dq.method.Expm

Expm()

Explicit matrix exponentiation to compute propagators.

Explicitly compute the propagators for all time intervals in tsave, then apply them iteratively. On GPU, all propagators are computed in one batch. On CPU, a propagator is computed only when it changes, at a discontinuity of the Hamiltonian or jump operators or for an interval of a different length, and is otherwise reused. The propagators are 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

If the Hamiltonian or jump operators are sparse qarrays, they will be silently converted to dense qarrays before computing their matrix exponentials.

Warning

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

Warning

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

Supported gradients

This method supports differentiation with dq.gradient.Direct (default).