One bound to rule them all: from Adiabatic to Zeno

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.

53 pages, 3 figures

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• Unification of Random Dynamical Decoupling and the Quantum Zeno Effect
• Optimal Zeno Dragging for Quantum Control: A Shortcut to Zeno with Action-based Scheduling Optimization
• Grover Speedup from Many Forms of the Zeno Effect
• A numerical approach for calculating exact non-adiabatic terms in quantum dynamics
• Numerical analysis of effective models for flux-tunable transmon systems
• Wave-particle correlations in multiphoton resonances of coherent light-matter interaction
• Generalized speed limits for classical stochastic systems and their applications to relaxation, annealing, and pumping processes
• Zeno-effect Computation: Opportunities and Challenges
• Efficient Quantum Cooling Algorithm for Fermionic Systems
• Coarse-grained effective Hamiltonian via the Magnus Expansion for a three-level system
• Ultrastrong coupling, nonselective measurement and quantum Zeno dynamics
• Error bounds for the Floquet-Magnus expansion and their application to the semiclassical quantum Rabi model
• Efficiency of Dynamical Decoupling for (Almost) Any Spin-Boson Model
• Bath Dynamical Decoupling with a Quantum Channel
• On a sharper bound on the stability of non-autonomous Schrödinger equations and applications to quantum control
• Long-time behavior of multi-level open systems interacting with non-vacuum reservoirs
• Robustness of quantum symmetries against perturbations
• Trajectory-independent speed limits for controlled open quantum systems
• Stability and convergence of dynamical decoupling with finite amplitude control
• Higher-order Zeno sequences
• Efficiency of optimal control for noisy spin qubits in diamond