docs/source/tutorials/mera_cn.ipynb

{
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  {
   "cell_type": "markdown",
   "source": [
    "#  MERA"
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "## 概述\n",
    "\n",
    "在本教程中，我们将不涉及物理层面的探讨，而将演示如何使用TensorCircuit实现MERA (多尺度纠缠重整化假设，multi-scale entangled renormalization ansatz)。"
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "## 背景\n",
    "\n",
    "MERA是VQE的其中一种。它由一个量子比特的 $\\ket{0}$ 态开始，逐层添加新的量子比特以扩张希尔伯特空间。\n",
    "在将要训练的MERA（记作 $U(\\theta)$ ）中，我们使用可变参量子门 $e^{i\\theta XX}$ 、$e^{i\\theta ZZ}$ 作为双比特门，以及 $e^{i\\theta X}$ 、$e^{i\\theta Z}$ 作为单比特门。\n",
    "在本教程中，我们使用的哈密顿量是横场伊辛模型的哈密顿量 $\\hat{H}_{Ising}=J\\sum_{i}{Z_{i}Z_{i+1}}-B_{x}\\sum_{i}{X_{i}}$。\n",
    "我们要减小的损失函数是 $\\mathcal{L}_{MERA}(\\rm{\\theta})=\\langle 0^n\\vert U(\\theta)^\\dagger \\hat{H} U(\\theta)\\vert 0^n\\rangle$。"
   ],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "## 设置"
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "source": [
    "import numpy as np\n",
    "import tensorflow as tf\n",
    "import tensorcircuit as tc\n",
    "\n",
    "tc.set_backend(\"tensorflow\")\n",
    "tc.set_dtype(\"complex128\")"
   ],
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "('complex128', 'float64')"
      ]
     },
     "metadata": {},
     "execution_count": 1
    }
   ],
   "metadata": {
    "scrolled": false
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## 能量\n",
    "我们先设计哈密顿量的能量期望函数作为损失函数。\n",
    "$$ \\hat{H}_{Ising}=J\\sum_{i}{Z_{i}Z_{i+1}}-B_{x}\\sum_{i}{X_{i}} $$"
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "source": [
    "def energy(c: tc.Circuit, j: float = 1.0, hx: float = 1.0):\n",
    "    e = 0.0\n",
    "    n = c._nqubits\n",
    "    # <Z_i Z_{i+1}>\n",
    "    for i in range(n - 1):\n",
    "        e += j * c.expectation((tc.gates.z(), [i]), (tc.gates.z(), [i + 1]))\n",
    "    # <X_i>\n",
    "    for i in range(n):\n",
    "        e -= hx * c.expectation((tc.gates.x(), [i]))\n",
    "    return tc.backend.real(e)"
   ],
   "outputs": [],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "## MERA 电路\n",
    "\n",
    "现在，我们设计电路。我们用 $\\theta$ 作为输入。"
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "source": [
    "def MERA(params, n):\n",
    "    params = tc.backend.cast(params, \"complex128\")\n",
    "    c = tc.Circuit(n)\n",
    "\n",
    "    idx = 0  # index of params\n",
    "\n",
    "    for i in range(n):\n",
    "        c.rx(i, theta=params[2 * i])\n",
    "        c.rz(i, theta=params[2 * i + 1])\n",
    "    idx += 2 * 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",
    "        # 偶数层\n",
    "        for i in range(step, n - step, 2 * step):\n",
    "            c.exp1(i, i + step, theta=params[idx], unitary=tc.gates._xx_matrix)\n",
    "            c.exp1(i, i + step, theta=params[idx + 1], unitary=tc.gates._zz_matrix)\n",
    "            idx += 2\n",
    "\n",
    "        # 奇数层\n",
    "        for i in range(0, n, 2 * step):\n",
    "            c.exp1(i, i + step, theta=params[idx], unitary=tc.gates._xx_matrix)\n",
    "            c.exp1(i, i + step, theta=params[idx + 1], unitary=tc.gates._zz_matrix)\n",
    "            idx += 2\n",
    "\n",
    "        # 单比特门\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",
    "    # 测量\n",
    "    e = energy(c)\n",
    "    return e\n",
    "    # return c, idx"
   ],
   "outputs": [],
   "metadata": {}
  },
  {
   "cell_type": "markdown",
   "source": [
    "我们可以将MERA电路可视化。 \n",
    "\n",
    "注意：请把return改为`return c, idx`。这个return只有在这儿会被用到。可视化完成后，请别忘了将return还原并重新运行上方代码块。"
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "source": [
    "n = 8\n",
    "cirq, idx = MERA(np.zeros(1000), n)\n",
    "print(\"The number of parameters is\", idx)\n",
    "cirq.draw()"
   ],
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "The number of parameters is 66\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
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     },
     "metadata": {},
     "execution_count": 4
    }
   ],
   "metadata": {
    "scrolled": true
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  },
  {
   "cell_type": "markdown",
   "source": [
    "## 训练\n",
    "\n",
    "现在，我们使用tensorflow训练MERA电路。"
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "source": [
    "MERA_tfim_vvag = tc.backend.jit(tc.backend.vectorized_value_and_grad(MERA))\n",
    "\n",
    "\n",
    "def batched_train(n, batch=10, maxiter=10000, lr=0.005):\n",
    "    params = tf.Variable(\n",
    "        initial_value=tf.random.normal(\n",
    "            shape=[batch, idx], stddev=1, dtype=getattr(tf, tc.rdtypestr)\n",
    "        )\n",
    "    )\n",
    "    opt = tf.keras.optimizers.Adam(lr)\n",
    "    lowest_energy = 1e5\n",
    "    for i in range(maxiter):\n",
    "        e, grad = MERA_tfim_vvag(params, n)\n",
    "        opt.apply_gradients([(grad, params)])\n",
    "        if tf.reduce_min(e) < lowest_energy:\n",
    "            lowest_energy = tf.reduce_min(e)\n",
    "        if i % 200 == 0:\n",
    "            print(e)\n",
    "    return lowest_energy\n",
    "\n",
    "\n",
    "n = 8\n",
    "lowest_energy = batched_train(n, batch=5, maxiter=2000, lr=0.007)"
   ],
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "tf.Tensor([-0.6449017   0.14083987  0.17227418  1.42731099  0.93767164], shape=(5,), dtype=float64)\n",
      "tf.Tensor([-9.57952648 -9.15354269 -9.53415983 -9.55291257 -9.46880555], shape=(5,), dtype=float64)\n",
      "tf.Tensor([-9.63166728 -9.60922826 -9.59883555 -9.66639936 -9.60174669], shape=(5,), dtype=float64)\n",
      "tf.Tensor([-9.65441326 -9.61830383 -9.6219077  -9.68289435 -9.61427165], shape=(5,), dtype=float64)\n",
      "tf.Tensor([-9.66991104 -9.6307931  -9.64993901 -9.71396225 -9.63848947], shape=(5,), dtype=float64)\n",
      "tf.Tensor([-9.67960751 -9.64303661 -9.67696885 -9.76317346 -9.6507455 ], shape=(5,), dtype=float64)\n",
      "tf.Tensor([-9.68303361 -9.6575349  -9.70118521 -9.7740601  -9.65751254], shape=(5,), dtype=float64)\n",
      "tf.Tensor([-9.68481667 -9.67473162 -9.71392119 -9.78200161 -9.66880068], shape=(5,), dtype=float64)\n",
      "tf.Tensor([-9.6864865  -9.67835678 -9.73033137 -9.79128949 -9.68317883], shape=(5,), dtype=float64)\n",
      "tf.Tensor([-9.68762425 -9.67928153 -9.77502182 -9.79465957 -9.69252806], shape=(5,), dtype=float64)\n"
     ]
    }
   ],
   "metadata": {
    "scrolled": false
   }
  },
  {
   "cell_type": "markdown",
   "source": [
    "## 对比\n",
    "\n",
    "我们可以把我们用MERA得到的基态能量和DMRG进行对比。"
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "source": [
    "# DMRG\n",
    "import quimb\n",
    "\n",
    "h = quimb.tensor.tensor_gen.MPO_ham_ising(n, j=4.0, bx=2.0, S=0.5, cyclic=False)\n",
    "dmrg = quimb.tensor.tensor_dmrg.DMRG(\n",
    "    h, bond_dims=[10, 20, 100, 100, 200], cutoffs=1e-13\n",
    ")\n",
    "dmrg.solve(tol=1e-9, verbosity=0)\n",
    "energy_DMRG = dmrg.energy\n",
    "\n",
    "# Compare\n",
    "print(\"DMRG solution: \", energy_DMRG)\n",
    "print(\"MERA solution: \", lowest_energy.numpy())"
   ],
   "outputs": [
    {
     "output_type": "stream",
     "name": "stdout",
     "text": [
      "DMRG solution:  -9.837951447459426\n",
      "MERA solution:  -9.795198473308487\n"
     ]
    }
   ],
   "metadata": {}
  }
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