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.
 L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, revised ed. (Dover, New York, 1987).
 P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A Quantum Engineer's Guide to Superconducting Qubits, Appl. Phys. Rev. 6, 021318 (2019).
 A. Laucht, S. Simmons, R. Kalra, G. Tosi, J. P. Dehollain, J. T. Muhonen, S. Freer, F. E. Hudson, K. M. Itoh, D. N. Jamieson, J. C. McCallum, A. S. Dzurak, and A. Morello, Breaking the Rotating Wave Approximation for a Strongly Driven Dressed Single-Electron Spin, Phys. Rev. B 94, 161302 (2016).
 T. Chambrion, Periodic Excitations of Bilinear Quantum Systems, Automatica 48, 2040 (2012).
 N. Augier, U. Boscain, and M. Sigalotti, On the Compatibility between the Adiabatic and the Rotating Wave Approximations in Quantum Control, in Proceedings of the 2019 IEEE 58th Conference on Decision and Control (CDC) (IEEE, New York, 2019), pp. 2292–2297.
 R. Robin, N. Augier, U. Boscain, and M. Sigalotti, Ensemble Qubit Controllability with a Single Control via Adiabatic and Rotating Wave Approximations, arXiv:2003.05831 [math-ph] (2020).
 N. Augier, U. Boscain, and M. Sigalotti, Effective Adiabatic Control of a Decoupled Hamiltonian Obtained by Rotating Wave Approximation, arXiv:2005.02737 [math.OC] (2020).
 A. Messiah, Quantum Mechanics (Dover, New York, 2017).
 P. Facchi and S. Pascazio, Quantum Zeno Dynamics: Mathematical and Physical Aspects, J. Phys. A: Math. Theor. 41, 493001 (2008).
 P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse, and D. A. Lidar, Control of Decoherence: Analysis and Comparison of Three Different Strategies, Phys. Rev. A 71, 022302 (2005).
 E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, Continuous and Pulsed Quantum Zeno Effect, Phys. Rev. Lett. 97, 260402 (2006).
 F. Schäfer, I. Herrera, S. Cherukattil, C. Lovecchio, F. S. Cataliotti, F. Caruso, and A. Smerzi, Experimental Realization of Quantum Zeno Dynamics, Nat. Commun. 5, 3194 (2014).
 Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, Universal Error Bound for Constrained Quantum Dynamics, Phys. Rev. Lett. 124, 210606 (2020).
 Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, Error Bounds for Constrained Dynamics in Gapped Quantum Systems: Rigorous Results and Generalizations, Phys. Rev. A 101, 052122 (2020).
 D. Burgarth, P. Facchi, H. Nakazato, S. Pascazio, and K. Yuasa, Kolmogorov-Arnold-Moser Stability for Conserved Quantities in Finite-Dimensional Quantum Systems, Phys. Rev. Lett. 126, 150401 (2021).
 B. Simon, Functional Integration and Quantum Physics (Academic Press, New York, 1979), Vol. 86.
 M. Suzuki, Decomposition Formulas of Exponential Operators and Lie Exponentials with Some Applications to Quantum Mechanics and Statistical Physics, J. Math. Phys. 26, 601 (1985).
 L. M. Sieberer, T. Olsacher, A. Elben, M. Heyl, P. Hauke, F. Haake, and P. Zoller, Digital Quantum Simulation, Trotter Errors, and Quantum Chaos of the Kicked Top, npj Quant. Inf. 5, 78 (2019).
 D. Burgarth, P. Facchi, V. Giovannetti, H. Nakazato, S. Pascazio, and K. Yuasa, Non-Abelian Phases from Quantum Zeno Dynamics, Phys. Rev. A 88, 042107 (2013).
 T. G. Kurtz, A Random Trotter Product Formula, Proc. Amer. Math. Soc. 35, 147 (1972).
 E. Campbell, Random Compiler for Fast Hamiltonian Simulation, Phys. Rev. Lett. 123, 070503 (2019).
 L. F. Santos and L. Viola, Advantages of Randomization in Coherent Quantum Dynamical Control, New J. Phys. 10, 083009 (2008).
 R. Hillier, C. Arenz, and D. Burgarth, A Continuous-Time Diffusion Limit Theorem for Dynamical Decoupling and Intrinsic Decoherence, J. Phys. A: Math. Theor. 48, 155301 (2015).
 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).
 Daniel Burgarth, Paolo Facchi, and Robin Hillier, "Stability and convergence of dynamical decoupling with finite amplitude control", arXiv:2205.00988.
 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).
The above citations are from Crossref's cited-by service (last updated successfully 2022-10-04 20:17:17) and SAO/NASA ADS (last updated successfully 2022-10-04 20:17:18). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.