Construction of arbitrary robust one-qubit operations using planar geometry

We show how to construct an arbitrary robust one-qubit unitary operation with a control Hamiltonian of $A_x\left(t\right){\sigma }_x+A_y\left(t\right){\sigma }_y$, where ${\sigma }_i$ is a Pauli matrix and $A_i\left(t\right)$ is piecewise constant. Our method, based on planar geometry, admits a simple and intuitive interpretation. Furthermore, the total execution time and the number of elementary gates of the obtained sequence are comparable to those of the shortest known concatenated composite pulses.

Figure 1

Planar geometric interpretation of $\mathit{m}=0$. The vectors $\overrightarrow{\mathrm{AB}}={\theta }_2\mathbf{n}\left({\varphi }_1\right),\overrightarrow{\mathrm{BC}}={\theta }_1\mathbf{n}({\varphi }_1+{\varphi }_2),\overrightarrow{\mathrm{CD}}=2\pi \mathbf{n}\left({\varphi }_3\right)$, and $\overrightarrow{\mathrm{DA}}=2\pi \mathbf{n}\left({\varphi }_4\right)$ are the summands in Eq. (17). ${\theta }_1,{\theta }_2$, and $2\pi$ are the lengths of these vectors, while $\mathit{n}\left({\varphi }_i\right)$ indicates their directions.

Figure 2

Infidelity for the composite $Z_{\pi }$ gate, robust against amplitude errors, as a function of the amplitude error $\epsilon$, for a fixed off-resonance error $f$. The different fixed off-resonant errors are given by $f=0$ (solid, red), $f=0.001$ (dashed, black), $f=0.01$ (dash-dotted, blue), and $f=0.1$ (dotted, green).

Figure 3

Infidelity for composite Hadamard gates as a function of the amplitude error $\epsilon$, for a fixed off-resonance error $f$. (a) $\Theta =U_H$ based on the decomposition (25). (b) $U_{\mathrm{H}}$ based on the decomposition (26). The different fixed off-resonant errors are given by $f=0$ (solid, red), $f=0.001$ (dashed, black), $f=0.01$ (dash-dotted, blue), and $f=0.1$ (dotted, green). Note that these Hadamard gates are robust only against amplitude errors.

Figure 4

Infidelity for the nested composite pulses, which have simultaneous robustness against amplitude and off-resonance errors, as a function of the amplitude error $\left(\epsilon \right)$, for a fixed ORE $\left(f\right)$. (a) $Z_{\pi }$ gate. (b) $\Theta =U_H$ based on the decomposition (25). (c) $U_H$ based on the decomposition (26). The different fixed off-resonant errors are given by $f=0$ (solid, red), $f=0.001$ (dashed, black), $f=0.01$ (dash-dotted, blue), and $f=0.1$ (dotted, green).