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Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds

Authors Avantika Agarwal, Sevag Gharibian , Venkata Koppula, Dorian Rudolph

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Author Details

Avantika Agarwal
  • David R. Cheriton School of Computer Science and Institute for Quantum Computing, University of Waterloo, Canada
Sevag Gharibian
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany
Venkata Koppula
  • Department of Computer Science and Engineering, Indian Institute of Technology Delhi, India
Dorian Rudolph
  • Department of Computer Science and Institute for Photonic Quantum Systems (PhoQS), Paderborn University, Germany

Funding

  • Gharibian, Sevag: Supported by the DFG under grant numbers 450041824 and 432788384, the BMBF within the funding program "Quantum Technologies - from Basic Research to Market" via project PhoQuant (grant number 13N16103), and the project "PhoQC" from the programme "Profilbildung 2020", an initiative of the Ministry of Culture and Science of the State of Northrhine Westphalia.
  • Koppula, Venkata: Supported by the Pankaj Gupta Fellowship at IIT Delhi.

Acknowledgements

We thank Chirag Falor, Shu Ge, Anand Natarajan, Sabee Grewal, and Justin Yirka for the pleasure of productive discussions during the concurrent development of our works. This work was completed in part while Avantika Agarwal was a student at Indian Institute of Technology Delhi and in part while visiting Paderborn University.

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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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.7

Abstract

The Polynomial-Time Hierarchy (PH) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to "quantum advantage" analyses for near-term quantum computers. Quantumly, however, despite the fact that at least four definitions of quantum PH exist, it has been challenging to prove analogues for these of even basic facts from PH. This work studies three quantum-verifier based generalizations of PH, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings (QCPH) and quantum mixed states (QPH) as proofs, and one of which is new to this work, utilizing quantum pure states (QPHpure) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for QCPH. Then, for our new class QPHpure, we show one-sided error reduction QPHpure, as well as the first bounds relating these quantum variants of PH, namely QCPH ⊆ QPHpure ⊆ EXP^PP.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum complexity
  • polynomial hierarchy

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