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The Quantum Smooth Label Cover Problem Is Undecidable

Authors Eric Culf , Kieran Mastel , Connor Paddock , Taro Spirig

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ACM Subject Classification
  • Theory of computation → Quantum complexity theory
Keywords
  • Complexity Theory
  • Constraint Satisfaction Problems
  • Hardness of Approximation
  • Quantum Computing

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Abstract

We show that the quantum smooth label cover problem is undecidable and RE-hard. This sharply contrasts the quantum unique label cover problem, which can be decided efficiently by a result of Kempe, Regev, and Toner (FOCS'08). On the other hand, our result aligns with the RE-hardness of the quantum label cover problem, which follows from the celebrated MIP^* = RE result of Ji, Natarajan, Vidick, Wright, and Yuen (ACM'21). Additionally, we show that the quantum oracularized smooth label cover problem is RE-hard. Our second result fits with the alternative quantum unique games conjecture recently proposed by Mousavi and Spirig (ITCS'25) on the RE-hardness of the quantum oracularized unique label cover problem. Our proof techniques include a quantum version of Feige’s reduction from 3SAT to 3SAT5 (STOC'96) for BCS-MIP^*-protocols, which may be of independent interest.

Cite As Get BibTex

Eric Culf, Kieran Mastel, Connor Paddock, and Taro Spirig. The Quantum Smooth Label Cover Problem Is Undecidable. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 71:1-71:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.71

Author Details

Eric Culf
  • Department of Applied Mathematics, University of Waterloo, Canada
  • Institute for Quantum Computing, University of Waterloo, Canada
  • Perimeter Institute for Theoretical Physics, Waterloo, Canada
Kieran Mastel
  • Department of Mathematics and Statistics, University of Ottawa, Canada
  • Department of Pure Mathematics, University of Waterloo, Canada
  • Institute for Quantum Computing, University of Waterloo, Canada
Connor Paddock
  • Department of Computer Science, University of Calgary, Canada
  • Department of Mathematics and Statistics, University of Ottawa, Canada
Taro Spirig
  • Department of Mathematical Sciences and QMATH, University of Copenhagen, Denmark

Funding

  • Culf, Eric: CGS D scholarship from Canada’s NSERC.
  • Mastel, Kieran: CGS D scholarship from Canada’s NSERC.
  • Paddock, Connor: NSERC ALLRP-578455-2022
  • Spirig, Taro: ERC grant QInteract (grant No 101078107), VILLUM FONDEN via the QMATH Centre of Excellence (grant No 10059), and Villum Young Investigator (grant No 37532).

Acknowledgements

The authors thank the anonymous ICALP reviewers for valuable comments and feedback. They also thank Anand Natarajan and Honghao Fu for helpful discussions. TS thanks Hamoon Mousavi for initial discussions that motivated this work.

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