dq.solver.Expm

Expm()

Explicit matrix exponentiation to compute propagators.

Explicitly batch-compute the propagators for all time intervals in tsave. These propagators are then iteratively applied:

• starting from the initial state for dq.sesolve() and dq.mesolve(), to compute states for all times in tsave,
• starting from the identity matrix for dq.sepropagator() and dq.mepropagator(), to compute propagators for all times in tsave.

For the Schrödinger equation with constant Hamiltonian $H$, the propagator from time $t_0$ to time $t_1$ is an $n\times n$ matrix given by $$U(t_0, t_1) = \exp(-i (t_1 - t_0) H).$$

For the Lindblad master equation with constant Liouvillian $\mathcal{L}$, the problem is vectorized and the propagator from time $t_0$ to time $t_1$ is an $n^2\times n^2$ matrix given by $$\mathcal{U}(t_0, t_1) = \exp((t_1 - t_0)\mathcal{L}).$$

Warning

This solver is not recommended for open systems of large dimension, due to the $\mathcal{O}(n^6)$ scaling of computing the Liouvillian exponential.

Warning

This solver only supports constant or piecewise constant Hamiltonian and jump operators.

Supported gradients

This solver supports differentiation with dq.gradient.Autograd (default).