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QMA Lower Bounds for Batch Verification via Approximate Degree

Authors Mark Bun , Mandar Juvekar , Samuel King

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ACM Subject Classification
  • Theory of computation → Quantum query complexity
Keywords
  • QMA
  • approximate degree
  • query complexity
  • communication complexity
  • quantum computation

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Abstract

We study batch verification in QMA query and communication complexity, where the goal is to understand how the resources needed to verify m copies of a Boolean function f depend on m. We give a general technique for proving lower bounds on the witness-query tradeoff needed to batch verify a function f in terms of its approximate degree. Applying this technique to an explicit family of DNF formulas f, we show that attempting to save even a constant factor on the witness length of the baseline approach to batch verifying f necessitates a large polynomial increase in the query cost. We also obtain new lower bounds on the QMA query complexity of read-once CNF formulas and on the surjectivity and k-element distinctness functions. Our lower bounds also lift to give communication analogs of these results.

Cite As Get BibTex

Mark Bun, Mandar Juvekar, and Samuel King. QMA Lower Bounds for Batch Verification via Approximate Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 69:1-69:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.69

Author Details

Mark Bun
  • Boston University, MA, USA
Mandar Juvekar
  • Boston University, MA, USA
Samuel King
  • Georgetown University, Washington, DC, USA

Funding

  • Bun, Mark: supported in part by NSF award CNS-2046425 and a Sloan Research Fellowship.
  • Juvekar, Mandar: supported in part by NSF award CNS-2046425 and an IBM PhD Fellowship.
  • King, Samuel: supported in part by NSF CAREER award CCF-2338730.

Acknowledgements

We would like to thank Nadezhda Voronova for her insightful comments on a draft of this paper, Sasha Golovnev for invaluable discussions during this work, and Robin Kothari and Justin Thaler for stimulating our interest in these problems. We would also like to thank Robin Kothari for spotting an error in a previous manuscript.

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