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Quantum Mass Production Theorems

Author William Kretschmer

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Author Details

William Kretschmer
  • University of Texas at Austin, TX, USA


Part of this work was done while the author attended the 2022 Extended Reunion for the Quantum Wave in Computing at the Simons Institute for the Theory of Computing. We thank Alex Meiburg for helpful discussions.

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William Kretschmer. Quantum Mass Production Theorems. In 18th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 266, pp. 10:1-10:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We prove that for any n-qubit unitary transformation U and for any r = 2^{o(n / log n)}, there exists a quantum circuit to implement U^{⊗ r} with at most O(4ⁿ) gates. This asymptotically equals the number of gates needed to implement just a single copy of a worst-case U. We also establish analogous results for quantum states and diagonal unitary transformations. Our techniques are based on the work of Uhlig [Math. Notes 1974], who proved a similar mass production theorem for Boolean functions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Circuit complexity
  • mass production
  • quantum circuit synthesis
  • quantum circuit complexity


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