New aspects of quantum topological data analysis: Betti number estimation, and testing and tracking of homology and cohomology classes

TL;DR

The paper introduces a structured-input quantum TDA framework that leverages classical boundary data to estimate Betti numbers and persistent Betti numbers, and to perform homology testing, with a dual cohomology perspective. By building block-encodings of boundary operators and Laplacians and applying stochastic rank estimation, it achieves polylogarithmic to near-linear runtimes in regimes where the simplicial complex is sparse or moderately sized, surpassing prior oracle-based quantum approaches and several classical methods in these regimes. It also develops homology testing (triviality and equivalence) and an alternative cycle-tracking method, plus a cohomology-based testing pipeline, highlighting rank-independent advantages and broader algorithmic flexibility. The work points to important open problems, including computing cup products and cohomology-ring operations, connections to higher invariants like Persistent Khovanov homology, and understanding approximation complexity for Betti-number estimation in realistic data settings. Overall, the results advance quantum TDA by exploiting structured input to obtain provable quantum advantages for key topological invariants and testing tasks with potential practical impact in data analysis contexts.

Abstract

We present new quantum algorithms for estimating homological invariants, specifically Betti and persistent Betti numbers, of a simplicial complex given through structured classical data. Our approach efficiently constructs block-encodings of (persistent) Laplacians, enabling estimation via stochastic rank methods with complexity polylogarithmic in the number of simplices across both sparse and dense regimes. Unlike prior spectral algorithms that suffer when Betti numbers are small, we introduce homology tracking and property testing techniques achieving exponential speedups under natural sparsity and structure assumptions. We also formulate homology triviality and equivalence testing as property testing problems, giving nearly linear-time quantum algorithms when the boundary rank is large. A cohomological formulation further yields rank-independent testing and polylog-time manipulation of $r$-cocycles via block-encoded projections. These results open a new direction in quantum topological data analysis and demonstrate provable quantum advantages in computing topological invariants.

Figures (4)

• Figure 1: Overview of our results. We explore quantum algorithms for topological data analysis given explicit classical access to a simplicial complex $K$ and its associated descriptions $\pazocal{S}_r$. Our results include algorithms for Betti number estimation $\beta_r$ and persistent Betti number estimation, as well as homology property testing tasks such as triviality and equivalence testing. We further demonstrate applications to homology tracking and introduce a new cohomological approach based on differential operators.
• Figure 2: Illustration of standard simplexes. Top left: a point ($0$-simplex); top right: a line segment ($1$-simplex); bottom left: a filled triangle ($2$-simplex); bottom right: a filled tetrahedron ($3$-simplex). Each $r$-simplex is formed by $(r{+}1)$-geometrically independent vertices in Euclidean space.
• Figure 3: Quantum algorithm for estimating Betti numbers (LGZ algorithm). The procedure begins with $n$ data points, from which a simplicial complex $K$ is constructed. Each $r$-simplex $\sigma_r \in K$ is encoded into an $n$-qubit basis state $\ket{\sigma_r}$. A uniform mixture over all such basis states yields a density matrix, while the combinatorial Laplacian $\Delta_r$ is computed using the boundary maps. Quantum dynamics governed by $\exp(-i \Delta_r)$ is simulated, and quantum phase estimation is used to estimate the fraction of zero eigenvalues of $\Delta_r$, which corresponds to the normalized $r$-th Betti number.
• Figure 4: Block structure of the boundary matrix $\partial_{r+1}^{K_2}$ and its adjoint $(\partial_{r+1}^{K_2})^\dagger$. The blue block $\pazocal{B}$ corresponds to the original boundary operator $\partial_{r+1}^{K_1}$, while the red block $\pazocal{R}$ represents the interaction between simplices in $K_1$ and those newly added in $K_2 \setminus K_1$. The green block $\pazocal{G}$ encodes the internal structure among new simplices. Dashed outlines in the adjoint matrix highlight the transpose-like dual roles of each sub-block.

Theorems & Definitions (21)

• Theorem 1.1: Time complexity of estimating (normalized) Betti numbers, see \ref{['sec: estimatingBettinumbers']}
• Theorem 1.2: Time complexity of estimating (normalized) persistent Betti numbers, see \ref{['sec: estimatingpersistentbettinumbers']}
• Theorem 1.3: Time complexity of testing homology triviality, see \ref{['sec: zerohomologyclass']}
• Theorem 1.4: Time complexity of testing homology equivalence, see \ref{['sec: nonzerohomologyclass']}
• Theorem 1.5: Time complexity of homology equivalence testing via cohomology, see \ref{['sec: hom_equiv_test_cohom']}
• Definition 2.1: Block-encoding unitary, see e.g. low2017optimallow2019hamiltoniangilyen2019quantum
• Remark 2.1: Properties of block-encoding unitary
• Lemma 2.1: Informal, product of block-encoded operators, see e.g. gilyen2019quantum
• Lemma 2.2: Informal, tensor product of block-encoded operators, see e.g. camps2020approximate
• Lemma 2.3: Informal, linear combination of block-encoded operators, see e.g. gilyen2019quantum