Universally robust dynamical decoupling (UR
n) sequences were proposed to compensate pulse imperfections arising from arbitrary experimental parameters while achieving high-order error suppression with only a linear increase in the number of pulses. Although their performance was supported by analytical arguments, numerical simulations, and experiments, a complete mathematical proof of the claimed order of error compensation has been absent. In this work, we present a rigorous proof for UR
n DD sequences with even
n. Using a series expansion of a quantity whose modulus is the fidelity
F, we derive necessary and sufficient conditions for the cancellation of its coefficients up to, but not including, order
n. The UR
n phase prescription satisfies these conditions, and therefore
1−F=O(εn). Our results establish the UR
n construction on firm analytical grounds and clarify the structure responsible for its high-order robustness.
Dynamical decoupling has been actively investigated since Viola first suggested using a pulse sequence to protect a qubit from decoherence. Since then, many schemes of dynamical decoupling have been proposed to achieve high-order suppression, both analytically and numerically. However, hitherto, there has not been a systematic framework to understand all existing uniform $\ensuremath{\pi}$-pulse dynamical decoupling schemes. In this report, we use the projection pulse sequences as basic building blocks and concatenation as a way to combine them. We derived a concatenated-projection dynamical decoupling, a framework in which we can systematically construct pulse sequences to achieve arbitrary high suppression order. All previously known uniform dynamical decoupling sequences using $\ensuremath{\pi}$ pulse can be fit into this framework. Understanding uniform dynamical decoupling as successive projections on the Hamiltonian will also give insights on how to invent new ways to construct better pulse sequences.