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Fine-Grained Complexity via Quantum Natural Proofs

Authors Yanlin Chen, Yilei Chen, Rajendra Kumar, Subhasree Patro, Florian Speelman

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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
Keywords
  • Fine-grained complexity
  • Quantum natural proofs
  • QSETH
  • Pseudorandom functions

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Abstract

Buhrman, Patro, and Speelman [Buhrman et al., 2021] presented a framework of conjectures that together form a quantum analogue of the strong exponential-time hypothesis and its variants. They called it the QSETH framework. In this paper, using a notion of quantum natural proofs (built from natural proofs introduced by Razborov and Rudich), we show how part of the QSETH conjecture that requires properties to be "compression oblivious" can in many cases be replaced by assuming the existence of quantum-secure pseudorandom functions, a standard hardness assumption. Combined with techniques from Fourier analysis of Boolean functions, we show that properties such as parity and majority are compression oblivious for certain circuit class Λ if subexponentially secure quantum pseudorandom functions exist in Λ, answering an open question in [Buhrman et al., 2021].

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Yanlin Chen, Yilei Chen, Rajendra Kumar, Subhasree Patro, and Florian Speelman. Fine-Grained Complexity via Quantum Natural Proofs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 60:1-60:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.60

Author Details

Yanlin Chen
  • University of Maryland (QuICS), College Park, MD, USA
Yilei Chen
  • Shanghai Qi Zhi Institute, Tsinghua University, Beijing, China
Rajendra Kumar
  • Indian Institute of Technology Delhi, New Delhi, India
Subhasree Patro
  • Eindhoven University of Technology, The Netherlands
Florian Speelman
  • University of Amsterdam and QuSoft, The Netherlands

Funding

  • Chen, Yanlin: Acknowledges the support from a QuICS Hartree Postdoctoral Fellowship.
  • Chen, Yilei: Supported by Tsinghua University startup funding, and Shanghai Qi Zhi Institute Innovation Program SQZ202405.
  • Kumar, Rajendra: Supported by the Chandrukha New Faculty Fellowship from IIT Delhi and the Prime Minister Early Career Research Grant from the Anusandhan National Research Foundation (ANRF).
  • Patro, Subhasree: Acknowledges the support from the Dutch Ministry of Education, Culture, and Science through Gravitation project "Challenges in Cyber Security - 024.006.037" for this work.
  • Speelman, Florian: Supported by the Dutch Ministry of Economic Affairs and Climate Policy (EZK), as part of the Quantum Delta NL program, and the project Divide and Quantum "D&Q" NWA.1389.20.241 of the program "NWA-ORC", which is partly funded by the Dutch Research Council (NWO).

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