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On the Complexity of the Circuit Width Problem

Authors Zhengfeng Ji , Yinchen Liu , Zhe'ou Zhou

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LIPIcs.TQC.2026.7.pdf
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ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • IQP circuits
  • circuit width
  • NP-completeness
  • approximation hardness
  • parameterized algorithms

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Abstract

Montanaro associated to every quantum circuit over the gate set {H,Z,CZ,CCZ} a degree-three polynomial over 𝔽₂, and showed that the strong simulation cost of the circuit is governed by the minimum number of qubit wires needed to realize that polynomial. We study this minimum-width problem. We prove that deciding whether a degree-three polynomial has width at most k is NP-complete. The reduction is parsimonious enough to imply that, for every α < 49/48, the corresponding gap version is NP-hard. We also prove the same inapproximability threshold for degree-two polynomials by a twin-copy replacement of the cubic constraints. On the positive side, we give a nondeterministic search algorithm using only O(klog(en/k)) nondeterministic bits, and a deterministic fixed-parameter algorithm running in k^{6k+o(k)}n+O(m) time on an input with n symbols and m monomials.

Cite As Get BibTex

Zhengfeng Ji, Yinchen Liu, and Zhe'ou Zhou. On the Complexity of the Circuit Width Problem. In 21st Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 389, pp. 7:1-7:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.TQC.2026.7

Author Details

Zhengfeng Ji
  • Department of Computer Science and Technology, Tsinghua University, Beijing, China
Yinchen Liu
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
Zhe'ou Zhou
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

Funding

  • Ji, Zhengfeng: National Natural Science Foundation of China (Grant No. 12347104), National Key Research and Development Program of China (Grant No. 2023YFA1009403), Beijing Science and Technology Planning Project (Grant No. Z25110100810000), and Beijing Natural Science Foundation (Grant No. Z220002).
  • Liu, Yinchen: Beijing Natural Science Foundation (Grant No. QY26408).

Acknowledgements

We thank the anonymous reviewers for their helpful comments and suggestions, especially on the importance of better characterizing the degree-2 instances used in our hardness reduction and of clarifying the role of potentially tractable special cases.

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