High-fidelity one-qubit operations under random telegraph noise

We address the problem of implementing high-fidelity one-qubit operations subject to time-dependent noise in the qubit energy splitting. We show with explicit numerical results that high-fidelity bit flip operations may be generated by imposing bounded control fields. For noise correlation times shorter than the time for a $\pi$ pulse, the time-optimal $\pi$ pulse itself yields the highest fidelity. For very long correlation times, fidelity loss is approximately due to systematic error, which is efficiently tackled by compensation for off resonance with a pulse sequence (CORPSE). For intermediate ranges of the noise correlation time, we find that short CORPSE, which is less accurate than CORPSE in correcting systematic errors, yields higher fidelities. Numerical optimization of the pulse sequences using gradient ascent pulse engineering results in noticeable improvement of the fidelity for a bit flip operation on the computational basis states and a small but still positive fidelity enhancement for the NOT gate.

Figure 1

Fidelities $\varphi ({\rho }_{\mathrm{f}},{\rho }_0)$ for bit flip on computational basis states as functions of the noise correlation time ${\tau }_{\mathrm{c}}$, for a $\pi$ pulse [Eq. (15), dotted line], CORPSE [Eq. (16), solid line], and SCORPSE [Eq. (17), dash-dotted line]. The optimized fidelity found using GRAPE is shown as the dashed line. The strength of the RTN in this example is $\Delta =0.125\times a_{\max }$.

Figure 2

Fidelity $\varphi ({\rho }_{\mathrm{f}},{\rho }_0)$ of the GRAPE pulse sequence as a function of the operation time $T$, for noise correlation time ${\tau }_{\mathrm{c}}=5\hslash ∕a_{\max }$ and noise strength $\Delta =0.25\times a_{\max }$.

Figure 3

Error $ϵ({\rho }_{\mathrm{f}},{\rho }_0)=1-\varphi ({\rho }_{\mathrm{f}},{\rho }_0)$ of the GRAPE optimized pulse sequence, shown as a function of the correlation time ${\tau }_{\mathrm{c}}$ for noise strengths $\Delta =0.25\times a_{\max }$ (dotted line), $\Delta =0.125\times a_{\max }$ (solid line), and $\Delta =0.0625\times a_{\max }$ (dash-dotted line).

Figure 4

Fidelities $\Phi$ for the NOT gate, shown as functions of the correlation time ${\tau }_{\mathrm{c}}$ for a $\pi$ pulse (dotted line), CORPSE (solid line), and SCORPSE (dash-dotted line). The figure also shows the optimized fidelity found using the GRAPE algorithm (circles). The RTN strength $\Delta$ is set at $0.125\times a_{\max }$.

Figure 5

Optimized fidelity $\Phi$ for the NOT gate obtained from GRAPE, shown as a function of the operation time $T$ for noise correlation time ${\tau }_{\mathrm{c}}=30\hslash ∕a_{\max }$ and noise strength $\Delta =0.125\times a_{\max }$.

Figure 6

Optimal operation time of pulse sequences obtained numerically with the GRAPE algorithm, shown as a function of the correlation time ${\tau }_{\mathrm{c}}$ for noise strength $\Delta =0.125\times a_{\max }$.