Dynamical decoupling

Dynamical decoupling (DD) is an open-loop quantum control technique employed in quantum computing to suppress decoherence by taking advantage of rapid, time-dependent control modulation. In its simplest form, DD is implemented by periodic sequences of instantaneous control pulses, whose net effect is to approximately average the unwanted system-environment coupling to zero. Different schemes exist for designing DD protocols that use realistic bounded-strength control pulses, as well as for achieving high-order error suppression, and for making DD compatible with quantum gates. Commonly used protocols range from the basic Carr-Purcell-Meiboom-Gill (CPMG) sequence to more advanced, non-periodic sequences like Uhrig Dynamical Decoupling (UDD) and recursive, high-order schemes like Concatenated Dynamical Decoupling (CDD). They are based on the Hahn spin echo technique of applying periodic pulses to enable refocusing and hence extend the coherence times of qubits.

Periodic repetition of suitable high-order DD sequences may be employed to engineer a 'stroboscopic saturation' of qubit coherence, or coherence plateau, that can persist in the presence of realistic noise spectra and experimental control imperfections. This permits device-independent, high-fidelity data storage for computationally useful periods with bounded error probability.

Dynamical decoupling has also been studied in a classical context for two coupled pendulums whose oscillation frequencies are modulated in time.

The foundation of most dynamical decoupling sequences is the Hahn spin echo, first discovered in 1950 by Erwin Hahn. The technique was originally developed in the context of nuclear magnetic resonance (NMR), but its principle is general. It is designed to reverse the effects of dephasing caused by slow or static inhomogeneities in the environment.

The process for a single qubit (or spin-1/2 particle) is as follows:

The crucial effect of the π-pulse is that it inverts the accumulated phase. The qubits that were precessing faster and had accumulated more phase now precess "backwards" relative to the slower ones. After the second evolution period of τ, the slower and faster components realign perfectly, leading to a recovery of the quantum coherence in the form of an "echo."

A common analogy is that of a group of runners on a track. They all start at the same line but run at slightly different speeds. After a time τ, they have spread out along the track. If the starter instructs all of them to instantly turn around and run back towards the start at their same individual speeds, the fastest runner, who is furthest away, will also cover the most ground on the return trip. All runners will cross the starting line at the exact same moment, at time 2τ, perfectly regrouped. The π-pulse is the "turn around" command.

The Hahn echo is effective at cancelling noise that is constant or varies very slowly on the timescale of 2τ. However, it is ineffective against noise that fluctuates on a faster timescale.