Bang-Bang Refocusing of a Qubit Exposed to Telegraph Noise

Solid state qubits promise the great advantage of being naturally scalable to large quantum computer architectures, but they also possess the significant disadvantage of being intrinsically exposed to many sources of noise in the macroscopic solid-state environment. With suitably chosen systems such as superconductors, many of sources of noise can be suppressed. However, imprecision in nanofabrication will inevitably induce defects and disorder, such as charged impurities in the device material or substrate. Such defects generically produce telegraph noise and can hence be modelled as bistable fluctuators. We demonstrate the possibility of the active suppression of such telegraph noise by bang-bang control through an exhaustive study of a qubit coupled to a single bistable fluctuator. We use a stochastic Schrodinger equation, which is solved both numerically and analytically. The resulting dynamics can be visualized as diffusion of a spin vector on the Bloch sphere. We find that bang-bang control suppresses the effect of a bistable fluctuator by a factor roughly equalling the ratio of the bang-bang period and the typical fluctuator period. Therefore, we show the bang-bang protocol works essentially as a high pass filter on the spectrum of such telegraph noise sources. This suggests how the influence of 1/f -noise ubiquitous to the solid state world could be reduced, as it is typically generated by an ensemble of bistable fluctuators. Finally, we develop random walk models that estimate the level of noise suppression resulting from imperfect bang-bang operations, such as those that cannot be treated as delta-function impulses and those that have phase and axis errors.