Controlling Quantum Systems by Embedded Dynamical Decoupling Schemes

A dynamical decoupling method is presented which is based on embedding a deterministic decoupling scheme into a stochastic one. This way it is possible to combine the advantages of both methods and to increase the suppression of undesired perturbations of quantum systems significantly even for long interaction times. As a first application the stabilization of a quantum memory is discussed which is perturbed by one- and two-qubit interactions.

Figure 1

Schematic representation of the deterministic bang-bang method (a), the PAREC method (b), and an associated embedded decoupling scheme: the time evolution between subsequent instantaneously applied unitary operations is governed by the perturbing Hamiltonian ${\widehat{H}}_0$. In the embedded decoupling scheme (c) the deterministic decoupling cycle of (a) is embedded between any two subsequent random unitary operations of the PAREC method with $\Delta t=T_c\equiv N_c\tau$.

Figure 2

Time evolution of the fidelity $f(T)$: without error correction (full lower curve), resulting from the PAREC method (full upper curve). The dashed curves indicate the approximations of (6, 8). The interaction time $T$ is plotted in units of the time interval $\tau$ between subsequent randomly applied uncorrelated Pauli operations. The rectangular array of the nine qubits constituting the quantum memory is depicted in the inset. The lines indicate the nonzero values of the static coupling parameters $J_{kl}\tau$. They are chosen randomly from the interval $[-\sqrt{3}\times {10}^{-3},\sqrt{3}\times {10}^{-3}]$.

Figure 3

Fidelity decays resulting from the deterministic bang-bang method (full middle curve), the PAREC method ($\Delta t\equiv \tau$; full lower curve), and from the associated embedded decoupling scheme ($\Delta t\equiv T_c=32\tau$; full upper curve) with static coupling parameters $J_{kl}\tau$ as in Fig. 2: the corresponding approximate fidelities are indicated by dashed lines. The interaction time $T$ is plotted in units of the time interval $\tau$ between subsequently applied unitary operations. The inset shows the corresponding double-logarithmic plot.