Trajectories

Trajectories in Piccolo.jl represent a complete quantum control scenario: a system, a control pulse, and a goal. They are the central object passed to problem templates.

Overview

A trajectory encapsulates:

Quantum system: The hardware being controlled
Pulse: How controls vary over time
Goal: The desired outcome (gate, state, etc.)
State evolution: The quantum state over time

Trajectory Types

Type	Use Case	State Variable
`UnitaryTrajectory`	Gate synthesis	Unitary matrix `U(t)`
`KetTrajectory`	State preparation	State vector `\|ψ(t)⟩`
`DensityTrajectory`	Open systems	Density matrix `ρ(t)`
`MultiKetTrajectory`	Multiple state transfers	Multiple `\|ψᵢ(t)⟩`
`SamplingTrajectory`	Robust optimization	States for multiple systems

UnitaryTrajectory

For synthesizing quantum gates.

Construction

using Piccolo

# Define system
H_drift = PAULIS[:Z]
H_drives = [PAULIS[:X], PAULIS[:Y]]
sys = QuantumSystem(H_drift, H_drives, [1.0, 1.0])

# Create pulse
T, N = 10.0, 100
times = collect(range(0, T, length = N))
pulse = ZeroOrderPulse(0.1 * randn(2, N), times)

# Create trajectory with goal
U_goal = GATES[:X]
qtraj = UnitaryTrajectory(sys, pulse, U_goal)

UnitaryTrajectory{ZeroOrderPulse{DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}}, SciMLBase.ODESolution{ComplexF64, 3, Vector{Matrix{ComplexF64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Matrix{ComplexF64}}}, Nothing, SciMLBase.ODEProblem{Matrix{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqLinear.MagnusGL4, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Matrix{ComplexF64}}, Vector{Float64}, Vector{Vector{Matrix{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Matrix{ComplexF64}, Matrix{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}, Matrix{ComplexF64}}(QuantumSystem: levels = 2, n_drives = 2, ZeroOrderPulse{DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}}(DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}([0.024125957739533625 0.31665899630172045 … -0.0223520741359005 0.06638949253475927; -0.07691172462803174 -0.0785166666937139 … 0.17445214121505342 0.11015383965749459], [0.0, 0.10101010101010101, 0.20202020202020202, 0.30303030303030304, 0.40404040404040403, 0.5050505050505051, 0.6060606060606061, 0.7070707070707071, 0.8080808080808081, 0.9090909090909091  …  9.090909090909092, 9.191919191919192, 9.292929292929292, 9.393939393939394, 9.494949494949495, 9.595959595959595, 9.696969696969697, 9.797979797979798, 9.8989898989899, 10.0], Union{}[], nothing, :left, DataInterpolations.ExtrapolationType.None, DataInterpolations.ExtrapolationType.None, FindFirstFunctions.Guesser{Vector{Float64}}([0.0, 0.10101010101010101, 0.20202020202020202, 0.30303030303030304, 0.40404040404040403, 0.5050505050505051, 0.6060606060606061, 0.7070707070707071, 0.8080808080808081, 0.9090909090909091  …  9.090909090909092, 9.191919191919192, 9.292929292929292, 9.393939393939394, 9.494949494949495, 9.595959595959595, 9.696969696969697, 9.797979797979798, 9.8989898989899, 10.0], Base.RefValue{Int64}(100), true), false, true), 10.0, 2, :u, [0.0, 0.0], [0.0, 0.0]), ComplexF64[1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im], ComplexF64[0.0 + 0.0im 1.0 + 0.0im; 1.0 + 0.0im 0.0 + 0.0im], SciMLBase.ODESolution{ComplexF64, 3, Vector{Matrix{ComplexF64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Matrix{ComplexF64}}}, Nothing, SciMLBase.ODEProblem{Matrix{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqLinear.MagnusGL4, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Matrix{ComplexF64}}, Vector{Float64}, Vector{Vector{Matrix{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Matrix{ComplexF64}, Matrix{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}(Matrix{ComplexF64}[[1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im], [0.9949717321962044 - 0.09983233482033259im 0.007678277044674883 - 0.002408550690914315im; -0.007678277044674883 - 0.002408550690914315im 0.9949717321962045 + 0.09983233482033257im], [0.9793712954516756 - 0.1983705330175472im 0.0183486698622555 - 0.033827852280730104im; -0.0183486698622555 - 0.03382785228073009im 0.9793712954516758 + 0.1983705330175472im], [0.9547953388137095 - 0.2956202114009613im 0.02103830673724913 - 0.023063851380334843im; -0.02103830673724913 - 0.02306385138033483im 0.9547953388137096 + 0.29562021140096123im], [0.920047695985192 - 0.38992502206947466im 0.03512958500554924 - 0.015382669917707321im; -0.035129585005549246 - 0.015382669917707315im 0.9200476959851922 + 0.3899250220694747im], [0.8763114190796807 - 0.4797370012904443im 0.03855014232852761 - 0.02108537194024548im; -0.03855014232852762 - 0.021085371940245474im 0.8763114190796808 + 0.4797370012904444im], [0.8231727329224509 - 0.5645794179960625im 0.05304450574828032 - 0.02868820242069043im; -0.05304450574828033 - 0.02868820242069043im 0.8231727329224512 + 0.5645794179960627im], [0.7634329281648423 - 0.6440470139981667im 0.03624828524435082 - 0.03255256933857614im; -0.03624828524435084 - 0.03255256933857615im 0.7634329281648428 + 0.644047013998167im], [0.6946686410099089 - 0.7167949185612756im 0.044821564588345186 - 0.040392465510686884im; -0.0448215645883452 - 0.0403924655106869im 0.6946686410099093 + 0.7167949185612761im], [0.6195992153618644 - 0.7822937116932729im 0.041393943239478266 - 0.04898879903825483im; -0.04139394323947827 - 0.04898879903825483im 0.6195992153618647 + 0.7822937116932733im]  …  [-0.9419910965171467 - 0.31340573657531234im -0.08988537572757598 + 0.07968837803861738im; 0.08988537572757585 + 0.07968837803861731im -0.9419910965171454 + 0.31340573657531146im], [-0.9693650022437484 - 0.21708033862108592im -0.0830898112056715 + 0.07939585809689267im; 0.08308981120567137 + 0.07939585809689255im -0.9693650022437469 + 0.2170803386210851im], [-0.9868826094266525 - 0.11735821428106535im -0.08390586058336114 + 0.0724539254407259im; 0.08390586058336107 + 0.07245392544072578im -0.986882609426651 + 0.11735821428106473im], [-0.9939220804823504 - 0.017773134520775762im -0.07778463981310185 + 0.07584565530135652im; 0.0777846398131018 + 0.0758456553013564im -0.9939220804823488 + 0.017773134520775277im], [-0.9914655473746162 + 0.08231086611465395im -0.06976803202219498 + 0.07316701030082912im; 0.06976803202219496 + 0.07316701030082898im -0.9914655473746147 - 0.08231086611465427im], [-0.9791119078580938 + 0.18086473799823638im -0.05576383734302224 + 0.07428467462470005im; 0.05576383734302223 + 0.0742846746246999im -0.9791119078580922 - 0.18086473799823655im], [-0.9562698520345991 + 0.2786624111339759im -0.053813021325251884 + 0.07070636072218855im; 0.05381302132525191 + 0.07070636072218839im -0.9562698520345976 - 0.2786624111339759im], [-0.9228808674189595 + 0.373024216889459im -0.05525895085503532 + 0.07804028777777991im; 0.05525895085503539 + 0.07804028777777974im -0.922880867418958 - 0.3730242168894589im], [-0.8795241505477998 + 0.464757797792036im -0.06731348742079526 + 0.07685279699478817im; 0.06731348742079538 + 0.076852796994788im -0.8795241505477982 - 0.4647577977920358im], [-0.8299421942834037 + 0.5489773454332255im -0.04294054213434883 + 0.08930810820048407im; 0.04294054213434893 + 0.08930810820048392im -0.8299421942834022 - 0.5489773454332253im]], nothing, nothing, [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], Vector{Matrix{ComplexF64}}[[[1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im]]], nothing, SciMLBase.ODEProblem{Matrix{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}(SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}(MatrixOperator(2 × 2), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(Dict{Symbol, Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(:U => CartesianIndices((2, 2)), :U_2_2 => CartesianIndex(2, 2), :U_1_1 => CartesianIndex(1, 1), :U_2_1 => CartesianIndex(2, 1), :U_1_2 => CartesianIndex(1, 2)), :t, Dict{Symbol, Float64}()), nothing, nothing), ComplexF64[1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im], (0.0, 10.0), SciMLBase.NullParameters(), Base.Pairs(:tstops => [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], :saveat => [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0]), SciMLBase.StandardODEProblem()), OrdinaryDiffEqLinear.MagnusGL4(false, 30, 0), OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Matrix{ComplexF64}}, Vector{Float64}, Vector{Vector{Matrix{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Matrix{ComplexF64}, Matrix{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}(SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}(MatrixOperator(2 × 2), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(Dict{Symbol, Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(:U => CartesianIndices((2, 2)), :U_2_2 => CartesianIndex(2, 2), :U_1_1 => CartesianIndex(1, 1), :U_2_1 => CartesianIndex(2, 1), :U_1_2 => CartesianIndex(1, 2)), :t, Dict{Symbol, Float64}()), nothing, nothing), Matrix{ComplexF64}[[1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im], [0.9949717321962044 - 0.09983233482033259im 0.007678277044674883 - 0.002408550690914315im; -0.007678277044674883 - 0.002408550690914315im 0.9949717321962045 + 0.09983233482033257im], [0.9793712954516756 - 0.1983705330175472im 0.0183486698622555 - 0.033827852280730104im; -0.0183486698622555 - 0.03382785228073009im 0.9793712954516758 + 0.1983705330175472im], [0.9547953388137095 - 0.2956202114009613im 0.02103830673724913 - 0.023063851380334843im; -0.02103830673724913 - 0.02306385138033483im 0.9547953388137096 + 0.29562021140096123im], [0.920047695985192 - 0.38992502206947466im 0.03512958500554924 - 0.015382669917707321im; -0.035129585005549246 - 0.015382669917707315im 0.9200476959851922 + 0.3899250220694747im], [0.8763114190796807 - 0.4797370012904443im 0.03855014232852761 - 0.02108537194024548im; -0.03855014232852762 - 0.021085371940245474im 0.8763114190796808 + 0.4797370012904444im], [0.8231727329224509 - 0.5645794179960625im 0.05304450574828032 - 0.02868820242069043im; -0.05304450574828033 - 0.02868820242069043im 0.8231727329224512 + 0.5645794179960627im], [0.7634329281648423 - 0.6440470139981667im 0.03624828524435082 - 0.03255256933857614im; -0.03624828524435084 - 0.03255256933857615im 0.7634329281648428 + 0.644047013998167im], [0.6946686410099089 - 0.7167949185612756im 0.044821564588345186 - 0.040392465510686884im; -0.0448215645883452 - 0.0403924655106869im 0.6946686410099093 + 0.7167949185612761im], [0.6195992153618644 - 0.7822937116932729im 0.041393943239478266 - 0.04898879903825483im; -0.04139394323947827 - 0.04898879903825483im 0.6195992153618647 + 0.7822937116932733im]  …  [-0.9419910965171467 - 0.31340573657531234im -0.08988537572757598 + 0.07968837803861738im; 0.08988537572757585 + 0.07968837803861731im -0.9419910965171454 + 0.31340573657531146im], [-0.9693650022437484 - 0.21708033862108592im -0.0830898112056715 + 0.07939585809689267im; 0.08308981120567137 + 0.07939585809689255im -0.9693650022437469 + 0.2170803386210851im], [-0.9868826094266525 - 0.11735821428106535im -0.08390586058336114 + 0.0724539254407259im; 0.08390586058336107 + 0.07245392544072578im -0.986882609426651 + 0.11735821428106473im], [-0.9939220804823504 - 0.017773134520775762im -0.07778463981310185 + 0.07584565530135652im; 0.0777846398131018 + 0.0758456553013564im -0.9939220804823488 + 0.017773134520775277im], [-0.9914655473746162 + 0.08231086611465395im -0.06976803202219498 + 0.07316701030082912im; 0.06976803202219496 + 0.07316701030082898im -0.9914655473746147 - 0.08231086611465427im], [-0.9791119078580938 + 0.18086473799823638im -0.05576383734302224 + 0.07428467462470005im; 0.05576383734302223 + 0.0742846746246999im -0.9791119078580922 - 0.18086473799823655im], [-0.9562698520345991 + 0.2786624111339759im -0.053813021325251884 + 0.07070636072218855im; 0.05381302132525191 + 0.07070636072218839im -0.9562698520345976 - 0.2786624111339759im], [-0.9228808674189595 + 0.373024216889459im -0.05525895085503532 + 0.07804028777777991im; 0.05525895085503539 + 0.07804028777777974im -0.922880867418958 - 0.3730242168894589im], [-0.8795241505477998 + 0.464757797792036im -0.06731348742079526 + 0.07685279699478817im; 0.06731348742079538 + 0.076852796994788im -0.8795241505477982 - 0.4647577977920358im], [-0.8299421942834037 + 0.5489773454332255im -0.04294054213434883 + 0.08930810820048407im; 0.04294054213434893 + 0.08930810820048392im -0.8299421942834022 - 0.5489773454332253im]], [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], Vector{Matrix{ComplexF64}}[[[1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im]]], nothing, false, OrdinaryDiffEqLinear.MagnusGL4Cache{Matrix{ComplexF64}, Matrix{ComplexF64}, Matrix{ComplexF64}, Nothing}(ComplexF64[-0.8299421942834037 + 0.5489773454332255im -0.04294054213434883 + 0.08930810820048407im; 0.04294054213434893 + 0.08930810820048392im -0.8299421942834022 - 0.5489773454332253im], ComplexF64[-0.8795241505477998 + 0.464757797792036im -0.06731348742079526 + 0.07685279699478817im; 0.06731348742079538 + 0.076852796994788im -0.8795241505477982 - 0.4647577977920358im], ComplexF64[-0.8795241505477998 + 0.464757797792036im -0.06731348742079526 + 0.07685279699478817im; 0.06731348742079538 + 0.076852796994788im -0.8795241505477982 - 0.4647577977920358im], ComplexF64[0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im], ComplexF64[0.45129699636284687 + 0.8676216116148688im 0.24067596905968802 + 0.12873229137464945im; -0.24067596905968813 + 0.12873229137464956im 0.4512969963628467 - 0.8676216116148672im], ComplexF64[0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im 0.0 + 0.0im], ComplexF64[0.5501763998228221 + 0.8172537624511065im 0.1442831002181761 + 0.15851194573045702im; -0.14428310021817609 + 0.15851194573045724im 0.5501763998228218 - 0.817253762451105im], nothing), nothing, false), false, 0, SciMLBase.DEStats(101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0.0), nothing, SciMLBase.ReturnCode.Success, nothing, nothing, nothing))

Solve and Analyze

qcp = SmoothPulseProblem(qtraj, N; Q = 100.0)
cached_solve!(qcp, "trajectories_unitary"; max_iter = 50)
fidelity(qcp)

0.9999787116455973

Extracting the Pulse

optimized_pulse = get_pulse(qcp.qtraj)
duration(optimized_pulse)

14.262180172049334

KetTrajectory

For state preparation (state-to-state transfer).

Construction

# Initial and goal states
ψ_init = ComplexF64[1, 0]  # |0⟩
ψ_goal = ComplexF64[0, 1]  # |1⟩

qtraj_ket = KetTrajectory(sys, pulse, ψ_init, ψ_goal)

KetTrajectory{ZeroOrderPulse{DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}}, SciMLBase.ODESolution{ComplexF64, 2, Vector{Vector{ComplexF64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqLinear.MagnusGL4, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}}(QuantumSystem: levels = 2, n_drives = 2, ZeroOrderPulse{DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}}(DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}([0.024125957739533625 0.31665899630172045 … -0.0223520741359005 0.06638949253475927; -0.07691172462803174 -0.0785166666937139 … 0.17445214121505342 0.11015383965749459], [0.0, 0.10101010101010101, 0.20202020202020202, 0.30303030303030304, 0.40404040404040403, 0.5050505050505051, 0.6060606060606061, 0.7070707070707071, 0.8080808080808081, 0.9090909090909091  …  9.090909090909092, 9.191919191919192, 9.292929292929292, 9.393939393939394, 9.494949494949495, 9.595959595959595, 9.696969696969697, 9.797979797979798, 9.8989898989899, 10.0], Union{}[], nothing, :left, DataInterpolations.ExtrapolationType.None, DataInterpolations.ExtrapolationType.None, FindFirstFunctions.Guesser{Vector{Float64}}([0.0, 0.10101010101010101, 0.20202020202020202, 0.30303030303030304, 0.40404040404040403, 0.5050505050505051, 0.6060606060606061, 0.7070707070707071, 0.8080808080808081, 0.9090909090909091  …  9.090909090909092, 9.191919191919192, 9.292929292929292, 9.393939393939394, 9.494949494949495, 9.595959595959595, 9.696969696969697, 9.797979797979798, 9.8989898989899, 10.0], Base.RefValue{Int64}(100), true), false, true), 10.0, 2, :u, [0.0, 0.0], [0.0, 0.0]), ComplexF64[1.0 + 0.0im, 0.0 + 0.0im], ComplexF64[0.0 + 0.0im, 1.0 + 0.0im], SciMLBase.ODESolution{ComplexF64, 2, Vector{Vector{ComplexF64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqLinear.MagnusGL4, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}(Vector{ComplexF64}[[1.0 + 0.0im, 0.0 + 0.0im], [0.9949717321962044 - 0.09983233482033259im, -0.007678277044674883 - 0.002408550690914315im], [0.9793712954516756 - 0.1983705330175472im, -0.0183486698622555 - 0.03382785228073009im], [0.9547953388137095 - 0.2956202114009613im, -0.02103830673724913 - 0.02306385138033483im], [0.920047695985192 - 0.38992502206947466im, -0.035129585005549246 - 0.015382669917707315im], [0.8763114190796807 - 0.4797370012904443im, -0.03855014232852762 - 0.021085371940245474im], [0.8231727329224509 - 0.5645794179960625im, -0.05304450574828033 - 0.02868820242069043im], [0.7634329281648423 - 0.6440470139981667im, -0.03624828524435084 - 0.03255256933857615im], [0.6946686410099089 - 0.7167949185612756im, -0.0448215645883452 - 0.0403924655106869im], [0.6195992153618644 - 0.7822937116932729im, -0.04139394323947827 - 0.04898879903825483im]  …  [-0.9419910965171467 - 0.31340573657531234im, 0.08988537572757585 + 0.07968837803861731im], [-0.9693650022437484 - 0.21708033862108592im, 0.08308981120567137 + 0.07939585809689255im], [-0.9868826094266525 - 0.11735821428106535im, 0.08390586058336107 + 0.07245392544072578im], [-0.9939220804823504 - 0.017773134520775762im, 0.0777846398131018 + 0.0758456553013564im], [-0.9914655473746162 + 0.08231086611465395im, 0.06976803202219496 + 0.07316701030082898im], [-0.9791119078580938 + 0.18086473799823638im, 0.05576383734302223 + 0.0742846746246999im], [-0.9562698520345991 + 0.2786624111339759im, 0.05381302132525191 + 0.07070636072218839im], [-0.9228808674189595 + 0.373024216889459im, 0.05525895085503539 + 0.07804028777777974im], [-0.8795241505477998 + 0.464757797792036im, 0.06731348742079538 + 0.076852796994788im], [-0.8299421942834037 + 0.5489773454332255im, 0.04294054213434893 + 0.08930810820048392im]], nothing, nothing, [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], Vector{Vector{ComplexF64}}[[[1.0 + 0.0im, 0.0 + 0.0im]]], nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}(SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}(MatrixOperator(2 × 2), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(Dict{Symbol, Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(:ψ_2 => 2, :ψ_1 => 1, :ψ => 1:2), :t, Dict{Symbol, Float64}()), nothing, nothing), ComplexF64[1.0 + 0.0im, 0.0 + 0.0im], (0.0, 10.0), SciMLBase.NullParameters(), Base.Pairs(:tstops => [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], :saveat => [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0]), SciMLBase.StandardODEProblem()), OrdinaryDiffEqLinear.MagnusGL4(false, 30, 0), OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}(SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}(MatrixOperator(2 × 2), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(Dict{Symbol, Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(:ψ_2 => 2, :ψ_1 => 1, :ψ => 1:2), :t, Dict{Symbol, Float64}()), nothing, nothing), Vector{ComplexF64}[[1.0 + 0.0im, 0.0 + 0.0im], [0.9949717321962044 - 0.09983233482033259im, -0.007678277044674883 - 0.002408550690914315im], [0.9793712954516756 - 0.1983705330175472im, -0.0183486698622555 - 0.03382785228073009im], [0.9547953388137095 - 0.2956202114009613im, -0.02103830673724913 - 0.02306385138033483im], [0.920047695985192 - 0.38992502206947466im, -0.035129585005549246 - 0.015382669917707315im], [0.8763114190796807 - 0.4797370012904443im, -0.03855014232852762 - 0.021085371940245474im], [0.8231727329224509 - 0.5645794179960625im, -0.05304450574828033 - 0.02868820242069043im], [0.7634329281648423 - 0.6440470139981667im, -0.03624828524435084 - 0.03255256933857615im], [0.6946686410099089 - 0.7167949185612756im, -0.0448215645883452 - 0.0403924655106869im], [0.6195992153618644 - 0.7822937116932729im, -0.04139394323947827 - 0.04898879903825483im]  …  [-0.9419910965171467 - 0.31340573657531234im, 0.08988537572757585 + 0.07968837803861731im], [-0.9693650022437484 - 0.21708033862108592im, 0.08308981120567137 + 0.07939585809689255im], [-0.9868826094266525 - 0.11735821428106535im, 0.08390586058336107 + 0.07245392544072578im], [-0.9939220804823504 - 0.017773134520775762im, 0.0777846398131018 + 0.0758456553013564im], [-0.9914655473746162 + 0.08231086611465395im, 0.06976803202219496 + 0.07316701030082898im], [-0.9791119078580938 + 0.18086473799823638im, 0.05576383734302223 + 0.0742846746246999im], [-0.9562698520345991 + 0.2786624111339759im, 0.05381302132525191 + 0.07070636072218839im], [-0.9228808674189595 + 0.373024216889459im, 0.05525895085503539 + 0.07804028777777974im], [-0.8795241505477998 + 0.464757797792036im, 0.06731348742079538 + 0.076852796994788im], [-0.8299421942834037 + 0.5489773454332255im, 0.04294054213434893 + 0.08930810820048392im]], [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], Vector{Vector{ComplexF64}}[[[1.0 + 0.0im, 0.0 + 0.0im]]], nothing, false, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}(ComplexF64[-0.8299421942834037 + 0.5489773454332255im, 0.04294054213434893 + 0.08930810820048392im], ComplexF64[-0.8795241505477998 + 0.464757797792036im, 0.06731348742079538 + 0.076852796994788im], ComplexF64[-0.8795241505477998 + 0.464757797792036im, 0.06731348742079538 + 0.076852796994788im], ComplexF64[0.0 + 0.0im, 0.0 + 0.0im], ComplexF64[0.45129699636284687 + 0.8676216116148688im, -0.24067596905968813 + 0.12873229137464956im], ComplexF64[0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im], ComplexF64[0.5501763998228221 + 0.8172537624511065im, -0.14428310021817609 + 0.15851194573045724im], nothing), nothing, false), false, 0, SciMLBase.DEStats(101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0.0), nothing, SciMLBase.ReturnCode.Success, nothing, nothing, nothing))

When to Use

Use KetTrajectory when:

Preparing a specific quantum state
Global phase doesn't matter
Single state transfer

qcp_ket = SmoothPulseProblem(qtraj_ket, N; Q = 100.0)
cached_solve!(qcp_ket, "trajectories_ket"; max_iter = 50)
fidelity(qcp_ket)

0.9999531491901026

MultiKetTrajectory

For gates defined by multiple state mappings with coherent phases.

Construction

# Define state pairs: X gate maps |0⟩ → |1⟩ and |1⟩ → |0⟩
ψ0 = ComplexF64[1, 0]
ψ1 = ComplexF64[0, 1]

initial_states = [ψ0, ψ1]
goal_states = [ψ1, ψ0]

qtraj_multi = MultiKetTrajectory(sys, pulse, initial_states, goal_states)

MultiKetTrajectory{ZeroOrderPulse{DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}}, SciMLBase.EnsembleSolution{ComplexF64, 3, Vector{SciMLBase.ODESolution{ComplexF64, 2, Vector{Vector{ComplexF64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqLinear.MagnusGL4, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}}}}(QuantumSystem: levels = 2, n_drives = 2, ZeroOrderPulse{DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}}(DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}([0.024125957739533625 0.31665899630172045 … -0.0223520741359005 0.06638949253475927; -0.07691172462803174 -0.0785166666937139 … 0.17445214121505342 0.11015383965749459], [0.0, 0.10101010101010101, 0.20202020202020202, 0.30303030303030304, 0.40404040404040403, 0.5050505050505051, 0.6060606060606061, 0.7070707070707071, 0.8080808080808081, 0.9090909090909091  …  9.090909090909092, 9.191919191919192, 9.292929292929292, 9.393939393939394, 9.494949494949495, 9.595959595959595, 9.696969696969697, 9.797979797979798, 9.8989898989899, 10.0], Union{}[], nothing, :left, DataInterpolations.ExtrapolationType.None, DataInterpolations.ExtrapolationType.None, FindFirstFunctions.Guesser{Vector{Float64}}([0.0, 0.10101010101010101, 0.20202020202020202, 0.30303030303030304, 0.40404040404040403, 0.5050505050505051, 0.6060606060606061, 0.7070707070707071, 0.8080808080808081, 0.9090909090909091  …  9.090909090909092, 9.191919191919192, 9.292929292929292, 9.393939393939394, 9.494949494949495, 9.595959595959595, 9.696969696969697, 9.797979797979798, 9.8989898989899, 10.0], Base.RefValue{Int64}(100), true), false, true), 10.0, 2, :u, [0.0, 0.0], [0.0, 0.0]), Vector{ComplexF64}[[1.0 + 0.0im, 0.0 + 0.0im], [0.0 + 0.0im, 1.0 + 0.0im]], Vector{ComplexF64}[[0.0 + 0.0im, 1.0 + 0.0im], [1.0 + 0.0im, 0.0 + 0.0im]], [0.5, 0.5], SciMLBase.EnsembleSolution{ComplexF64, 3, Vector{SciMLBase.ODESolution{ComplexF64, 2, Vector{Vector{ComplexF64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqLinear.MagnusGL4, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}}}(SciMLBase.ODESolution{ComplexF64, 2, Vector{Vector{ComplexF64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqLinear.MagnusGL4, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}[SciMLBase.ODESolution{ComplexF64, 2, Vector{Vector{ComplexF64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqLinear.MagnusGL4, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}(Vector{ComplexF64}[[1.0 + 0.0im, 0.0 + 0.0im], [0.9949717321962044 - 0.09983233482033259im, -0.007678277044674883 - 0.002408550690914315im], [0.9793712954516756 - 0.1983705330175472im, -0.0183486698622555 - 0.03382785228073009im], [0.9547953388137095 - 0.2956202114009613im, -0.02103830673724913 - 0.02306385138033483im], [0.920047695985192 - 0.38992502206947466im, -0.035129585005549246 - 0.015382669917707315im], [0.8763114190796807 - 0.4797370012904443im, -0.03855014232852762 - 0.021085371940245474im], [0.8231727329224509 - 0.5645794179960625im, -0.05304450574828033 - 0.02868820242069043im], [0.7634329281648423 - 0.6440470139981667im, -0.03624828524435084 - 0.03255256933857615im], [0.6946686410099089 - 0.7167949185612756im, -0.0448215645883452 - 0.0403924655106869im], [0.6195992153618644 - 0.7822937116932729im, -0.04139394323947827 - 0.04898879903825483im]  …  [-0.9419910965171467 - 0.31340573657531234im, 0.08988537572757585 + 0.07968837803861731im], [-0.9693650022437484 - 0.21708033862108592im, 0.08308981120567137 + 0.07939585809689255im], [-0.9868826094266525 - 0.11735821428106535im, 0.08390586058336107 + 0.07245392544072578im], [-0.9939220804823504 - 0.017773134520775762im, 0.0777846398131018 + 0.0758456553013564im], [-0.9914655473746162 + 0.08231086611465395im, 0.06976803202219496 + 0.07316701030082898im], [-0.9791119078580938 + 0.18086473799823638im, 0.05576383734302223 + 0.0742846746246999im], [-0.9562698520345991 + 0.2786624111339759im, 0.05381302132525191 + 0.07070636072218839im], [-0.9228808674189595 + 0.373024216889459im, 0.05525895085503539 + 0.07804028777777974im], [-0.8795241505477998 + 0.464757797792036im, 0.06731348742079538 + 0.076852796994788im], [-0.8299421942834037 + 0.5489773454332255im, 0.04294054213434893 + 0.08930810820048392im]], nothing, nothing, [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], Vector{Vector{ComplexF64}}[[[1.0 + 0.0im, 0.0 + 0.0im]]], nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}(SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}(MatrixOperator(2 × 2), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(Dict{Symbol, Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(:ψ_2 => 2, :ψ_1 => 1, :ψ => 1:2), :t, Dict{Symbol, Float64}()), nothing, nothing), ComplexF64[1.0 + 0.0im, 0.0 + 0.0im], (0.0, 10.0), SciMLBase.NullParameters(), Base.Pairs(:tstops => [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], :saveat => [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0]), SciMLBase.StandardODEProblem()), OrdinaryDiffEqLinear.MagnusGL4(false, 30, 0), OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}(SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}(MatrixOperator(2 × 2), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(Dict{Symbol, Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(:ψ_2 => 2, :ψ_1 => 1, :ψ => 1:2), :t, Dict{Symbol, Float64}()), nothing, nothing), Vector{ComplexF64}[[1.0 + 0.0im, 0.0 + 0.0im], [0.9949717321962044 - 0.09983233482033259im, -0.007678277044674883 - 0.002408550690914315im], [0.9793712954516756 - 0.1983705330175472im, -0.0183486698622555 - 0.03382785228073009im], [0.9547953388137095 - 0.2956202114009613im, -0.02103830673724913 - 0.02306385138033483im], [0.920047695985192 - 0.38992502206947466im, -0.035129585005549246 - 0.015382669917707315im], [0.8763114190796807 - 0.4797370012904443im, -0.03855014232852762 - 0.021085371940245474im], [0.8231727329224509 - 0.5645794179960625im, -0.05304450574828033 - 0.02868820242069043im], [0.7634329281648423 - 0.6440470139981667im, -0.03624828524435084 - 0.03255256933857615im], [0.6946686410099089 - 0.7167949185612756im, -0.0448215645883452 - 0.0403924655106869im], [0.6195992153618644 - 0.7822937116932729im, -0.04139394323947827 - 0.04898879903825483im]  …  [-0.9419910965171467 - 0.31340573657531234im, 0.08988537572757585 + 0.07968837803861731im], [-0.9693650022437484 - 0.21708033862108592im, 0.08308981120567137 + 0.07939585809689255im], [-0.9868826094266525 - 0.11735821428106535im, 0.08390586058336107 + 0.07245392544072578im], [-0.9939220804823504 - 0.017773134520775762im, 0.0777846398131018 + 0.0758456553013564im], [-0.9914655473746162 + 0.08231086611465395im, 0.06976803202219496 + 0.07316701030082898im], [-0.9791119078580938 + 0.18086473799823638im, 0.05576383734302223 + 0.0742846746246999im], [-0.9562698520345991 + 0.2786624111339759im, 0.05381302132525191 + 0.07070636072218839im], [-0.9228808674189595 + 0.373024216889459im, 0.05525895085503539 + 0.07804028777777974im], [-0.8795241505477998 + 0.464757797792036im, 0.06731348742079538 + 0.076852796994788im], [-0.8299421942834037 + 0.5489773454332255im, 0.04294054213434893 + 0.08930810820048392im]], [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], Vector{Vector{ComplexF64}}[[[1.0 + 0.0im, 0.0 + 0.0im]]], nothing, false, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}(ComplexF64[-0.8299421942834037 + 0.5489773454332255im, 0.04294054213434893 + 0.08930810820048392im], ComplexF64[-0.8795241505477998 + 0.464757797792036im, 0.06731348742079538 + 0.076852796994788im], ComplexF64[-0.8795241505477998 + 0.464757797792036im, 0.06731348742079538 + 0.076852796994788im], ComplexF64[0.0 + 0.0im, 0.0 + 0.0im], ComplexF64[0.45129699636284687 + 0.8676216116148688im, -0.24067596905968813 + 0.12873229137464956im], ComplexF64[0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im], ComplexF64[0.5501763998228221 + 0.8172537624511065im, -0.14428310021817609 + 0.15851194573045724im], nothing), nothing, false), false, 0, SciMLBase.DEStats(101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0.0), nothing, SciMLBase.ReturnCode.Success, nothing, nothing, nothing), SciMLBase.ODESolution{ComplexF64, 2, Vector{Vector{ComplexF64}}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqLinear.MagnusGL4, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}, SciMLBase.DEStats, Nothing, Nothing, Nothing, Nothing}(Vector{ComplexF64}[[0.0 + 0.0im, 1.0 + 0.0im], [0.007678277044674883 - 0.002408550690914315im, 0.9949717321962045 + 0.09983233482033257im], [0.0183486698622555 - 0.033827852280730104im, 0.9793712954516758 + 0.1983705330175472im], [0.02103830673724913 - 0.023063851380334843im, 0.9547953388137096 + 0.29562021140096123im], [0.03512958500554924 - 0.015382669917707321im, 0.9200476959851922 + 0.3899250220694747im], [0.03855014232852761 - 0.02108537194024548im, 0.8763114190796808 + 0.4797370012904444im], [0.05304450574828032 - 0.02868820242069043im, 0.8231727329224512 + 0.5645794179960627im], [0.03624828524435082 - 0.03255256933857614im, 0.7634329281648428 + 0.644047013998167im], [0.044821564588345186 - 0.040392465510686884im, 0.6946686410099093 + 0.7167949185612761im], [0.041393943239478266 - 0.04898879903825483im, 0.6195992153618647 + 0.7822937116932733im]  …  [-0.08988537572757598 + 0.07968837803861738im, -0.9419910965171454 + 0.31340573657531146im], [-0.0830898112056715 + 0.07939585809689267im, -0.9693650022437469 + 0.2170803386210851im], [-0.08390586058336114 + 0.0724539254407259im, -0.986882609426651 + 0.11735821428106473im], [-0.07778463981310185 + 0.07584565530135652im, -0.9939220804823488 + 0.017773134520775277im], [-0.06976803202219498 + 0.07316701030082912im, -0.9914655473746147 - 0.08231086611465427im], [-0.05576383734302224 + 0.07428467462470005im, -0.9791119078580922 - 0.18086473799823655im], [-0.053813021325251884 + 0.07070636072218855im, -0.9562698520345976 - 0.2786624111339759im], [-0.05525895085503532 + 0.07804028777777991im, -0.922880867418958 - 0.3730242168894589im], [-0.06731348742079526 + 0.07685279699478817im, -0.8795241505477982 - 0.4647577977920358im], [-0.04294054213434883 + 0.08930810820048407im, -0.8299421942834022 - 0.5489773454332253im]], nothing, nothing, [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], Vector{Vector{ComplexF64}}[[[0.0 + 0.0im, 1.0 + 0.0im]]], nothing, SciMLBase.ODEProblem{Vector{ComplexF64}, Tuple{Float64, Float64}, true, SciMLBase.NullParameters, SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Base.Pairs{Symbol, Vector{Float64}, Nothing, @NamedTuple{tstops::Vector{Float64}, saveat::Vector{Float64}}}, SciMLBase.StandardODEProblem}(SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}(MatrixOperator(2 × 2), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(Dict{Symbol, Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(:ψ_2 => 2, :ψ_1 => 1, :ψ => 1:2), :t, Dict{Symbol, Float64}()), nothing, nothing), ComplexF64[0.0 + 0.0im, 1.0 + 0.0im], (0.0, 10.0), SciMLBase.NullParameters(), Base.Pairs(:tstops => [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], :saveat => [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0]), SciMLBase.StandardODEProblem()), OrdinaryDiffEqLinear.MagnusGL4(false, 30, 0), OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}, Vector{Vector{ComplexF64}}, Vector{Float64}, Vector{Vector{Vector{ComplexF64}}}, Nothing, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}, Nothing}(SciMLBase.ODEFunction{true, SciMLBase.FullSpecialize, SciMLOperators.MatrixOperator{ComplexF64, Matrix{ComplexF64}, SciMLOperators.FilterKwargs{Nothing, Val{()}}, SciMLOperators.FilterKwargs{Piccolo.Quantum.Rollouts.var"#update!#_construct_operator##2"{QuantumSystem{Piccolo.Quantum.QuantumSystems.var"#26#27"{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}, Vector{SparseArrays.SparseMatrixCSC{ComplexF64, Int64}}, Int64}, Piccolo.Quantum.QuantumSystems.var"#28#29"{Vector{SparseArrays.SparseMatrixCSC{Float64, Int64}}, Int64, SparseArrays.SparseMatrixCSC{Float64, Int64}}, @NamedTuple{}}}, Val{()}}}, LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}, Nothing, Nothing}(MatrixOperator(2 × 2), LinearAlgebra.UniformScaling{Bool}(true), nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, SciMLBase.DEFAULT_OBSERVED, nothing, Piccolo.Quantum.Rollouts.PiccoloRolloutSystem{Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(Dict{Symbol, Union{Int64, AbstractVector{Int64}, CartesianIndex, CartesianIndices}}(:ψ_2 => 2, :ψ_1 => 1, :ψ => 1:2), :t, Dict{Symbol, Float64}()), nothing, nothing), Vector{ComplexF64}[[0.0 + 0.0im, 1.0 + 0.0im], [0.007678277044674883 - 0.002408550690914315im, 0.9949717321962045 + 0.09983233482033257im], [0.0183486698622555 - 0.033827852280730104im, 0.9793712954516758 + 0.1983705330175472im], [0.02103830673724913 - 0.023063851380334843im, 0.9547953388137096 + 0.29562021140096123im], [0.03512958500554924 - 0.015382669917707321im, 0.9200476959851922 + 0.3899250220694747im], [0.03855014232852761 - 0.02108537194024548im, 0.8763114190796808 + 0.4797370012904444im], [0.05304450574828032 - 0.02868820242069043im, 0.8231727329224512 + 0.5645794179960627im], [0.03624828524435082 - 0.03255256933857614im, 0.7634329281648428 + 0.644047013998167im], [0.044821564588345186 - 0.040392465510686884im, 0.6946686410099093 + 0.7167949185612761im], [0.041393943239478266 - 0.04898879903825483im, 0.6195992153618647 + 0.7822937116932733im]  …  [-0.08988537572757598 + 0.07968837803861738im, -0.9419910965171454 + 0.31340573657531146im], [-0.0830898112056715 + 0.07939585809689267im, -0.9693650022437469 + 0.2170803386210851im], [-0.08390586058336114 + 0.0724539254407259im, -0.986882609426651 + 0.11735821428106473im], [-0.07778463981310185 + 0.07584565530135652im, -0.9939220804823488 + 0.017773134520775277im], [-0.06976803202219498 + 0.07316701030082912im, -0.9914655473746147 - 0.08231086611465427im], [-0.05576383734302224 + 0.07428467462470005im, -0.9791119078580922 - 0.18086473799823655im], [-0.053813021325251884 + 0.07070636072218855im, -0.9562698520345976 - 0.2786624111339759im], [-0.05525895085503532 + 0.07804028777777991im, -0.922880867418958 - 0.3730242168894589im], [-0.06731348742079526 + 0.07685279699478817im, -0.8795241505477982 - 0.4647577977920358im], [-0.04294054213434883 + 0.08930810820048407im, -0.8299421942834022 - 0.5489773454332253im]], [0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9  …  9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 9.9, 10.0], Vector{Vector{ComplexF64}}[[[0.0 + 0.0im, 1.0 + 0.0im]]], nothing, false, OrdinaryDiffEqLinear.MagnusGL4Cache{Vector{ComplexF64}, Vector{ComplexF64}, Matrix{ComplexF64}, Nothing}(ComplexF64[-0.04294054213434883 + 0.08930810820048407im, -0.8299421942834022 - 0.5489773454332253im], ComplexF64[-0.06731348742079526 + 0.07685279699478817im, -0.8795241505477982 - 0.4647577977920358im], ComplexF64[-0.06731348742079526 + 0.07685279699478817im, -0.8795241505477982 - 0.4647577977920358im], ComplexF64[0.0 + 0.0im, 0.0 + 0.0im], ComplexF64[0.24067596905968802 + 0.12873229137464945im, 0.4512969963628467 - 0.8676216116148672im], ComplexF64[0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im], ComplexF64[0.1442831002181761 + 0.15851194573045702im, 0.5501763998228218 - 0.817253762451105im], nothing), nothing, false), false, 0, SciMLBase.DEStats(101, 0, 0, 0, 0, 0, 0, 0, 0, 0, 100, 0, 0.0), nothing, SciMLBase.ReturnCode.Success, nothing, nothing, nothing)], 0.333908142, true, SciMLBase.DEStats(202, 0, 0, 0, 0, 0, 0, 0, 0, 0, 200, 0, 0.0)))

Coherent Fidelity

MultiKetTrajectory uses CoherentKetInfidelityObjective, which ensures:

Each state reaches its target
Relative phases between states are preserved

This is important for gates where phase relationships matter (unlike UnitaryTrajectory which tracks the full unitary).

qcp_multi = SmoothPulseProblem(qtraj_multi, N; Q = 100.0)
cached_solve!(qcp_multi, "trajectories_multi"; max_iter = 50)
fidelity(qcp_multi)

0.999953052858623

DensityTrajectory

For open quantum systems with dissipation.

# Open system with collapse operators
open_sys = OpenQuantumSystem(H_drift, H_drives, bounds, c_ops)

# Initial density matrix
ρ_init = [1.0 0.0; 0.0 0.0]  # Pure |0⟩
ρ_goal = [0.0 0.0; 0.0 1.0]  # Pure |1⟩

qtraj = DensityTrajectory(open_sys, pulse, ρ_init, ρ_goal)

Note

DensityTrajectory support for fidelity objectives is still in development. For most open-system problems, consider using KetTrajectory with an effective Hamiltonian.

SamplingTrajectory

For robust optimization over parameter variations. This is created internally by SamplingProblem.

How It Works

SamplingTrajectory wraps multiple system variants:

# Created internally by SamplingProblem
systems = [sys_nominal, sys_high, sys_low]
qcp_robust = SamplingProblem(qcp_base, systems)
# This creates a SamplingTrajectory internally

The resulting trajectory has:

Single shared control pulse
Multiple state trajectories (one per system)

Common Operations

Named Trajectory Integration

After solving, the NamedTrajectory stores discrete time points:

traj = get_trajectory(qcp)

# Controls at timestep k
u_1 = traj[1][:u]
u_1

# All timesteps
Δts = get_timesteps(traj)
length(Δts)

Internal Representation

Piccolo.jl uses real isomorphisms internally for optimization. Complex states are converted to real vectors:

# Complex ket |ψ⟩ → real vector ψ̃
ψ = ComplexF64[1, im] / √2
ψ̃ = ket_to_iso(ψ)  # [Re(ψ); Im(ψ)]

# Unitary U → real vector Ũ⃗
U = GATES[:H]
Ũ = operator_to_iso_vec(U)

See Isomorphisms for details.

Best Practices

1. Match Pulse Type to Problem

# For SmoothPulseProblem
pulse = ZeroOrderPulse(controls, times)
qtraj = UnitaryTrajectory(sys, pulse, U_goal)
qcp = SmoothPulseProblem(qtraj, N)  # ✓

# For SplinePulseProblem
pulse = CubicSplinePulse(controls, tangents, times)
qtraj = UnitaryTrajectory(sys, pulse, U_goal)
qcp = SplinePulseProblem(qtraj)  # ✓

2. Initialize with Reasonable Controls

# Scale by drive bounds
max_amp = 0.1 * 1.0
initial_controls = max_amp * randn(2, N)
extrema(initial_controls)

(-0.2985029931360587, 0.27958334653220407)