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 $1-F=O(ε^n)$. Our results establish the UR$n$ construction on firm analytical grounds and clarify the structure responsible for its high-order robustness.

翻译：通用鲁棒动力学去耦（UR$n$）序列被提出，用于补偿任意实验参数导致的脉冲不完美性，同时仅通过线性增加脉冲数量实现高阶误差抑制。尽管其性能已通过解析论证、数值模拟和实验得到支持，但关于其宣称的误差补偿阶数的完整数学证明一直缺失。本文针对偶数$n$的UR$n$ DD序列给出了严格证明。通过展开一个模为保真度$F$的量的级数，我们推导出将其系数消去至$n$阶（不含该阶）的充要条件。UR$n$相位方案满足这些条件，因此$1-F=O(ε^n)$。我们的成果为UR$n$构造奠定了坚实的解析基础，并阐明了其高阶鲁棒性背后的结构。