- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Improving success probability in Lechner-Hauke-Zoller parity embedding by computing with quantum walks

Jemma Bennett1,*, Nicholas Chancellor2, Viv Kendon3, and Wolfgang Lechner1,4

  • 1Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria
  • 2School of Computing, Newcastle University, 1 Science Square, Newcastle upon Tyne NE4 5TG, United Kingdom
  • 3Department of Physics, University of Strathclyde, Glasgow G4 0NG, United Kingdom
  • 4Parity Quantum Computing GmbH, Rennweg 1/Top 314, 6020 Innsbruck, Austria

  • *Contact author: jemma.bennett@uibk.ac.at
PDF

Phys. Rev. A 112, 042616 – Published 23 October, 2025

DOI: https://doi.org/10.1103/my7q-8tbj

Abstract

Lechner-Hauke-Zoller (LHZ) parity embedding is one of the front-running methods for implementing difficult-to-engineer long-range interactions in quantum optimization problems. Continuous-time quantum walks are a leading approach for solving quantum optimization problems. Because they populate excited states, quantum walks can avoid the exponential gap-closing problems seen in other continuous-time techniques such as quantum annealing and adiabatic quantum computation. An important question, therefore, is how continuous-time quantum walks perform in combination with the LHZ parity embedding. By numerically simulating continuous-time quantum walks on four-, five-, and six-logical-qubit Sherrington-Kirkpatrick Ising spin glass instances embedded onto the LHZ parity embedding, we are able to verify the continued efficacy of heuristics used to estimate the optimal hopping rate and the numerical agreement with the theory behind the location of the lower bound of the LHZ parity constraint strength. In addition, by comparing several different LHZ-based decoding methods, we identified postreadout error correction techniques which improve the success probability of the quantum walk.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (38)

  1. K. Boothby, A. D. King, and J. Raymond, Zephyr topology of D-wave quantum processors, Technical Report Number: 14-1056A-A, Company: D-Wave (Canada, 2021).
  2. C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, Topological and subsystem codes on low-degree graphs with flag qubits, Phys. Rev. X 10, 011022 (2020).
  3. V. Choi, Minor-embedding in adiabatic quantum computation: I. The parameter setting problem, Quantum Inf. Process. 7, 193 (2008).
  4. V. Choi, Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design, Quantum Inf. Process. 10, 343 (2011).
  5. minorminer, https://github.com/dwavesystems/minorminer (2018).
  6. E. Mozgunov, Precision of quantum simulation of all-to-all coupling in a local architecture, arXiv:2302.02458.
  7. X. Qiu, P. Zoller, and X. Li, Programmable quantum annealing architectures with Ising quantum wires, PRX Quantum 1, 020311 (2020).
  8. M. T. Nguyen, J. G. Liu, J. Wurtz, M. D. Lukin, S. T. Wang, and H. Pichler, Quantum optimization with arbitrary connectivity using Rydberg atom arrays, PRX Quantum 4, 010316 (2023).
  9. A. Palacios, A. Garcia-Saez, and M. P. Estarellas, Scalable 2-local architecture for quantum annealing of Ising models with arbitrary dimensions, Phys. Rev. Applied 23, 054070 (2025).
  10. W. Lechner, P. Hauke, and P. Zoller, A quantum annealing architecture with all-to-all connectivity from local interactions, Sci. Adv. 1, e1500838 (2015).
  11. M. Lanthaler and W. Lechner, Minimal constraints in the parity formulation of optimization problems, New J. Phys. 23, 083039 (2021).
  12. A. Weidinger, G. B. Mbeng, and W. Lechner, Error mitigation for quantum approximate optimization, Phys. Rev. A 108, 032408 (2023).
  13. A. Weidinger, G. B. Mbeng, M. Fellner, D. Khachatryan, and W. Lechner, Performance of parity QAOA for the signed max-cut problem, arXiv:2409.14786.
  14. T. Albash, W. Vinci, and D. A. Lidar, Simulated-quantum-annealing comparison between all-to-all connectivity schemes, Phys. Rev. A 94, 022327 (2016).
  15. J. King, M. Mohseni, W. Bernoudy, A. Fréchette, H. Sadeghi, S. V. Isakov, H. Neven, and M. H. Amin, Quantum-assisted genetic algorithm, arXiv:1907.00707.
  16. F. Pastawski and J. Preskill, Error correction for encoded quantum annealing, Phys. Rev. A 93, 052325 (2016).
  17. A. Messinger, V. Torggler, B. Klaver, M. Fellner, and W. Lechner, Fault-tolerant quantum computing with the parity code and biased-noise qubits, Phys. Rev. Applied 23, 044032 (2025).
  18. Y. Nambu, Error correction of parity-encoding-based annealing through post-readout decoding, arXiv:2402.08839.
  19. Y. Nambu, Error correction for encoded quantum annealing revisited, arXiv:2407.15480.
  20. Y. Nambu, Effcient classical error correction for parity encoded spin systems, arXiv:2502.07170.
  21. M. Fellner, A. Messinger, K. Ender, and W. Lechner, Universal parity quantum computing, Phys. Rev. Lett. 129, 180503 (2022).
  22. A. Messinger, M. Fellner, and W. Lechner, Constant depth code deformations in the parity architecture, in Proceedings of the 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) (IEEE, New York, 2023), pp. 120–130.
  23. I. D. Smith, H. P. Nautrup, and H. J. Briegel, Parity quantum computing as YZ-plane measurement-based quantum computing, Phys. Rev. Lett. 132, 220602 (2024).
  24. E. Farhi and S. Gutmann, Quantum computation and decision trees, Phys. Rev. A 58, 915 (1998).
  25. A. M. Childs and J. Goldstone, Spatial search by quantum walk, Phys. Rev. A 70, 022314 (2004).
  26. A. Callison, N. Chancellor, F. Mintert, and V. Kendon, Finding spin glass ground states using quantum walks, New J. Phys. 21, 123022 (2019).
  27. J. G. Morley, N. Chancellor, S. Bose, and V. Kendon, Quantum search with hybrid adiabatic–quantum-walk algorithms and realistic noise, Phys. Rev. A 99, 022339 (2019).
  28. A. Callison, M. Festenstein, J. Chen, L. Nita, V. Kendon, and N. Chancellor, An energetic perspective on rapid quenches in quantum annealing, Phys. Rev. X Quantum 2, 010338 (2021).
  29. R. J. Banks, E. Haque, F. Nazef, F. Fethallah, F. Ruqaya, H. Ahsan, H. Vora, H. Tahir, I. Ahmad, I. Hewins, I. Shah, K. Baranwal, M. Arora, M. Asad, M. Khan, N. Hasan, N. Azad, S. Fedaiee, S. Majeed, S. Bhuyan et al., Continuous-time quantum walks for max-cut are hot, Quantum 8, 1254 (2024).
  30. S. Schulz, D. Willsch, and K. Michielsen, Guided quantum walk, Phys. Rev. Res. 6, 013312 (2024).
  31. L. Gerblich, T. Dasanjh, H. Q. X. Wong, D. Ross, L. Novo, N. Chancellor, and V. Kendon, Advantages of multistage quantum walks over QAOA, arXiv:2407.06663.
  32. R. Ghosh, L. A. Nutricati, N. Feinstein, P. A. Warburton, and S. Bose, Exponential speed-up of quantum annealing via n-local catalysts, arXiv:2409.13029.
  33. J. Bennett, Suppressing precision errors by connecting copies of Ising models for continuous-time quantum computing, Ph.D. thesis, Durham University, 2023.
  34. Y. Nambu, Rejection-free Monte Carlo simulation of QUBO and Lechner–Hauke–Zoller optimization problems, IEEE Acc. 10, 84279 (2022).
  35. N. Chancellor, A. Callison, V. Kendon, and F. Mintert, Finding Spin-Glass Ground States Using Quantum Walks, data archive at Durham University, UK (2019), for spin glass instances used in [26], doi: 10.15128/r21544bp097.
  36. V. Choi, Adiabatic quantum algorithms for the NP-complete maximum-weight independent set, exact cover and 3SAT problems, arXiv:1004.2226.
  37. D. Sherrington and S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett. 35, 1792 (1975).
  38. A. Rocchetto, S. C. Benjamin, and Y. Li, Stabilizers as a design tool for new forms of the Lechner-Hauke-Zoller annealer, Sci. Adv. 2, e1601246 (2016).

Outline

Information

Phys. Rev. A 112, 042616– Published 23 October, 2025
  • Received 22 April 2025
  • Revised 29 August 2025
  • Accepted 17 September 2025
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation