Conditions allowing error correction in driven qubits

TL;DR

This paper investigates conditions under which error correction is possible for driven qubits affected by noise, revealing that error cancellation depends on the functional form of the noise and proposing pulse sequences for correction.

Contribution

It identifies specific functional forms of noise where error correction is feasible and designs pulse sequences to correct errors in driven qubits with quasistatic noise.

Findings

Error correction possible for odd integer powers of the noise function, excluding 1.

A sequence of four square pulses can correct errors for certain rotations.

Error correction schemes are effective even with finite rise time pulses.

Abstract

We consider a qubit that is driven along its logical $z$ axis, with noise along the $z$ axis in the driving field $\Omega$ proportional to some function $f(\Omega)$, as well as noise along the logical $x$ axis. We establish that whether or not errors due to both types of noise can be canceled out, even approximately, depends on the explicit functional form of $f(\Omega)$ by considering a power-law form, $f(\Omega)\propto\Omega^k$. In particular, we show that such cancellation is impossible for $k=0$, $1$, or any even integer. However, any other odd integer value of $k$ besides $1$ does permit cancellation; in fact, we show that both types of errors can be corrected with a sequence of four square pulses of equal duration. We provide sets of parameters that correct for errors for various rotations and evaluate the error, measured by the infidelity, for the corrected rotations versus the na\"ive rotations, i.e., the operations that, in the complete absence of noise, would produce the desired rotations (in this case a single pulse of appropriate duration and magnitude). We also consider a train of four trapezoidal pulses, which take into account the fact that there will be, in real experimental systems, a finite rise time, again providing parameters for error-corrected rotations that employ such pulse sequences. Our dynamical decoupling error correction scheme works for any qubit platform as long as the errors are quasistatic.