docs/source/tutorials/nnvqe.ipynb

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    "# <center> Neural Network encoded Variational Quantum Eigensolver (NN-VQE)"
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    "## Overview\n",
    "\n",
    "In this tutorial, we'll show you a general framework called neural network encoded variational quantum algorithms (NN-VQAs) with TensorCircuit. NN-VQA feeds input (such as parameters of a Hamiltonian) from a given problem to a neural network and uses its outputs to parameterize an ansatz circuit for the standard VQA. Here, we take NN-variational quantum eigensolver (VQE) to illustrate concretely."
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   "source": [
    "## Setup"
   ]
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    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import tensorflow as tf\n",
    "import tensorcircuit as tc\n",
    "import cotengra\n",
    "import quimb\n",
    "from tqdm.notebook import tqdm\n",
    "from functools import partial\n",
    "\n",
    "optc = cotengra.ReusableHyperOptimizer(\n",
    "    methods=[\"greedy\"],\n",
    "    parallel=\"ray\",\n",
    "    minimize=\"combo\",\n",
    "    max_time=30,\n",
    "    max_repeats=1024,\n",
    "    progbar=True,\n",
    ")\n",
    "tc.set_contractor(\"custom\", optimizer=optc, preprocessing=True)\n",
    "\n",
    "K = tc.set_backend(\"tensorflow\")\n",
    "tc.set_dtype(\"complex128\")"
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    "## Energy\n",
    "\n",
    "The Hamiltonian used in this tutorial is a 1D XXZ model with periodic boundary conditions, with the transverse field strength $\\lambda$ and the anisotropy parameter $\\Delta$. The Hamiltonian energy expectation function is chosen as loss.\n",
    "\n",
    "$$ \\hat{H}_{XXZ}=\\sum_{i}{ \\left( X_{i}X_{i+1}+Y_{i}Y_{i+1}+\\Delta Z_{i}Z_{i+1} \\right) } + \\lambda \\sum_{i}{Z_{i}} $$"
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    "def energy(c: tc.Circuit, lamb: float = 1.0, delta: float = 1.0):\n",
    "    e = 0.0\n",
    "    n = c._nqubits\n",
    "    for i in range(n):\n",
    "        e += lamb * c.expectation((tc.gates.z(), [i]))  # <Z_i>\n",
    "    for i in range(n):\n",
    "        e += c.expectation(\n",
    "            (tc.gates.x(), [i]), (tc.gates.x(), [(i + 1) % n])\n",
    "        )  # <X_i X_{i+1}>\n",
    "        e += c.expectation(\n",
    "            (tc.gates.y(), [i]), (tc.gates.y(), [(i + 1) % n])\n",
    "        )  # <Y_i Y_{i+1}>\n",
    "        e += delta * c.expectation(\n",
    "            (tc.gates.z(), [i]), (tc.gates.z(), [(i + 1) % n])\n",
    "        )  # <Z_i Z_{i+1}>\n",
    "    return K.real(e)"
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    "## Ansatz circuit\n",
    "\n",
    "Now we design the circuit. We choose multi-scale entangled renormalization ansatz (MERA) as the ansatz here, $d$ is the circuit depth. (see [MERA tutorial](https://tensorcircuit.readthedocs.io/en/latest/tutorials/mera.html))"
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    "def MERA(\n",
    "    inp, n, d=1, lamb=1.0, energy_flag=False\n",
    "):  # for single-parameter 1D XXZ model, we fix lamb\n",
    "    params = K.cast(inp[\"params\"], \"complex128\")\n",
    "    delta = K.cast(inp[\"delta\"], \"complex128\")\n",
    "    c = tc.Circuit(n)\n",
    "\n",
    "    idx = 0\n",
    "\n",
    "    for i in range(n):\n",
    "        c.rx(i, theta=params[3 * i])\n",
    "        c.rz(i, theta=params[3 * i + 1])\n",
    "        c.rx(i, theta=params[3 * i + 2])\n",
    "    idx += 3 * n\n",
    "\n",
    "    for n_layer in range(1, int(np.log2(n)) + 1):\n",
    "        n_qubit = 2**n_layer  # number of qubits involving\n",
    "        step = int(n / n_qubit)\n",
    "\n",
    "        for _ in range(d):  # circuit depth\n",
    "            # even\n",
    "            for i in range(step, n - step, 2 * step):\n",
    "                c.rxx(i, i + step, theta=params[idx])\n",
    "                c.rzz(i, i + step, theta=params[idx + 1])\n",
    "                idx += 2\n",
    "\n",
    "            # odd\n",
    "            for i in range(0, n, 2 * step):\n",
    "                c.rxx(i, i + step, theta=params[idx])\n",
    "                c.rzz(i, i + step, theta=params[idx + 1])\n",
    "                idx += 2\n",
    "\n",
    "            # single qubit\n",
    "            for i in range(0, n, step):\n",
    "                c.rx(i, theta=params[idx])\n",
    "                c.rz(i, theta=params[idx + 1])\n",
    "                idx += 2\n",
    "\n",
    "    if energy_flag:\n",
    "        return energy(c, lamb, delta)  # return Hamiltonian energy expectation\n",
    "    else:\n",
    "        return c, idx  # return the circuit & number of circuit parameters"
   ]
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     "text": [
      "The number of parameters is 74\n"
     ]
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     },
     "execution_count": 9,
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   "source": [
    "# circuit visulization\n",
    "n = 8\n",
    "d = 1\n",
    "cirq, idx = MERA({\"params\": np.zeros(3000), \"delta\": 0.0}, n, d, 1.0)\n",
    "print(\"The number of parameters is\", idx)\n",
    "cirq.draw()"
   ]
  },
  {
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   "id": "bfe2fbee",
   "metadata": {},
   "source": [
    "## NN-VQE\n",
    "\n",
    "Design the NN-VQE. We use a neural network to transform the Hamiltonian parameters to the optimized parameters in the parameterized quantum circuit (PQC) for VQE."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "1ac050a8",
   "metadata": {},
   "outputs": [],
   "source": [
    "def NN_MERA(n, d, lamb, NN_shape, stddev):\n",
    "    input = tf.keras.layers.Input(shape=[1])  # input layer\n",
    "\n",
    "    x = tf.keras.layers.Dense(\n",
    "        units=NN_shape,\n",
    "        kernel_initializer=tf.keras.initializers.RandomNormal(stddev=stddev),\n",
    "        activation=\"ReLU\",\n",
    "    )(\n",
    "        input\n",
    "    )  # hidden layer\n",
    "\n",
    "    x = tf.keras.layers.Dropout(0.05)(x)  # dropout layer\n",
    "\n",
    "    _, idx = MERA(\n",
    "        {\"params\": np.zeros(3000), \"delta\": 0.0}, n, d, 1.0, energy_flag=False\n",
    "    )\n",
    "    params = tf.keras.layers.Dense(\n",
    "        units=idx,\n",
    "        kernel_initializer=tf.keras.initializers.RandomNormal(stddev=stddev),\n",
    "        activation=\"sigmoid\",\n",
    "    )(\n",
    "        x\n",
    "    )  # output layer\n",
    "\n",
    "    qlayer = tc.KerasLayer(partial(MERA, n=n, d=d, lamb=lamb, energy_flag=True))  # PQC\n",
    "\n",
    "    output = qlayer({\"params\": 6.3 * params, \"delta\": input})  # NN-VQE output\n",
    "\n",
    "    m = tf.keras.Model(inputs=input, outputs=output)\n",
    "\n",
    "    return m"
   ]
  },
  {
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   "id": "e9afb1a5",
   "metadata": {},
   "source": [
    "## Train\n",
    "\n",
    "Now we can train the NN-VQE with TensorFlow."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "873ebd5e",
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "def train(n, d, lamb, delta, NN_shape, maxiter=10000, lr=0.005, stddev=1.0):\n",
    "    exp_lr = tf.keras.optimizers.schedules.ExponentialDecay(\n",
    "        initial_learning_rate=lr, decay_steps=1000, decay_rate=0.7\n",
    "    )\n",
    "    opt = tf.keras.optimizers.Adam(exp_lr)  # optimizer\n",
    "\n",
    "    m = NN_MERA(n, d, lamb, NN_shape, stddev)\n",
    "    for i in range(maxiter):\n",
    "        with tf.GradientTape() as tape:\n",
    "            e = tf.zeros([1], dtype=tf.float64)\n",
    "            for de in delta:\n",
    "                e += m(K.reshape(de, [1]))  # sum up energies of all training points\n",
    "        grads = tape.gradient(e, m.variables)\n",
    "        opt.apply_gradients(zip(grads, m.variables))\n",
    "        if i % 500 == 0:\n",
    "            print(\"epoch\", i, \":\", e)\n",
    "\n",
    "    m.save_weights(\"NN-VQE.weights.h5\")  # save the trained model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "e8df3d67",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 0 : tf.Tensor([[117.53523392]], shape=(1, 1), dtype=float64)\n",
      "epoch 500 : tf.Tensor([[-361.85937039]], shape=(1, 1), dtype=float64)\n",
      "epoch 1000 : tf.Tensor([[-365.35288984]], shape=(1, 1), dtype=float64)\n",
      "epoch 1500 : tf.Tensor([[-366.65891358]], shape=(1, 1), dtype=float64)\n",
      "epoch 2000 : tf.Tensor([[-366.94258369]], shape=(1, 1), dtype=float64)\n"
     ]
    }
   ],
   "source": [
    "n = 8  # number of qubits\n",
    "d = 2  # circuit depth\n",
    "lamb = 0.75  # fixed\n",
    "delta = np.linspace(-3.0, 3.0, 20, dtype=\"complex128\")  # training set\n",
    "NN_shape = 20  # node number of the hidden layer\n",
    "maxiter = 2500  # maximum iteration for the optimization\n",
    "lr = 0.009  # learning rate\n",
    "stddev = 0.1  # the initial standard deviation of the NN\n",
    "\n",
    "with tf.device(\"/cpu:0\"):\n",
    "    train(n, d, lamb, delta, NN_shape=NN_shape, maxiter=maxiter, lr=lr, stddev=stddev)"
   ]
  },
  {
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   "id": "c5a7cdb0",
   "metadata": {},
   "source": [
    "## Test\n",
    "\n",
    "We use a test set beyond the training set to test the accuracy and generalization capability of NN-VQE."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "f15f4f68",
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "0it [00:00, ?it/s]"
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   ],
   "source": [
    "test_delta = np.linspace(-4.0, 4.0, 201)  # test set\n",
    "test_energies = tf.zeros_like(test_delta).numpy()\n",
    "m = NN_MERA(n, d, lamb, NN_shape, stddev)\n",
    "m.load_weights(\"NN-VQE.weights.h5\")\n",
    "for i, de in tqdm(enumerate(test_delta)):\n",
    "    test_energies[i] = m(K.reshape(de, [1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "924027f8",
   "metadata": {},
   "source": [
    "## Compare\n",
    "\n",
    "We compare the results of NN-VQE with the analytical ones to calculate the ground-state energy relative error. From the figure, we can see that NN-VQE is able to estimate the ground-state energies of parameterized Hamiltonians with high precision without fine-tuning and has a favorable generalization capability."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "c8668a13",
   "metadata": {},
   "outputs": [],
   "source": [
    "analytical_energies = []  # analytical result\n",
    "for i in test_delta:\n",
    "    h = quimb.tensor.tensor_builder.MPO_ham_XXZ(\n",
    "        n, i * 4, jxy=4.0, bz=2.0 * 0.75, S=0.5, cyclic=True\n",
    "    )\n",
    "    h = h.to_dense()\n",
    "    analytical_energies.append(np.min(quimb.eigvalsh(h)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "42799e3e",
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
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     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# relative error\n",
    "plt.plot(\n",
    "    test_delta,\n",
    "    (test_energies - analytical_energies) / np.abs(analytical_energies),\n",
    "    \"-\",\n",
    "    color=\"b\",\n",
    ")\n",
    "plt.xlabel(\"Delta\", fontsize=14)\n",
    "plt.ylabel(\"GS Relative Error\", fontsize=14)\n",
    "plt.axvspan(-3.0, 3.0, color=\"darkgrey\", alpha=0.5)  # training set span\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f9bda8a",
   "metadata": {},
   "source": [
    "To get more detailed information or further study, please refer to our [paper](https://arxiv.org/abs/2308.01068) and [GitHub](https://github.com/JachyMeow/NN-VQA)."
   ]
  }
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