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arXiv:2512.19028

Classical and Quantum Algorithms for Topological Invariants of Torus Bundles

Nelson Abdiel Colón Vargas, Carlos Ortiz Marrero

quant-phmath.GTmath.QASubmitted Dec 22, 2025• Announced Dec 23, 2025• 0 citations
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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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31 pages, 3 figures

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