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A Complete Equational Theory for Real-Clifford+CH Quantum Circuits

Author Alexandre Clément

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LIPIcs.LICS.2026.28.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Equational logic and rewriting
Keywords
  • Quantum circuits
  • completeness
  • graphical language

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Abstract

We introduce a complete equational theory for the fragment of quantum circuits generated by the real Clifford gates plus the two-qubit controlled-Hadamard gate. That is, we give a simple set of equalities between circuits of this fragment, and prove that any other true equation can be derived from these. This is the first such completeness result for a finitely-generated, universal fragment of quantum circuits, with no parameterized gates and no need for ancillas.

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Alexandre Clément. A Complete Equational Theory for Real-Clifford+CH Quantum Circuits. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 28:1-28:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.LICS.2026.28

Author Details

Alexandre Clément
  • Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMF, 91190, Gif-sur-Yvette, France

Funding

This work has been partially funded by the French National Research Agency (ANR) within the framework of "Plan France 2030", under the research projects EPIQ ANR-22-PETQ-0007, HQI-Acquisition ANR-22-PNCQ-0001 and HQI-R&D ANR-22-PNCQ-0002.

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