Towards a complexity-theoretic dichotomy for TQFT invariants
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| Format: | Preprint |
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2025
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| _version_ | 1866909842857263104 |
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| author | Bridges, Nicolas Samperton, Eric |
| author_facet | Bridges, Nicolas Samperton, Eric |
| contents | We show that for any fixed $(2+1)$-dimensional TQFT over $\mathbb{C}$ of either Turaev-Viro-Barrett-Westbury or Reshetikhin-Turaev type, the problem of (exactly) computing its invariants on closed 3-manifolds is either solvable in polynomial time, or else it is $\#\mathsf{P}$-hard to (exactly) contract certain tensors that are built from the TQFT's fusion category. Our proof is an application of a dichotomy result of Cai and Chen [J. ACM, 2017] concerning weighted constraint satisfaction problems over $\mathbb{C}$. We leave for future work the issue of reinterpreting the conditions of Cai and Chen that distinguish between the two cases (i.e. $\#\mathsf{P}$-hard tensor contractions vs. polynomial time invariants) in terms of fusion categories. We expect that with more effort, our reduction can be improved so that one gets a dichotomy directly for TQFTs' invariants of 3-manifolds rather than more general tensors built from the TQFT's fusion category. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02945 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards a complexity-theoretic dichotomy for TQFT invariants Bridges, Nicolas Samperton, Eric Quantum Algebra Computational Complexity Geometric Topology Quantum Physics We show that for any fixed $(2+1)$-dimensional TQFT over $\mathbb{C}$ of either Turaev-Viro-Barrett-Westbury or Reshetikhin-Turaev type, the problem of (exactly) computing its invariants on closed 3-manifolds is either solvable in polynomial time, or else it is $\#\mathsf{P}$-hard to (exactly) contract certain tensors that are built from the TQFT's fusion category. Our proof is an application of a dichotomy result of Cai and Chen [J. ACM, 2017] concerning weighted constraint satisfaction problems over $\mathbb{C}$. We leave for future work the issue of reinterpreting the conditions of Cai and Chen that distinguish between the two cases (i.e. $\#\mathsf{P}$-hard tensor contractions vs. polynomial time invariants) in terms of fusion categories. We expect that with more effort, our reduction can be improved so that one gets a dichotomy directly for TQFTs' invariants of 3-manifolds rather than more general tensors built from the TQFT's fusion category. |
| title | Towards a complexity-theoretic dichotomy for TQFT invariants |
| topic | Quantum Algebra Computational Complexity Geometric Topology Quantum Physics |
| url | https://arxiv.org/abs/2503.02945 |