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Two Bases Suffice for QMA ₁-Completeness

Authors Henry Ma , Anand Natarajan

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LIPIcs.ITCS.2026.101.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum complexity theory
  • Hamiltonian complexity
  • Quantum Merlin Arthur (QMA)
  • QMA₁
  • quantum satisfiability problem

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Abstract

We introduce a basis-restricted variant of the Quantum-k-Sat problem, in which each term in the input Hamiltonian is required to be diagonal in either the standard or Hadamard basis. Our main result is that the Quantum-6-Sat  problem with this basis restriction is already  QMA₁-complete, defined with respect to a natural gateset. Our construction is based on the Feynman-Kitaev circuit-to-Hamiltonian construction, with a modified clock encoding that interleaves two clocks in the standard and Hadamard bases. In light of the central role played by CSS codes and the uncertainty principle in the proof of the NLTS theorem of Anshu, Breuckmann, and Nirkhe (STOC '23), we hope that the CSS-like structure of our Hamiltonians will make them useful for progress towards a quantum PCP theorem.

Cite As Get BibTex

Henry Ma and Anand Natarajan. Two Bases Suffice for QMA ₁-Completeness. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 101:1-101:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.101

Author Details

Henry Ma
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Anand Natarajan
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Ma, Henry: HM was supported by the NSF Graduate Research Fellowship Program.
  • Natarajan, Anand: AN was supported by NSF CAREER grant number 2339948.

Acknowledgements

Part of this work was done while both authors were visiting the Simons Institute for the Theory of Computing as part of the 2025 Summer Cluster on Quantum Computing, and the Challenge Institute for Quantum Computation at UC Berkeley. We thank Chinmay Nirkhe for several helpful discussions.

References

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