Triangular Lattice Fidelity - Working.ipynb

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     "data": {
      "text/plain": [
       "H_odf (generic function with 2 methods)"
      ]
     },
     "execution_count": 45,
     "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 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, U, ψ, μ, ω)\n",
    "    total = 0\n",
    "    for (i, coeff) in enumerate(zernike_coefficients_even)\n",
    "        total += coeff * (R(Int64(2*i - 2), 0, ρ)) \n",
    "    end\n",
    "    U * cos(total) \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "reconstruct_gaussian (generic function with 1 method)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "using Combinatorics\n",
    "function Z_polynomial(n::Int64)\n",
    "    \"\"\"Coefficients for Z_j^0, from low degree to high degree.\"\"\"\n",
    "    function summand(k)\n",
    "        n = big(n)\n",
    "        k = big(k)\n",
    "        # The normalization in the front seems right, according to Wikipedia.\n",
    "        (-1)^k * factorial(n-k)/(factorial(k)*factorial(Int(n/2) - k)*factorial(Int((n)/2) - k))\n",
    "    end\n",
    "    res = []\n",
    "    for i in Array(range(0, stop=Int(n/2), step=1))\n",
    "        push!(res, summand(i))\n",
    "    end\n",
    "    reverse(res)\n",
    "end\n",
    "function gaussian_coefficients(n, σ)\n",
    "    \"\"\"Integral of a Gaussian against an nth order polynomial from 0 to 1, times 2pi. n must be odd.\"\"\"\n",
    "    total = BigFloat(0)\n",
    "    n = big(n)\n",
    "    total = 1/2 * σ^2 * doublefactorial(n - 1) * (σ^2/2)^((n-1)/2)\n",
    "    for i in range(0, Int64((n-1)/2), step = 1)\n",
    "        total += -1/2 * σ^2 * exp(-1/σ^2) * (σ^2/2)^(i) * doublefactorial(n-1)/doublefactorial(n - 1 - 2 * i)\n",
    "    end\n",
    "    2 * π * total\n",
    "end\n",
    "function zernike_gaussian_coefficients(σ, m)\n",
    "    \"\"\"Try to reconstruct a gaussian, using Zernike polynomials up to order m\"\"\"\n",
    "    coeffs = []\n",
    "    for n in collect(2 * range(0, m/2, step=1))\n",
    "        coeff = 0\n",
    "        for (i, coefficient) in enumerate(Z_polynomial(Int64(n)))\n",
    "            scalar = (2 * n + 2)/π\n",
    "            coeff += coefficient/2 * gaussian_coefficients(2 * i - 1, σ) * scalar #/2 because m=0\n",
    "        end\n",
    "        push!(coeffs, coeff)\n",
    "    end\n",
    "    coeffs\n",
    "end\n",
    "function reconstruct_gaussian(ρ, σ, m)\n",
    "    \"\"\"Try to reconstruct a gaussian, using Zernike polynomials up to order m\"\"\"\n",
    "    total = BigFloat(0)\n",
    "    coeffs = zernike_gaussian_coefficients(σ, m)\n",
    "    for (i, coeff) in enumerate(coeffs)\n",
    "        total += coeff * (R(Int64(2*i - 2), 0, ρ)) # THIS WAS THE PROBLEM. MAKE SURE YOU'RE DOING THE MATH RIGHT.\n",
    "    end\n",
    "    total\n",
    "end\n",
    "\n",
    "# The way I was/am reconstructing is b0rked."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using Plots\n",
    "\n",
    "# Including too many terms, esp. at high σ, breaks it.\n",
    "Γ = 1/62\n",
    "ω = 2*π*180E3\n",
    "θ = 0.;\n",
    "σ = .1\n",
    "b = SpinBasis(1//2)\n",
    "ψ0 = 1/sqrt(2) * (spindown(b) + spinup(b))\n",
    "max_n = 15\n",
    "evolution_time =  50E-6\n",
    "U = π/(2*evolution_time)\n",
    "step_size = evolution_time\n",
    "T = [0.0:step_size:evolution_time;]\n",
    "order =  0\n",
    "μ = order * ω\n",
    "zernikeeven = zernike_gaussian_coefficients(σ, max_n)\n",
    "\n",
    "\n",
    "\n",
    "max_ρ = 1\n",
    "min_ρ = 0\n",
    "num_samples = 50\n",
    "ρs = range(min_ρ, max_ρ, step=max_ρ/num_samples)\n",
    "ys = [reconstruct_gaussian(ρ, σ, max_n) for ρ in ρs];\n",
    "target = [exp(-ρ^2/σ^2) for ρ in ρs];\n",
    "display(plot(ρs, [target, ys], labels=[\"σ=.4\" \"n=8 reconstruction\"]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "infidelity_across_disk (generic function with 1 method)"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function evolution_evaluator_factory(ψ0, T, zernikeeven, U, ψ, μ, ω, b)\n",
    "    function evaluator(ρ, ϕ)\n",
    "        H(t, _) = H_odf(ρ, ϕ, t, zernikeeven, U, ψ, μ, ω)*sigmaz(b), [sigmam(b)], [dagger(sigmam(b))]\n",
    "        _, ψTARGET = timeevolution.master_dynamic(T, ψ0, H; rates=[Γ])\n",
    "        last(ψTARGET)\n",
    "    end\n",
    "end\n",
    "exact_evolution_evaluator = evolution_evaluator_factory(ψ0, T, zernikeeven, zernikeodd, U, θ, μ, ω, b)\n",
    "\n",
    "function gaussian_spin_profile_evaluator(scale)\n",
    "    function gaussian_spin_profile(ρ, ϕ)\n",
    "    ψ0 = 1/sqrt(2) * (spindown(b) + spinup(b))\n",
    "    H(t, _) = scale * U * cos(exp(-ρ^2/σ^2)) * sigmaz(b), [sigmam(b)], [dagger(sigmam(b))]\n",
    "    _, ψ = timeevolution.master_dynamic(T, ψ0, H; rates=[Γ])\n",
    "    last(ψ)\n",
    "    end\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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
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       "  2232.76,1406.92 2232.76,1158.44 2256.76,1158.44 2232.76,1158.44 2232.76,909.962 2256.76,909.962 2232.76,909.962 2232.76,661.483 2256.76,661.483 2232.76,661.483 \n",
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "discretization = 10\n",
    "max_dim = .5\n",
    "function plot_across_unit_disk(f, discretization)\n",
    "    x = y = Array(range(-max_dim, max_dim, 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",
    "exact_evolution_evaluator = evolution_evaluator_factory(ψ0, T, zernikeeven, U, θ, μ, ω, b)\n",
    "plot(heatmap(plot_across_unit_disk(infidelity_across_disk(gaussian_spin_profile_evaluator(1), exact_evolution_evaluator), discretization)), aspectratio = true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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