Proof of the Error Scaling for Universally Robust Dynamical Decoupling Sequences

2026-04-28

AI Summary

This work addresses the absence of rigorous mathematical proof regarding the higher-order error suppression capability of URₙ dynamical decoupling sequences. By constructing a series expansion of the fidelity modulus, the authors derive necessary and sufficient algebraic conditions under which error coefficients cancel exactly up to order n. For the first time, they establish a rigorous error scaling theory for URₙ sequences with even n, demonstrating that the fidelity under pulse errors satisfies 1 − F = O(εⁿ). This result reveals the intrinsic mathematical structure underlying the phase design of URₙ sequences that enables high-order robustness and provides a solid theoretical foundation for their application in quantum control error suppression.

Abstract

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.

Problem

• dynamical decoupling
• error scaling
• universal robustness
• fidelity
• pulse imperfections

Innovation

• dynamical decoupling
• error suppression
• universal robustness
• fidelity
• pulse imperfections