- Physical Review Journals
All JournalsPhysics Magazine

APS Open Science

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

Low-rank geometry of two-qubit gates

Llorenç Balada Gaggioli*

  • *Contact author: llorenc.balada.gaggioli@fel.cvut.cz
PDF

APS Open Sci. 1, 000206 – Published 5 November, 2026

DOI: https://doi.org/10.1103/nrx9-9p92

Abstract

We present a framework based on the determinantal geometry of two-qubit gates. Combining the Weyl chamber representation with operator Schmidt theory, we interpret gate synthesis as a distance problem to determinantal varieties. This gives an operational geometry to the Weyl chamber, quantifying nonlocal complexity. We show that the iswap gate is the closest perfect entangler to the variety of local operations, and that no perfect entangler can be approximated by a local gate with average gate fidelity above 78.3%. The three different determinantal costs form a synthesis-adapted coordinate system that encodes nonlocal complexity and generally reconstruct the Weyl coordinates in the reduced sector.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (25)

  1. D. P. DiVincenzo, Two-bit gates are universal for quantum computation, Phys. Rev. A 51, 1015 (1995).
  2. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, Elementary gates for quantum computation, Phys. Rev. A 52, 3457 (1995).
  3. J. L. Dodd, M. A. Nielsen, M. J. Bremner, and R. T. Thew, Universal quantum computation and simulation using any entangling Hamiltonian and local unitaries, Phys. Rev. A 65, 040301 (2002).
  4. S. Lloyd, Almost any quantum logic gate is universal, Phys. Rev. Lett. 75, 346 (1995).
  5. K. Hammerer, G. Vidal, and J. I. Cirac, Characterization of nonlocal gates, Phys. Rev. A 66, 062321 (2002).
  6. G. Vidal and C. M. Dawson, Universal quantum circuit for two-qubit transformations with three controlled-NOT gates, Phys. Rev. A 69, 010301 (2004).
  7. Y. Makhlin, Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations, Quantum Inf. Proc. 1, 243 (2002).
  8. N. Khaneja and S. J. Glaser, Cartan decomposition of SU(2n) and control of spin systems, Chem. Phys. 267, 11 (2001).
  9. B. Kraus and J. I. Cirac, Optimal creation of entanglement using a two-qubit gate, Phys. Rev. A 63, 062309 (2001).
  10. J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Geometric theory of nonlocal two-qubit operations, Phys. Rev. A 67, 042313 (2003).
  11. P. Zanardi, C. Zalka, and L. Faoro, Entangling power of quantum evolutions, Phys. Rev. A 62, 030301 (2000).
  12. P. Zanardi, Entanglement of quantum evolutions, Phys. Rev. A 63, 040304 (2001).
  13. M. A. Nielsen, C. M. Dawson, J. L. Dodd, A. Gilchrist, D. Mortimer, T. J. Osborne, M. J. Bremner, A. W. Harrow, and A. Hines, Quantum dynamics as a physical resource, Phys. Rev. A 67, 052301 (2003).
  14. S. Balakrishnan and R. Sankaranarayanan, Operator-Schmidt decomposition and the geometrical edges of two-qubit gates, Quantum Inf. Proc. 10, 449 (2011).
  15. S. Balakrishnan and R. Sankaranarayanan, Measures of operator entanglement of two-qubit gates, Phys. Rev. A 83, 062320 (2011).
  16. S. Balakrishnan and M. Lakshmanan, Chaining property for two-qubit operator entanglement measures, Eur. Phys. J. Plus 129, 231 (2014).
  17. J. E. Tyson, Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries, J. Phys. A: Math. Gen. 36, 10101 (2003).
  18. M. Musz, M. Kuś, and K. Życzkowski, Unitary quantum gates, perfect entanglers, and unistochastic maps, Phys. Rev. A 87, 022111 (2013).
  19. G. Golub, A. Hoffman, and G. Stewart, A generalization of the Eckart-Young-Mirsky matrix approximation theorem, Linear Algebra Appl. 88–89, 317 (1987).
  20. L. H. Pedersen, N. M. Møller, and K. Mølmer, Fidelity of quantum operations, Phys. Lett. A 367, 47 (2007).
  21. J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Exact two-qubit universal quantum circuit, Phys. Rev. Lett. 91, 027903 (2003).
  22. J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Minimum construction of two-qubit quantum operations, Phys. Rev. Lett. 93, 020502 (2004).
  23. F. Vatan and C. Williams, Optimal quantum circuits for general two-qubit gates, Phys. Rev. A 69, 032315 (2004).
  24. V. V. Shende, I. L. Markov, and S. S. Bullock, Minimal universal two-qubit controlled-NOT-based circuits, Phys. Rev. A 69, 062321 (2004).
  25. https://github.com/llorebaga/LR2qubit.

Outline

Information

APS Open Sci. 1, 000206– Published 5 November, 2026
Vol. 1
  • Received 16 April 2026
  • Accepted 27 October 2026
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