One bound to rule them all: from Adiabatic to Zeno

Daniel Burgarth1, Paolo Facchi2,3, Giovanni Gramegna4,5,6, 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

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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.

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.

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[8] Philippe Lewalle, Yipei Zhang, and K. Birgitta Whaley, "Optimal Zeno Dragging for Quantum Control: A Shortcut to Zeno with Action-based Scheduling Optimization", arXiv:2311.01631, (2023).

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