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arXiv:2609.28112

Quantum Topological Data Analysis Beyond Betti Numbers: Complexity Hardness $\&$ An Algorithm for Torsion Witness

Nhat A. Nghiem, Dominic W. Berry, Trung V. Phan

quant-phSubmitted Sep 23, 2026• Announced Sep 24, 2026• 0 citations• Revised Oct 8, 2026 (v2)
v1 · Sep 23, 2026v2 · Oct 8, 2026
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Abstract

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 $G$ and its clique complex $K = \mathrm{Cl}(G)$, we first prove that, for fixed $r$ and prime $p$, deciding whether $H_r(K,\mathbb{Z})$ contains $p$-torsion is NP-hard. As a corollary, when a homological rotor code is specified by $G$, 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 $P$, we develop a quantum algorithm that serves as a one-sided torsion witness. For fixed $r$, it outputs WITNESS or INCONCLUSIVE, i.e. WITNESS certifies that either $H_r(K,\mathbb{Z})$ or $H_{r-1}(K,\mathbb{Z})$ contains $p$-torsion for some $p\in P$ 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.

Related papers

Quantum Advantage in Topological Data Analysis via Mayer Homology

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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.

quant-pharXiv:2609.28058•Sep 24, 2026•0 citations

Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis

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Quantum algorithms for topological data analysis (TDA) seem to provide an exponential advantage over the best classical approach while remaining immune to dequantization procedures and the data-loading problem.In this paper, we give complexity-theoretic evidence that the central task of TDA-estimating Betti numbers-is intractable even for quantum computers.Specifically, we prove that the problem of computing Betti numbers exactly is #P-hard, while the problem of approximating Betti numbers up to multiplicative error is NP-hard.Moreover, both problems retain their hardness if restricted to the regime where quantum algorithms for TDA perform best.Because quantum computers are not expected to solve #P-hard or NP-hard problems in subexponential time, our results imply that quantum algorithms for TDA offer only a polynomial advantage in the worst case.We support our claim by showing that the seminal quantum algorithm for TDA developed by Lloyd, Garnerone, and Zanardi achieves a quadratic speedup over the best-known classical approach in asymptotically almost all cases.Finally, we argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices rather than as a list of vertices and edges.

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Alexander Schmidhuber, Seth Lloyd

Quantum algorithms for topological data analysis (TDA) seem to provide an exponential advantage over the best classical approach while remaining immune to dequantization procedures and the data-loading problem. In this paper, we give complexity-theoretic evidence that the central task of TDA -- estimating Betti numbers -- is intractable even for quantum computers. Specifically, we prove that the problem of computing Betti numbers exactly is #P-hard, while the problem of approximating Betti numbers up to multiplicative error is NP-hard. Moreover, both problems retain their hardness if restricted to the regime where quantum algorithms for TDA perform best. Because quantum computers are not expected to solve #P-hard or NP-hard problems in subexponential time, our results imply that quantum algorithms for TDA offer only a polynomial advantage in the worst case. We support our claim by showing that the seminal quantum algorithm for TDA developed by Lloyd, Garnerone and Zanardi achieves a quadratic speedup over the best known classical approach in asymptotically almost all cases. Finally, we argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices rather than as a list of vertices and edges.

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Average-case hardness of Betti number estimation

Sergii Strelchuk, Sathyawageeswar Subramanian, Adam Wesołowski

We establish the average-case hardness of Betti number estimation on random clique complexes via a reduction from the planted clique problem. We further show that our reduction implies a series of hardness results for many problems in both classical and quantum Topological Data Analysis (qTDA). Under the classical planted clique conjecture, no randomized polynomial-time Betti number estimator achieves additive error below $\tfrac12$ with constant advantage. Under a new quantum planted clique conjecture that we introduce, the same conclusion holds for quantum polynomial-time algorithms. We also obtain related conditional hardness results for homology vanishing, additive approximations with larger error tolerances, preparation of simplex and harmonic states, cycle recovery, and counting eigenvalues at low energy. Our reduction clarifies the structural requirements for quantum advantage in TDA and provides a new lens to investigate the classical and quantum complexity of related problems.

quant-pharXiv:2609.12777•Sep 14, 2026•0 citations
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.
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 NNN-nilpotent boundary operators (∂N=0\partial^N =0∂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.
Quantum algorithms for topological data analysis (TDA) seem to provide an exponential advantage over the best classical approach while remaining immune to dequantization procedures and the data-loading problem.In this paper, we give complexity-theoretic evidence that the central task of TDA-estimating Betti numbers-is intractable even for quantum computers.Specifically, we prove that the problem of computing Betti numbers exactly is #P-hard, while the problem of approximating Betti numbers up to multiplicative error is NP-hard.Moreover, both problems retain their hardness if restricted to the regime where quantum algorithms for TDA perform best.Because quantum computers are not expected to solve #P-hard or NP-hard problems in subexponential time, our results imply that quantum algorithms for TDA offer only a polynomial advantage in the worst case.We support our claim by showing that the seminal quantum algorithm for TDA developed by Lloyd, Garnerone, and Zanardi achieves a quadratic speedup over the best-known classical approach in asymptotically almost all cases.Finally, we argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices rather than as a list of vertices and edges.
Betti numbers are central to topological data analysis but fail to capture torsion in integral homology. Torsion can distinguish topological structures with identical Betti numbers, and examples from classical topological data analysis show that it can affect the topology inferred from data. Related torsion-sensitive quantities are also known to have practical applications in graph-based machine learning, including biomedical applications. We investigate how quantum spectral methods can access this additional information. Log pseudodeterminants, obtained by summing the logarithms of the nonzero boundary-Laplacian eigenvalues, summarize spectral information about a simplicial complex. Under explicit topological assumptions, an alternating combination of these quantities gives the size of a higher critical group, the higher-dimensional analogue of a graph sandpile group. With stronger certification, a related construction recovers the size of the torsion subgroup in homology itself. This is, to our knowledge, the first provably correct quantum estimator of integral-homology torsion order. We give a new quantum algorithm for estimating these quantities, with explicit dependence on how the input is queried, the required accuracy, and the spectral gaps. Exact spectra for balanced multipartite clique complexes provide favorable gaps, and a related construction gives an explicit torsion-bearing benchmark. With compact query access, the algorithm need not construct the exponentially large boundary matrices explicitly. For a restricted torsion-bearing family, a tailored amplitude-estimation routine uses quadratically fewer queries than any randomized classical algorithm given comparable access when estimating the log pseudodeterminant. Estimating the full, unnormalized quantity to fixed additive error still costs in proportion to the candidate-space dimension DDD.
Torsion detection in clique complexes is conditionally QMA1QMA_1QMA1​-hard
Quantum algorithms for topological data analysis compute Betti numbers, the ranks of the homology groups of a simplicial complex, which can be read off from the kernel of a combinatorial Laplacian. Deciding whether a Betti number of a clique complex is nonzero is QMA1QMA_1QMA1​-hard, and remains so under a spectral gap promise on vertex-weighted graphs. Integral homology, however, contains information inaccessible to the Laplacian spectrum. A new part that appears in integral homology is \emph{torsion}: cycles that become boundaries only after being traversed several times, as in a projective plane or a Klein bottle. We provide a simple reduction that allows us to establish the first hardness results for the problem of detecting torsion. We attach to an arbitrary clique complex a fixed 313131-vertex triangulation of the projective plane. The kkkth mod-222 Betti number of the input then reappears as 222-torsion two degrees up, while all rational homology disappears and every combinatorial Laplacian acquires a constant spectral gap. We conclude that detecting torsion in clique complexes of \emph{unweighted} graphs is NPNPNP-hard, even under a constant gap promise, and that it is QMA1QMA_1QMA1​-hard if mod-222 clique homology is.
Quantum algorithms for topological data analysis (TDA) seem to provide an exponential advantage over the best classical approach while remaining immune to dequantization procedures and the data-loading problem. In this paper, we give complexity-theoretic evidence that the central task of TDA -- estimating Betti numbers -- is intractable even for quantum computers. Specifically, we prove that the problem of computing Betti numbers exactly is #P-hard, while the problem of approximating Betti numbers up to multiplicative error is NP-hard. Moreover, both problems retain their hardness if restricted to the regime where quantum algorithms for TDA perform best. Because quantum computers are not expected to solve #P-hard or NP-hard problems in subexponential time, our results imply that quantum algorithms for TDA offer only a polynomial advantage in the worst case. We support our claim by showing that the seminal quantum algorithm for TDA developed by Lloyd, Garnerone and Zanardi achieves a quadratic speedup over the best known classical approach in asymptotically almost all cases. Finally, we argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices rather than as a list of vertices and edges.
We establish the average-case hardness of Betti number estimation on random clique complexes via a reduction from the planted clique problem. We further show that our reduction implies a series of hardness results for many problems in both classical and quantum Topological Data Analysis (qTDA). Under the classical planted clique conjecture, no randomized polynomial-time Betti number estimator achieves additive error below 12\tfrac1221​ with constant advantage. Under a new quantum planted clique conjecture that we introduce, the same conclusion holds for quantum polynomial-time algorithms. We also obtain related conditional hardness results for homology vanishing, additive approximations with larger error tolerances, preparation of simplex and harmonic states, cycle recovery, and counting eigenvalues at low energy. Our reduction clarifies the structural requirements for quantum advantage in TDA and provides a new lens to investigate the classical and quantum complexity of related problems.