Effective Hamiltonian models of the cross-resonance gate

Effective Hamiltonian methods are utilized to model the two-qubit cross-resonance gate for both the ideal two-qubit case and when higher levels are included. Analytic expressions are obtained in the qubit case and the higher-level model is solved both perturbatively and numerically, with the solutions agreeing well in the weak-drive limit. The methods are applied to parameters from recent experiments and, accounting for classical cross-talk effects, results in good agreement between theory and experimental results.

Figure 1

All Pauli coefficients excluding $ZI$ as a function of CR drive amplitude for model including higher levels.

Figure 2

$ZI$ Pauli coefficient as a function of CR drive amplitude for model including higher levels.

Figure 3

$IY$, $IZ$, $ZY$, and $ZZ$ Pauli coefficients as a function of CR drive amplitude for model including higher levels.

Figure 4

Perturbative, qubit model, and numerical $ZX$ values as a function of CR drive amplitude.

Figure 5

Two-dimensional $ZX$ coefficient sweep (color scale in MHz).

Figure 6

All Pauli coefficients excluding $ZI$ as a function of CR drive amplitude for model with higher levels and a drive on target representing classical cross talk.

Figure 7

$I\left(H_{\text{eff}}\right)$ parameter as a function of transmon detuning and drive power.