We analyze and compare three different strategies, all aimed at controlling and eventually halting decoherence. The first strategy hinges upon the quantum Zeno effect, the second makes use of frequent unitary interruptions (``bang-bang'' pulses and their generalization, quantum dynamical decoupling), and the third uses a strong, continuous coupling. Decoherence is shown to be suppressed only if the frequency
N of the measurements or pulses is large enough or if the coupling
K is sufficiently strong. Otherwise, if
N or
K is large, but not extremely large, all these control procedures accelerate decoherence. We investigate the problem in a general setting and then consider some practical examples, relevant for quantum computation.
We construct efficient deterministic dynamical decoupling schemes protecting continuous-variable degrees of freedom. Our schemes target decoherence induced by quadratic system-bath interactions with analytic time dependence. We show how to suppress such interactions to Nth order using only N pulses. Furthermore, we show how to homogenize a 2^{m}-mode bosonic system using only (N+1)^{2m+1} pulses, yielding-up to the Nth order-an effective evolution described by noninteracting harmonic oscillators with identical frequencies. The decoupled and homogenized system provides natural decoherence-free subspaces for encoding quantum information. Our schemes only require pulses which are tensor products of single-mode passive Gaussian unitaries and swap gates between pairs of modes.
We introduce a high-order dynamical decoupling (DD) scheme for arbitrary bounded system-bath interactions in the weak-coupling regime. Given any decoupling group
G that averages the interaction to zero, our construction {guarantees the existence of pulse sequences with at most
(∣G∣−1)K pulses}, while canceling all error terms linear in the system-bath coupling strength up to order
K in the total evolution time. As a corollary, for an
n-qubit system with
k-local system-bath interactions, we obtain an
O(nk−1K)-pulse sequence, a significant improvement over existing schemes with
O(exp(n)) pulses (for
k=O(1)). The construction is obtained via a mapping to the continuous necklace-splitting problem, which asks how to cut a multi-colored interval into pieces that give each party the same share of every color. We provide explicit pulse sequences for suppressing general single-qubit decoherence, prove that the pulse count is asymptotically optimal, and verify the predicted error scaling in numerical simulations. For the same number of pulses, we observe that our sequences outperform the state-of-the-art Quadratic DD in the weak-coupling regime. We also construct explicit high-order sequences for suppressing representative 2-local noise models. Finally, the same construction extends to suppress slow, time-dependent classical noise and to filter-function design.
In the framework of nuclear magnetic resonance, we consider the general problem of the coherent control of a spin coupled to a bath by means of composite or continuous pulses of duration ${\ensuremath{\tau}}_{\text{p}}$. We show explicitly that it is possible to design the pulse in order to achieve a decoupling of the spin from the bath up to the third order in ${\ensuremath{\tau}}_{\text{p}}$. The evolution of the system is separated in the evolution of the spin under the action of the pulse and of the bath times correction terms. We derive the correction terms for a general time dependent axis of rotation and for a general coupling between the spin and the environment. The resulting corrections can be made to vanish by an appropriate design of the pulse. For $\ensuremath{\pi}$ and $\ensuremath{\pi}/2$ pulses, we demonstrate explicitly that pulses exist which annihilate the first- and the second-order corrections even if the bath is fully quantum mechanical, i.e., it displays internal dynamics. Such pulses will also be useful for quantum information processing.