Complexity of Supersymmetric Systems and the Cohomology Problem

Chris Cade1 and P. Marcos Crichigno2

1QuSoft & CWI, Amsterdam, the Netherlands.
2Blackett Laboratory, Imperial College Prince Consort Rd., London, SW7 2AZ, U.K.

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Abstract

We consider the complexity of the local Hamiltonian problem in the context of fermionic Hamiltonians with $\mathcal N=2 $ supersymmetry and show that the problem remains $\mathsf{QMA}$-complete. Our main motivation for studying this is the well-known fact that the ground state energy of a supersymmetric system is exactly zero if and only if a certain cohomology group is nontrivial. This opens the door to bringing the tools of Hamiltonian complexity to study the computational complexity of a large number of algorithmic problems that arise in homological algebra, including problems in algebraic topology, algebraic geometry, and group theory. We take the first steps in this direction by introducing the $k$-local Cohomology problem and showing that it is $\mathsf{QMA}_1$-hard and, for a large class of instances, is contained in $\mathsf{QMA}$. We then consider the complexity of estimating normalized Betti numbers and show that this problem is hard for the quantum complexity class $\mathsf{DQC}1$, and for a large class of instances is contained in $\mathsf{BQP}$. In light of these results, we argue that it is natural to frame many of these homological problems in terms of finding ground states of supersymmetric fermionic systems. As an illustration of this perspective we discuss in some detail the model of Fendley, Schoutens, and de Boer consisting of hard-core fermions on a graph, whose ground state structure encodes $l$-dimensional holes in the independence complex of the graph. This offers a new perspective on existing quantum algorithms for topological data analysis and suggests new ones.

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[1] 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).

[2] Ismail Yunus Akhalwaya, Shashanka Ubaru, Kenneth L. Clarkson, Mark S. Squillante, Vishnu Jejjala, Yang-Hui He, Kugendran Naidoo, Vasileios Kalantzis, and Lior Horesh, "Topological data analysis on noisy quantum computers", arXiv:2209.09371, (2022).

[3] 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).

[4] Alexander Schmidhuber and Seth Lloyd, "Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis", PRX Quantum 4 4, 040349 (2023).

[5] Sergey Bravyi, Anirban Chowdhury, David Gosset, Vojtěch Havlíček, and Guanyu Zhu, "Quantum Complexity of the Kronecker Coefficients", PRX Quantum 5 1, 010329 (2024).

[6] Sam McArdle, András Gilyén, and Mario Berta, "A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits", arXiv:2209.12887, (2022).

[7] Casper Gyurik, Chris Cade, and Vedran Dunjko, "Towards quantum advantage via topological data analysis", arXiv:2005.02607, (2020).

[8] Simon Apers, Sander Gribling, Sayantan Sen, and Dániel Szabó, "A (simple) classical algorithm for estimating Betti numbers", Quantum 7, 1202 (2023).

[9] Ismail Yunus Akhalwaya, Yang-Hui He, Lior Horesh, Vishnu Jejjala, William Kirby, Kugendran Naidoo, and Shashanka Ubaru, "Representation of the Fermionic Boundary Operator", arXiv:2201.11510, (2022).

[10] Casper Gyurik and Vedran Dunjko, "Exponential separations between classical and quantum learners", arXiv:2306.16028, (2023).

[11] Marcos Crichigno and Tamara Kohler, "Clique Homology is QMA1-hard", arXiv:2209.11793, (2022).

[12] Casper Gyurik, Chris Cade, and Vedran Dunjko, "Towards quantum advantage via topological data analysis", Quantum 6, 855 (2022).

[13] Sevag Gharibian, "The 7 faces of quantum NP", arXiv:2310.18010, (2023).

[14] Ismail Yunus Akhalwaya, Yang-Hui He, Lior Horesh, Vishnu Jejjala, William Kirby, Kugendran Naidoo, and Shashanka Ubaru, "Representation of the fermionic boundary operator", Physical Review A 106 2, 022407 (2022).

[15] 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).

[16] Stefano Scali, Chukwudubem Umeano, and Oleksandr Kyriienko, "The topology of data hides in quantum thermal states", arXiv:2402.15633, (2024).

[17] Nhat A. Nghiem, "Alternative Method for Estimating Betti Numbers", arXiv:2403.04686, (2024).

[18] Ryu Hayakawa, Kuo-Chin Chen, and Min-Hsiu Hsieh, "Quantum Walks on Simplicial Complexes and Harmonic Homology: Application to Topological Data Analysis with Superpolynomial Speedups", arXiv:2404.15407, (2024).

[19] Roberto Zucchini, "A new quantum computational set-up for algebraic topology via simplicial sets", arXiv:2309.11304, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-05-26 09:21:44). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2024-05-26 09:21:43).