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Space-Bounded Quantum Interactive Proof Systems

Authors François Le Gall , Yupan Liu , Harumichi Nishimura , Qisheng Wang

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  • Theory of computation → Interactive proof systems
  • Theory of computation → Complexity classes
  • Theory of computation → Quantum complexity theory
Keywords
  • Intermediate measurements
  • Quantum interactive proofs
  • Space-bounded quantum computation

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Abstract

We introduce two models of space-bounded quantum interactive proof systems, QIPL and QIP_{U}L. The QIP_{U}L model, a space-bounded variant of quantum interactive proofs (QIP) introduced by Watrous (CC 2003) and Kitaev and Watrous (STOC 2000), restricts verifier actions to unitary circuits. In contrast, QIPL allows logarithmically many pinching intermediate measurements per verifier action, making it the weakest model that encompasses the classical model of Condon and Ladner (JCSS 1995). 
We characterize the computational power of QIPL and QIP_{U}L. When the message number m is polynomially bounded, QIP_{U}L ⊊ QIPL unless P = NP:  
- QIPL^HC, a subclass of QIPL defined by a high-concentration condition on yes instances, exactly characterizes NP. 
- QIP_{U}L is contained in P and contains SAC¹ ∪ BQL, where SAC¹ denotes problems solvable by classical logarithmic-depth, semi-unbounded fan-in circuits.  However, this distinction vanishes when m is constant. Our results further indicate that (pinching) intermediate measurements uniquely impact space-bounded quantum interactive proofs, unlike in space-bounded quantum computation, where BQL = BQ_{U}L. 
We also introduce space-bounded unitary quantum statistical zero-knowledge (QSZK_{U}L), a specific form of QIP_{U}L proof systems with statistical zero-knowledge against any verifier. This class is a space-bounded variant of quantum statistical zero-knowledge (QSZK) defined by Watrous (SICOMP 2009). We prove that QSZK_{U}L = BQL, implying that the statistical zero-knowledge property negates the computational advantage typically gained from the interaction.

Cite As Get BibTex

François Le Gall, Yupan Liu, Harumichi Nishimura, and Qisheng Wang. Space-Bounded Quantum Interactive Proof Systems. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.17

Author Details

François Le Gall
  • Graduate School of Mathematics, Nagoya University, Japan
Yupan Liu
  • Graduate School of Mathematics, Nagoya University, Japan
Harumichi Nishimura
  • Graduate School of Informatics, Nagoya University, Japan
Qisheng Wang
  • School of Informatics, University of Edinburgh, UK
  • Graduate School of Mathematics, Nagoya University, Japan

Funding

This work was partially supported by MEXT Q-LEAP grant No. JPMXS0120319794.
  • Le Gall, François: Supported in part by JSPS KAKENHI grants Nos. JP20H05966, 20H00579, 24H00071, and by MEXT JST CREST grant No. JPMJCR24I4.
  • Liu, Yupan: Supported in part by JST SPRING grant No. JPMJSP2125 and the "THERS Make New Standards Program for the Next Generation Researchers."
  • Nishimura, Harumichi: Supported in part by JSPS KAKENHI grants Nos. JP19H04066, JP20H05966, JP21H04879, and JP22H00522.
  • Wang, Qisheng: Supported in part by the Engineering and Physical Sciences Research Council under Grant No. EP/X026167/1.

Acknowledgements

The authors appreciate Dorian Rudolph for pointing out a gap in the earlier argument that QIPL is contained in AM, specifically noting that Lemma 3.14 in the full version [Le Gall et al., 2024] does not directly apply to QIPL, as it does not verify the consistency of the prover strategies across different measurement outcome branches. YL is grateful to Uma Girish for a helpful discussion that inspired Question c. The authors also thank the anonymous reviewers for their helpful comments, which made the introduction more accessible to non-expert readers.

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