Pulse Parametrization for Krotov's Method

$\gdef\op#1{\hat{#1}}$ $\gdef\init{\text{init}}$ $\gdef\tgt{\text{tgt}}$

This example illustrates the parametrization of control pulses as a form of amplitude constraint.

datadir(names...) = joinpath(@__DIR__, names...);
using QuantumControl
using QuantumControl.Shapes: flattop
using QuantumControl.Generators
using QuantumControl.Controls
using QuantumControl.PulseParametrizations:
    SquareParametrization,
    TanhParametrization,
    TanhSqParametrization,
    LogisticParametrization,
    LogisticSqParametrization,
    ParametrizedAmplitude
using QuantumPropagators: ExpProp
using LinearAlgebra
using Plots
Plots.default(
    linewidth               = 3,
    size                    = (550, 300),
    legend                  = :top,
    foreground_color_legend = nothing,
    background_color_legend = RGBA(1, 1, 1, 0.8),
)

Parametrizations

Symmetric Bounded Controls

Positive (Bounded) Controls

Two-level Hamiltonian

We consider the Hamiltonian $\op{H}_{0} = - \frac{\omega}{2} \op{\sigma}_{z}$, representing a simple qubit with energy level splitting $\omega$ in the basis $\{\ket{0},\ket{1}\}$. The control field $\epsilon(t)$ is assumed to couple via the Hamiltonian $\op{H}_{1}(t) = \epsilon(t) \op{\sigma}_{x}$ to the qubit, i.e., the control field effectively drives transitions between both qubit states.

We we will use

ϵ(t) = 0.2 * flattop(t, T=5, t_rise=0.3, func=:blackman);
"""Two-level-system Hamiltonian."""
function tls_hamiltonian(; Ω=1.0, ampl=ϵ)
    σ̂_z = ComplexF64[
        1  0
        0 -1
    ]
    σ̂_x = ComplexF64[
        0  1
        1  0
    ]
    Ĥ₀ = -0.5 * Ω * σ̂_z
    Ĥ₁ = σ̂_x
    return hamiltonian(Ĥ₀, (Ĥ₁, ampl))
end;
H = tls_hamiltonian();

The control field here switches on from zero at $t=0$ to it's maximum amplitude 0.2 within the time period 0.3 (the switch-on shape is half a Blackman pulse). It switches off again in the time period 0.3 before the final time $T=5$). We use a time grid with 500 time steps between 0 and $T$:

tlist = collect(range(0, 5, length=500));
function plot_amplitude(ampl, tlist)
    plot(tlist, discretize(ampl, tlist), xlabel="time", ylabel="amplitude", legend=false)
end

fig = plot_amplitude(ϵ, tlist)

Optimization target

The Krotov package requires the optimization to be expressed over a set of "trajectories". In this example, there is only a single trajectory: the state-to-state transfer from initial state $\ket{\Psi_{\init}} = \ket{0}$ to the target state $\ket{\Psi_{\tgt}} = \ket{1}$, under the dynamics of the Hamiltonian $\op{H}(t)$:

function ket(label)
    result = Dict("0" => Vector{ComplexF64}([1, 0]), "1" => Vector{ComplexF64}([0, 1]),)
    return result[string(label)]
end;
trajectories = [Trajectory(ket(0), H, target_state=ket(1))]
1-element Vector{QuantumControlBase.Trajectory{Vector{ComplexF64}, QuantumPropagators.Generators.Generator{Matrix{ComplexF64}, typeof(Main.var"##225".ϵ)}}}:
 Trajectory with 2-element Vector{ComplexF64} initial state, Generator with 2 ops and 1 amplitudes, 2-element Vector{ComplexF64} target state

Square-parametrization for positive pulses

a = ParametrizedAmplitude(
    ϵ,
    tlist;
    parametrization=SquareParametrization(),
    parameterize=true
)
ParametrizedAmplitude(::Vector{Float64}; parametrization=SquareParametrization())
function plot_amplitude(ampl::ParametrizedAmplitude, tlist)
    plot(
        tlist,
        discretize(Array(ampl), tlist),
        xlabel="time",
        ylabel="amplitude",
        legend=false
    )
end

fig = plot_amplitude(a, tlist)

problem = ControlProblem(
    trajectories=substitute(trajectories, IdDict(ϵ => a)),
    prop_method=ExpProp,
    lambda_a=5,
    update_shape=(t -> flattop(t, T=5, t_rise=0.3, func=:blackman)),
    tlist=tlist,
    iter_stop=50,
    J_T=QuantumControl.Functionals.J_T_ss,
    check_convergence=res -> begin
        ((res.J_T < 1e-3) && (res.converged = true) && (res.message = "J_T < 10⁻³"))
    end
);
using Krotov
opt_result_positive = @optimize_or_load(
    datadir("parametrization#opt_result_positive.jld2"),
    problem;
    method=Krotov
);
 iter.        J_T   ∫gₐ(t)dt          J       ΔJ_T         ΔJ    secs
     0   9.51e-01   0.00e+00   9.51e-01        n/a        n/a     3.5
[ Info: Set callback to store result in parametrization#opt_result_positive.jld2 on unexpected exit.
     1   9.31e-01   9.29e-03   9.40e-01  -2.05e-02  -1.12e-02     1.6
     2   9.01e-01   1.34e-02   9.15e-01  -2.96e-02  -1.63e-02     0.0
     3   8.59e-01   1.91e-02   8.78e-01  -4.25e-02  -2.34e-02     0.1
     4   7.99e-01   2.67e-02   8.26e-01  -5.96e-02  -3.29e-02     0.0
     5   7.20e-01   3.56e-02   7.56e-01  -7.91e-02  -4.35e-02     0.0
     6   6.25e-01   4.33e-02   6.68e-01  -9.52e-02  -5.19e-02     0.0
     7   5.25e-01   4.62e-02   5.72e-01  -9.96e-02  -5.33e-02     0.0
     8   4.36e-01   4.25e-02   4.79e-01  -8.92e-02  -4.66e-02     0.0
     9   3.65e-01   3.49e-02   4.00e-01  -7.14e-02  -3.64e-02     0.0
    10   3.10e-01   2.74e-02   3.37e-01  -5.48e-02  -2.74e-02     0.0
    11   2.68e-01   2.14e-02   2.89e-01  -4.24e-02  -2.09e-02     0.0
    12 …
    37   5.05e-02   8.59e-04   5.13e-02  -1.70e-03  -8.45e-04     0.0
    38   4.89e-02   8.05e-04   4.97e-02  -1.60e-03  -7.93e-04     0.0
    39   4.74e-02   7.56e-04   4.81e-02  -1.50e-03  -7.45e-04     0.0
    40   4.60e-02   7.12e-04   4.67e-02  -1.41e-03  -7.02e-04     0.0
    41   4.46e-02   6.71e-04   4.53e-02  -1.33e-03  -6.62e-04     0.0
    42   4.34e-02   6.34e-04   4.40e-02  -1.26e-03  -6.26e-04     0.0
    43   4.22e-02   5.99e-04   4.28e-02  -1.19e-03  -5.92e-04     0.0
    44   4.11e-02   5.68e-04   4.16e-02  -1.13e-03  -5.61e-04     0.0
    45   4.00e-02   5.38e-04   4.05e-02  -1.07e-03  -5.32e-04     0.0
    46   3.90e-02   5.11e-04   3.95e-02  -1.02e-03  -5.06e-04     0.0
    47   3.80e-02   4.86e-04   3.85e-02  -9.67e-04  -4.81e-04     0.0
    48   3.71e-02   4.63e-04   3.75e-02  -9.21e-04  -4.58e-04     0.0
    49   3.62e-02   4.41e-04   3.66e-02  -8.78e-04  -4.37e-04     0.0
    50   3.54e-02   4.21e-04   3.58e-02  -8.38e-04  -4.17e-04     0.0

opt_result_positive
Krotov Optimization Result
--------------------------
- Started at 2024-05-23T12:07:34.649
- Number of trajectories: 1
- Number of iterations: 50
- Value of functional: 3.53585e-02
- Reason for termination: Reached maximum number of iterations
- Ended at 2024-05-23T12:07:40.377 (5 seconds, 728 milliseconds)

We can plot the optimized field:

fig = plot_amplitude(
    substitute(a, IdDict(a.control => opt_result_positive.optimized_controls[1])),
    tlist
)

Tanh-Square-Parametrization for positive amplitude-constrained pulses

a = ParametrizedAmplitude(
    ϵ,
    tlist;
    parametrization=TanhSqParametrization(3),
    parameterize=true
)

problem_tanhsq = ControlProblem(
    trajectories=substitute(trajectories, IdDict(ϵ => a)),
    prop_method=ExpProp,
    lambda_a=10,
    update_shape=(t -> flattop(t, T=5, t_rise=0.3, func=:blackman)),
    tlist=tlist,
    iter_stop=50,
    J_T=QuantumControl.Functionals.J_T_ss,
    check_convergence=res -> begin
        ((res.J_T < 1e-3) && (res.converged = true) && (res.message = "J_T < 10⁻³"))
    end
);
opt_result_tanhsq = @optimize_or_load(
    datadir("parametrization#opt_result_tanhsq.jld2"),
    problem_tanhsq;
    method=Krotov
);
 iter.        J_T   ∫gₐ(t)dt          J       ΔJ_T         ΔJ    secs
     0   9.51e-01   0.00e+00   9.51e-01        n/a        n/a     0.1
[ Info: Set callback to store result in parametrization#opt_result_tanhsq.jld2 on unexpected exit.
     1   9.24e-01   1.21e-02   9.36e-01  -2.76e-02  -1.54e-02     0.0
     2   8.81e-01   1.90e-02   9.00e-01  -4.32e-02  -2.42e-02     0.0
     3   8.15e-01   2.88e-02   8.44e-01  -6.54e-02  -3.65e-02     0.0
     4   7.24e-01   4.06e-02   7.65e-01  -9.08e-02  -5.02e-02     0.0
     5   6.16e-01   4.99e-02   6.65e-01  -1.09e-01  -5.90e-02     0.0
     6   5.08e-01   5.11e-02   5.59e-01  -1.08e-01  -5.66e-02     0.0
     7   4.19e-01   4.38e-02   4.62e-01  -8.92e-02  -4.54e-02     0.0
     8   3.52e-01   3.37e-02   3.85e-01  -6.69e-02  -3.33e-02     0.0
     9   3.02e-01   2.51e-02   3.28e-01  -4.92e-02  -2.41e-02     0.0
    10   2.66e-01   1.89e-02   2.84e-01  -3.69e-02  -1.79e-02     0.0
    11   2.37e-01   1.46e-02   2.52e-01  -2.84e-02  -1.38e-02     0.0
    12   …
    37   8.06e-02   7.52e-04   8.14e-02  -1.49e-03  -7.37e-04     0.0
    38   7.92e-02   7.10e-04   8.00e-02  -1.41e-03  -6.96e-04     0.0
    39   7.79e-02   6.72e-04   7.86e-02  -1.33e-03  -6.59e-04     0.0
    40   7.67e-02   6.36e-04   7.73e-02  -1.26e-03  -6.24e-04     0.0
    41   7.55e-02   6.03e-04   7.61e-02  -1.20e-03  -5.92e-04     0.0
    42   7.43e-02   5.73e-04   7.49e-02  -1.14e-03  -5.63e-04     0.0
    43   7.32e-02   5.45e-04   7.38e-02  -1.08e-03  -5.35e-04     0.0
    44   7.22e-02   5.19e-04   7.27e-02  -1.03e-03  -5.10e-04     0.0
    45   7.12e-02   4.95e-04   7.17e-02  -9.81e-04  -4.87e-04     0.0
    46   7.03e-02   4.72e-04   7.08e-02  -9.37e-04  -4.65e-04     0.0
    47   6.94e-02   4.52e-04   6.98e-02  -8.96e-04  -4.44e-04     0.0
    48   6.85e-02   4.32e-04   6.90e-02  -8.57e-04  -4.25e-04     0.0
    49   6.77e-02   4.14e-04   6.81e-02  -8.21e-04  -4.07e-04     0.0
    50   6.69e-02   3.97e-04   6.73e-02  -7.87e-04  -3.91e-04     0.0

opt_result_tanhsq
Krotov Optimization Result
--------------------------
- Started at 2024-05-23T12:07:43.054
- Number of trajectories: 1
- Number of iterations: 50
- Value of functional: 6.69321e-02
- Reason for termination: Reached maximum number of iterations
- Ended at 2024-05-23T12:07:43.613 (559 milliseconds)

We can plot the optimized field:

fig = plot_amplitude(
    substitute(a, IdDict(a.control => opt_result_tanhsq.optimized_controls[1])),
    tlist
)

Logistic-Square-Parametrization for positive amplitude-constrained pulses

a = ParametrizedAmplitude(
    ϵ,
    tlist;
    parametrization=LogisticSqParametrization(3, k=1.0),
    parameterize=true
)

problem_logisticsq = ControlProblem(
    trajectories=substitute(trajectories, IdDict(ϵ => a)),
    prop_method=ExpProp,
    lambda_a=1,
    update_shape=(t -> flattop(t, T=5, t_rise=0.3, func=:blackman)),
    tlist=tlist,
    iter_stop=50,
    J_T=QuantumControl.Functionals.J_T_ss,
    check_convergence=res -> begin
        ((res.J_T < 1e-3) && (res.converged = true) && (res.message = "J_T < 10⁻³"))
    end
);
opt_result_logisticsq = @optimize_or_load(
    datadir("parametrization#opt_result_logisticsq.jld2"),
    problem_logisticsq;
    method=Krotov
);
 iter.        J_T   ∫gₐ(t)dt          J       ΔJ_T         ΔJ    secs
     0   9.51e-01   0.00e+00   9.51e-01        n/a        n/a     0.1
[ Info: Set callback to store result in parametrization#opt_result_logisticsq.jld2 on unexpected exit.
     1   8.72e-01   2.97e-02   9.02e-01  -7.91e-02  -4.93e-02     0.0
     2   6.93e-01   6.94e-02   7.63e-01  -1.79e-01  -1.10e-01     0.0
     3   4.57e-01   1.03e-01   5.60e-01  -2.36e-01  -1.33e-01     0.0
     4   3.07e-01   7.51e-02   3.82e-01  -1.50e-01  -7.49e-02     0.0
     5   2.31e-01   4.00e-02   2.71e-01  -7.60e-02  -3.60e-02     0.0
     6   1.87e-01   2.31e-02   2.10e-01  -4.37e-02  -2.06e-02     0.0
     7   1.59e-01   1.48e-02   1.74e-01  -2.82e-02  -1.34e-02     0.0
     8   1.39e-01   1.03e-02   1.49e-01  -1.97e-02  -9.38e-03     0.0
     9   1.25e-01   7.53e-03   1.32e-01  -1.45e-02  -6.93e-03     0.0
    10   1.14e-01   5.74e-03   1.19e-01  -1.11e-02  -5.33e-03     0.0
    11   1.05e-01   4.52e-03   1.09e-01  -8.75e-03  -4.23e-03     0.0
    1…
    37   4.98e-02   3.05e-04   5.01e-02  -6.03e-04  -2.99e-04     0.0
    38   4.92e-02   2.88e-04   4.95e-02  -5.71e-04  -2.83e-04     0.0
    39   4.87e-02   2.74e-04   4.89e-02  -5.42e-04  -2.68e-04     0.0
    40   4.81e-02   2.60e-04   4.84e-02  -5.15e-04  -2.55e-04     0.0
    41   4.76e-02   2.47e-04   4.79e-02  -4.90e-04  -2.43e-04     0.0
    42   4.72e-02   2.35e-04   4.74e-02  -4.66e-04  -2.31e-04     0.0
    43   4.67e-02   2.24e-04   4.70e-02  -4.45e-04  -2.20e-04     0.0
    44   4.63e-02   2.14e-04   4.65e-02  -4.25e-04  -2.11e-04     0.0
    45   4.59e-02   2.05e-04   4.61e-02  -4.06e-04  -2.01e-04     0.0
    46   4.55e-02   1.96e-04   4.57e-02  -3.88e-04  -1.93e-04     0.0
    47   4.51e-02   1.87e-04   4.53e-02  -3.72e-04  -1.84e-04     0.0
    48   4.48e-02   1.80e-04   4.50e-02  -3.57e-04  -1.77e-04     0.0
    49   4.44e-02   1.72e-04   4.46e-02  -3.42e-04  -1.70e-04     0.0
    50   4.41e-02   1.66e-04   4.43e-02  -3.29e-04  -1.63e-04     0.0

We can plot the optimized field:

fig = plot_amplitude(
    substitute(a, IdDict(a.control => opt_result_logisticsq.optimized_controls[1])),
    tlist
)

Tanh-parametrization for amplitude-constrained pulses

a = ParametrizedAmplitude(
    ϵ,
    tlist;
    parametrization=TanhParametrization(-0.5, 0.5),
    parameterize=true
)

problem_tanh = ControlProblem(
    trajectories=substitute(trajectories, IdDict(ϵ => a)),
    prop_method=ExpProp,
    lambda_a=1,
    update_shape=(t -> flattop(t, T=5, t_rise=0.3, func=:blackman)),
    tlist=tlist,
    iter_stop=50,
    J_T=QuantumControl.Functionals.J_T_ss,
    check_convergence=res -> begin
        ((res.J_T < 1e-3) && (res.converged = true) && (res.message = "J_T < 10⁻³"))
    end
);
opt_result_tanh = @optimize_or_load(
    datadir("parametrization#opt_result_tanh.jld2"),
    problem_tanh;
    method=Krotov
);
 iter.        J_T   ∫gₐ(t)dt          J       ΔJ_T         ΔJ    secs
     0   9.51e-01   0.00e+00   9.51e-01        n/a        n/a     0.1
[ Info: Set callback to store result in parametrization#opt_result_tanh.jld2 on unexpected exit.
     1   9.27e-01   1.09e-02   9.38e-01  -2.40e-02  -1.31e-02     0.0
     2   8.93e-01   1.56e-02   9.09e-01  -3.42e-02  -1.87e-02     0.0
     3   8.46e-01   2.18e-02   8.67e-01  -4.77e-02  -2.59e-02     0.0
     4   7.82e-01   2.94e-02   8.11e-01  -6.37e-02  -3.43e-02     0.0
     5   7.03e-01   3.70e-02   7.40e-01  -7.89e-02  -4.18e-02     0.0
     6   6.16e-01   4.18e-02   6.58e-01  -8.68e-02  -4.50e-02     0.0
     7   5.33e-01   4.12e-02   5.74e-01  -8.33e-02  -4.21e-02     0.0
     8   4.62e-01   3.59e-02   4.98e-01  -7.11e-02  -3.52e-02     0.0
     9   4.05e-01   2.89e-02   4.34e-01  -5.66e-02  -2.77e-02     0.0
    10   3.61e-01   2.26e-02   3.84e-01  -4.40e-02  -2.14e-02     0.0
    11   3.27e-01   1.76e-02   3.44e-01  -3.43e-02  -1.67e-02     0.0
    12   3.…
    37   1.35e-01   9.69e-04   1.36e-01  -1.92e-03  -9.52e-04     0.0
    38   1.33e-01   9.17e-04   1.34e-01  -1.82e-03  -9.02e-04     0.0
    39   1.31e-01   8.70e-04   1.32e-01  -1.73e-03  -8.56e-04     0.0
    40   1.29e-01   8.26e-04   1.30e-01  -1.64e-03  -8.13e-04     0.0
    41   1.28e-01   7.86e-04   1.29e-01  -1.56e-03  -7.74e-04     0.0
    42   1.26e-01   7.49e-04   1.27e-01  -1.49e-03  -7.38e-04     0.0
    43   1.25e-01   7.14e-04   1.26e-01  -1.42e-03  -7.04e-04     0.0
    44   1.24e-01   6.82e-04   1.24e-01  -1.35e-03  -6.73e-04     0.0
    45   1.22e-01   6.52e-04   1.23e-01  -1.30e-03  -6.43e-04     0.0
    46   1.21e-01   6.24e-04   1.22e-01  -1.24e-03  -6.16e-04     0.0
    47   1.20e-01   5.98e-04   1.21e-01  -1.19e-03  -5.91e-04     0.0
    48   1.19e-01   5.74e-04   1.19e-01  -1.14e-03  -5.67e-04     0.0
    49   1.18e-01   5.51e-04   1.18e-01  -1.10e-03  -5.45e-04     0.0
    50   1.17e-01   5.30e-04   1.17e-01  -1.05e-03  -5.24e-04     0.0

fig = plot_amplitude(
    substitute(a, IdDict(a.control => opt_result_tanh.optimized_controls[1])),
    tlist
)