contrasting it with prior homology-based LGZ methods. By working in the space of forms and using block encoding/QSVT, the algorithm computes coexact and exact components, reconstructs harmonic forms, and reads off Betti numbers from the harmonic space, achieving favorable scaling in regimes where Betti numbers are small relative to the simplex count. The paper provides concrete complexity bounds for normalized and multiplicative accuracy estimates and demonstrates potential exponential speedups over classical and homology-based quantum methods in many practical regimes, while discussing the regime where speedups may be limited. It also clarifies input requirements (triangulated, near-uniform manifolds with explicit $F_r$ and $G_r$ data) and positions the cohomology framework as a powerful, complementary tool for quantum topological data analysis with broad applicability to high-dimensional data geometry.

Abstract

Topological data analysis has emerged as a powerful tool for analyzing large-scale data. An abstract simplicial complex, in principle, can be built from data points, and by using tools from homology, topological features could be identified. Given a simplex, an important feature is called the Betti numbers, which roughly count the number of `holes' in different dimensions. Calculating Betti numbers exactly can be $\#$P-hard, and approximating them can be NP-hard, which rules out the possibility of any generic efficient algorithms and unconditional exponential quantum speedup. Here, we explore the specific setting of a triangulated manifold. In contrast to most known methods to estimate Betti numbers, which rely on homology, we exploit the `dual' approach, namely, cohomology, combining the insight of the Hodge theory and de Rham cohomology. Our proposed algorithm can calculate its $r$-th normalized Betti number $β_r/|S_r|$ up to some additive error $ε$ with running time $\mathcal{O}\Big(\frac{\log(|S_r^K| |S_{r+1}^K|)}{ε^2} \log (\log |S_r^K|) \big( r\log |S_r^K| \big) \Big)$, where $|S_r|$ is the number of $r$-simplexes in the given complex. For the estimation of $r$-th Betti number $β_r$ to a chosen multiplicative accuracy $ε'$, our algorithm has complexity $ \mathcal{O}\Big(\frac{\log(|S_r^K| |S_{r+1}^K|)}{ε'^2} \big( \frac{ Γ}{β_r}\big)^2 (\log |S_r^K|) \log \big( r\log |S_r^K| \big) \Big)$, where $Γ\leq |S_r^K|$ can be chosen. A detailed analysis is provided, showing that our cohomology framework can even perform exponentially faster than previous homology methods in several regimes. In particular, our method is most effective when $β_r \ll |S_r^K|$, which can offer more flexibility and practicability than existing quantum algorithms that achieve the best performance in the regime $β_r \approx |S_r^K|$.

Figures (11)

• Figure 1: (a) Application of TDA in medical imaging: performing topological denoise. (b) An example of point cloud. Each point might be a vector in a very high dimension. Upon an appropriate metric, one can define distance between two points, and set a threshold for connectivity, e.g., two points are connected if their distance is less than some value. Roughly speaking, a point is called 0-simplex, an edge connecting two points is called 1-simplex, a triangle having 3 points mutually connected is called 2-simplex. Higher dimensional simplex is generalized in a straightforward manner. Then the resultant configuration forms a so-called simplicial complex. The goal of TDA is to analyze the underlying structure of given data points by using tools from algebraic topology, more concretely, homology theory.
• Figure 2: A triangulated sphere
• Figure 3: A part of a 2-D triangulated manifold, which contains two triangles (or 2-simplex) glued together. From a graph perspective, we have a set of vertices $v_1,v_2,v_3,v_4$, edges (1-simplex) $e_1,e_2,e_3,e_4$, and faces (2-simplex) $f_1,f_2$. The above figure could be specified by some functions that describe the local connectivity between points/edges/faces as well as the inclusion relation between point-edge or edge-face.
• Figure 4: Illustration of the key difference between homology and cohomology. In homology, the main working objects are simplexes, as will be formally defined in Sec. \ref{['sec: crashcoursehomology']}. A central recipe within homology theory is the chain group/space, for which the simplexes serve as the basis. The cohomology theory is based on the dual of such a chain group/space, called form space, with the main working objects being linear functionals, or forms, that map simplexes to real numbers. As illustrated above, a 1-form maps an edge, or 1-simplex, to a real value; a 2-form maps a triangle, or 2-simplex, to a real value. Higher-dimensional forms are further extended and defined for higher-dimensional objects in a similar manner.
• Figure 5: A diagram capturing the essential steps and main flow of quantum algorithms for estimating Betti numbers using the cohomology and homology frameworks, respectively.

Theorems & Definitions (26)

• Theorem 1
• Definition 1: Block Encoding Unitary
• Lemma 1: Appendix \ref{['sec: elaboration']}
• Lemma 2: Appendix \ref{['sec: stochasticrankestimation']}
• Theorem 2: Estimating the 1st Betti number
• Lemma 3
• Lemma 4: gilyen2019quantum Product
• Lemma 5: camps2020approximate Tensor Product
• Lemma 6: gilyen2019quantum (Lemma 48) Sparse-Access Matrix
• Lemma 7: Linear Combination gilyen2019quantum