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Your First Gate

This tutorial walks through synthesizing your first quantum gate with Piccolo.jl. We'll implement an X gate (NOT gate) on a single qubit.

What We're Doing

We want to find control pulses that implement:

\[X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\]

Our qubit has Hamiltonian:

\[H(t) = \frac{\omega}{2}\sigma_z + u_x(t)\sigma_x + u_y(t)\sigma_y\]

The optimizer will find $u_x(t)$ and $u_y(t)$ that produce the X gate.

Setup

First, load the required packages:

using Piccolo
using CairoMakie
using Random
Random.seed!(42)  # For reproducibility
Random.TaskLocalRNG()

Step 1: Define the Quantum System

A QuantumSystem needs:

  • Drift Hamiltonian: Always-on terms (qubit frequency)
  • Drive Hamiltonians: Controllable interactions
  • Drive bounds: Maximum control amplitudes
# The drift Hamiltonian: ω/2 σ_z (qubit frequency)
# We set ω = 1.0 for simplicity
H_drift = 0.5 * PAULIS[:Z]

# The drive Hamiltonians: σ_x and σ_y controls
H_drives = [PAULIS[:X], PAULIS[:Y]]

# Maximum amplitude for each drive (in same units as H_drift)
drive_bounds = [1.0, 1.0]

# Create the system
sys = QuantumSystem(H_drift, H_drives, drive_bounds)
QuantumSystem: levels = 2, n_drives = 2

Let's check what we created:

sys.levels, sys.n_drives
(2, 2)

Step 2: Create an Initial Pulse

We need an initial guess for the control pulse. ZeroOrderPulse represents piecewise constant controls - the standard choice for most problems.

# Gate duration and discretization
T = 10.0   # Total time (in units where ω = 1)
N = 100    # Number of timesteps

# Time vector
times = collect(range(0, T, length = N))

# Random initial controls (small amplitude)
# Shape: (n_drives, N) = (2, 100)
initial_controls = 0.1 * randn(2, N)

# Create the pulse
pulse = ZeroOrderPulse(initial_controls, times)
ZeroOrderPulse{DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}}(DataInterpolations.ConstantInterpolation{Matrix{Float64}, Vector{Float64}, Vector{Union{}}, Float64}([-0.03633574814517775 -0.031498797116895606 … 0.17995350308617175 -0.1529323847225266; 0.02517372155742292 -0.03112524013244207 … -0.0844068143386927 -0.019509907821117518], [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}(1), true), false, true), 10.0, 2, :u, [0.0, 0.0], [0.0, 0.0])

Check the pulse:

duration(pulse)
10.0
n_drives(pulse)
2
pulse(5.0)
2-element Vector{Float64}:
  0.036820693581548374
 -0.004656094092083756

Step 3: Define the Goal

A UnitaryTrajectory combines the system, pulse, and target gate.

# Our target: the X gate
U_goal = GATES[:X]

U_goal

# Create the trajectory
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}([-0.03633574814517775 -0.031498797116895606 … 0.17995350308617175 -0.1529323847225266; 0.02517372155742292 -0.03112524013244207 … -0.0844068143386927 -0.019509907821117518], [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], [0.9987404944669499 - 0.049979006477971294im -0.002516315185586171 + 0.003632049183859544im; 0.0025163151855861713 + 0.003632049183859544im 0.9987404944669498 + 0.04997900647797128im], [0.9949810326156695 - 0.0998128901148409im 0.000618317190123448 + 0.007053324452425682im; -0.0006183171901234484 + 0.007053324452425682im 0.9949810326156693 + 0.09981289011484087im], [0.9887651340956775 - 0.14943984295092333im -0.0029569509531839483 - 0.001580941688088154im; 0.0029569509531839496 - 0.0015809416880881548im 0.9887651340956773 + 0.1494398429509233im], [0.9799747340913558 - 0.19864452647497763im 0.010205296881408978 + 0.009258755838166377im; -0.010205296881408978 + 0.00925875583816638im 0.9799747340913557 + 0.1986445264749776im], [0.9684124344840319 - 0.2475462641464486im 0.006026186154200027 + 0.02935794482388511im; -0.006026186154200024 + 0.029357944823885117im 0.9684124344840316 + 0.24754626414644848im], [0.954571577620917 - 0.29591535958187054im -0.0026905777648045855 + 0.034927982380001006im; 0.0026905777648045907 + 0.03492798238000102im 0.9545715776209169 + 0.2959153595818705im], [0.9387347697463967 - 0.3431224741834879im 0.004115047313186222 + 0.03204787301821318im; -0.0041150473131862176 + 0.03204787301821321im 0.9387347697463968 + 0.3431224741834879im], [0.9205276647108096 - 0.3900108166967894im -0.006094734215670173 + 0.021982619871316816im; 0.006094734215670175 + 0.02198261987131685im 0.9205276647108098 + 0.3900108166967894im], [0.8998675265877255 - 0.43521233432193496im 0.006358472942583045 + 0.028075406831952972im; -0.006358472942583041 + 0.028075406831953007im 0.8998675265877257 + 0.43521233432193496im]  …  [-0.15648685930229364 + 0.980231622083645im 0.08938839371552486 + 0.08165503659863035im; -0.0893883937155247 + 0.08165503659863027im -0.15648685930229342 - 0.9802316220836469im], [-0.1062020698992996 + 0.9862472347642309im 0.09207417041155784 + 0.08694745201574915im; -0.09207417041155767 + 0.08694745201574904im -0.10620206989929928 - 0.9862472347642327im], [-0.055457928069234406 + 0.9893318748804325im 0.0957688073980352 + 0.09473750624430216im; -0.09576880739803505 + 0.09473750624430204im -0.05545792806923401 - 0.9893318748804343im], [-0.005069586540250117 + 0.9906817788550469im 0.0970357066123597 + 0.09543575838827797im; -0.09703570661235958 + 0.09543575838827785im -0.005069586540249638 - 0.9906817788550488im], [0.04384945505659482 + 0.9909383652577827im 0.09796698818217227 + 0.08075178498325397im; -0.09796698818217214 + 0.08075178498325392im 0.04384945505659539 - 0.9909383652577848im], [0.09419276928652728 + 0.9879693198879221im 0.09445275379439133 + 0.0782497442486703im; -0.09445275379439122 + 0.07824974424867025im 0.09419276928652794 - 0.987969319887924im], [0.14235459429456435 + 0.9794179639922784im 0.11799643595026194 + 0.08094728158177968im; -0.11799643595026181 + 0.0809472815817796im 0.14235459429456512 - 0.9794179639922804im], [0.1903805029323721 + 0.9727412767337852im 0.11644111065185543 + 0.06301698494611356im; -0.11644111065185533 + 0.06301698494611355im 0.19038050293237302 - 0.9727412767337874im], [0.23729640140889482 + 0.9615172202573865im 0.12882380706888938 + 0.05078857902292991im; -0.1288238070688893 + 0.05078857902292992im 0.23729640140889588 - 0.961517220257389im], [0.28483233686338644 + 0.9510122210154612im 0.11588306385955965 + 0.03189687863564486im; -0.11588306385955958 + 0.031896878635644915im 0.28483233686338755 - 0.9510122210154633im]], 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], [0.9987404944669499 - 0.049979006477971294im -0.002516315185586171 + 0.003632049183859544im; 0.0025163151855861713 + 0.003632049183859544im 0.9987404944669498 + 0.04997900647797128im], [0.9949810326156695 - 0.0998128901148409im 0.000618317190123448 + 0.007053324452425682im; -0.0006183171901234484 + 0.007053324452425682im 0.9949810326156693 + 0.09981289011484087im], [0.9887651340956775 - 0.14943984295092333im -0.0029569509531839483 - 0.001580941688088154im; 0.0029569509531839496 - 0.0015809416880881548im 0.9887651340956773 + 0.1494398429509233im], [0.9799747340913558 - 0.19864452647497763im 0.010205296881408978 + 0.009258755838166377im; -0.010205296881408978 + 0.00925875583816638im 0.9799747340913557 + 0.1986445264749776im], [0.9684124344840319 - 0.2475462641464486im 0.006026186154200027 + 0.02935794482388511im; -0.006026186154200024 + 0.029357944823885117im 0.9684124344840316 + 0.24754626414644848im], [0.954571577620917 - 0.29591535958187054im -0.0026905777648045855 + 0.034927982380001006im; 0.0026905777648045907 + 0.03492798238000102im 0.9545715776209169 + 0.2959153595818705im], [0.9387347697463967 - 0.3431224741834879im 0.004115047313186222 + 0.03204787301821318im; -0.0041150473131862176 + 0.03204787301821321im 0.9387347697463968 + 0.3431224741834879im], [0.9205276647108096 - 0.3900108166967894im -0.006094734215670173 + 0.021982619871316816im; 0.006094734215670175 + 0.02198261987131685im 0.9205276647108098 + 0.3900108166967894im], [0.8998675265877255 - 0.43521233432193496im 0.006358472942583045 + 0.028075406831952972im; -0.006358472942583041 + 0.028075406831953007im 0.8998675265877257 + 0.43521233432193496im]  …  [-0.15648685930229364 + 0.980231622083645im 0.08938839371552486 + 0.08165503659863035im; -0.0893883937155247 + 0.08165503659863027im -0.15648685930229342 - 0.9802316220836469im], [-0.1062020698992996 + 0.9862472347642309im 0.09207417041155784 + 0.08694745201574915im; -0.09207417041155767 + 0.08694745201574904im -0.10620206989929928 - 0.9862472347642327im], [-0.055457928069234406 + 0.9893318748804325im 0.0957688073980352 + 0.09473750624430216im; -0.09576880739803505 + 0.09473750624430204im -0.05545792806923401 - 0.9893318748804343im], [-0.005069586540250117 + 0.9906817788550469im 0.0970357066123597 + 0.09543575838827797im; -0.09703570661235958 + 0.09543575838827785im -0.005069586540249638 - 0.9906817788550488im], [0.04384945505659482 + 0.9909383652577827im 0.09796698818217227 + 0.08075178498325397im; -0.09796698818217214 + 0.08075178498325392im 0.04384945505659539 - 0.9909383652577848im], [0.09419276928652728 + 0.9879693198879221im 0.09445275379439133 + 0.0782497442486703im; -0.09445275379439122 + 0.07824974424867025im 0.09419276928652794 - 0.987969319887924im], [0.14235459429456435 + 0.9794179639922784im 0.11799643595026194 + 0.08094728158177968im; -0.11799643595026181 + 0.0809472815817796im 0.14235459429456512 - 0.9794179639922804im], [0.1903805029323721 + 0.9727412767337852im 0.11644111065185543 + 0.06301698494611356im; -0.11644111065185533 + 0.06301698494611355im 0.19038050293237302 - 0.9727412767337874im], [0.23729640140889482 + 0.9615172202573865im 0.12882380706888938 + 0.05078857902292991im; -0.1288238070688893 + 0.05078857902292992im 0.23729640140889588 - 0.961517220257389im], [0.28483233686338644 + 0.9510122210154612im 0.11588306385955965 + 0.03189687863564486im; -0.11588306385955958 + 0.031896878635644915im 0.28483233686338755 - 0.9510122210154633im]], [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[0.28483233686338644 + 0.9510122210154612im 0.11588306385955965 + 0.03189687863564486im; -0.11588306385955958 + 0.031896878635644915im 0.28483233686338755 - 0.9510122210154633im], ComplexF64[0.23729640140889482 + 0.9615172202573865im 0.12882380706888938 + 0.05078857902292991im; -0.1288238070688893 + 0.05078857902292992im 0.23729640140889588 - 0.961517220257389im], ComplexF64[0.23729640140889482 + 0.9615172202573865im 0.12882380706888938 + 0.05078857902292991im; -0.1288238070688893 + 0.05078857902292992im 0.23729640140889588 - 0.961517220257389im], ComplexF64[0.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 0.0 + 0.0im], ComplexF64[0.479024585674971 - 0.09117900318138918im -0.1276046692545698 - 0.18827282773143922im; 0.12760466925456945 - 0.18827282773143877im 0.47902458567497225 + 0.09117900318138969im], 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[0.46836717689884777 - 0.1595161365747257im 0.16694605881466196 - 0.03293560417593988im; -0.16694605881466162 - 0.03293560417593997im 0.4683671768988488 + 0.15951613657472627im], 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))

Step 4: Set Up the Optimization Problem

SmoothPulseProblem creates an optimization problem with:

  • Fidelity objective (weight Q)
  • Control regularization (weight R)
  • Smoothness via derivative bounds
qcp = SmoothPulseProblem(
    qtraj,
    N;
    Q = 100.0,       # Fidelity weight (higher = prioritize fidelity)
    R = 1e-2,        # Regularization weight (higher = smoother controls)
    ddu_bound = 1.0,  # Limit on control acceleration
)
QuantumControlProblem{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}}}
  System: 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{}}
  Goal: Matrix{ComplexF64}
  Trajectory: 100 knots
  State: Ũ⃗
  Controls: u

Step 5: Solve!

The solve! function runs the optimizer:

cached_solve!(qcp, "first_gate"; max_iter = 20, verbose = false, print_level = 1)

Step 6: Analyze the Results

First, check the fidelity:

fidelity(qcp)
0.9993911615795472

Get the optimized trajectory:

traj = get_trajectory(qcp)

# Check the final unitary
U_final = iso_vec_to_operator(traj[:Ũ⃗][:, end])
round.(U_final, digits = 3)
2×2 Matrix{ComplexF64}:
 0.017+0.015im  -0.017+1.0im
 0.017+1.0im     0.017-0.015im

Step 7: Visualize

Plot the optimized control pulses:

fig = Figure(size = (800, 400))

# Time axis
plot_times = cumsum([0; get_timesteps(traj)])[1:(end-1)]

# Control pulses
ax1 = Axis(
    fig[1, 1],
    xlabel = "Time",
    ylabel = "Control Amplitude",
    title = "Optimized Controls",
)
lines!(ax1, plot_times, traj[:u][1, :], label = "u_x (σ_x drive)", linewidth = 2)
lines!(ax1, plot_times, traj[:u][2, :], label = "u_y (σ_y drive)", linewidth = 2)
axislegend(ax1, position = :rt)

fig
Example block output

Understanding the Solution

The optimizer found control pulses that:

  1. Start and end smoothly (due to derivative regularization)
  2. Stay within bounds (due to drive_bounds)
  3. Achieve high fidelity (due to the Q-weighted objective)

The X gate rotates the qubit state around the X-axis by π radians. You can see the controls create the right rotation!

What's Next?

Now that you've synthesized your first gate, try:

  1. Different gates: Change U_goal to GATES[:H] (Hadamard) or GATES[:T]
  2. Faster gates: Reduce T and see how fidelity changes
  3. Smoother pulses: Increase R or decrease ddu_bound
  4. Time-optimal: Add Δt_bounds and use MinimumTimeProblem

Continue to the State Transfer tutorial to learn about preparing specific quantum states.

This page was generated using Literate.jl.