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Parity flow formalism: Tracking quantum information throughout computation

Berend Klaver1,2,*, Katharina Ludwig3, Anette Messinger2, Stefan M. A. Rombouts3, Michael Fellner1,2, Kilian Ender1, and Wolfgang Lechner1,2,3

  • 1Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
  • 2Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria
  • 3Parity Quantum Computing Germany GmbH, Schauenburgerstraße 6, 20095 Hamburg, Germany

  • *Contact author: berend.klaver@uibk.ac.at
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Phys. Rev. Research 8, 013095 – Published 29 January, 2026

DOI: https://doi.org/10.1103/6xlb-l92j

Abstract

We propose the parity flow formalism, a method for tracking the information flow in quantum circuits. This method adds labels to quantum circuit diagrams such that the action of Clifford gates can be understood as a recoding of quantum information. The action of non-Clifford gates in the encoded space can be directly deduced from those labels without backtracking. An application of flow tracking is to design resource-efficient quantum circuits by changing any present encoding via a simple set of rules. Finally, the parity flow formalism can be used in combination with stabilizer codes to further reduce quantum circuit depth and to reveal additional operations that can be implemented in parallel.

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See Also

SWAP-less implementation of quantum algorithms

Berend Klaver, Stefan M. A. Rombouts, Michael Fellner, Anette Messinger, Kilian Ender, Katharina Ludwig, and Wolfgang Lechner
Phys. Rev. A 113, 012443 (2026)

Article Text

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Phys. Rev. Research 8, 013095– Published 29 January, 2026
  • Received 26 May 2025
  • Accepted 5 December 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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