For and , the Magnus expansion writes the logarithm of the propagator as iterated integrals of commutators. Its first terms are and . For a Quantum Hamiltonian, , so a noncommuting time-dependent control has corrections beyond the exponential of its integrated Quantum Hamiltonian. This formula is a local convergent expansion under suitable smallness conditions, or a formal expansion; it must not be assumed globally convergent for arbitrary pulse durations.
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The Magnus expansion is a mathematical technique used in the field of differential equations and quantum mechanics to solve time-dependent problems involving linear differential equations. Specifically, it is often applied to systems governed by operators that evolve in time, which is particularly relevant in quantum mechanics for the evolution of state vectors and operators. In essence, the Magnus expansion provides a way to express the time-evolution operator \( U(t) \), which describes how a state changes over time under the influence of a Hamiltonian or other operator.