One bound to rule them all: from Adiabatic to Zeno

Daniel Burgarth; Paolo Facchi; Giovanni Gramegna; Kazuya Yuasa

doi:10.22331/q-2022-06-14-737

One bound to rule them all: from Adiabatic to Zeno

Daniel Burgarth¹, Paolo Facchi^2,3, Giovanni Gramegna^4,5,6, and Kazuya Yuasa⁷

¹Center for Engineered Quantum Systems, Macquarie University, 2109 NSW, Australia
²Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy
³INFN, Sezione di Bari, I-70126 Bari, Italy
⁴Dipartimento di Fisica, Università di Trieste, I-34151 Trieste, Italy
⁵INFN, Sezione di Trieste, I-34151 Trieste, Italy
⁶Eberhard-Karls-Universität Tübingen, Institut für Theoretische Physik, 72076 Tübingen, Germany
⁷Department of Physics, Waseda University, Tokyo 169-8555, Japan

Published:	2022-06-14, volume 6, page 737
Eprint:	arXiv:2111.08961v2
Doi:	https://doi.org/10.22331/q-2022-06-14-737
Citation:	Quantum 6, 737 (2022).

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Abstract

We derive a universal nonperturbative bound on the distance between unitary evolutions generated by time-dependent Hamiltonians in terms of the difference of their integral actions. We apply our result to provide explicit error bounds for the rotating-wave approximation and generalize it beyond the qubit case. We discuss the error of the rotating-wave approximation over long time and in the presence of time-dependent amplitude modulation. We also show how our universal bound can be used to derive and to generalize other known theorems such as the strong-coupling limit, the adiabatic theorem, and product formulas, which are relevant to quantum-control strategies including the Zeno control and the dynamical decoupling. Finally, we prove generalized versions of the Trotter product formula, extending its validity beyond the standard scaling assumption.

Featured image: One bound to bring them all and in mathematics bind them.

Popular summary

The quantum dynamics of time-dependent systems is extraordinarily complex even for the simplest examples. Approximations are therefore the key to understanding this rich dynamics, with applications ranging across all areas of quantum physics, and important consequences for quantum information processing and control. However, such approximations are often ad hoc or do not provide a good handle to bound the error made in simplifying the dynamics. Here, we describe a simple tool which we use to bound a surprisingly wide range of time-dependent phenomena, ranging from the famous Adiabatic Theorems, via the commonly used Rotating-Wave Approximation and the Trotter Product Formulas with importance in quantum simulation, to the conceptually puzzling Zeno Paradox. One bound to bring them all, and in mathematics bind them.

► BibTeX data

@article{Burgarth2022oneboundtorulethem,
  doi = {10.22331/q-2022-06-14-737},
  url = {https://doi.org/10.22331/q-2022-06-14-737},
  title = {One bound to rule them all: from {A}diabatic to {Z}eno},
  author = {Burgarth, Daniel and Facchi, Paolo and Gramegna, Giovanni and Yuasa, Kazuya},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {6},
  pages = {737},
  month = jun,
  year = {2022}
}

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[1] Daniel Burgarth, Paolo Facchi, and Robin Hillier, "Stability and convergence of dynamical decoupling with finite amplitude controls", Journal of Mathematical Physics 63 11, 112206 (2022).

[2] Daniel Burgarth, Niklas Galke, Alexander Hahn, and Lauritz van Luijk, "State-dependent Trotter limits and their approximations", Physical Review A 107 4, L040201 (2023).

[3] Nicola Macrì, Luigi Giannelli, Elisabetta Paladino, and Giuseppe Falci, "Coarse-Grained Effective Hamiltonian via the Magnus Expansion for a Three-Level System", Entropy 25 2, 234 (2023).

[4] Hsin-Yuan Huang, Yu Tong, Di Fang, and Yuan Su, "Learning Many-Body Hamiltonians with Heisenberg-Limited Scaling", Physical Review Letters 130 20, 200403 (2023).

[5] H. Lagemann, D. Willsch, M. Willsch, F. Jin, H. De Raedt, and K. Michielsen, "Numerical analysis of effective models for flux-tunable transmon systems", Physical Review A 106 2, 022615 (2022).

[6] Kazutaka Takahashi and Yasuhiro Utsumi, "Generalized speed limits for classical stochastic systems and their applications to relaxation, annealing, and pumping processes", Physical Review Research 5 1, 013217 (2023).

[7] Alexander Hahn, Daniel Burgarth, and Kazuya Yuasa, "Unification of random dynamical decoupling and the quantum Zeno effect", New Journal of Physics 24 6, 063027 (2022).

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One bound to rule them all: from Adiabatic to Zeno

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