34.4 A Cryogenic Controller IC for Superconducting Qubits with DRAG Pulse Generation by Direct Synthesis without Using Memory

Recent achievements in the physical size of quantum computers based on superconducting qubits foresee reaching the next milestones of the exponentially growing number of qubits. The promises in the scalability present opportunities in integrated control electronics operating at 4K stage in the dilution refrigerator. In addition to the number of physical qubits, however, commercialization necessitates a further increase in the number of consecutive high-fidelity gate operations within a coherence time of $\sim 100\mu \mathsf{s}$. It leads to the stringent requirement on the duration of qubit-driving microwave pulse $(\mathsf{i.e}.,\ < 20\mathsf{ns})$. The microwave pulses should only drive $\vert 0\rangle\leftrightarrow\vert 1\rangle$ for single qubit gate operations while avoiding leakage to $\vert 1\rangle\leftrightarrow\vert 2\rangle$ which is the major factor that degrades the gate fidelity. The frequency for $\vert 1\rangle\leftrightarrow\vert 2\rangle, \omega_{12}$, is typically lower than the frequency for $\vert 0\rangle\leftrightarrow\vert 1\rangle, \omega_{01}$, by $2\pi\times(160\mathsf{MHz}-\mathsf{to}-360\mathsf{MHz})$. The frequency difference $(\omega_{01}-\omega_{12})$, called anharmonicity, is an individual qubit characteristic and becomes the margin for spectral management. But, a short driving pulse even with Gaussian, sine, or raised-cosine shapes eventually widens the spectral width. It would range a few hundred MHz and result in a considerable spectral leakage at $\omega_{12}$. To mitigate this spectral leakage, the derivative removal by adiabatic gate (DRAG) has been adopted in the pulse shaping [1], [2]. As shown in Fig. 34.4.1, the first-order DRAG pulse shaping is realized by adding two parts as \begin{equation*}\mathsf{O}(\mathrm{t})=\mathsf{S}_{\mathsf{M}}(\mathsf{t})\cdot\mathsf{cos}(\omega_{01}\mathsf{t}+\varphi)+\mathsf{q}_{\mathsf{scale}}\mathsf{S}_{\mathsf{M}}(\mathsf{t})^{\prime}\cdot\mathsf{sin}(\omega_{01}\mathsf{t}+\varphi)\end{equation*} where $\mathsf{S}_{\mathsf{M}}(\mathsf{t})$ is the main pulse shape and $\mathsf{q}_{\mathsf{scale}}$ is a coefficient inversely proportional to the anharmonicity. This first-order DRAG makes an asymmetrical spectrum that lowers the left frequency side and raises the right frequency side. Since the $\omega_{12}$ is always on the left side of $\omega_{01}$, the DRAG can effectively minimize the leakage.