Classical and Quantum Algorithms for Topological Invariants of Torus Bundles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909973307457536 |
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| author | Vargas, Nelson Abdiel Colón Marrero, Carlos Ortiz |
| author_facet | Vargas, Nelson Abdiel Colón Marrero, Carlos Ortiz |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_19028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classical and Quantum Algorithms for Topological Invariants of Torus Bundles Vargas, Nelson Abdiel Colón Marrero, Carlos Ortiz Quantum Physics Geometric Topology Quantum Algebra 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. |
| title | Classical and Quantum Algorithms for Topological Invariants of Torus Bundles |
| topic | Quantum Physics Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2512.19028 |