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Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits

Authors Neil J. Ross , Scott Wesley

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  • Hardware → Quantum computation
  • Hardware → Equivalence checking
Keywords
  • Quantum Circuits
  • Parameterized Equivalence Checking

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Abstract

Many promising quantum algorithms in economics, medical science, and material science rely on circuits that are parameterized by a large number of angles. To ensure that these algorithms are efficient, these parameterized circuits must be heavily optimized. However, most quantum circuit optimizers are not verified, so this procedure is known to be error-prone. For this reason, there is growing interest in the design of equivalence checking algorithms for parameterized quantum circuits. In this paper, we define a generalized class of parameterized circuits with arbitrary rotations and show that this problem is decidable for cyclotomic gate sets. We propose a cutoff-based procedure which reduces the problem of verifying the equivalence of parameterized quantum circuits to the problem of verifying the equivalence of finitely many parameter-free quantum circuits. Because the number of parameter-free circuits grows exponentially with the number of parameters, we also propose a probabilistic variant of the algorithm for cases when the number of parameters is intractably large. We show that our techniques extend to equivalence modulo global phase, and describe an efficient angle sampling procedure for cyclotomic gate sets.

Cite As Get BibTex

Neil J. Ross and Scott Wesley. Cutoff Theorems for the Equivalence of Parameterized Quantum Circuits. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 85:1-85:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.85

Author Details

Neil J. Ross
  • Dalhousie University, Halifax, Canada
Scott Wesley
  • Dalhousie University, Halifax, Canada

Acknowledgements

We thank Mingkuan Xu for his feedback on the paper and our initial experiments, Linh Dinh for sharing their insights on the number-theoretic aspects of this paper, and Mingsheng Ying for his feedback on an early draft of the paper.

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