Classical and Quantum Algorithms for Topological Invariants of Torus Bundles

Nelson Abdiel Colón Vargas; Carlos Ortiz Marrero

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arXiv · 2512.19028

Classical and Quantum Algorithms for Topological Invariants of Torus Bundles

Nelson Abdiel Colón Vargas·Carlos Ortiz Marrero

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Abstract

Computing topological invariants of 3-manifolds is generally intractable, yet specialized algebraic structures can enable efficient algorithms. For Witten-Reshetikhin-Turaev (WRT) invariants of torus bundles, we exploit the non-commutative torus structure to embed the skein algebra of the closed torus into its symmetric subalgebra at roots of unity. This yields a fixed $N^2$-dimensional representation that supports polynomial-time classical computation with $O(N^2)$ space, and a quantum algorithm using only $O(\log N)$ qubits -- an exponential space advantage. We further prove that extracting individual expansion coefficients is #P-complete, yet there is a quantum algorithm that can efficiently approximate these coefficients for a non-negligible fraction of configurations.

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Nelson Abdiel Colón Vargas, Carlos Ortiz Marrero. 2025-12-22. Classical and Quantum Algorithms for Topological Invariants of Torus Bundles. https://arxiv.org/abs/2512.19028

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