Generalized cross-resonance scheme for maximally-entangling two-qutrit gates

To utilize higher-dimensional quantum systems, in this Letter, we derive a generalized cross-resonance (GCR) scheme for realizing maximally entangling two-qutrit gates on fixed-frequency transmons beyond the 0-1 subspace. Our two-qutrit gates, namely, $U_{CR}^{01}$ and $U_{CR}^{12}$, acting on the $0{\text -}1$ and $1{\text -}2$ energy transitions of transmons, respectively, directly allow for entanglement on the $1{\text -}2$ levels. Unlike the known works, our gate is parametric in nature, enabling us to construct multiple entangling gates of interest. By performing simulations in Qiskit, we demonstrate two-qutrit generalized controlled-$X$ ($U_{CX}^{01}$ and $U_{CX}^{12}$) and controlled-$H$ ($U_{CH}^{01}$ and $U_{CH}^{12}$) gates, which are instances of the proposed $U_{CR}$ gates, with reported gate fidelities of $86.14\%~(99.73\%),~84.6\%~(97.88\%),~92.35\%~(99.39\%)$, and $91.99\%~(98.99\%)$, respectively with (and without) noise. We also reveal a two-qutrit Bell state with a fidelity of $99.06 \pm 0.01\%$, with a complete Bell state preparation in a $\sim514$ ns pulse sequence, which is less than the gate time of the known scheme by cross-Kerr-based entangling gates.