Grape liouv.m

Gradient Ascent Pulse Engineering (GRAPE) objective function, gradient, and Hessian. Propagates the system through a user-supplied shaped pulse from a given initial state and projects the result onto the given final state. The fidelity is returned, along with its gradient and Hessian with respect to amplitudes of all control operators at every time step of the shaped pulse. Uses Liouville-space formalism.

Syntax

       [traj_data,fidelity,...
        grad,hess]=grape_liouv(spin_system,drifts,controls,...
                               waveform,rho_init,rho_targ,...
                               fidelity_type)

Parameters

  spin_system         - Spinach data object that has been through 
                        the optimcon.m problem setup function.

  drifts              - the drift Liouvillians: a cell array con-
                        taining one matrix (for time-independent 
                        drift) or multiple matrices (one per time
                        slice / point, for time-dependent drift).

  controls            - control operators in Liouville space (cell 
                        array of matrices).

  waveform            - control coefficients for each control ope-
                        rator (in vertical dimension) at each time
                        slice / point (horizonal dimension), rad/s

  rho_init            - initial state of the system as a vector in
                        Liouville space, ignored in stroboscopic
                        steady state optimisations.

  rho_targ            - target state of the system as a vector in
                        Liouville space.

  fidelity_type       - 'real'   (real part of the overlap)
                        'imag'   (imaginary part of the overlap)
                        'square' (absolute square of the overlap)

Returns

  fidelity            - fidelity of the control sequence

  grad                - gradient of the fidelity with respect to
                        the control sequence

  hess                - Hessian of the fidelity with respect to 
                        the control sequence, not available for
                        piecewise-linear and stroboscopic stea-
                        dy state optimisations

  traj_data.forward   - forward trajectory from the initial con-
                        dition or stroboscopic steady state (a 
                        stack of state vectors); this is returned
                        only when requested by the control settings

Notes

This is a low level function that is not designed to be called directly. Use grape_xy.m, grape_phase.m, or other wrapper functions instead.

Trajectory cost terms are read from spin_system.control: when fid_type is 'average', the fidelity is averaged over the pulse nodes 1..N instead of being taken at the last node; traj_pen operators are summed, their expectation value is averaged over the same nodes and subtracted from the fidelity. Both terms use costates that ride on the backward sweep, the trajectory never leaves the worker. Hessians are not available with these terms.

See also

dirdiff.m, step.m, optimcon.m, grape_xy.m, grape_phase.m, penalty.m, grape_coop.m, grape_curv.m, grape_hilb.m, tgrape.m, Optimal_control_module

Version 2.2, authors: Ilya Kuprov, David Goodwin, Uluk Rasulov, Maxi Keitel