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arXiv:2607.03278

Complexity of Normalized Persistence Problems for Topological Data Analysis and Local Hamiltonians

Dominic Lowe, M. S. Kim, Roberto Bondesan, Ryu Hayakawa

quant-phcs.CCcs.LGSubmitted Jul 3, 2026• Announced Jul 6, 2026• 0 citations• Revised Sep 29, 2026 (v2)
v1 · Jul 3, 2026v2 · Sep 29, 2026
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Abstract

Topological data analysis (TDA) is a machine learning technique that uses topology to extract patterns from data and has shown the potential to exhibit quantum advantage. A key concept in TDA is persistent homology, which measures the robustness of topological information at different lengthscales. In this paper, we introduce and study the problem of normalized persistence, a practically motivated and easily interpretable version of persistent homology that counts the fraction of holes that persist at different lengthscales. We prove that a variant of normalized persistence is $\mathsf{DQC}_1$-hard and contained in $\mathsf{BQP}$, giving evidence of an exponential quantum speedup for TDA under the standard assumption that $\mathsf{DQC}_1 \not\subseteq \mathsf{BPP}$. These are the first $\mathsf{DQC}_1$-hardness results for clique complexes, making them directly applicable to TDA instances. We also find a close connection between normalized persistence and the complexity of estimating spectral quantities in the low-energy subspace of local Hamiltonians. We study a family of such problems, including a low-energy normalized subtrace and spectral density. We show that these are $\mathsf{DQC}_1$-hard for $O(1)$-local Hamiltonians, strengthening previous results that required log-local interactions. We also introduce a variant of $\mathsf{DQC}_1$ with perfect completeness ($\mathsf{SDQC}_1$) to characterize the hardness of problems normalized by an exact kernel. This includes normalized persistence for $O(1)$-local Hamiltonians, which we show is $\mathsf{SDQC}_1$-hard.

Comments
45 pages, 5 figures

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Topological data analysis (TDA) aims to extract noise-robust features from a dataset by examining the number and persistence of holes in its topology. We provide an efficient quantum algorithm for a computational problem closely related to a core task in TDA—determining whether a given hole persists across different length scales. Further, we prove the problem itself is B Q P 1 -hard, implying that a classical solution is extremely unlikely; this stands in contrast to all previous quantum approaches to TDA, where the problems were also intractable for quantum computers, or where a rigorous proof of classical hardness still remains open. This result implies an exponential quantum speedup for this problem under standard complexity-theoretic assumptions. Our approach relies on encoding the persistence of a hole in a variant of the guided sparse Hamiltonian problem, where the guiding state is constructed from a harmonic representative of the hole.
Quantum Topological Data Analysis Beyond Betti Numbers: Complexity Hardness &\&& An Algorithm for Torsion Witness
Recent advances have revealed an interplay between quantum computing and topological data analysis (TDA). Most quantum TDA has focused on Betti numbers, which characterize the connectivity and ``holes'' of a dataset. Homology, however, contains additional information in the form of torsion: a nontrivial cycle can become trivial after being repeated finitely, revealing global constraints on how cycles combine and wrap around one another. Beyond applications in biomolecular studies, torsion appears in physical settings including homological quantum rotor codes, discrete charges, and gauge sectors. We study torsion from both classical and quantum perspectives. Given a graph GGG and its clique complex K=Cl(G)K = \mathrm{Cl}(G)K=Cl(G), we first prove that, for fixed rrr and prime ppp, deciding whether Hr(K,Z)H_r(K,\mathbb{Z})Hr​(K,Z) contains ppp-torsion is NP-hard. As a corollary, when a homological rotor code is specified by GGG, deciding whether the code has a finite-dimensional logical sector of a given order is NP-hard. We discuss related problems, including the Bockstein homomorphism, Smith normal form, lattice saturation, and cohomology. Second, for a finite set of primes PPP, we develop a quantum algorithm that serves as a one-sided torsion witness. For fixed rrr, it outputs WITNESS or INCONCLUSIVE, i.e. WITNESS certifies that either Hr(K,Z)H_r(K,\mathbb{Z})Hr​(K,Z) or Hr−1(K,Z)H_{r-1}(K,\mathbb{Z})Hr−1​(K,Z) contains ppp-torsion for some p∈Pp\in Pp∈P while INCONCLUSIVE makes no claim about its presence or absence. We identify a regime in which the algorithm achieves a near-quadratic quantum speedup over the corresponding classical algorithm under the same input model. Our NP-hardness result complements recent hardness results for estimating Betti numbers, adding a complexity-theoretic perspective to quantum TDA. Together, these results demonstrate that integral homology, beyond its Betti numbers, can be computationally challenging.
Gapped Clique Homology on weighted graphs is QMA1\text{QMA}_1QMA1​-hard and contained in QMA\text{QMA}QMA
We study the complexity of a classic problem in computational topology, the homology problem: given a description of some space XXX and an integer kkk, decide if XXX contains a kkk-dimensional hole. The setting and statement of the homology problem are completely classical, yet we find that the complexity is characterized by quantum complexity classes. Our result can be seen as an aspect of a connection between homology and supersymmetric quantum mechanics. We consider clique complexes, motivated by the practical application of topological data analysis (TDA). The clique complex of a graph is the simplicial complex formed by declaring every k+1k+1k+1-clique in the graph to be a kkk-simplex. Our main result is that deciding whether the clique complex of a weighted graph has a hole or not, given a suitable promise on the gap, is QMA1\text{QMA}_1QMA1​-hard and contained in QMA\text{QMA}QMA. Our main innovation is a technique to lower bound the eigenvalues of the combinatorial Laplacian operator. For this, we invoke a tool from algebraic topology known as \emph{spectral sequences}. In particular, we exploit a connection between spectral sequences and Hodge theory. Spectral sequences will play a role analogous to perturbation theory for combinatorial Laplacians. In addition, we develop the simplicial surgery technique used in prior work. Our result provides some suggestion that the quantum TDA algorithm \emph{cannot} be dequantized. More broadly, we hope that our results will open up new possibilities for quantum advantage in topological data analysis.
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Gapped Clique Homology on Weighted Graphs Is QMA1{\text{QMA}_1}QMA1​-Hard and Contained in QMA
Abstract. We study the complexity of a classic problem in computational topology, the homology problem: given a description of some space [Formula: see text] and an integer [Formula: see text], decide if [Formula: see text] contains a [Formula: see text]-dimensional hole. The setting and statement of the homology problem are completely classical, yet we find that the complexity is characterized by quantum complexity classes. Our result can be seen as an aspect of a connection between homology and supersymmetric quantum mechanics established by Witten (1982). We consider clique complexes, motivated by the practical application of topological data analysis (TDA). The clique complex of a graph is the simplicial complex formed by declaring every [Formula: see text]-clique in the graph to be a [Formula: see text]-simplex. Our main result is that deciding whether the clique complex of a weighted graph has a hole or not, given a suitable promise on the gap, is [Formula: see text]-hard and contained in QMA. Our main innovation is a technique to lower bound the eigenvalues of the combinatorial Laplacian operator. For this, we invoke a tool from algebraic topology known as spectral sequences. In particular, we exploit a connection between spectral sequences and Hodge theory outlined by Forman (1994). Spectral sequences will play a role analogous to perturbation theory for combinatorial Laplacians. In addition, we develop the simplicial surgery technique used in prior work by Crichigno and Kohler (2022). Our result provides some suggestion that the quantum TDA algorithm due to Lloyd, Garnerone, and Zanardi (2016) cannot be dequantized, although proving this would require studying the complexity of a different problem related to the combinatorial Laplacian of clique complexes.
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The computational complexity of simplicial homology has been studied since the question was raised by Kaibel and Pfetsch, and clique homology was later shown to be QMA1\mathsf{QMA}_1QMA1​-hard by Crichigno and Kohler. King and Kohler subsequently introduced the weighted, spectrally gapped version considered here and developed a reduction intended to establish the same hardness result in this setting. We give a complete proof of the hard direction. The earlier King-Kohler argument contains a gap in the analysis of the nontrivially folded local gadgets, so the required topological and spectral properties have to be re-established on the clique complexes actually produced by the reduction. This is not a local repair of the published proof: the argument requires a new analysis of the local topology, the filtered cochain complexes, and the interaction of many overlapping gadgets. Our proof uses exact finite certificates for the local combinatorial data and relative homology to determine the effect of each gadget on the encoded qubit space. We then prove the finite-dimensional filtered-Hodge statement needed to recover the corresponding rank-one low-energy penalty and derive a quantitative global estimate that transfers the source-Hamiltonian promise gap to the final weighted Hodge Laplacian. Consequently, for fixed efficiently computable k(n)k(n)k(n) and inverse-polynomial γ(n)γ(n)γ(n), weighted gapped clique homology is QMA1(G)\mathsf{QMA}_1(\mathcal G)QMA1​(G)-hard. The previously known QMA\mathsf{QMA}QMA-containment argument is logically independent of the gap and remains unaffected.