Zernike Transform - Step Function.ipynb

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      "text/plain": [
       "toarray (generic function with 2 methods)"
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   ],
   "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",
    "toarray(s::Union{Set, Array}) = [toarray.(s)...]\n",
    "toarray(v::Number) = v"
   ]
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   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [],
   "source": [
    "max_order = 15\n",
    "discretization = 50\n",
    "zernike_samples = []\n",
    "ordering = []\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",
    "            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",
    "            end\n",
    "        end\n",
    "    end\n",
    "end\n",
    "Z = reduce(hcat, zernike_samples);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "amp = .01 # Used to make the small angle approximation work better.\n",
    "function heaviside(ρ, ϕ)\n",
    "    if  -π/2 < ϕ < π/2\n",
    "        amp\n",
    "    else\n",
    "        -amp\n",
    "    end\n",
    "end\n",
    "xx, yy, zz = sample_across_unit_disk(heaviside, discretization)\n",
    "G = zz[:];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "function least_squares_solution(mat, y)\n",
    "    inv(transpose(mat) * mat)*transpose(mat) * y\n",
    "end\n",
    "coefficients = least_squares_solution(Z, G);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
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       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reconstruction = zeros(discretization*discretization)\n",
    "for (i, zernike_sample) in enumerate(zernike_samples)\n",
    "    reconstruction += coefficients[i] * zernike_sample\n",
    "end\n",
    "using Plots\n",
    "plot(heatmap(xx, yy, reshape(reconstruction, (discretization, discretization))), aspectratio = true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "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",
    "            if m == 0\n",
    "                push!(curr_even, lookup[(n, m)]/2)\n",
    "                push!(curr_odd, lookup[(n, -m)]/2)\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": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sequential_exact_evolution_evaluator_factory (generic function with 1 method)"
      ]
     },
     "execution_count": 90,
     "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "Γ = 1/62\n",
    "ω = 2*π*180E3\n",
    "U = 2*π*10E3/amp # We had to increase this, for faster gates.\n",
    "θ = -π/2;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::var\"#evaluator#75\"{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": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = SpinBasis(1//2)\n",
    "ψ0 = 1/sqrt(2) * (spindown(b) + spinup(b))\n",
    "evolution_time =  π / (2.5*U)\n",
    "step_size = evolution_time/1\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": "markdown",
   "metadata": {},
   "source": [
    "# n = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "max_ρ = 1\n",
    "min_ρ = .0\n",
    "ρs = range(min_ρ, max_ρ, step=max_ρ/20)\n",
    "y1s = [Real.(expect(sigmax(b), sequential_exact_evolution(ρ, 0))) for ρ in ρs];\n",
    "y2s = [Real.(expect(sigmax(b), sequential_exact_evolution(ρ, π))) for ρ in ρs];\n",
    "plot(vcat(reverse(-ρs[2:end]), ρs), vcat(reverse(y1s[2:end]), y2s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# n=15"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "max_ρ = 1\n",
    "min_ρ = .0\n",
    "ρs = range(min_ρ, max_ρ, step=max_ρ/20)\n",
    "y1s = [Real.(expect(sigmax(b), sequential_exact_evolution(ρ, 0))) for ρ in ρs];\n",
    "y2s = [Real.(expect(sigmax(b), sequential_exact_evolution(ρ, π))) for ρ in ρs];\n",
    "plot(vcat(reverse(-ρs[2:end]), ρs), vcat(reverse(y1s[2:end]), y2s))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#  Conclusion for thr day of 9/8. Between this and the other Gaussian notebook, it's clear that my code for numerically finding zernike coefficients works. Main take aways are that DC terms cause some trouble, because you need a small angle, but we can make that work. What needs to be fixed? Above, there is a 2.5 in the evolution time that is a fudge factor - you'll notice that the above plots aren't going to 1, althought the wiggles seem to get better. This is qualitatively what we want, but I think I need to fix other factors first. But really it's hiding other problems. First - does the sequential application code even work? I have yet to very this. It could be doing the wrong sequence of beat notes, additionally the way different R's are applied could also be wrong right now. Second, there is a pi/2 phase shift somewhere because I had to shift the 0th order zernike polynomials to get the right pattern - this is probably why the reconstruction is phase shifted from the above plot (notice that I'm doing 0 and \\pi. This phase shift needs to be applied for the 0th order, but not for the higher order terms. Third - there is an extra factor of two for the 0th order term which should be correctly applied right now, but we shhould double check. Fourth - the even and odd zernike terms probably need to be switched, I've correctly decomposed the basis, but they switch roles in the rotating frame. Those are the four problems I can think of right now. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Consider doing exp(-(x^2/\\sigma_x^2 + y^2/\\sigma_y^2) for the Gaussian to make it less symmetric."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plot \\sigma as a function of Zernike polynomials that we need to converge."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Note 9/14 Most of these problems are addressed by considering a different problem - the gaussian problem. We will return to this. :)"
   ]
  }
 ],
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