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On Polynomially Many Queries to NP or QMA Oracles

Authors Sevag Gharibian , Dorian Rudolph

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


We thank Eric Allender, Johannes Bausch, Stephen Piddock, James Watson, and Justin Yirka for helpful discussions.

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Sevag Gharibian and Dorian Rudolph. On Polynomially Many Queries to NP or QMA Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 75:1-75:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as P^NP and P^QMA, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes P^NP[log] and P^QMA[log], defined identically to P^NP and P^QMA, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a P^NP machine have a "query graph" which is a tree, then this computation can be simulated in P^NP[log]. In this work, we first show that for any verification class C ∈ {NP, MA, QCMA, QMA, QMA(2), NEXP, QMA_exp}, any P^C machine with a query graph of "separator number" s can be simulated using deterministic time exp(slog n) and slog n queries to a C-oracle. When s ∈ O(1) (which includes the case of O(1)-treewidth, and thus also of trees), this gives an upper bound of P^C[log], and when s ∈ O(log^k(n)), this yields bound QP^{C[log^{k+1}]} (QP meaning quasi-polynomial time). We next show how to combine Gottlob’s "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding P^C computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial p specified via an arithmetic circuit, if one can "weakly compress" p so that its optimal value requires m bits to represent, then P^NP can be decided with only m queries to an NP-oracle.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Complexity classes
  • admissible weighting function
  • oracle complexity class
  • quantum complexity theory
  • Quantum Merlin Arthur (QMA)
  • simulation of local measurement


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  1. Scott Aaronson. BQP and the polynomial hierarchy. In Proceedings of the Forty-Second ACM Symposium on Theory of Computing, STOC '10, pages 141-150, New York, NY, USA, 2010. Association for Computing Machinery. URL:
  2. Dorit Aharonov, Daniel Gottesman, Sandy Irani, and Julia Kempe. The power of quantum systems on a line. Communications in Mathematical Physics, 287(1):41-65, April 2009. URL:
  3. Dorit Aharonov, Alex B. Grilo, and Yupan Liu. StoqMA vs. MA: the power of error reduction, 2021. URL:
  4. Dorit Aharonov and Sandy Irani. Hamiltonian complexity in the thermodynamic limit, 2021. URL:
  5. A. Ambainis. On physical problems that are slightly more difficult than QMA. In 2014 IEEE 29th Conference on Computational Complexity (CCC), pages 32-43, 2014. Google Scholar
  6. Richard Beigel, L.A. Hemachandra, and G. Wechsung. On the power of probabilistic polynomial time: P^NP[log] ⊆ PP. In [1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference, pages 225-227, June 1989. URL:
  7. Hans L. Bodlaender, John R. Gilbert, Hjálmtýr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238-255, March 1995. URL:
  8. Adam D. Bookatz. QMA-complete problems. Quantum Information & Computation, 14(5&6):361-383, April 2014. Google Scholar
  9. Sergey Bravyi, Arvid J. Bessen, and Barbara M. Terhal. Merlin-Arthur Games and Stoquastic Complexity, December 2006. URL:
  10. Samuel R. Buss and Louise Hay. On truth-table reducibility to SAT. Information and Computation, 91(1):86-102, 1991. URL:
  11. Jorge Castro and Carlos Seara. Characterizations of some complexity classes between ω₂^p and δ₂^p. In Alain Finkel and Matthias Jantzen, editors, STACS 92, Lecture Notes in Computer Science, pages 303-317, Berlin, Heidelberg, 1992. Springer. URL:
  12. André Chailloux and Or Sattath. The Complexity of the Separable Hamiltonian Problem. In 2012 IEEE 27th Conference on Computational Complexity, pages 32-41, June 2012. ISSN: 1093-0159. URL:
  13. Stephen A. Cook. The complexity of theorem-proving procedures. In Proceedings of the third annual ACM symposium on Theory of computing, STOC '71, pages 151-158, Shaker Heights, Ohio, USA, May 1971. Association for Computing Machinery. URL:
  14. Toby Cubitt and Ashley Montanaro. Complexity classification of local hamiltonian problems. SIAM Journal on Computing, 45(2):268-316, 2016. URL:
  15. Bill Fefferman and Cedric Lin. Quantum Merlin Arthur with Exponentially Small Gap, January 2016. URL:
  16. Michael R. Garey and David. S. Johnson. COMPUTERS and INTRACTABILITY: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979. Google Scholar
  17. Sevag Gharibian, Yichen Huang, Zeph Landau, and Seung Woo Shin. Quantum Hamiltonian Complexity. Foundations and Trends® in Theoretical Computer Science, 10(3):159-282, October 2015. URL:
  18. Sevag Gharibian, Stephen Piddock, and Justin Yirka. Oracle Complexity Classes and Local Measurements on Physical Hamiltonians. In Christophe Paul and Markus Bläser, editors, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020), volume 154 of Leibniz International Proceedings in Informatics (LIPIcs), pages 20:1-20:37, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL:
  19. Sevag Gharibian and Dorian Rudolph. On polynomially many queries to NP or QMA oracles, 2021. URL:
  20. Sevag Gharibian and Justin Yirka. The complexity of simulating local measurements on quantum systems. Quantum, 3:189, September 2019. URL:
  21. Daniel Gottesman and Sandy Irani. The quantum and classical complexity of translationally invariant tiling and hamiltonian problems. 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 95-104, 2009. Google Scholar
  22. Georg Gottlob. NP trees and Carnap’s modal logic. Journal of the ACM, 42(2):421-457, March 1995. URL:
  23. Hermann Gruber. On balanced separators, treewidth, and cycle rank. Journal of Combinatorics, 3(4):669-681, 2012. URL:
  24. Sean Hallgren, Daniel Nagaj, and Sandeep Narayanaswami. The local hamiltonian problem on a line with eight states is QMA-complete. Quantum Info. Comput., 13(9–10):721-750, September 2013. Google Scholar
  25. Aram W. Harrow and Ashley Montanaro. Testing product states, quantum Merlin-Arthur games and tensor optimisation. Journal of the ACM, 60(1):1-43, February 2013. URL:
  26. Juris Hartmanis. Sparse complete sets for NP and the optimal collapse of the polynomial hierarchy. In Current Trends in Theoretical Computer Science, pages 403-411. WORLD SCIENTIFIC, June 1993. URL:
  27. Lane A. Hemachandra. The strong exponential hierarchy collapses. Journal of Computer and System Sciences, 39(3):299-322, December 1989. URL:
  28. Edith Hemaspaandra, Lane A. Hemaspaandra, and Jörg Rothe. Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP. In Pierpaolo Degano, Roberto Gorrieri, and Alberto Marchetti-Spaccamela, editors, Automata, Languages and Programming, Lecture Notes in Computer Science, pages 214-224, Berlin, Heidelberg, 1997. Springer. URL:
  29. Richard M. Karp. Reducibility among Combinatorial Problems. In Raymond E. Miller, James W. Thatcher, and Jean D. Bohlinger, editors, Complexity of Computer Computations, The IBM Research Symposia Series, pages 85-103. Springer US, Boston, MA, 1972. URL:
  30. Julia Kempe, Alexei Kitaev, and Oded Regev. The complexity of the local hamiltonian problem. SIAM Journal on Computing, 35(5):1070-1097, January 2006. URL:
  31. Julia Kempe and Oded Regev. 3-local Hamitonian is QMA-complete. Quantum Information & Computation, 3(3):258-264, May 2003. Google Scholar
  32. A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi. Classical and Quantum Computation. American Mathematical Society, USA, 2002. Google Scholar
  33. Mark W. Krentel. The complexity of optimization problems. Journal of Computer and System Sciences, 36(3):490-509, 1988. URL:
  34. Mark W. Krentel. Generalizations of Opt P to the polynomial hierarchy. Theoretical Computer Science, 97(2):183-198, April 1992. URL:
  35. L. A. Levin. Universal sequential search problems. Problems of Information Transmission, 9(3):265-266, 1973. Google Scholar
  36. Tobias J. Osborne. Hamiltonian complexity. Reports on Progress in Physics, 75(2):022001, January 2012. URL:
  37. Christos H. Papadimitriou. On the complexity of unique solutions. In 23rd Annual Symposium on Foundations of Computer Science (SFCS 1982), pages 14-20, 1982. URL:
  38. Christos H. Papadimitriou and Stathis K. Zachos. Two remarks on the power of counting. In Armin B. Cremers and Hans-Peter Kriegel, editors, Theoretical Computer Science, Lecture Notes in Computer Science, pages 269-275, Berlin, Heidelberg, 1982. Springer. URL:
  39. Ran Raz and Avishay Tal. Oracle separation of bqp and ph. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, pages 13-23, New York, NY, USA, 2019. Association for Computing Machinery. URL:
  40. Neil Robertson and P.D Seymour. Graph minors. ii. algorithmic aspects of tree-width. Journal of Algorithms, 7(3):309-322, 1986. URL:
  41. Philippe Schnoebelen. Oracle circuits for branching-time model checking. In Jos C. M. Baeten, Jan Karel Lenstra, Joachim Parrow, and Gerhard J. Woeginger, editors, Automata, Languages and Programming: 30th International Colloquium, ICALP 2003 Eindhoven, The Netherlands, June 30 - July 4, 2003 Proceedings, pages 790-801, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg. URL:
  42. Larry J. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3(1):1-22, October 1976. URL:
  43. Klaus W. Wagner. More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science, 51(1):53-80, January 1987. URL:
  44. Klaus W. Wagner. Bounded query computations. [1988] Proceedings. Structure in Complexity Theory Third Annual Conference, pages 260-277, 1988 . URL:
  45. James D. Watson and Johannes Bausch. The complexity of approximating critical points of quantum phase transitions, 2021. URL:
  46. James D. Watson, Johannes Bausch, and Sevag Gharibian. The complexity of translationally invariant problems beyond ground state energies, 2020. URL:
  47. James D. Watson and Toby S. Cubitt. Computational complexity of the ground state energy density problem, 2021. URL:
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