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  • Open Access

SWAP-less implementation of quantum algorithms

Berend Klaver1,2, Stefan M. A. Rombouts3, Michael Fellner1,2, Anette Messinger2, Kilian Ender1, Katharina Ludwig3, 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, 20095 Hamburg, Germany

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Phys. Rev. A 113, 012443 – Published 29 January, 2026

DOI: https://doi.org/10.1103/2wzk-fnhx

Abstract

We present a formalism based on tracking the flow of parity quantum information to implement algorithms on devices with limited connectivity without qubit overhead, SWAP operations, or shuttling. Instead, we leverage the fact that entangling gates not only manipulate quantum states but can also be exploited to transport quantum information. We demonstrate the effectiveness of this method by applying it to the quantum Fourier transform (QFT) and the quantum approximate optimization algorithm (QAOA) with n qubits. This improves upon all state-of-the-art implementations of the QFT on a linear nearest-neighbor architecture, resulting in a total circuit depth of 5n−3 and requiring n2−1 cnot gates. For the QAOA, our method outperforms SWAP networks, which are currently the most efficient implementation of the QAOA on a linear architecture. We further demonstrate the potential to balance qubit count against circuit depth by implementing the QAOA on twice the number of qubits using bilinear connectivity, which approximately halves the circuit depth.

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Parity flow formalism: Tracking quantum information throughout computation

Berend Klaver, Katharina Ludwig, Anette Messinger, Stefan M. A. Rombouts, Michael Fellner, Kilian Ender, and Wolfgang Lechner
Phys. Rev. Research 8, 013095 (2026)

Article Text

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Phys. Rev. A 113, 012443– Published 29 January, 2026
  • Received 26 May 2025
  • Accepted 9 September 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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