Proof of the Error Scaling for Universally Robust Dynamical Decoupling Sequences

Universally robust dynamical decoupling (URn) 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 URn 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 URn phase prescription satisfies these conditions, and therefore 1-F=O\(εⁿ\). Our results establish the URn construction on firm analytical grounds and clarify the structure responsible for its high-order robustness.