Proof of the Error Scaling for Universally Robust Dynamical Decoupling Sequences

TL;DR

This paper provides a rigorous mathematical proof confirming that UR$n$ dynamical decoupling sequences achieve high-order error suppression, validating their effectiveness in compensating pulse imperfections.

Contribution

It offers the first complete proof of the error scaling order for UR$n$ sequences, confirming their high-order robustness analytically.

Findings

UR$n$ sequences achieve error suppression of order n

The phase prescription in UR$n$ satisfies necessary conditions for error cancellation

The proof clarifies the structure responsible for high-order robustness

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