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Towards a Universal Gateset for QMA₁

Author Dorian Rudolph

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LIPIcs.MFCS.2026.98.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Quantum complexity theory
Keywords
  • QMA with perfect completeness
  • quantum satisfiability
  • universal gatesets
  • local Hamiltonian
  • clique homology

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Abstract

QMA₁ is QMA with perfect completeness: in the YES-case the verifier must accept some proof with probability exactly 1. Whether QMA₁ = QMA remains open, and even the definition of QMA₁ has a gateset issue, since Solovay-Kitaev only gives approximate synthesis and may destroy perfect completeness. For a gateset 𝒢, we write QMA₁^𝒢 for QMA₁ restricted to verifiers using gates from 𝒢.
Using the gatesets 𝒢_{2^k} of Amy et al. (RC 2024), we prove that QMA₁^𝒢 ⊆ QMA₁^{𝒢_{2^k}} for every finite gateset 𝒢 whose entries lie in the cyclotomic field ℚ(ζ_{2^k}), ζ_{2^k} = e^{2πi/2^k}. For BQP₁ (aka coRQP), the rational gateset 𝒢₂ already suffices for all these fields. We also give complete problems for the resulting classes: quantum 𝓁-SAT over ℚ(ζ_{2^k}) is complete for QMA₁^{𝒢_{2^k}} for all 𝓁 ≥ 4, and also for 𝓁 = 3 when k ≥ 3.
The main technical tool is to use linear combinations of unitaries and postselection to exactly apply operators whose entries lie in the relevant field, and then use oblivious amplitude amplification to make the postselection failure probability negligible. This also gives an exact kernel test for Hamiltonians. As a consequence, we prove the first QMA₁-complete 2-local Hamiltonian problem: for k ≥ 3, deciding whether a 2-local Hamiltonian H over ℚ(ζ_{2^k}) has σ₁(H) = 0 or σ₁(H) ≥ 1/poly is complete for QMA₁^{𝒢_{2^k}}. The same ideas extend to sparse Hamiltonians and yield the first QMA₁(2)-complete Hamiltonian problem.
Finally, we apply the gateset framework to clique homology. We prove that the Gapped Clique Homology problem on weighted graphs defined by King and Kohler (FOCS 2024) is QMA₁^𝒢₂-complete, and the Clique Homology problem (Kaibel and Pfetsch, 2002) without promise gap is PSPACE-complete, resolving a conjecture of Crichigno and Kohler (Nat. Commun. 2024).

Cite As Get BibTex

Dorian Rudolph. Towards a Universal Gateset for QMA₁. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 98:1-98:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.98

Author Details

Dorian Rudolph
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany

Funding

I was supported by the DFG under grant number 432788384.

Acknowledgements

I thank Tamara Kohler and Marcos Crichigno for helpful discussions, and especially Tamara Kohler for helping me understand Reference [King and Kohler, 2024]. I also thank my supervisor Sevag Gharibian for his support and many discussions.

Supplementary Materials

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