Shor's_Squad/Nelder_Mead.ipynb

{
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   "execution_count": 2,
   "id": "fe2aaf1f-eb1f-4666-9410-d004ffed5a3d",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "using Bloqade\n",
    "using Random\n",
    "Random.seed!(4)\n",
    "using PythonCall\n",
    "plt = pyimport(\"matplotlib.pyplot\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "64f4aa55-43aa-4a7f-8db6-b13b39e6e6b3",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(8.0, 4.0)"
     ]
    }
   ],
   "source": [
    "Random.seed!(10)\n",
    "a=4\n",
    "atoms = generate_sites(SquareLattice(), 4,4; scale = a) |> random_dropout(0.2)\n",
    "print(atoms[5])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "61cc64de-631b-41db-b233-195478daa551",
   "metadata": {
    "tags": []
   },
   "outputs": [
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      ],
      "text/plain": []
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rb=sqrt(sqrt(20))*a\n",
    "Bloqade.plot(atoms, blockade_radius = Rb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "ec69d947-c538-4a5f-ac1a-761af1b07ca6",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-3.0710793975846857"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# based on methodology from Bloqade tutorial\n",
    "# define a loss function that takes the weighted average number of excited atoms in a graph\n",
    "# we would like to maximize this\n",
    "# doesn't explicitly check for optimality or independence, but we can do this in post-processing\n",
    "\n",
    "function loss_piecewise_linear(a, b, atoms::AtomList, x::AbstractVector{T}) where {T}\n",
    "    @assert length(x) == 3\n",
    "    C6 = 2π * 862690;\n",
    "    Ω_max = 4 * 2π\n",
    "    Δ_start = (-13+a/2)*2π\n",
    "    Δ_end = (11+b)*2π\n",
    "    Δ0 = Δ_end\n",
    "    T_max = 0.6\n",
    "\n",
    "    # the strength of the detunings at each step takes the optimizing x as their input\n",
    "    # we start with a piecewise linear function with some parameters, we want to fine-tune these\n",
    "    Δs = smooth(\n",
    "        piecewise_linear(\n",
    "            clocks = T[0.0, 0.05, 0.2, 0.3, 0.4, 0.55, T_max],\n",
    "            values = T[Δ_start, Δ_start, Δ0*x[1], Δ0*x[2], Δ0*x[3], Δ_end, Δ_end],\n",
    "        );\n",
    "        kernel_radius = 0.02,\n",
    "    )\n",
    "    Ωs = smooth(\n",
    "        piecewise_linear(clocks = T[0.0, 0.05, 0.1, 0.5, 0.55, T_max], values = T[0, 0, Ω_max, Ω_max, 0, 0]);\n",
    "        kernel_radius = 0.02,\n",
    "    )\n",
    "\n",
    "    hamiltonian = rydberg_h(atoms; Ω = Ωs, Δ = Δs)\n",
    "    subspace = blockade_subspace(atoms, Rb)\n",
    "    prob = SchrodingerProblem(zero_state(Complex{T}, subspace), T_max, hamiltonian)\n",
    "    emulate!(prob)\n",
    "    #return -cost_pedro(atoms, probs(prob.reg)), prob.reg, Δs\n",
    "    return -rydberg_density_sum(prob.reg), prob.reg, Δs\n",
    "end\n",
    "\n",
    "x0 = [0.1, 0.8, 0.8]; # initial point for the optimization\n",
    "\n",
    "Δ_start = -13*2π\n",
    "Δ_end = 11*2π\n",
    "Δ0 = Δ_end\n",
    "T_max = 0.6\n",
    "Δ_initial = piecewise_linear(\n",
    "    clocks = [0.0, 0.05, 0.2, 0.3, 0.4, 0.55, T_max],\n",
    "    values = [Δ_start, Δ_start, Δ0 * x0[1], Δ0 * x0[2], Δ0 * x0[3], Δ_end, Δ_end],\n",
    ")\n",
    "\n",
    "# outputs the initial loss\n",
    "rydberg_density, reg2, Δ_initial_smooth = loss_piecewise_linear(0, 0, atoms, x0)\n",
    "rydberg_density"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "66668e1f-d955-48e5-8b81-3af7d2b09fe9",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-3.1984704331720732"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# final optimized loss output\n",
    "optresult = Optim.optimize(x -> loss_piecewise_linear(0, 0, atoms, x)[1], x0)\n",
    "\n",
    "rydberg_density_final, reg_final, Δ_final = loss_piecewise_linear(0, 0, atoms, optresult.minimizer)\n",
    "rydberg_density_final"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "ecd307ee-e806-4703-a454-41aec35b5ac9",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "Python: <Figure size 640x480 with 1 Axes>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# how did the optimization modify the graph? blue = original, orange = final\n",
    "fig, ax = plt.subplots()\n",
    "Bloqade.plot!(ax, Δ_initial)\n",
    "Bloqade.plot!(ax, Δ_initial_smooth)\n",
    "ax.set_ylabel(\"Δ/2π (MHz)\")\n",
    "ax.legend([\"piecewise linear\", \"smoothened piecewise linear\"], loc = \"lower right\")\n",
    "fig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "a35d3c67-e74b-4c5a-bee3-290d83fd325c",
   "metadata": {
    "tags": []
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   "outputs": [
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      "text/plain": [
       "Python: <Figure size 640x480 with 1 Axes>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# first and third items valid but not optimal, second optimal\n",
    "# optimal solution occurs ~20% of the time\n",
    "bitstring_hist(reg_final; nlargest = 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "d1e12e3c-144e-44eb-b94b-024ce0edf615",
   "metadata": {
    "tags": []
   },
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      "text/plain": []
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#valid but not optimal\n",
    "best_bit_strings = most_probable(reg_final, 3)\n",
    "\n",
    "Bloqade.plot(atoms, blockade_radius = Rb; colors = [iszero(b) ? \"white\" : \"red\" for b in best_bit_strings[1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "189fb5d4-5971-421c-8abd-a78ebc6dcd9e",
   "metadata": {
    "tags": []
   },
   "outputs": [
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      "text/plain": []
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# optimal, occurs 20% of the time, which is pretty good for one of the hardest Rb values!\n",
    "best_bit_strings = most_probable(reg_final, 3)\n",
    "\n",
    "Bloqade.plot(atoms, blockade_radius = Rb; colors = [iszero(b) ? \"white\" : \"red\" for b in best_bit_strings[2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "1e1fa5ee-706e-4baa-8956-3adffad7cf14",
   "metadata": {},
   "outputs": [
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      "text/plain": []
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# valid but not optimal\n",
    "best_bit_strings = most_probable(reg_final, 3)\n",
    "\n",
    "Bloqade.plot(atoms, blockade_radius = 7.5; colors = [iszero(b) ? \"white\" : \"red\" for b in best_bit_strings[3]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "id": "bf49b416-aa9a-4cc5-91bc-e6d0351a4f6a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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   ],
   "source": [
    "# old function, grid searches along non optimized function for best delta/omega values\n",
    "function grid_search(min_mult, max_mult, delta, steps)\n",
    "    for i in 0:steps\n",
    "        #Total evolution time\n",
    "        T_max = 1\n",
    "\n",
    "        #Fix Δ according to the desired Rydberg radius\n",
    "        Ω_max = min_mult*delta + (max_mult - min_mult)*delta*i/steps\n",
    "        Δ_max = delta\n",
    "\n",
    "        #Build time traces for adiabatic algorithm\n",
    "        Ω = piecewise_linear(clocks = [0.0, 0.2, 0.8, T_max], values = [0.0, Ω_max, Ω_max, 0])\n",
    "        Δ_start = -Δ_max\n",
    "        Δ_end = Δ_max\n",
    "        Δ = piecewise_linear(clocks = [0.0, 0.2, 0.5, T_max], values = [Δ_start, Δ_start, Δ_end, Δ_end])\n",
    "        \n",
    "        h_kings = rydberg_h(atoms; Ω = Ω, Δ = Δ)\n",
    "        prob = SchrodingerProblem(zero_state(nqubits(h_kings)), T_max, h_kings);\n",
    "        emulate!(prob)\n",
    "        \n",
    "        println(Ω_max, \" \", Δ_max, \" \", rydberg_density_sum(prob.reg))\n",
    "    end\n",
    "end\n",
    "grid_search(0, 6, C6/Rb^6, 30)"
   ]
  },
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   "execution_count": null,
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    {
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     "output_type": "stream",
     "text": [
      "[0.24341433235117596, 0.622321493126969, 1.1831245389205918]\n",
      "-87.96459430051421 62.83185307179586 -3.1243942720876294\n",
      "[0.21850021553866528, 0.590141701339195, 0.949656351319748]\n",
      "-87.96459430051421 64.40264939859075 -3.1868362868931954\n",
      "[0.21045349025629895, 0.5611163296750671, 0.7853934787501148]\n",
      "-87.96459430051421 65.97344572538566 -3.2377304798770106\n",
      "[0.14025099185972253, 0.5950170169735297, 1.3519770951686958]\n",
      "-87.96459430051421 67.54424205218055 -3.193617536980704\n",
      "[0.19530140925448675, 0.49660372863452823, 0.5905873664469281]\n",
      "-87.96459430051421 69.11503837897544 -3.297735246427469\n",
      "[0.1864573912932655, 0.4655481205845056, 0.5315472572723667]\n",
      "-87.96459430051421 70.68583470577035 -3.3100875442482587\n",
      "[0.17699908280421117, 0.43676434791255764, 0.48689758174088765]\n",
      "-87.96459430051421 72.25663103256524 -3.314946927975257\n",
      "[0.16722056045901193, 0.4104820415200814, 0.4523074619111448]\n",
      "-87.96459430051421 73.82742735936014 -3.3153586879709365\n",
      "[0.15756522997391045, 0.38661414577518266, 0.4245825385651495]\n",
      "-87.96459430051421 75.39822368615503 -3.3135935275321984\n",
      "[0.2776327916782563, 0.6383370156814012, 1.1829134211818642]\n",
      "-86.39379797371932 62.83185307179586 -3.123283885454135\n",
      "[0.23949950378891907, 0.6000051588008022, 0.9481766736213613]\n",
      "-86.39379797371932 64.40264939859075 -3.1859796451753652\n",
      "[0.22316207820232664, 0.5668017241297858, 0.7838564296229481]\n",
      "-86.39379797371932 65.97344572538566 -3.2370446195544242\n",
      "[0.14405260035362427, 0.5958076457748964, 1.3539050946715998]\n",
      "-86.39379797371932 67.54424205218055 -3.1934335228035313\n",
      "[0.20153378694096805, 0.49909178912883384, 0.5897869180806747]\n",
      "-86.39379797371932 69.11503837897544 -3.2970249360965433\n",
      "[0.1913065092068617, 0.46735262813370737, 0.5309965224216063]\n",
      "-86.39379797371932 70.68583470577035 -3.3092984081787926\n",
      "[0.18093538411908996, 0.43812399123285406, 0.48663881739980297]\n",
      "-86.39379797371932 72.25663103256524 -3.3140839756212785\n"
     ]
    }
   ],
   "source": [
    "# grid search for the best delta and omega\n",
    "function opt_grid_search()\n",
    "    for a in -4:4\n",
    "        for b in -4:4\n",
    "            x0 = [0.1, 0.8, 0.8]; # initial point for the optimization\n",
    "\n",
    "            Δ_start = (-13+a/4) * 2π\n",
    "            Δ_end = (11+b/4) * 2π\n",
    "            Δ0 = Δ_end\n",
    "            T_max = 0.6\n",
    "            Δ_initial = piecewise_linear(\n",
    "                clocks = [0.0, 0.05, 0.2, 0.3, 0.4, 0.55, T_max],\n",
    "                values = [Δ_start, Δ_start, Δ0 * x0[1], Δ0 * x0[2], Δ0 * x0[3], Δ_end, Δ_end],\n",
    "            )\n",
    "            optresult = Optim.optimize(x -> loss_piecewise_linear(a, b, atoms, x)[1], x0)\n",
    "\n",
    "            rydberg_density_final, reg_final, Δ_final = loss_piecewise_linear(a, b, atoms, optresult.minimizer)\n",
    "            println(optresult.minimizer)\n",
    "            println(Δ_start, \" \", Δ_end, \" \", rydberg_density_final)\n",
    "        end\n",
    "    end\n",
    "end\n",
    "\n",
    "opt_grid_search()"
   ]
  },
  {
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   "execution_count": 120,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "cost_pedro (generic function with 1 method)"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Cost functions\n",
    "# the cost function that we used did not check for optimality or independence\n",
    "# we implemented a cost function that does check for these, but gradient descent did not converge well for this\n",
    "# likely because the function is less continuous or too strict\n",
    "function is_IS(atoms, str)\n",
    "    n_atoms = length(atoms)\n",
    "    for i in 1:n_atoms\n",
    "        # print(\"Checking: \")\n",
    "        # println(i)\n",
    "        if str[n_atoms-i+1] == '1'\n",
    "            for j in 1:n_atoms\n",
    "                # println(j)\n",
    "                # println(str[j])\n",
    "                if (sqrt((atoms[i][1] - atoms[j][1])^2 + (atoms[i][2] - atoms[j][2])^2) <= Rb) && (str[n_atoms-j+1] == '1') && (i != j)\n",
    "                    return false\n",
    "                end\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    return true\n",
    "end\n",
    "\n",
    "function int_to_bitstring(num, n_bits)\n",
    "    str = \"\"\n",
    "    while num > 0\n",
    "        str = string(num % 2, str)\n",
    "        num = num ÷ 2\n",
    "    end\n",
    "    while length(str) < n_bits\n",
    "        str = string('0', str)\n",
    "    end\n",
    "    return str\n",
    "end\n",
    "\n",
    "function count_in_string(haystack, needle)\n",
    "    count = 0\n",
    "    for c in haystack\n",
    "        if c == needle\n",
    "            count += 1\n",
    "        end\n",
    "    end\n",
    "    return count\n",
    "end\n",
    "\n",
    "function cost_R(atoms, probabilities)\n",
    "    n_strings = length(probabilities)\n",
    "    string_size = length(atoms)\n",
    "    sum = 0\n",
    "    n_mis = 0\n",
    "    mis_size = 0\n",
    "    for i in 1:n_strings\n",
    "        str = int_to_bitstring(i-1, string_size)\n",
    "        if is_IS(atoms, str)\n",
    "            n_ones = count_in_string(str, '1')\n",
    "            sum += n_ones\n",
    "            n_mis += 1\n",
    "            mis_size = max(mis_size, n_ones)\n",
    "        end\n",
    "    end\n",
    "    return (sum / n_mis) / mis_size\n",
    "end\n",
    "\n",
    "function get_mis_size(atoms, probabilities)\n",
    "    n_strings = length(probabilities)\n",
    "    string_size = length(atoms)\n",
    "    max_mis_size = 0\n",
    "    for i in 1:n_strings\n",
    "        str = int_to_bitstring(i-1, string_size)\n",
    "        if is_IS(atoms, str)\n",
    "            max_mis_size = max(max_mis_size, count_in_string(str, '1'))\n",
    "        end\n",
    "    end\n",
    "    return max_mis_size\n",
    "end\n",
    "\n",
    "function cost_pedro(atoms, probabilities)\n",
    "    n_strings = length(probabilities)\n",
    "    string_size = length(atoms)\n",
    "    mis_size = get_mis_size(atoms, probabilities)\n",
    "    cost = 0\n",
    "    for i in 1:n_strings\n",
    "        str = int_to_bitstring(i-1, string_size)\n",
    "        if is_IS(atoms, str) && count_in_string(str, '1') == mis_size\n",
    "            cost += probabilities[i]\n",
    "        end\n",
    "    end\n",
    "    return cost\n",
    "end"
   ]
  }
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