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On the Pure Quantum Polynomial Hierarchy and Quantified Hamiltonian Complexity

Authors Sabee Grewal , Dorian Rudolph

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LIPIcs.ICALP.2026.103.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Problems, reductions and completeness
Keywords
  • quantum complexity theory
  • quantum polynomial hierarchy
  • pure quantum polynomial hierarchy
  • QPH
  • QMA(2)
  • quantum proof systems
  • interactive proofs
  • quantified Hamiltonian complexity
  • local Hamiltonian problem
  • sparse Hamiltonians
  • disentanglers

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Abstract

We prove several new results concerning the pure quantum polynomial hierarchy pureQPH. First, we show that QMA(2) ⊆ pureQΣ_2, i.e., two unentangled existential provers can be simulated by competing existential and universal provers. We further prove that pureQΣ_2 ⊆ QΣ_3 ⊆ NEXP. Second, we give an error reduction result for pureQPH, and, as a consequence, prove that pureQPH = QPH. A key ingredient in this result is an improved dimension-independent disentangler. Finally, we initiate the study of quantified Hamiltonian complexity, the quantum analogue of quantified Boolean formulae. We prove that the quantified pure sparse Hamiltonian problem is pureQΣ_i-complete. By contrast, other natural variants (pure/local, mixed/local, and mixed/sparse) admit nontrivial containments but fail to be complete under known techniques. For example, we show that the ∃∀-mixed local Hamiltonian problem lies in NP^QMA ∩ coNP^QMA.

Cite As Get BibTex

Sabee Grewal and Dorian Rudolph. On the Pure Quantum Polynomial Hierarchy and Quantified Hamiltonian Complexity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 103:1-103:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.103

Author Details

Sabee Grewal
  • The University of Texas at Austin, TX, USA
Dorian Rudolph
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany

Funding

  • Grewal, Sabee: SG was supported in part by an IBM PhD Fellowship.
  • Rudolph, Dorian: DR was supported in part by the DFG under grant number 432788384.

Acknowledgements

We thank Justin Yirka, Sevag Gharibian, and William Kretschmer for helpful conversations. This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing.

References

  1. Scott Aaronson and Alex Arkhipov. The Computational Complexity of Linear Optics. In Forty-Third Annual ACM Symposium on Theory of Computing, STOC '11, pages 333-342. ACM, 2011. URL: https://doi.org/10.1145/1993636.1993682.
  2. Avantika Agarwal, Sevag Gharibian, Venkata Koppula, and Dorian Rudolph. Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024), volume 306, pages 7:1-7:17, 2024. URL: https://doi.org/10.4230/LIPIcs.MFCS.2024.7.
  3. Avantika Agarwal and Srijita Kundu. A Cautionary Note on Quantum Oracles, 2025. URL: https://doi.org/10.48550/arXiv.2504.19470.
  4. Dorit Aharonov and Amnon Ta‐Shma. Adiabatic Quantum State Generation. SIAM Journal on Computing, 37(1):47-82, 2007. URL: https://doi.org/10.1137/060648829.
  5. Kartik Anand, Kabgyun Jeong, and Junseo Lee. Collapses in quantum-classical probabilistically checkable proofs and the quantum polynomial hierarchy, 2025. URL: https://doi.org/10.48550/arXiv.2506.19792.
  6. Sanjeev Arora and Boaz Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009. URL: https://doi.org/10.1017/CBO9780511804090.
  7. Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani. On the Complexity and Verification of Quantum Random Circuit Sampling. Nature Physics, 15(2):159-163, 2019. URL: https://doi.org/10.1038/s41567-018-0318-2.
  8. Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd. Classical Simulation of Commuting Quantum Computations Implies Collapse of the Polynomial Hierarchy. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467(2126):459-472, 2010. URL: https://doi.org/10.1098/rspa.2010.0301.
  9. Anne Broadbent and Alex Bredariol Grilo. Qma-hardness of consistency of local density matrices with applications to quantum zero-knowledge. SIAM Journal on Computing, 51(4):1400-1450, August 2022. URL: https://doi.org/10.1137/21m140729x.
  10. Harry Buhrman, Richard Cleve, John Watrous, and Ronald De Wolf. Quantum Fingerprinting. Physical Review Letters, 87(16):167902, 2001. URL: https://doi.org/10.1103/PhysRevLett.87.167902.
  11. C. Carathéodory. Über den variabilitätsbereich der fourier’schen konstanten von positiven harmonischen funktionen. Rendiconti del Circolo Matematico di Palermo (1884-1940), 32(1):193-217, 1911. URL: https://doi.org/10.1007/BF03014795.
  12. Andre Chailloux and Or Sattath. The Complexity of the Separable Hamiltonian Problem. In Proceedings of the 2012 IEEE Conference on Computational Complexity, CCC '12, pages 32-41, 2012. URL: https://doi.org/10.1109/CCC.2012.42.
  13. Chirag Falor, Shu Ge, and Anand Natarajan. A Collapsible Polynomial Hierarchy for Promise Problems, 2023. URL: https://doi.org/10.48550/arXiv.2311.12228.
  14. Merrick Furst, James B. Saxe, and Michael Sipser. Parity, Circuits, and the Polynomial-Time Hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984. URL: https://doi.org/10.1007/BF01744431.
  15. Sevag Gharibian, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin Yirka. Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2). computational complexity, 31(2):13, 2022. URL: https://doi.org/10.1007/s00037-022-00231-8.
  16. Oded Goldreich. On Promise Problems: A Survey, pages 254-290. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006. URL: https://doi.org/10.1007/11685654_12.
  17. Sabee Grewal and Dorian Rudolph. On the pure quantum polynomial hierarchy and quantified hamiltonian complexity, 2025. URL: https://doi.org/10.48550/arXiv.2510.06522.
  18. Sabee Grewal and Justin Yirka. The Entangled Quantum Polynomial Hierarchy Collapses. In 39th Computational Complexity Conference (CCC 2024), volume 300, pages 6:1-6:23, 2024. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.6.
  19. Aram W. Harrow and Ashley Montanaro. Testing Product States, Quantum Merlin-Arthur Games and Tensor Optimization. Journal of the ACM, 60(1), 2013. URL: https://doi.org/10.1145/2432622.2432625.
  20. Rahul Jain and John Watrous. Parallel Approximation of Non-interactive Zero-sum Quantum Games. In 24th Annual IEEE Conference on Computational Complexity, pages 243-253, 2009. URL: https://doi.org/10.1109/CCC.2009.26.
  21. Fernando Granha Jeronimo and Pei Wu. Dimension Independent Disentanglers from Unentanglement and Applications. In 39th Computational Complexity Conference (CCC 2024), volume 300 of CCC '24, pages 26:1-26:28, 2024. URL: https://doi.org/10.4230/LIPIcs.CCC.2024.26.
  22. Joshua Lockhart and Carlos E. González-Guillén. Quantum State Isomorphism, 2017. URL: https://arxiv.org/abs/1709.09622.
  23. Jonas Kamminga and Dorian Rudolph. On the Complexity of Pure-State Consistency of Local Density Matrices, 2025. URL: https://arxiv.org/abs/2411.03096.
  24. Richard M. Karp and Richard J. Lipton. Some Connections Between Nonuniform and Uniform Complexity Classes. In Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, STOC '80, pages 302-309, 1980. URL: https://doi.org/10.1145/800141.804678.
  25. Julia Kempe, Alexei Kitaev, and Oded Regev. The Complexity of the Local Hamiltonian Problem. SIAM Journal on Computing, 35(5):1070-1097, 2006. URL: https://doi.org/10.1137/S0097539704445226.
  26. A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi. Classical and Quantum Computation. American Mathematical Society, USA, 2002. Google Scholar
  27. Hirotada Kobayashi, Keiji Matsumoto, and Tomoyuki Yamakami. Quantum certificate verification: Single versus multiple quantum certificates, 2001. URL: https://arxiv.org/abs/quant-ph/0110006.
  28. Clemens Lautemann. BPP and the Polynomial Time Hierarchy. Information Processing Letters, 17:215-218, 1983. Google Scholar
  29. Yi-Kai Liu. Consistency of Local Density Matrices is QMA-Complete. In Proceedings of the 9th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th International Conference on Randomization and Computation, APPROX'06/RANDOM'06, pages 438-449, 2006. URL: https://doi.org/10.1007/11830924_40.
  30. Dorian Rudolph, Sevag Gharibian, and Daniel Nagaj. Quantum 2-SAT on Low Dimensional Systems Is QMA₁-Complete: Direct Embeddings and Black-Box Simulation. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025), volume 325, pages 85:1-85:24, 2025. URL: https://doi.org/10.4230/LIPIcs.ITCS.2025.85.
  31. Michael Sipser. A Complexity Theoretic Approach to Randomness. In 15th Symposium on Theory of Computing, pages 330-335. ACM Press, 1983. URL: https://doi.org/10.1145/800061.808762.
  32. Larry J. Stockmeyer. The Polynomial-Time Hierarchy. Theoretical Computer Science, 3(1):1-22, 1976. URL: https://doi.org/10.1016/0304-3975(76)90061-X.
  33. Larry J. Stockmeyer and Albert R. Meyer. Word Problems Requiring Exponential Time (Preliminary Report). In Proceedings of the Fifth Annual ACM Symposium on Theory of Computing, STOC '73, pages 1-9. Association for Computing Machinery, 1973. URL: https://doi.org/10.1145/800125.804029.
  34. Seinosuke Toda. PP Is as Hard as the Polynomial-Time Hierarchy. SIAM Journal on Computing, 20(5):865-877, 1991. URL: https://doi.org/10.1137/0220053.
  35. Andreas Winter. Coding Theorem and Strong Converse for Quantum Channels. IEEE Transactions on Information Theory, 45(7):2481-2485, 1999. URL: https://doi.org/10.1109/18.796385.
  36. Tomoyuki Yamakami. Quantum NP and a Quantum Hierarchy. In 2nd IFIP International Conference on Theoretical Computer Science, pages 323-336. Kluwer Academic Publishers, 2002. Google Scholar
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