Universal dynamical decoupling of multiqubit states from environment

We study the dynamical decoupling of multiqubit states from environment. For a system of $m$ qubits, the nested Uhrig dynamical decoupling (NUDD) sequence can efficiently suppress generic decoherence induced by the system-environment interaction to order $N$ using ${(N+1)}^{2m}$ pulses. We prove that the NUDD sequence is universal, i.e., it can restore the coherence of an $m$-qubit quantum system independent of the details of the system-environment interaction. We also construct a general mapping between dynamical decoupling problems and discrete quantum walks in certain functional spaces.

Figure 1

NUDD scheme with $2m$ nesting levels and suppression order $N=2$. (a) The timings of NUDD pulses have a self-similar structure, determined by Eqs. (2) and (10). The set of pulses associated with the $r$th level is ${\sigma }_x^{\left(j\right)}$ for $r=2j-1$, and ${\sigma }_z^{\left(j\right)}$ for $r=2j$. (b) The time-dependent modulation functions $F_{\alpha }\left(t\right)$ of the toggling frame Hamiltonian, and the corresponding pulses for the $m$th qubit.

Figure 2

Evolution under discrete quantum walks for $N=4$. The symbols represent different types of states: ‘S' for the initial state, ‘X' for states explored by a quantum walk, ‘#' for unexplored target states, ‘@' for explored target states, and ‘$·$' for the remaining unexplored functional basis states. At least $N=4$ steps are necessary to reach the target states, which is an illustration of Eq. (14). (a) For UDD, a discrete quantum walk occurs in a one-dimensional functional basis, with initial state $|1\rangle$, target state $|N+1\rangle$, and quantum walk matrix $D_1$. (b) For QDD, a discrete quantum walk occurs in a two-dimensional functional basis, with initial state $|11\rangle$, target states $\left\{\right|{\kappa }^{\#}\rangle \}$, and a quantum walk matrix $D_{\left(01\right)}$.