A (simple) classical algorithm for estimating Betti numbers

Simon Apers1, Sander Gribling2, Sayantan Sen3, and Dániel Szabó1

1Université Paris Cité, CNRS, IRIF, Paris, France
2Tilburg University, The Netherlands
3National University of Singapore, Singapore

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We describe a simple algorithm for estimating the $k$-th normalized Betti number of a simplicial complex over $n$ elements using the path integral Monte Carlo method. For a general simplicial complex, the running time of our algorithm is $n^{O\left(\frac{1}{\sqrt{\gamma}}\log\frac{1}{\varepsilon}\right)}$ with $\gamma$ measuring the spectral gap of the combinatorial Laplacian and $\varepsilon \in (0,1)$ the additive precision. In the case of a clique complex, the running time of our algorithm improves to $\left(n/\lambda_{\max}\right)^{O\left(\frac{1}{\sqrt{\gamma}}\log\frac{1}{\varepsilon}\right)}$ with $\lambda_{\max} \geq k$, where $\lambda_{\max}$ is the maximum eigenvalue of the combinatorial Laplacian. Our algorithm provides a classical benchmark for a line of quantum algorithms for estimating Betti numbers. On clique complexes it matches their running time when, for example, $\gamma \in \Omega(1)$ and $k \in \Omega(n)$.

We describe a simple algorithm for estimating the $k$-th normalized Betti number of a simplicial complex over $n$ elements using the path integral Monte Carlo method. Our algorithm provides a classical benchmark for a line of quantum algorithms for estimating Betti numbers.
While the runtime of naïve classical algorithms is exponential in $k$, the quantum algorithm of Lloyd, Garnerone and Zanardi runs in polynomial time if the spectral gap of the combinatorial Laplacian and the precision are at least inverse polynomial in $n$. This suggests that there can be an exponential quantum advantage for this problem.
Our algorithm further pins down the region where we can expect a quantum advantage: it runs in polynomial time if the spectral gap and the precision are constant. In the special case of clique complexes – that is particularly interesting for applications in Topological Data Analysis – we can go further: for example if $k \in \Omega(n)$ then we can afford inverse polynomial precision (if the gap is constant) or gap $\Omega(1/\log^2 n)$ (if the precision is constant).

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Cited by

[1] Dominic W. Berry, Yuan Su, Casper Gyurik, Robbie King, Joao Basso, Alexander Del Toro Barba, Abhishek Rajput, Nathan Wiebe, Vedran Dunjko, and Ryan Babbush, "Analyzing Prospects for Quantum Advantage in Topological Data Analysis", PRX Quantum 5 1, 010319 (2024).

[2] Massimiliano Incudini, Francesco Martini, and Alessandra Di Pierro, "Toward Useful Quantum Kernels", Advanced Quantum Technologies 2300298 (2024).

[3] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[4] Daniel Leykam and Dimitris G. Angelakis, "Topological data analysis and machine learning", Advances in Physics X 8 1, 2202331 (2023).

[5] Samson Wang, Sam McArdle, and Mario Berta, "Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra", arXiv:2302.01873, (2023).

[6] Ashley Montanaro and Changpeng Shao, "Quantum and classical query complexities of functions of matrices", arXiv:2311.06999, (2023).

[7] Robbie King and Tamara Kohler, "Promise Clique Homology on weighted graphs is $\text{QMA}_1$-hard and contained in $\text{QMA}$", arXiv:2311.17234, (2023).

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