State-to-state transfer in a two-level quantum system

The goal of this example is misimise the cost

\[\mathcal{C} = |\langle \uparrow |\psi(t_f)\rangle |^2 + \frac{\gamma}2 \int_0^{t_f}u(t)^2\mathrm{d}t\]

while we steer the system bettwen to states, $|\uparrow\;\rangle$ to $|\downarrow\;\rangle$, described by following Hamiltonian:

\[H(t) = \frac{\Delta}{2} \sigma_z + \frac{u(t)}{2} \sigma_x\]

where

The final time $t_f$ is set equalt to $2\pi/\sqrt{1+\Delta^2}$;
\[\psi(t_f)\]
is the state at the final time $t_f$;
\[\gamma\]
is a factor to adjust the relative weight of the two terms in the cost
\[\Delta\]
represents the frequency offset;
\[u(t) \in [-u_0,u_0]\]
is the control;
\[\sigma_z,\sigma_x\]
are Pauli matrices.

The dynamics is governed by the Schrödinger equation $ i\hbar |\dot{\psi}\rangle = H(t) |\psi\rangle. $ However, ths states here belong to a Hilber space $\mathcal{H} = \mathbb{C}^2$, which is not easy to visuzalize. To improve that, we can tranform in Bloch representation to have a real system. It can be shown applinng $\dot x_j = [H,\sigma_j ]$, where $[ \cdot,\cdot]$ denote the Lie bracket. In other words,

\[ \begin{cases} \dot x = -\Delta \cdot y\\ \dot y = \Delta \cdot x - u \cdot z\\ \dot z = u \cdot y \end{cases}\]

with $|\uparrow\;\rangle = (0,0,1)$ to $|\downarrow\;\rangle= (0,0,-1)$. Furthermore, we will set $\Delta = 0.5$ and $p_0= 0.1$.

using OptimalControl
using OrdinaryDiffEq
using LinearAlgebra
using NLPModelsIpopt


q0 = [0.0, 0.0, 1.0]
qf = [0.0, 0.0, -1.0]
Δ = 0.5
tf = 2 * π / (√( 1 + Δ^2))
γ = 0.1


ocp1 = @def begin
    t ∈ [0, tf], time
    q = [x, y, z] ∈ R³, state
    u ∈ R, control

    q(0) == q0
    ∂(x)(t) == - Δ * y(t)
    ∂(y)(t) == Δ * x(t) - u(t) * z(t)
    ∂(z)(t) == u(t) * y(t)
    sum((q(tf) - qf).^2) + (γ / 2) * ∫(u(t)^2) → min
end

Direct solve

N = 100
direct_sol1 = solve(ocp1, grid_size=N)

▫ This is OptimalControl version v1.1.6 running with: direct, adnlp, ipopt.

▫ The optimal control problem is solved with CTDirect version v0.17.4.

   ┌─ The NLP is modelled with ADNLPModels and solved with NLPModelsIpopt.
   │
   ├─ Number of time steps⋅: 100
   └─ Discretisation scheme: midpoint

▫ This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.1.

Number of nonzeros in equality constraint Jacobian...:     1603
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:      503

Total number of variables............................:      403
                     variables with only lower bounds:        0
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        0
Total number of equality constraints.................:      303
Total number of inequality constraints...............:        0
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        0

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.2328099e+00 9.00e-01 1.17e-02   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  3.4017483e+00 2.13e-02 1.85e+00 -11.0 9.00e-01   0.0 1.00e+00 1.00e+00h  1
   2  3.9540847e+00 1.32e-03 1.14e+00 -11.0 4.27e-01   0.4 1.00e+00 1.00e+00h  1
   3  3.9449068e+00 1.79e-05 3.91e-02 -11.0 2.96e-02  -0.1 1.00e+00 1.00e+00h  1
   4  3.9192381e+00 1.13e-04 1.57e-02 -11.0 5.31e-02  -0.5 1.00e+00 1.00e+00h  1
   5  3.8962038e+00 5.08e-03 4.32e-02 -11.0 9.82e+00  -1.0 1.00e+00 3.12e-02f  6
   6  3.4793145e+00 2.47e-03 6.02e-02 -11.0 2.30e-01  -0.6 1.00e+00 1.00e+00f  1
   7  2.9665605e+00 2.38e-02 1.22e-01 -11.0 2.87e+00  -1.1 1.00e+00 2.50e-01f  3
   8  2.1466343e+00 5.14e-02 1.49e-01 -11.0 1.27e+01  -1.5 1.00e+00 6.25e-02f  5
   9  4.7586995e-01 1.71e-02 8.43e-02 -11.0 8.95e-01  -1.1 1.00e+00 1.00e+00f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  2.5464439e-01 8.87e-03 1.22e-02 -11.0 5.49e-01    -  1.00e+00 1.00e+00h  1
  11  2.1583590e-01 3.96e-02 1.25e-02 -11.0 1.47e+00    -  1.00e+00 1.00e+00h  1
  12  1.9565085e-01 3.84e-03 1.18e-03 -11.0 3.85e-01    -  1.00e+00 1.00e+00h  1
  13  1.9743698e-01 1.43e-04 2.70e-04 -11.0 7.77e-02    -  1.00e+00 1.00e+00h  1
  14  1.9748158e-01 1.72e-05 8.34e-06 -11.0 3.49e-02    -  1.00e+00 1.00e+00h  1
  15  1.9747207e-01 2.56e-09 8.79e-10 -11.0 2.52e-04    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 15

                                   (scaled)                 (unscaled)
Objective...............:   1.9747206792824548e-01    1.9747206792824548e-01
Dual infeasibility......:   8.7927574943247322e-10    8.7927574943247322e-10
Constraint violation....:   2.5610386034102817e-09    2.5610386034102817e-09
Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   0.0000000000000000e+00    0.0000000000000000e+00
Overall NLP error.......:   2.5610386034102817e-09    2.5610386034102817e-09


Number of objective function evaluations             = 29
Number of objective gradient evaluations             = 16
Number of equality constraint evaluations            = 30
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 16
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 15
Total seconds in IPOPT                               = 5.262

EXIT: Optimal Solution Found.

using Plots
plt = plot(direct_sol1)

Indirect solve

We can also solve the problem with shooting thechincs. Using the Pontryagin’s Maximum Principle, the pseudo-Hamiltonian is given by

\[H_p(x, p, u) = \Delta(p_yx - p_x y) + u(p_z y - p_yz)+ \mathcal{V} \frac{\gamma}{2} u^2,\]

where $p = (p_x, p_y,p_z)$ is the costate vector. The optimal control is given by the maximization of $H_p$:

\[ u = \frac{p_zy - p_yz}{\gamma}.\]

Define the packages:

using MINPACK

Define the control and Hamiltonian flow:

# Control
u(q, p) = (p[3] * q[2] - p[2] * q[3]) / γ

# Hamiltonian flow
f = Flow(ocp1, u)

The shooting function enforces the conditions:

\[ S : \mathbb{R}^3 \longrightarrow \mathbb{R}^3, \\ S(p_0) := p(t_f,q_0,p_0) + 2(q(t_f,x_0,p_f)-q_f)\]

p0 = direct_sol1.costate(0);

function shoot!(s, p0)
    qqf, pf = f(0, q0, p0, tf)
    s[1:3] .= pf + 2(qqf - qf)
    return nothing
end
s = similar(p0, 3)
shoot!(s, p0)
println("\nNorm of the shooting function: ‖s‖ = ", norm(s), "\n")


Norm of the shooting function: ‖s‖ = 0.14202210597785073

We are now ready to solve the shooting equations:

using DifferentiationInterface
import ForwardDiff
backend = AutoForwardDiff();

ξ = p0 # initial guess
nle! = (s, ξ) -> shoot!(s, ξ)
jnle! = (js, ξ) -> jacobian!(nle!, similar(ξ), js, backend, ξ)

indirect_sol = fsolve(nle!, jnle!, ξ; show_trace=true)
p0 = indirect_sol.x

Iter     f(x) inf-norm    Step 2-norm      Step time
------   --------------   --------------   --------------
     1     1.106012e-01     0.000000e+00         0.221316
     2     7.370664e-03     1.073115e-05         5.646616
     3     3.931570e-03     6.277312e-06         0.006357
     4     3.418731e-05     6.585040e-06         0.007321
     5     5.918382e-07     2.865815e-10         0.006292
     6     1.451672e-08     1.679182e-13         0.006444
     7     1.157234e-10     1.531461e-16         0.006877
     8     9.989128e-13     1.323170e-20         0.006242

shoot!(s, p0)
println("\nNorm of the shooting function: ‖s‖ = ", norm(s), "\n")


Norm of the shooting function: ‖s‖ = 1.3886406424534495e-12

Finally, we reconstruct and plot the solution obtained by the indirect method:

flow_sol = f((0, tf), q0, p0)
plot!(plt, flow_sol, solution_label="(indirect)")

Trjajectorie in the Bloch sphere

using Plots
x = []
y = []
z = []

for t in time_grid(flow_sol)
    push!(x, state(flow_sol)(t)[1])
    push!(y, state(flow_sol)(t)[2])
    push!(z, state(flow_sol)(t)[3])
end
gr()
θ = 0:0.01:π
φ = 0:0.01:2π
xs = [sin(t) * cos(p) for t in θ, p in φ]
ys = [sin(t) * sin(p) for t in θ, p in φ]
zs = [cos(t) for t in θ, p in φ]



p = plot(xs, ys, zs,
    st=:surface,
    color=:lightblue,
    alpha=0.5,
    legend=false,
    axis = nothing,
    background_color=:transparent,
    grid=false,
)

plot!(x, y, z, lw=2, color=:blue, label="Trajectory")
scatter!([x[1]], [y[1]], [z[1]], markersize=2, color=:green, label="Start")
scatter!([x[end]], [y[end]], [z[end]], markersize=2, color=:red, label="End")

Reproducibility

_downloads_toml(".") # hider

You can download the exact environment used to build this documentation:

📦 Project.toml - Package dependencies
📋 Manifest.toml - Complete dependency tree with versions

ℹ️ Version info

Julia Version 1.12.1
Commit ba1e628ee49 (2025-10-17 13:02 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver3)
  GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 4 virtual cores)
Environment:
  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager

📦 Package status

Status `~/work/sts2levelqs.jl/sts2levelqs.jl/docs/Project.toml`
  [a0c0ee7d] DifferentiationInterface v0.7.10
  [e30172f5] Documenter v1.15.0
  [f6369f11] ForwardDiff v1.2.2
  [4854310b] MINPACK v1.3.0
  [f4238b75] NLPModelsIpopt v0.11.0
  [5f98b655] OptimalControl v1.1.6
  [1dea7af3] OrdinaryDiffEq v6.103.0
  [91a5bcdd] Plots v1.41.1
  [d1f5d2d8] sts2levelqs v0.1.1 `~/work/sts2levelqs.jl/sts2levelqs.jl`
  [37e2e46d] LinearAlgebra v1.12.0

📚 Complete manifest

Status `~/work/sts2levelqs.jl/sts2levelqs.jl/docs/Manifest.toml`
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  [41fe7b60] Unzip v0.2.0
  [efce3f68] WoodburyMatrices v1.0.0
  [d1f5d2d8] sts2levelqs v0.1.1 `~/work/sts2levelqs.jl/sts2levelqs.jl`
  [ae81ac8f] ASL_jll v0.1.3+0
  [6e34b625] Bzip2_jll v1.0.9+0
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  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
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  [b22a6f82] FFMPEG_jll v8.0.0+0
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  [559328eb] FriBidi_jll v1.0.17+0
  [0656b61e] GLFW_jll v3.4.0+2
  [d2c73de3] GR_jll v0.73.18+0
  [b0724c58] GettextRuntime_jll v0.22.4+0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [020c3dae] Git_LFS_jll v3.7.0+0
  [f8c6e375] Git_jll v2.51.3+0
  [7746bdde] Glib_jll v2.86.0+0
  [3b182d85] Graphite2_jll v1.3.15+0
  [017b0a0e] HSL_jll v4.0.4+0
  [2e76f6c2] HarfBuzz_jll v8.5.1+0
  [e33a78d0] Hwloc_jll v2.12.2+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [9cc047cb] Ipopt_jll v300.1400.1900+0
  [aacddb02] JpegTurbo_jll v3.1.3+0
  [c1c5ebd0] LAME_jll v3.100.3+0
  [88015f11] LERC_jll v4.0.1+0
  [1d63c593] LLVMOpenMP_jll v18.1.8+0
  [dd4b983a] LZO_jll v2.10.3+0
  [e9f186c6] Libffi_jll v3.4.7+0
  [7e76a0d4] Libglvnd_jll v1.7.1+1
  [94ce4f54] Libiconv_jll v1.18.0+0
  [4b2f31a3] Libmount_jll v2.41.2+0
  [89763e89] Libtiff_jll v4.7.2+0
  [38a345b3] Libuuid_jll v2.41.2+0
  [d00139f3] METIS_jll v5.1.3+0
  [856f044c] MKL_jll v2025.2.0+0
  [d7ed1dd3] MUMPS_seq_jll v500.800.100+0
  [c8ffd9c3] MbedTLS_jll v2.28.10+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [656ef2d0] OpenBLAS32_jll v0.3.29+0
  [9bd350c2] OpenSSH_jll v10.2.1+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.5.2+0
  [36c8627f] Pango_jll v1.56.4+0
⌅ [30392449] Pixman_jll v0.44.2+0
  [c0090381] Qt6Base_jll v6.8.2+2
  [629bc702] Qt6Declarative_jll v6.8.2+1
  [ce943373] Qt6ShaderTools_jll v6.8.2+1
  [e99dba38] Qt6Wayland_jll v6.8.2+2
⌅ [319450e9] SPRAL_jll v2025.5.20+0
  [a44049a8] Vulkan_Loader_jll v1.3.243+0
  [a2964d1f] Wayland_jll v1.24.0+0
⌅ [02c8fc9c] XML2_jll v2.13.9+0
  [ffd25f8a] XZ_jll v5.8.1+0
  [f67eecfb] Xorg_libICE_jll v1.1.2+0
  [c834827a] Xorg_libSM_jll v1.2.6+0
  [4f6342f7] Xorg_libX11_jll v1.8.12+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.13+0
  [935fb764] Xorg_libXcursor_jll v1.2.4+0
  [a3789734] Xorg_libXdmcp_jll v1.1.6+0
  [1082639a] Xorg_libXext_jll v1.3.7+0
  [d091e8ba] Xorg_libXfixes_jll v6.0.2+0
  [a51aa0fd] Xorg_libXi_jll v1.8.3+0
  [d1454406] Xorg_libXinerama_jll v1.1.6+0
  [ec84b674] Xorg_libXrandr_jll v1.5.5+0
  [ea2f1a96] Xorg_libXrender_jll v0.9.12+0
  [a65dc6b1] Xorg_libpciaccess_jll v0.18.1+0
  [c7cfdc94] Xorg_libxcb_jll v1.17.1+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.3+0
  [e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
  [12413925] Xorg_xcb_util_image_jll v0.4.1+0
  [2def613f] Xorg_xcb_util_jll v0.4.1+0
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
  [35661453] Xorg_xkbcomp_jll v1.4.7+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
  [c5fb5394] Xorg_xtrans_jll v1.6.0+0
  [3161d3a3] Zstd_jll v1.5.7+1
  [b792d7bf] cminpack_jll v1.3.12+0
  [35ca27e7] eudev_jll v3.2.14+0
  [214eeab7] fzf_jll v0.61.1+0
  [a4ae2306] libaom_jll v3.13.1+0
  [0ac62f75] libass_jll v0.17.4+0
  [1183f4f0] libdecor_jll v0.2.2+0
  [2db6ffa8] libevdev_jll v1.13.4+0
  [f638f0a6] libfdk_aac_jll v2.0.4+0
  [36db933b] libinput_jll v1.28.1+0
  [b53b4c65] libpng_jll v1.6.50+0
  [f27f6e37] libvorbis_jll v1.3.8+0
  [009596ad] mtdev_jll v1.1.7+0
  [1317d2d5] oneTBB_jll v2022.0.0+1
  [1270edf5] x264_jll v10164.0.1+0
  [dfaa095f] x265_jll v4.1.0+0
  [d8fb68d0] xkbcommon_jll v1.9.2+0
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [a63ad114] Mmap v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.12.0
  [de0858da] Printf v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [1a1011a3] SharedArrays v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.12.0
  [f489334b] StyledStrings v1.11.0
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test v1.11.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.3.0+1
  [deac9b47] LibCURL_jll v8.11.1+1
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.5.20
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.1+0
  [efcefdf7] PCRE2_jll v10.44.0+1
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.5.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`