Quadratic dynamical decoupling: Universality proof and error analysis

Abstract

We prove the universality of the generalized QDD${}_{{N}_{1}{N}_{2}}$ (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 ${N}_{1}$ and ${N}_{2}$ 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 $N={N}_{1}={N}_{2}$. A proof of the universality of QDD has been given for even ${N}_{1}$. Here we give a general universality proof of QDD for arbitrary ${N}_{1}$ and ${N}_{2}$. 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 ${N}_{1}$ and ${N}_{2}$. We prove that using QDD${}_{{N}_{1}{N}_{2}}$ decoherence can always be eliminated to $O({T}^{\mathrm{min}{{N}_{1},{N}_{2}}})$.