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  • Open Access

Minimally universal parity quantum computing

Isaac D. Smith1,*, Berend Klaver1,2, Hendrik Poulsen Nautrup1, Wolfgang Lechner1,2,3, and Hans J. Briegel1

  • 1University of Innsbruck, Department of Theoretical Physics, Technikerstr. 21A, Innsbruck A-6020, Austria
  • 2Parity Quantum Computing GmbH, A-6020 Innsbruck, Austria
  • 3Parity Quantum Computing Germany GmbH, 20095 Hamburg, Germany

  • *Contact author: isaac.smith@uibk.ac.at
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Phys. Rev. A 112, 032606 – Published 8 September, 2025

DOI: https://doi.org/10.1103/9q8k-5378

Abstract

In parity quantum computing, multiqubit logical gates are implemented by single-qubit rotations on a suitably encoded state involving auxiliary qubits. Consequently, there is a correspondence between qubit count and the size of the native gate set. One might then wonder, what is the smallest number of auxiliary qubits that still allows for universal parity computing? Here, we demonstrate that the answer is 1 if the number of logical qubits is even, and 2 otherwise. Furthermore, we present a sufficient condition for a given parity gate set to be universal. This leads to a variety of different universal parity gate sets corresponding to different numbers of auxiliary qubits and more generally contributes to the understanding of which entangling gates are required to augment the set of single-qubit unitaries to perform universal quantum computing. As a consequence, we obtain (i) minimal implementations of the parity framework on, e.g., a triangular lattice, (ii) hardware-specific implementations of the parity flow framework on, e.g., a heavy-hex lattice, and (iii) novel universal resources for measurement-based quantum computation.

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Phys. Rev. A 112, 032606– Published 8 September, 2025
  • Received 22 April 2025
  • Accepted 25 August 2025
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