Towards quantum topological data analysis: torsion detection

TL;DR

This work extends quantum topological data analysis beyond Betti numbers by targeting torsion detection in simplicial complexes. It leverages the universal coefficient theorem to relate torsion to dimension differences of homology across coefficient fields, and designs a quantum procedure that combines block-encoding of boundary operators with finite-field rank estimation to detect torsion efficiently. The method computes ranks over $\mathbb{R}$ (yielding Betti numbers) and over $\mathbb{F}_p$ for several primes, then compares results to reveal the presence of torsion, with detailed non-oracle and oracle-case resource estimates. This approach opens a path to richer topological insights in quantum data analysis, while highlighting future work to recover full torsion structure and invariant factors.

Abstract

Topological data analysis (TDA) has become an attractive area for the application of quantum computing. Recent advances have uncovered many interesting connections between the two fields. On one hand, complexity theoretic results show that estimating Betti numbers, a central task in TDA, is NP hard, indicating that a generic quantum speedup is unlikely. On the other hand, several recent studies have explored structured, less generic settings and demonstrated that quantum algorithms can still achieve significant speedups under certain conditions. To date, most of these efforts have focused on Betti numbers, which are topological invariants capturing the intrinsic connectivity and holes in a dataset. However, there is another important feature of topological spaces: torsion. Torsion represents a distinct component of homology that can reveal richer structural information. In this work, we introduce a quantum algorithm for torsion detection, that is, determining whether a given simplicial complex contains torsion. Our algorithm, assisted by a low complexity classical procedure, can succeed with high probability and potentially offer exponential speedup over the classical counterpart.

Figures (2)

• Figure 1: Illustration of standard simplexes. From left to right: a point ($0$-simplex), a line segment ($1$-simplex), a filled triangle ($2$-simplex), a filled tetrahedron ($3$-simplex). Each $r$-simplex is formed by $(r{+}1)$-geometrically independent vertices in Euclidean space.
• Figure 2: Simple illustration of the general idea behind our state preparation for $\ket{\Phi}$ indicated at the beginning. We promote $x_i$ to a point on the $(x,y)$ plan with its $x$ coordinate being the index $i$ of $x_i$, and $y$ being the value $x_i$. Es each amplitude $x_i$ is randomly chosen from $\mathbb{F}_p$, it can be seen that $\frac{-p+1}{2} \leq x_i \leq \frac{p-1}{2}$. By connecting these points $\{x_i\}_{i=1}^6$, a piecewise-linear function is formed.

Theorems & Definitions (35)

• Lemma 1: Finite-field state preparation marin2023quantummcardle2022quantumnakaji2022approximate)-- Appendix \ref{['sec: proofofstatepreparation']}
• Lemma 2: Block-encoding of boundary operator -- Appendix \ref{['sec: proofoflemmaentrycomputablematrix']}
• Theorem 1: Torsion Detection
• Definition 1: Block-encoding unitary, see e.g. low2017optimallow2019hamiltoniangilyen2019quantum
• Remark 1: Properties of block-encoding unitary
• Lemma 3: Informal, product of block-encoded operators, see e.g. gilyen2019quantum
• Lemma 4: Informal, tensor product of block-encoded operators, see e.g. camps2020approximate
• Lemma 5: Informal, linear combination of block-encoded operators, see e.g. gilyen2019quantum
• Lemma 6: Informal, Scaling multiplication of block-encoded operators
• Lemma 7: Matrix inversion, see e.g. gilyen2019quantumchilds2017quantum