Conditions allowing error correction in driven qubits

We consider a qubit that is driven along its logical $z$ axis, with noise along the $z$ axis in the driving field $\mathrm{\Omega }$ proportional to some function $f\left(\mathrm{\Omega }\right)$, 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 $f\left(\mathrm{\Omega }\right)$ by considering a power-law form, $f\left(\mathrm{\Omega }\right)\propto {\mathrm{\Omega }}^k$. 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 naïve 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.

Figure 1

Plot of the dimensionless pulse paramaters ${\omega }_k$ as a function of the rotation angle $\varphi$ for $f\left(\mathrm{\Omega }\right)\propto {\mathrm{\Omega }}^{2n+1}$ and for $n=-2$ (a), $-1$ (b), 1 (c), and 2 (d). These parameters are obtained by solving Eqs. (33, 34, 35) numerically.

Figure 2

Plot of the infidelity in rotations by $\frac{\pi }{2}$ for $\delta \epsilon =0$ and as a function of $\delta \beta$ (left) and for $\delta \epsilon =\delta \beta$ and as a function of $\sqrt{{\left(\delta \beta \right)}^2+{\left(\delta \epsilon \right)}^2}$ (right) for $f\left(\mathrm{\Omega }\right)={\left(\frac{\mathrm{\Omega }}{{\mathrm{\Omega }}_0}\right)}^{-3}$ and ${\mathrm{\Omega }}_0=0.05\frac{\hslash }{T}$. The blue curves represent the naïve pulse sequences, while the red curves represent the corrected sequences.

Figure 3

Plot of the infidelity in rotations by $\frac{\pi }{2}$ for $\delta \epsilon =0$ and as a function of $\delta \beta$ (left) and for $\delta \epsilon =\delta \beta$ and as a function of $\sqrt{{\left(\delta \beta \right)}^2+{\left(\delta \epsilon \right)}^2}$ (right) for $f\left(\mathrm{\Omega }\right)={\left(\frac{\mathrm{\Omega }}{{\mathrm{\Omega }}_0}\right)}^{-1}$ and ${\mathrm{\Omega }}_0=0.05\frac{\hslash }{T}$. The blue curves represent the naïve pulse sequences, while the red curves represent the corrected sequences.

Figure 4

Plot of the infidelity in rotations by $\frac{\pi }{2}$ for $\delta \epsilon =0$ and as a function of $\delta \beta$ (left) and for $\delta \epsilon =\delta \beta$ and as a function of $\sqrt{{\left(\delta \beta \right)}^2+{\left(\delta \epsilon \right)}^2}$ (right) for $f\left(\mathrm{\Omega }\right)={\left(\frac{\mathrm{\Omega }}{{\mathrm{\Omega }}_0}\right)}^3$ and ${\mathrm{\Omega }}_0=8\frac{\hslash }{T}$. The blue curves represent the naïve pulse sequences, while the red curves represent the corrected sequences.

Figure 5

Plot of the infidelity in rotations by $\frac{\pi }{2}$ for $\delta \epsilon =0$ and as a function of $\delta \beta$ (left) and for $\delta \epsilon =\delta \beta$ and as a function of $\sqrt{{\left(\delta \beta \right)}^2+{\left(\delta \epsilon \right)}^2}$ (right) for $f\left(\mathrm{\Omega }\right)={\left(\frac{\mathrm{\Omega }}{{\mathrm{\Omega }}_0}\right)}^5$ and ${\mathrm{\Omega }}_0=8\frac{\hslash }{T}$. The blue curves represent the naïve pulse sequences, while the red curves represent the corrected sequences.

Figure 6

Plot of the infidelity in rotations by $\frac{\pi }{2}$ for $\delta \epsilon =0$ and as a function of $\delta \beta$ (left) and for $\delta \epsilon =\delta \beta$ and as a function of $\sqrt{{\left(\delta \beta \right)}^2+{\left(\delta \epsilon \right)}^2}$ (right) for the trapezoidal pulse sequences with $\alpha =0.05$. Here, we take $f\left(\mathrm{\Omega }\right)={\left(\frac{\mathrm{\Omega }}{{\mathrm{\Omega }}_0}\right)}^3$ and ${\mathrm{\Omega }}_0=1.975\frac{\hslash }{T^{\prime }}$. The blue curves represent the naïve pulse sequences, while the red curves represent the corrected sequences.