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 $O\left(T^N\right)$ using $O\left(N^2\right)$ 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 $O\left(T^{\min \{N_1,N_2\}}\right)$.

• Received 8 June 2011

DOI:https://doi.org/10.1103/PhysRevA.84.042329