Abstract: The present invention is a method for constructing fully-compensating composite pulses of arbitrary accuracy that can correct pulse power errors to any desired order. The invention has two preferred embodiments, the Trotter-Suzuki (TS) embodiment and the Solovay-Kitaev (SK) embodiment. Each preferred embodiment can be represented by a mathematical algorithm, which can be easily implemented on a computer, for constructing a list of phases {?1, . . . , ?l} and rotation angles {?1, . . . , ?l} that amounts to a composite pulse. The pulses can then be used for nuclear magnetic resonance, quantum computing, or any other application based on pulsed excitations. The present invention, as embodied in the pulse sequences SK n and TS n, allows the user to transform a pulse sequence of error ? to a robust pulse sequence with error O(?n) for arbitrary n. The sequences can be implemented using any existing quantum computer or NMR hardware.