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Quadratic Dynamical Decoupling: Universality Proof and Error Analysis

We prove the universality of the generalized QDD_{N1,N2} (quadratic dynamical decoupling) pulse sequence for near-optimal suppression of general single-qubit decoherence. Earlier work showed numerically that this dynamical decoupling sequence, which consists of an inner Uhrig DD (UDD) and outer UDD sequence using N1 and N2 pulses respectively, can eliminate decoherence to O(T^N) using O(N^2) unequally spaced "ideal" (zero-width) pulses, where T is the total evolution time and N=N1=N2. A proof of the universality of QDD has been given for even N1. Here we give a general universality proof of QDD for arbitrary N1 and N2. As in earlier proofs, our result holds for arbitrary bounded environments. Furthermore, we explore the single-axis (polarization) error suppression abilities of the inner and outer UDD sequences. We analyze both the single-axis QDD performance and how the overall performance of QDD depends on the single-axis errors. We identify various performance effects related to the parities and relative magnitudes of N1 and N2. We prove that using QDD_{N1,N2} decoherence can always be eliminated to O(T^min[N1,N2]).