Analytic Displaced Gaussian (Since apparently I don't know how fine the mesh has to be)-.1 displacement.ipynb

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  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "toarray (generic function with 2 methods)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using SpecialFunctions\n",
    "using QuantumOptics\n",
    "\n",
    "function fidelity(ρ, σ)\n",
    "    tr(sqrt(sqrt(ρ) * σ * sqrt(ρ)))^(1/2)\n",
    "end\n",
    "\n",
    "function R(n::Int64, m::Int64, ρ::Float64)\n",
    "    if (n - m) % 2 != 0\n",
    "        0\n",
    "    else\n",
    "        function summand(k)\n",
    "            n = big(n)\n",
    "            k = big(k)\n",
    "            (-1)^k * factorial(n-k)/(factorial(k)*factorial(Int((n+m)/2) - k)*factorial(Int((n-m)/2) - k))*(ρ)^(n-2*k)\n",
    "        end\n",
    "        mapreduce(summand, +, Array(range(0, stop=Int((n-m)/2), step=1)))\n",
    "    end\n",
    "end\n",
    "\n",
    "function sample_across_unit_disk(f, discretization)\n",
    "    x = y = Array(range(-1, 1, length=discretization))\n",
    "    z = Array{typeof(f(0.,0.))}(UndefInitializer(), length(x), length(y))\n",
    "    for (i, xx) in enumerate(x)\n",
    "        for (j, yy) in enumerate(y)\n",
    "            if xx^2 + yy^2 > 1\n",
    "                if typeof(f(0.,0.)) == Tuple{Float64,Float64}\n",
    "                    z[i, j] = (0.,0.)\n",
    "                else\n",
    "                    z[i, j] = 0.\n",
    "                end\n",
    "            else\n",
    "                ρ = sqrt(xx^2 + yy^2)\n",
    "                ϕ = atan(yy, xx)\n",
    "                res = f(ρ, ϕ)\n",
    "                z[i, j] = res\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    x, y, z\n",
    "end\n",
    "\n",
    "function unpack_zernike(zernike_coefficients_even, zernike_coefficients_odd, ρ, ω, t)\n",
    "    \"\"\"Each Zernike array is of arbitrary length, but they are ordered by order. The first is Z00 to Zn0 where\n",
    "    n is the number of 0th order zernike polynomials being considered.\"\"\"\n",
    "    Pevens = []\n",
    "    eventuples = []\n",
    "    for (n, zeven) in enumerate(zernike_coefficients_even)\n",
    "        neven = n - 1\n",
    "        push!(Pevens, [zeven[i] * R(neven, i-1, ρ) for i in Array(range(1, length(zeven), step=1))])\n",
    "    end\n",
    "    for i in range(1, length(Pevens), step=1)\n",
    "        rotationeven = - (i - 1) * ω * t\n",
    "        total = 0\n",
    "        for peven in Pevens\n",
    "            if length(peven) >= i\n",
    "                total += peven[i]\n",
    "            end\n",
    "        end    \n",
    "        push!(eventuples, (i - 1, total, rotationeven))\n",
    "    end\n",
    "\n",
    "    Podds = []\n",
    "    oddtuples = []\n",
    "    for (n, zodd) in enumerate(zernike_coefficients_odd)\n",
    "        nodd = n - 1\n",
    "        push!(Podds, [zodd[i] * R(nodd, i-1, ρ) for i in Array(range(1, length(zodd), step=1))])\n",
    "    end\n",
    "    for i in range(1, length(Podds), step=1)\n",
    "        rotationodd = - (i - 1) * ω * t\n",
    "        total = 0\n",
    "        for podd in Podds\n",
    "            if length(podd) >= i\n",
    "                total += podd[i]\n",
    "            end\n",
    "        end    \n",
    "        push!(oddtuples, (i - 1, total, rotationodd))\n",
    "    end\n",
    "    eventuples, oddtuples\n",
    "end\n",
    "\n",
    "function infidelity_across_disk(F1, F2)\n",
    "    function infidelity_polar(ρ, ϕ)\n",
    "        ψ1 = F1(ρ, ϕ).data\n",
    "        ψ2 = F2(ρ, ϕ).data\n",
    "        1 - real(fidelity(ψ1, ψ2))\n",
    "    end\n",
    "end\n",
    "\n",
    "function H_odf(ρ, ϕ, t, zernike_coefficients_even, zernike_coefficients_odd, U, ψ, μ, ω)\n",
    "    eventuples, oddtuples = unpack_zernike(zernike_coefficients_even, zernike_coefficients_odd, ρ, ω, t)\n",
    "    eventerm = oddterm = 0\n",
    "    if length(eventuples) > 0\n",
    "        eventerm = mapreduce(x->x[2]*cos(x[1]*ϕ + x[3]), +, eventuples)\n",
    "    end\n",
    "    if length(oddtuples) > 0\n",
    "        oddterm = mapreduce(x->x[2]*sin(x[1]*ϕ + x[3]), +, oddtuples)\n",
    "    end\n",
    "    U * cos(-μ*t + ψ + eventerm + oddterm) \n",
    "end\n",
    "\n",
    "function least_squares_solution(mat, y)\n",
    "    inv(transpose(mat) * mat)*transpose(mat) * y\n",
    "end\n",
    "\n",
    "toarray(s::Union{Set, Array}) = [toarray.(s)...]\n",
    "toarray(v::Number) = v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "integrand (generic function with 1 method)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function R(n::Int64, m::Int64, ρ::Float64)\n",
    "    if (n - m) % 2 != 0\n",
    "        0\n",
    "    else\n",
    "        function summand(k)\n",
    "            n = big(n)\n",
    "            k = big(k)\n",
    "            (-1)^k * factorial(n-k)/(factorial(k)*factorial(Int((n+m)/2) - k)*factorial(Int((n-m)/2) - k))*(ρ)^(n-2*k)\n",
    "        end\n",
    "        mapreduce(summand, +, Array(range(0, stop=Int((n-m)/2), step=1)))\n",
    "    end\n",
    "end\n",
    "\n",
    "function Z(n, m, ρ, θ)\n",
    "    if m < 0\n",
    "        R(n, abs(m), ρ) * sin(abs(m) * θ)\n",
    "    else\n",
    "        R(n, m, ρ) * cos(m * θ)\n",
    "    end\n",
    "end\n",
    "function neumann(m)\n",
    "    if m == 0\n",
    "        2\n",
    "    else \n",
    "        1\n",
    "    end\n",
    "end\n",
    "σ1 = .1\n",
    "σ2 = .1\n",
    "x_displacement = 0\n",
    "amp = 1 # Used to make the small angle approximation work better.\n",
    "function gaussian(σ1, σ2, x_displacement)\n",
    "    function func(ρ, ϕ)\n",
    "        x = (x_displacement + ρ*cos(ϕ))\n",
    "        y = ρ*sin(ϕ)\n",
    "        amp * exp(-x^2/σ1^2 + -y^2/σ2^2)\n",
    "    end\n",
    "end\n",
    "\n",
    "function integrand(n, m)\n",
    "    function rtn(coor)\n",
    "        ρ = coor[1]\n",
    "        θ = coor[2]\n",
    "        x = ρ * cos(θ) + x_displacement\n",
    "        y = ρ * sin(θ)\n",
    "        Z(n, m, ρ, θ) * exp(-x^2/σ1^2 - y^2/σ2^2) * ρ\n",
    "    end\n",
    "    rtn\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Cubature\n",
    "max_order = 20\n",
    "discretization = 50\n",
    "# I think the discretization may have been the problem...\n",
    "zernike_samples = []\n",
    "ordering = []\n",
    "coefficients = []\n",
    "for n in range(0, max_order, step=1)\n",
    "    for m in range(0, n, step=1)\n",
    "        if (n-m)%2 == 0\n",
    "            Zeven(ρ, ϕ) = R(n, m, ρ) * cos(m * ϕ)\n",
    "            push!(zernike_samples, sample_across_unit_disk(Zeven, discretization)[3][:])\n",
    "            push!(ordering, (n, m))\n",
    "            push!(coefficients, (2*n+2)/(π*neumann(m)) * hcubature(integrand(n, m), [0., 0.], [1., 2*π], maxevals=10000)[1])\n",
    "            if m !=0\n",
    "                Zodd(ρ, ϕ) = R(n, m, ρ) * sin(m * ϕ)\n",
    "                push!(zernike_samples, sample_across_unit_disk(Zodd, discretization)[3][:])\n",
    "                push!(ordering, (n, -m))\n",
    "                push!(coefficients, (2*n+2)/(π*neumann(m)) * hcubature(integrand(n, -m), [0., 0.], [1., 2*π], maxevals=10000)[1])\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "end\n",
    "ZZ = reduce(hcat, zernike_samples);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "xx, yy, zz = sample_across_unit_disk(gaussian(σ1, σ2, x_displacement), discretization)\n",
    "G = zz[:];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "reconstruction = zeros(discretization*discretization)\n",
    "for (i, zernike_sample) in enumerate(zernike_samples)\n",
    "    reconstruction += coefficients[i] * zernike_sample\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "plot(heatmap(xx, yy, reshape(reconstruction, (discretization, discretization))), aspectratio = true, xlabel=\"x\", ylabel=\"y\", title=\"Reconstruction of a Gaussian using n <= 10\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
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       "  2232.76,1349.85 2232.76,1349.85 2256.76,1349.85 2232.76,1349.85 2232.76,1210.53 2256.76,1210.53 2232.76,1210.53 2232.76,1071.21 2256.76,1071.21 2232.76,1071.21 \n",
       "  2232.76,931.887 2256.76,931.887 2232.76,931.887 2232.76,792.565 2256.76,792.565 2232.76,792.565 2232.76,653.242 2256.76,653.242 2232.76,653.242 2232.76,513.92 \n",
       "  2256.76,513.92 2232.76,513.92 2232.76,374.598 2256.76,374.598 2232.76,374.598 2232.76,235.275 2256.76,235.275 2232.76,235.275 2232.76,196.797 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "diff = sample_across_unit_disk(gaussian(σ1,σ2, x_displacement), discretization)[3] - reshape(reconstruction, (discretization, discretization))\n",
    "diff = diff.*diff\n",
    "plot(heatmap(xx, yy, diff), aspectratio = true, xlabel=\"x\", ylabel=\"y\", title=\"Error in Reconstruction of a Gaussian using n <= 10\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "lookup = Dict(zip(ordering, coefficients));\n",
    "zernike_even_coefficients = []\n",
    "zernike_odd_coefficients = []\n",
    "curr_even = []\n",
    "curr_odd = []\n",
    "for n in range(0, max_order, step=1)\n",
    "    for m in range(0, n, step=1)\n",
    "        if (n-m)%2 != 0\n",
    "            push!(curr_even, 0)\n",
    "            push!(curr_odd, 0)\n",
    "        else\n",
    "            # I'm not computing the overlap here, I'm computing the coefficient numerically - so no factor of two.\n",
    "            if m == 0\n",
    "                push!(curr_even, lookup[(n, m)])\n",
    "                push!(curr_odd, lookup[(n, -m)])\n",
    "            else\n",
    "                push!(curr_even, lookup[(n, m)])\n",
    "                push!(curr_odd, lookup[(n, -m)])\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    push!(zernike_even_coefficients, curr_even)\n",
    "    push!(zernike_odd_coefficients, curr_odd)\n",
    "    curr_even = []\n",
    "    curr_odd = []\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "gaussian_spin_profile (generic function with 1 method)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sequential_exact_evolution_evaluator_factory(ψ0, T, zernikeeven, zernikeodd, U, θ, ω, b)\n",
    "    \"\"\"Apply all the zernike coefficients given, in order, for time T each.\"\"\"\n",
    "    orders = range(0, max(length(zernikeeven), length(zernikeodd)) - 1, step=1)\n",
    "    function evaluator(ρ, ϕ)\n",
    "        ψ = ψ0\n",
    "        for order in orders\n",
    "            μ = order * ω\n",
    "            H(t, _) = H_odf(ρ, ϕ, t, zernikeeven, zernikeodd, U, θ, μ, ω)*sigmaz(b), [sigmam(b)], [dagger(sigmam(b))]\n",
    "            _, ψ = timeevolution.master_dynamic(T, ψ, H; rates=[Γ])\n",
    "            ψ = last(ψ)\n",
    "        end\n",
    "        ψ\n",
    "    end\n",
    "end\n",
    "\n",
    "function gaussian_spin_profile(ρ, ϕ)\n",
    "    ψ0 = 1/sqrt(2) * (spindown(b) + spinup(b))\n",
    "    H(t, _) = gaussian(σ1, σ2, x_displacement)(ρ, ϕ) * sigmaz(b), [sigmam(b)], [dagger(sigmam(b))]\n",
    "    evolution_time = π/(2*amp)\n",
    "    step_size = evolution_time/1\n",
    "    T = [0.0:step_size:evolution_time;];\n",
    "    _, ψ = timeevolution.master_dynamic(T, ψ0, H; rates=[Γ])\n",
    "    last(ψ)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "Γ = 1/62\n",
    "ω = 2*π*180E3\n",
    "θ = -π/2;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::var\"#evaluator#16\"{Ket{SpinBasis{1//2,Int64},Array{Complex{Float64},1}},Array{Float64,1},Array{Any,1},Array{Any,1},Float64,Float64,Float64,SpinBasis{1//2,Int64},StepRange{Int64,Int64}}) (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = SpinBasis(1//2)\n",
    "ψ0 = 1/sqrt(2) * (spindown(b) + spinup(b))\n",
    "evolution_time =  50E-6\n",
    "U = π/(2*amp*evolution_time) * 3\n",
    "step_size = evolution_time\n",
    "T = [0.0:step_size:evolution_time;];\n",
    "sequential_exact_evolution = sequential_exact_evolution_evaluator_factory(ψ0, T, zernike_even_coefficients, zernike_odd_coefficients, U, θ, ω, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Precompiling Plots [91a5bcdd-55d7-5caf-9e0b-520d859cae80]\n",
      "└ @ Base loading.jl:1278\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Your GR installation is incomplete. Rerun build step for GR package.\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "ERROR: LoadError: LoadError: InitError: Evaluation into the closed module `GR` breaks incremental compilation because the side effects will not be permanent. This is likely due to some other module mutating `GR` with `eval` during precompilation - don't do this.\n",
      "Stacktrace:\n",
      " [1] eval at ./boot.jl:331 [inlined]\n",
      " [2] __init__() at /home/ampolloreno/.julia/packages/GR/RlE5Y/src/GR.jl:274\n",
      " [3] _include_from_serialized(::String, ::Array{Any,1}) at ./loading.jl:697\n",
      " [4] _require_search_from_serialized(::Base.PkgId, ::String) at ./loading.jl:782\n",
      " [5] _require(::Base.PkgId) at ./loading.jl:1007\n",
      " [6] require(::Base.PkgId) at ./loading.jl:928\n",
      " [7] require(::Module, ::Symbol) at ./loading.jl:923\n",
      " [8] include(::Function, ::Module, ::String) at ./Base.jl:380\n",
      " [9] include at ./Base.jl:368 [inlined]\n",
      " [10] include(::String) at /home/ampolloreno/.julia/packages/Plots/oZheM/src/Plots.jl:1\n",
      " [11] top-level scope at /home/ampolloreno/.julia/packages/Plots/oZheM/src/Plots.jl:216\n",
      " [12] include(::Function, ::Module, ::String) at ./Base.jl:380\n",
      " [13] include(::Module, ::String) at ./Base.jl:368\n",
      " [14] top-level scope at none:2\n",
      " [15] eval at ./boot.jl:331 [inlined]\n",
      " [16] eval(::Expr) at ./client.jl:467\n",
      " [17] top-level scope at ./none:3\n",
      "during initialization of module GR\n",
      "in expression starting at /home/ampolloreno/.julia/packages/Plots/oZheM/src/backends/gr.jl:6\n",
      "in expression starting at /home/ampolloreno/.julia/packages/Plots/oZheM/src/Plots.jl:216\n"
     ]
    },
    {
     "ename": "LoadError",
     "evalue": "Failed to precompile Plots [91a5bcdd-55d7-5caf-9e0b-520d859cae80] to /home/ampolloreno/.julia/compiled/v1.5/Plots/ld3vC_hKmSG.ji.",
     "output_type": "error",
     "traceback": [
      "Failed to precompile Plots [91a5bcdd-55d7-5caf-9e0b-520d859cae80] to /home/ampolloreno/.julia/compiled/v1.5/Plots/ld3vC_hKmSG.ji.",
      "",
      "Stacktrace:",
      " [1] error(::String) at ./error.jl:33",
      " [2] compilecache(::Base.PkgId, ::String) at ./loading.jl:1305",
      " [3] _require(::Base.PkgId) at ./loading.jl:1030",
      " [4] require(::Base.PkgId) at ./loading.jl:928",
      " [5] require(::Module, ::Symbol) at ./loading.jl:923",
      " [6] include_string(::Function, ::Module, ::String, ::String) at ./loading.jl:1091"
     ]
    }
   ],
   "source": [
    "using Plots\n",
    "max_ρ = 1\n",
    "min_ρ = .0\n",
    "ρs = range(-max_ρ, -min_ρ, step=max_ρ/5)\n",
    "angle = 0\n",
    "target2 = [Real.(expect(sigmax(b), gaussian_spin_profile(ρ, angle))) for ρ in ρs]\n",
    "y2s = [Real.(expect(sigmax(b), sequential_exact_evolution(ρ, angle))) for ρ in ρs];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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     "execution_count": 67,
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     "output_type": "execute_result"
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   ],
   "source": [
    "plot(ρs, [target2, y2s], labels=[\"Gaussian Hamiltonian\" \"Zernike Reconstruction (Exact Evolution)\"], legend=:bottomleft, ylabel=\"<σx>\", xlabel=\"x\", title=\"Comparing Evolution Under a Gaussian Hamiltonian\\n to Evolution Under a Zernike-decomposed Hamiltonian\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
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     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(ρs, [target2, y2s], labels=[\"Gaussian Hamiltonian\" \"Zernike Reconstruction (Exact Evolution)\"], legend=:bottomright, ylabel=\"<σx>\", xlabel=\"x\", title=\"Comparing Evolution Under a Gaussian Hamiltonian\\n to Evolution Under a Zernike-decomposed Hamiltonian\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rec = reshape(-2 * reconstruction .+ 1, (discretization, discretization))[Int64(discretization/2):discretization,Int64(discretization/2)]\n",
    "exact = [-2 * gaussian(σ1, σ2, x_displacement)(2/discretization * i , 0) + 1 for i in range(0., discretization/2, step=1)]\n",
    "plot([exact, rec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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