Usage

using GRAPE
result = optimize(problem; method=GRAPE, kwargs...)

optimizes the given control problem via the GRAPE method, by minimizing the functional

\[J(\{ϵ_{nl}\}) = J_T(\{|Ψ_k(T)⟩\}) + λ_a J_a(\{ϵ_{nl}\})\,,\]

where the final time functional $J_T$ depends explicitly on the forward-propagated states $|Ψ_k(T)⟩$, where $|Ψ_k(t)⟩$ is the time evolution of the initial_state in the $k$th' trajectory in problem.trajectories, and the running cost $J_a$ depends explicitly on pulse values $ϵ_{nl}$ of the l'th control discretized on the n'th interval of the time grid.

It does this by calculating the gradient of the final-time functional

\[\nabla J_T \equiv \frac{\partial J_T}{\partial ϵ_{nl}} = -2 \Re \underbrace{% \underbrace{\bigg\langle χ(T) \bigg\vert \hat{U}^{(k)}_{N_T} \dots \hat{U}^{(k)}_{n+1} \bigg \vert}_{\equiv \bra{\chi(t_n)}\;\text{(bw. prop.)}} \frac{\partial \hat{U}^{(k)}_n}{\partial ϵ_{nl}} }_{\equiv \bra{χ_k^\prime(t_{n-1})}} \underbrace{\bigg \vert \hat{U}^{(k)}_{n-1} \dots \hat{U}^{(k)}_1 \bigg\vert Ψ_k(t=0) \bigg\rangle}_{\equiv |\Psi(t_{n-1})⟩\;\text{(fw. prop.)}}\,,\]

where $\hat{U}^{(k)}_n$ is the time evolution operator for the $n$ the interval, generally assumed to be $\hat{U}^{(k)}_n = \exp[-i \hat{H}_{kn} dt_n]$, where $\hat{H}_{kn}$ is the operator obtained by evaluating problem.trajectories[k].generator on the $n$'th time interval.

The backward-propagation of $|\chi_k(t)⟩$ has the boundary condition

\[ |\chi_k(T)⟩ \equiv - \frac{\partial J_T}{\partial ⟨\Psi_k(T)|}\,.\]

The final-time gradient $\nabla J_T$ is combined with the gradient for the running costs, and the total gradient is then fed into an optimizer (L-BFGS-B by default) that iteratively changes the values $\{ϵ_{nl}\}$ to minimize $J$.

See Background for details.

Returns a GrapeResult.

Keyword arguments that control the optimization are taken from the keyword arguments used in the instantiation of problem; any of these can be overridden with explicit keyword arguments to optimize.

Required problem keyword arguments

• J_T: A function J_T(Ψ, trajectories) that evaluates the final time functional from a list Ψ of forward-propagated states and problem.trajectories. The function J_T may also take a keyword argument tau. If it does, a vector containing the complex overlaps of the target states (target_state property of each trajectory in problem.trajectories) with the propagated states will be passed to J_T.

Optional problem keyword arguments

• chi: A function chi(Ψ, trajectories) that receives a list Ψ of the forward propagated states and returns a vector of states $|χₖ⟩ = -∂J_T/∂⟨Ψₖ|$. If not given, it will be automatically determined from J_T via QuantumControl.Functionals.make_chi with the default parameters. Similarly to J_T, if chi accepts a keyword argument tau, it will be passed a vector of complex overlaps.

• chi_min_norm=1e-100: The minimum allowable norm for any $|χₖ(T)⟩$. Smaller norms would mean that the gradient is zero, and will abort the optimization with an error.

• J_a: A function J_a(pulsevals, tlist) that evaluates running costs over the pulse values, where pulsevals are the vectorized values $ϵ_{nl}$, where n are in indices of the time intervals and l are the indices over the controls, i.e., [ϵ₁₁, ϵ₂₁, …, ϵ₁₂, ϵ₂₂, …] (the pulse values for each control are contiguous). If not given, the optimization will not include a running cost.

• gradient_method=:gradgen: One of :gradgen (default) or :taylor. With gradient_method=:gradgen, the gradient is calculated using QuantumGradientGenerators. With gradient_method=:taylor, it is evaluated via a Taylor series, see Eq. (20) in Kuprov and Rogers, J. Chem. Phys. 131, 234108 (2009) [22].

• taylor_grad_max_order=100: If given with gradient_method=:taylor, the maximum number of terms in the Taylor series. If taylor_grad_check_convergence=true (default), if the Taylor series does not convergence within the given number of terms, throw an an error. With taylor_grad_check_convergence=true, this is the exact order of the Taylor series.

• taylor_grad_tolerance=1e-16: If given with gradient_method=:taylor and taylor_grad_check_convergence=true, stop the Taylor series when the norm of the term falls below the given tolerance. Ignored if taylor_grad_check_convergence=false.

• taylor_grad_check_convergence=true: If given as true (default), check the convergence after each term in the Taylor series an stop as soon as the norm of the term drops below the given number. If false, stop after exactly taylor_grad_max_order terms.

• lambda_a=1: A weight for the running cost J_a.

• grad_J_a: A function to calculate the gradient of J_a. If not given, it will be automatically determined. See make_grad_J_a for the required interface.

• upper_bound: An upper bound for the value of any optimized control. Time-dependent upper bounds can be specified via pulse_options.

• lower_bound: A lower bound for the value of any optimized control. Time-dependent lower bounds can be specified via pulse_options.

• pulse_options: A dictionary that maps every control (as obtained by get_controls from the problem.trajectories) to a dict with the following possible keys:

  • :upper_bounds: A vector of upper bound values, one for each intervals of the time grid. Values of Inf indicate an unconstrained upper bound for that time interval, respectively the global upper_bound, if given.
  • :lower_bounds: A vector of lower bound values. Values of -Inf indicate an unconstrained lower bound for that time interval,

• print_iters=true: Whether to print information after each iteration.

• print_iter_info=["iter.", "J_T", "|∇J|", "|Δϵ|", "ΔJ", "FG(F)", "secs"]: Which fields to print if print_iters=true. If given, must be a list of header labels (strings), which can be any of the following:

  • "iter.": The iteration number

  • "J_T": The value of the final-time functional for the dynamics under the optimized pulses

  • "J_a": The value of the pulse-dependent running cost for the optimized pulses

  • "λ_a⋅J_a": The total contribution of J_a to the full functional J

  • "J": The value of the optimization functional for the optimized pulses

  • "ǁ∇J_Tǁ": The ℓ²-norm of the current gradient of the final-time functional. Note that this is usually the gradient of the optimize pulse, not the guess pulse.

  • "ǁ∇J_aǁ": The ℓ²-norm of the the current gradient of the pulse-dependent running cost. For comparison with "ǁ∇J_Tǁ".

  • "λ_aǁ∇J_aǁ": The ℓ²-norm of the the current gradient of the complete pulse-dependent running cost term. For comparison with "ǁ∇J_Tǁ".

  • "ǁ∇Jǁ": The norm of the guess pulse gradient. Note that the guess pulse gradient is not the same the current gradient.

  • "ǁΔϵǁ": The ℓ²-norm of the pulse update

  • "ǁϵǁ": The ℓ²-norm of optimized pulse values

  • "max|Δϵ|" The maximum value of the pulse update (infinity norm)

  • "max|ϵ|": The maximum value of the pulse values (infinity norm)

  • "ǁΔϵǁ/ǁϵǁ": The ratio of the pulse update tothe optimized pulse values

  • "∫Δϵ²dt": The L²-norm of the pulse update, summed over all pulses. A convergence measure comparable (proportional) to the running cost in Krotov's method

  • "ǁsǁ": The norm of the search direction. Should be ǁΔϵǁ scaled by the step with α.

  • "∠°": The angle (in degrees) between the negative gradient -∇J and the search direction s.

  • "α": The step width as determined by the line search (Δϵ = α⋅s)

  • "ΔJ_T": The change in the final time functional relative to the previous iteration

  • "ΔJ_a": The change in the control-dependent running cost relative to the previous iteration

  • "λ_a⋅ΔJ_a": The change in the control-dependent running cost term relative to the previous iteration.

  • "ΔJ": The change in the total optimization functional relative to the previous iteration.

  • "FG(F)": The number of functional/gradient evaluation (FG), or pure functional (F) evaluations

  • "secs": The number of seconds of wallclock time spent on the iteration.

  • store_iter_info=[]: Which fields to store in result.records, given as

  a list of header labels, see print_iter_info.

• callback: A function (or tuple of functions) that receives the GRAPE workspace and the iteration number. The function may return a tuple of values which are stored in the GrapeResult object result.records. The function can also mutate the workspace, in particular the updated pulsevals. This may be used, e.g., to apply a spectral filter to the updated pulses or to perform similar manipulations. Note that print_iters=true (default) adds an automatic callback to print information after each iteration. With store_iter_info, that callback automatically stores a subset of the available information.

• check_convergence: A function to check whether convergence has been reached. Receives a GrapeResult object result, and should set result.converged to true and result.message to an appropriate string in case of convergence. Multiple convergence checks can be performed by chaining functions with ∘. The convergence check is performed after any callback.

• prop_method: The propagation method to use for each trajectory, see below.

• verbose=false: If true, print information during initialization

• rethrow_exceptions: By default, any exception ends the optimization, but still returns a GrapeResult that captures the message associated with the exception. This is to avoid losing results from a long-running optimization when an exception occurs in a later iteration. If rethrow_exceptions=true, instead of capturing the exception, it will be thrown normally.

Experimental keyword arguments

The following keyword arguments may change in non-breaking releases:

• x_tol: Parameter for Optim.jl
• f_tol: Parameter for Optim.jl
• g_tol: Parameter for Optim.jl
• show_trace: Parameter for Optim.jl
• extended_trace: Parameter for Optim.jl
• show_every: Parameter for Optim.jl
• allow_f_increases: Parameter for Optim.jl
• optimizer: An optional Optim.jl optimizer (Optim.AbstractOptimizer instance). If not given, an L-BFGS-B optimizer will be used.

Trajectory propagation

GRAPE may involve three types of propagation:

• A forward propagation for every Trajectory in the problem
• A backward propagation for every trajectory
• A backward propagation of a gradient generator for every trajectory.

The keyword arguments for each propagation (see propagate) are determined from any properties of each Trajectory that have a prop_ prefix, cf. init_prop_trajectory.

In situations where different parameters are required for the forward and backward propagation, instead of the prop_ prefix, the fw_prop_ and bw_prop_ prefix can be used, respectively. These override any setting with the prop_ prefix. Similarly, properties for the backward propagation of the gradient generators can be set with properties that have a grad_prop_ prefix. These prefixes apply both to the properties of each Trajectory and the problem keyword arguments.

Note that the propagation method for each propagation must be specified. In most cases, it is sufficient (and recommended) to pass a global prop_method problem keyword argument.