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A Slightly Improved Upper Bound for Quantum Statistical Zero-Knowledge

Authors François Le Gall , Yupan Liu , Qisheng Wang

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ACM Subject Classification
  • Theory of computation → Interactive proof systems
  • Theory of computation → Complexity classes
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum statistical zero-knowledge
  • Algorithmic Holevo-Helstrom measurement
  • Algorithmic Uhlmann transform

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Abstract

The complexity class Quantum Statistical Zero-Knowledge (QSZK), introduced by Watrous (FOCS 2002) and later refined in Watrous (SICOMP, 2009), has the best known upper bound QIP(2) ∩ co-QIP(2), which was simplified following the inclusion QIP(2) ⊆ PSPACE established in Jain, Upadhyay, and Watrous (FOCS 2009). Here, QIP(2) denotes the class of promise problems that admit two-message quantum interactive proof systems in which the honest prover is typically computationally unbounded, and co-QIP(2) denotes the complement of QIP(2).
We slightly improve this upper bound to QIP(2) ∩ co-QIP(2) with a quantum linear-space honest prover. Specifically, the honest prover uses space linear in the size of the transcript of the original QSZK proof system. A similar improvement also applies to the upper bound for the non-interactive variant NIQSZK. Our main techniques are algorithmic versions of the Holevo-Helstrom measurement and the Uhlmann transform, both implementable in quantum linear space, implying polynomial-time complexity in the state dimension, using the recent space-efficient quantum singular value transformation of Le Gall, Liu, and Wang (CC, to appear).

Cite As Get BibTex

François Le Gall, Yupan Liu, and Qisheng Wang. A Slightly Improved Upper Bound for Quantum Statistical Zero-Knowledge. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.MFCS.2026.12

Author Details

François Le Gall
  • Graduate School of Mathematics, Nagoya University, Japan
Yupan Liu
  • School of Computer and Communication Sciences, École Polytechnique Fédérale de Lausanne, Switzerland
  • Graduate School of Mathematics, Nagoya University, Japan
Qisheng Wang
  • School of Computer Science, Shanghai Jiao Tong University, China

Funding

This work was partially supported by MEXT Q-LEAP grant No. JPMXS0120319794.
  • Le Gall, François: Supported in part by JSPS KAKENHI grant No. JP24H00071, JST ASPIRE grant No. JPMJAP2302, and JST CREST grant No. JPMJCR24I4.
  • Liu, Yupan: Supported in part by funding from the Swiss State Secretariat for Education, Research and Innovation (SERI), and in part by JSPS KAKENHI grant No. JP24H00071.
  • Wang, Qisheng: Supported in part by a startup funding from Shanghai Jiao Tong University and in part by the Engineering and Physical Sciences Research Council under Grant No. EP/X026167/1.

Acknowledgements

A preliminary version of Section 3 (whose informal theorems are summarized in Theorems 2 and 6) appeared in Section 5 of the second arXiv version of [Le Gall et al., 2025] and in Section 6.3 of the second-named author’s PhD thesis [Liu, 2025]. Part of the work of Qisheng Wang was done when the author was with the School of Informatics, University of Edinburgh, United Kingdom. The authors thank Harumichi Nishimura and Thomas Vidick for helpful feedback on a preliminary version of the manuscript. The circuit diagrams were drawn using the Quantikz package [Kay, 2018].

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