Effective Hamiltonian of a constant drive

The full time evolution operator \(U_{H}(t) = T\left[\exp(-i \int_0^t dt' H(t'))\right]\) of a driven qudit system is time-dependent and highly nontrivial. However, when the drive amplitude is a constant, at a longer time scale, it should be approximatable with a time evolution by a constant Hamiltonian (= effective Hamiltonian) \(U_{\mathrm{eff}}(t) = \exp(-i H_{\mathrm{eff}} t)\).

Identification of this \(H_{\mathrm{eff}}\) is essentially a linear fit to the time evolution of Pauli components of the generator \(i \mathrm{log} (U_{H}(t))\). However, it’s not possible to actually determine \(H_{\mathrm{eff}}\) from the generator because the latter is multi-valued; there are infinitely many Hermitian matrices whose eigenvalues are separated by multiples of \(2 \pi\) but exponentiate into the same unitary. Therefore, we instead compose an \(H_{\mathrm{eff}}\) ansatz first, multiply it by time and exponentiate it to form the effective time evolution \(U_{\mathrm{eff}}(t)\), and maximize the fidelity

\[\mathcal{F} = \sum_{i} \big| \mathrm{tr} \left[ U_{\mathrm{eff}}(t)^{\dagger} U_{H}(t_i) \right] \big|^2.\]

Ring-up

Applying the full-amplitude drive from the first instant may lead to unnatural oscillations of the generator components and produce spurious terms in \(H_{\mathrm{eff}}\). By increasing the amplitude adiabatically (practically speaking, just slowly), the trajectory of the generator in the space of Hermitians can be placed in a more stable path. The resulting evolution operators is more reminiscent of the actual pulse gates, which must also start from zero amplitude. We therefore use a one-sided GaussianSquare pulse with a very slow turn-on, and perform the \(H_{\mathrm{eff}}\) fit in the plateau of the pulse.