Topological data analysis (TDA) is a fast-growing field that utilizes advanced tools from topology to analyze large-scale data. A central problem in topological data analysis is estimating the so-called Betti numbers of the underlying simplicial complex. While the difficulty of this problem has been established as NP-hard, previous works have showcased appealing quantum speedup. In this article, we provide an alternative method for estimating Betti numbers and normalized Betti numbers of given simplicial complex, based on some recent advances in quantum algorithm, specifically, quantum singular value transformation. Our method can be faster than the best-known classical method for finding Betti numbers, and interestingly, it can also find the Betti numbers of the complement graph to our original one. Comparing to the best known quantum algorithm, our method generally requires lower depth circuit, in trade-off for longer running time. Regarding normalized Betti numbers, our method could match the running time of best-known quantum method in the case of dense simplices.
Topological data analysis (TDA) is a rapidly growing area that applies techniques from algebraic topology to extract robust features from large-scale data. A key task in TDA is the estimation of (normalized) Betti numbers, which capture essential topological invariants. While recent work has led to quantum algorithms for this problem, we explore an alternative direction: combining classical and quantum resources to estimate the Betti numbers of a simplicial complex more efficiently. Assuming the classical description of a simplicial complex, that is, its set of vertices and edges, we propose a hybrid quantum-classical algorithm. The classical component enumerates all simplices, and this combinatorial structure is subsequently processed by a quantum algorithm to estimate the Betti numbers. We analyze the performance of our approach and identify regimes where it potentially achieves polynomial to exponential speedups over existing quantum methods, at the trade-off of using more ancilla qubits. We further demonstrate the utility of normalized Betti numbers in concrete applications, highlighting the broader potential of hybrid quantum algorithms in topological data analysis.
Topological data analysis (TDA) is an emergent field of data analysis. The critical step of TDA is computing the persistent Betti numbers. Existing classical algorithms for TDA are limited if we want to learn from high-dimensional topological features because the number of high-dimensional simplices grows exponentially in the size of the data. In the context of quantum computation, it has been previously shown that there exists an efficient quantum algorithm for estimating the Betti numbers even in high dimensions. However, the Betti numbers are less general than the persistent Betti numbers, and there have been no quantum algorithms that can estimate the persistent Betti numbers of arbitrary dimensions. This paper shows the first quantum algorithm that can estimate the (normalized) persistent Betti numbers of arbitrary dimensions. Our algorithm is efficient for simplicial complexes such as the Vietoris-Rips complex and demonstrates exponential speedup over the known classical algorithms.
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 (
∂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.
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 O ( log ( | S r K | | S r + 1 K | ) ϵ 2 log ( log | S r K | ) ( r log | S r K | ) ) , 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 O ( log ( | S r K | | S r <mm
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=Cl(G), we first prove that, for fixed
r and prime
p, deciding whether
Hr(K,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
Hr(K,Z) or
Hr−1(K,Z) contains
p-torsion for some
p∈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.
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.