Classical and Quantum Algorithms for Topological Invariants of Torus Bundles

• URL: http://arxiv.org/abs/2512.19028v1
• Date: Mon, 22 Dec 2025 04:58:14 GMT
• Title: Classical and Quantum Algorithms for Topological Invariants of Torus Bundles
• Authors: Nelson Abdiel Colón Vargas, Carlos Ortiz Marrero,
• Abstract summary: We exploit the non-commutative torus structure to embed the skein algebra of the closed torus into its symmetric subalgebra at roots of unity.<n>We 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.
• Score: 0.42970700836450476
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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