qiskit_dynamics.solvers.solve_lmde

solve_lmde(generator, t_span, y0, method='DOP853', t_eval=None, **kwargs)[source]

General interface for solving Linear Matrix Differential Equations (LMDEs) in standard form.

LMDEs in standard form are differential equations of the form:

\[\dot{y}(t) = G(t)y(t).\]

where \(G(t)\) is a square matrix valued-function called the generator, and \(y(t)\) is an array of appropriate shape.

Thus function accepts \(G(t)\) as a qiskit_dynamics model class, or as an arbitrary callable.

Note

Not all model classes are by-default in standard form. E.g. LindbladModel represents an LMDE which is not typically written in standard form. As such, using LMDE-specific methods with this generator requires the equation to be vectorized.

The method argument exposes solvers specialized to both LMDEs, as well as general ODE solvers. If the method is not specific to LMDEs, the problem will be passed to solve_ode() by automatically setting up the RHS function \(f(t, y) = G(t)y\).

Optional arguments for any of the solver routines can be passed via kwargs. Available LMDE-specific methods are:

  • 'scipy_expm': A fixed-step matrix-exponential solver using scipy.linalg.expm. Requires additional kwarg max_dt indicating the maximum step size to take. This solver will break integration periods into even sub-intervals no larger than max_dt and solve over each sub-interval. The optional kwarg magnus_order controls the integration rule: if magnus_order==1, the generator is sampled at the interval midpoint and exponentiated, and if magnus_order==2 or magnus_order==3, higher-order exponentiation rules are adopted from [1]. The magnus_order parameter defaults to 1.

  • 'lanczos_diag': A fixed-step matrix-exponential solver, similar to 'scipy_expm' but restricted to anti-Hermitian generators. The matrix exponential is performed by diagonalizing an approximate projection of the generator to a small subspace (the Krylov Subspace), obtained via the Lanczos algorithm, and then exponentiating the eigenvalues. Requires additional kwargs max_dt and k_dim indicating the maximum step size to take and Krylov subspace dimension, respectively. k_dim acts as an adjustable accuracy parameter and can be no larger than the dimension of the generator. The method is recommended for sparse systems with large dimension.

  • 'jax_lanczos_diag': JAX implementation of 'lanczos_diag', with the same arguments and behaviour. Note that this method contains calls to jax.numpy.eigh, which may have limited validity when automatically differentiated.

  • 'jax_expm': JAX-implemented version of 'scipy_expm', with the same arguments and behaviour. Note that this method cannot be used for a model using a sparse array library.

  • 'jax_expm_parallel': Same as 'jax_expm', however all loops are implemented using parallel operations. I.e. all matrix-exponentials for taking a single step are computed in parallel using jax.vmap, and are subsequently multiplied together in parallel using jax.lax.associative_scan. This method is only recommended for use with GPU execution. Note that this method cannot be used for a model using a sparse array library.

  • 'jax_RK4_parallel': 4th order Runge-Kutta fixed step solver. Under the assumption of the structure of an LMDE, utilizes the same parallelization approach as 'jax_expm_parallel', however the single step rule is the standard 4th order Runge-Kutta rule, rather than matrix-exponentiation. Requires and utilizes the max_dt kwarg in the same manner as method='scipy_expm'. This method is only recommended for use with GPU execution.

Results are returned as a OdeResult object.

Parameters:

  • generator (Union[Callable, BaseGeneratorModel]) – Representation of generator function \(G(t)\).

  • t_span (Union[ndarray, number, int, float, complex, Tracer, Array, spmatrix, BCOO, list]) – Tuple or list of initial and final time.

  • y0 (Union[ndarray, number, int, float, complex, Tracer, Array, spmatrix, BCOO, list]) – State at initial time.

  • method (Union[str, OdeSolver, TypeVar(AbstractSolver), None]) – Solving method to use.

  • t_eval (Union[ndarray, number, int, float, complex, Tracer, Array, spmatrix, BCOO, list, None]) – Times at which to return the solution. Must lie within t_span. If unspecified, the solution will be returned at the points in t_span.

  • **kwargs – Additional arguments to pass to the solver.

Returns:

Results object.

Return type:

OdeResult

Raises:

QiskitError – If specified method does not exist, if dimension of y0 is incompatible with generator dimension, or if an LMDE-specific method is passed with a LindbladModel.

Additional Information:

While all BaseGeneratorModel subclasses represent LMDEs, they are not all in standard form by defualt. Using an LMDE-specific models like LindbladModel requires first setting the model to be vectorized.