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Towards a Complexity-Theoretic Dichotomy for TQFT Invariants

Authors Nicolas Bridges , Eric Samperton

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ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Quantum complexity theory
Keywords
  • Complexity
  • topological quantum field theory
  • dichotomy theorems
  • constraint satisfaction problems
  • tensor categories

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Abstract

We show that for any fixed (2+1)-dimensional TQFT over 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 #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 ℂ. We leave for future work the issue of reinterpreting the conditions of Cai and Chen that distinguish between the two cases (i.e. #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 TQFTs' fusion categories.

Cite As Get BibTex

Nicolas Bridges and Eric Samperton. Towards a Complexity-Theoretic Dichotomy for TQFT Invariants. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.TQC.2025.5

Author Details

Nicolas Bridges
  • Department of Mathematics, Purdue University, West Lafayette, IN, USA
Eric Samperton
  • Departments of Mathematics and Computer Science, Purdue University, West Lafayette, IN, USA

Funding

Both authors supported in part by NSF CCF grant 2330130.

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