docs/source/tutorials/tfim_vqe_cn.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 一维横场 Ising 模型 VQE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 概述\n",
    "\n",
    "本教程的主要目的不是关于 VQE 物理层面的讨论，而是我们通过演示\n",
    "这个简单的 VQE 玩具模型来了解张量电路的主要技术组件和用法。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 背景\n",
    "\n",
    "基本上，我们训练一个参数化的量子电路，其线路结构为重复的 $e^{i\\theta} ZZ$ 和 $e^{i\\theta X}$ 层的 $U(\\rm{\\theta})$。 而要最小化的目标是这个任务 $\\mathcal{L}(\\rm{\\theta})=\\langle 0^n\\vert U(\\theta)^\\dagger HU(\\theta)\\vert 0^n \\rangle$。 哈密顿量来自 TFIM，$H = \\sum_{i} Z_iZ_{i+1} -\\sum_i X_i$。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 设置"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from functools import partial\n",
    "import numpy as np\n",
    "import tensorflow as tf\n",
    "import jax\n",
    "from jax.config import config\n",
    "\n",
    "config.update(\"jax_enable_x64\", True)\n",
    "from jax import numpy as jnp\n",
    "from jax.experimental import optimizers\n",
    "import tensorcircuit as tc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "为了启用自动微分支持，我们应该将 TensorCircuit 设置为非 “numpy” 后端。\n",
    "而且我们还可以设置高精度 complex128 进行模拟。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "tc.set_backend(\"tensorflow\")\n",
    "tc.set_dtype(\"complex128\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "complex dtype of simulation: complex128 \n",
      "real dtype of simulation: float64 \n",
      "backend package of simulation: tensorflow\n"
     ]
    }
   ],
   "source": [
    "print(\n",
    "    \"complex dtype of simulation:\",\n",
    "    tc.dtypestr,\n",
    "    \"\\nreal dtype of simulation:\",\n",
    "    tc.rdtypestr,\n",
    "    \"\\nbackend package of simulation:\",\n",
    "    tc.backend.name,\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  0.  0.  0.]\n",
      " [ 0. -1.  0. -0.]\n",
      " [ 0.  0. -1. -0.]\n",
      " [ 0. -0. -0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "# 要使用的 zz 门矩阵\n",
    "zz = np.kron(tc.gates._z_matrix, tc.gates._z_matrix)\n",
    "print(zz)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 更高层的 API"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "我们首先设计了以量子电路为输入的哈密顿能量期望函数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def tfi_energy(c: tc.Circuit, j: float = 1.0, h: float = -1.0):\n",
    "    e = 0.0\n",
    "    n = c._nqubits\n",
    "    for i in range(n):\n",
    "        e += h * c.expectation((tc.gates.x(), [i]))  # <X_i>\n",
    "    for i in range(n - 1):  # OBC\n",
    "        e += j * c.expectation(\n",
    "            (tc.gates.z(), [i]), (tc.gates.z(), [(i + 1) % n])\n",
    "        )  # <Z_iZ_{i+1}>\n",
    "    return tc.backend.real(e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "现在我们以 $\\rm{\\theta}$ 作为输入；并将能量期望 $\\mathcal{L}$ 作为输出来制作量子函数。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def vqe_tfim(param, n, nlayers):\n",
    "    c = tc.Circuit(n)\n",
    "    paramc = tc.backend.cast(param, tc.dtypestr)  # 我们假设输入参数的 dtype 为 float64\n",
    "    for i in range(n):\n",
    "        c.H(i)\n",
    "    for j in range(nlayers):\n",
    "        for i in range(n - 1):\n",
    "            c.exp1(i, i + 1, unitary=zz, theta=paramc[2 * j, i])\n",
    "        for i in range(n):\n",
    "            c.rx(i, theta=paramc[2 * j + 1, i])\n",
    "    e = tfi_energy(c)\n",
    "    return e"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "为了训练参数化电路，我们应该利用梯度下降的梯度信息 $\\frac{\\partial \\mathcal{L}}{\\partial \\rm{\\theta}}$。\n",
    "我们还使用 ``jit`` 来包装 value 和 grad 函数以显著加快速度。 注意 ``vqe_tfim`` 的 (1, 2) args 是如何被标记为静态的，因为它们只是量子比特数和层数的整数，而不是张量。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "vqe_tfim_vag = tc.backend.jit(\n",
    "    tc.backend.value_and_grad(vqe_tfim), static_argnums=(1, 2)\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tf.Tensor(-5.2044235531710905, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.168460907768175, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.229007202330065, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.2368387790165105, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.246179597523659, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.262673966580785, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.286129364991173, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.291252895716095, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.2930457160020765, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.293225326335964, shape=(), dtype=float64)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<tf.Tensor: shape=(), dtype=float64, numpy=-7.293297606006469>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def train_step_tf(n, nlayers, maxiter=10000):\n",
    "    param = tf.Variable(\n",
    "        initial_value=tf.random.normal(\n",
    "            shape=[nlayers * 2, n], stddev=0.1, dtype=getattr(tf, tc.rdtypestr)\n",
    "        )\n",
    "    )\n",
    "    opt = tf.keras.optimizers.Adam(1e-2)\n",
    "    for i in range(maxiter):\n",
    "        e, grad = vqe_tfim_vag(param, n, nlayers)\n",
    "        opt.apply_gradients([(grad, param)])\n",
    "        if i % 200 == 0:\n",
    "            print(e)\n",
    "    return e\n",
    "\n",
    "\n",
    "train_step_tf(6, 3, 2000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 批处理 VQE 示例\n",
    "\n",
    "我们甚至可以运行批量版本的 VQE 优化，即我们针对不同的随机初始化同时优化参数化电路，这样我们就可以尽量避免局部最小值，从而找到收敛能量的最佳值。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "vqe_tfim_vvag = tc.backend.jit(\n",
    "    tc.backend.vectorized_value_and_grad(vqe_tfim), static_argnums=(1, 2)\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tf.Tensor(\n",
      "[-4.56780182 -5.32411397 -5.34948039 -5.49728838 -5.51974631 -4.89464895\n",
      " -5.23113926 -5.70097167 -5.4384308  -5.27898261 -4.73926061 -5.43748391\n",
      " -5.02246224 -4.46749643 -5.34320604 -5.29828815], shape=(16,), dtype=float64)\n",
      "tf.Tensor(\n",
      "[-7.15906597 -7.20867528 -7.16615816 -7.16164269 -7.15427498 -7.17176534\n",
      " -7.15677645 -7.19769858 -7.1876547  -7.17160745 -7.14313137 -7.16458417\n",
      " -7.12556993 -7.1043696  -7.17233218 -7.17955502], shape=(16,), dtype=float64)\n",
      "tf.Tensor(\n",
      "[-7.22332735 -7.28775096 -7.22854626 -7.28800389 -7.22006811 -7.2773814\n",
      " -7.22241623 -7.23446324 -7.23115651 -7.23081143 -7.25399986 -7.26564648\n",
      " -7.16463543 -7.27854832 -7.23574558 -7.28935649], shape=(16,), dtype=float64)\n",
      "tf.Tensor(\n",
      "[-7.23956454 -7.29093555 -7.23464822 -7.2914774  -7.22326999 -7.29014637\n",
      " -7.24891067 -7.2505597  -7.23879431 -7.23826618 -7.28737831 -7.29193732\n",
      " -7.22649018 -7.29136679 -7.25276205 -7.29214669], shape=(16,), dtype=float64)\n",
      "tf.Tensor(\n",
      "[-7.24561853 -7.29413883 -7.23950499 -7.29230127 -7.22749993 -7.29051998\n",
      " -7.28702174 -7.289441   -7.25016979 -7.26370483 -7.29320874 -7.29451577\n",
      " -7.22882824 -7.29213765 -7.27040912 -7.29358236], shape=(16,), dtype=float64)\n",
      "tf.Tensor(\n",
      "[-7.24997971 -7.294748   -7.25420008 -7.29271584 -7.24577837 -7.29082466\n",
      " -7.29171805 -7.29016935 -7.28645108 -7.29170429 -7.29499124 -7.29520514\n",
      " -7.23115011 -7.29305292 -7.28793637 -7.2949226 ], shape=(16,), dtype=float64)\n",
      "tf.Tensor(\n",
      "[-7.25300306 -7.29512508 -7.28240557 -7.29287622 -7.28264095 -7.29125472\n",
      " -7.29399162 -7.29066326 -7.29233232 -7.29290676 -7.29521188 -7.29530935\n",
      " -7.23475933 -7.29429836 -7.29053038 -7.29559969], shape=(16,), dtype=float64)\n",
      "tf.Tensor(\n",
      "[-7.25706762 -7.29527205 -7.29168082 -7.29292216 -7.29221412 -7.29183888\n",
      " -7.29474989 -7.29119684 -7.29306204 -7.29300514 -7.29525079 -7.29538907\n",
      " -7.24226472 -7.29544118 -7.29086393 -7.29576418], shape=(16,), dtype=float64)\n",
      "tf.Tensor(\n",
      "[-7.26218683 -7.29529443 -7.29438024 -7.29294969 -7.29426403 -7.29243576\n",
      " -7.29491073 -7.29171828 -7.29384631 -7.29301505 -7.29527304 -7.29545727\n",
      " -7.25772904 -7.29581457 -7.2910381  -7.29582192], shape=(16,), dtype=float64)\n",
      "tf.Tensor(\n",
      "[-7.26654138 -7.29529938 -7.29515899 -7.29297612 -7.29499748 -7.29281659\n",
      " -7.29500531 -7.29227399 -7.29455691 -7.29302087 -7.29529063 -7.29551808\n",
      " -7.2892432  -7.29593328 -7.29120759 -7.29585771], shape=(16,), dtype=float64)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<tf.Tensor: shape=(16,), dtype=float64, numpy=\n",
       "array([-7.29011428, -7.29530356, -7.29549915, -7.29300424, -7.29529224,\n",
       "       -7.29296047, -7.29508027, -7.29270021, -7.29499648, -7.29302725,\n",
       "       -7.29530574, -7.2955733 , -7.29593608, -7.29604964, -7.29138482,\n",
       "       -7.29589095])>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def batched_train_step_tf(batch, n, nlayers, maxiter=10000):\n",
    "    param = tf.Variable(\n",
    "        initial_value=tf.random.normal(\n",
    "            shape=[batch, nlayers * 2, n], stddev=0.1, dtype=getattr(tf, tc.rdtypestr)\n",
    "        )\n",
    "    )\n",
    "    opt = tf.keras.optimizers.Adam(1e-2)\n",
    "    for i in range(maxiter):\n",
    "        e, grad = vqe_tfim_vvag(param, n, nlayers)\n",
    "        opt.apply_gradients([(grad, param)])\n",
    "        if i % 200 == 0:\n",
    "            print(e)\n",
    "    return e\n",
    "\n",
    "\n",
    "batched_train_step_tf(16, 6, 3, 2000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 不同的后端\n",
    "\n",
    "我们可以在运行时更改后端，甚至无需更改一行代码！\n",
    "但是，在普通用户情况下，我们强烈建议用户在一个 jupyter 或 python 脚本中坚持使用一个后端。\n",
    "通过更改``set_backend``行并再次运行相同的脚本，可以享受其他后端提供的便利。 这种方法比在同一个文件中使用多个后端更安全，除非你足够了解 TensorCircuit 的底层细节。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "tc.set_backend(\"jax\")  # 更改为 jax 后端"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-5.67575948 -5.44768444 -5.7821556  -5.36699503 -5.00485098 -5.59416181\n",
      " -5.13421084 -5.70462279 -5.73699416 -5.25819658 -4.70729299 -5.82823766\n",
      " -5.69154358 -5.51112311 -5.46091316 -5.31649863]\n",
      "[-7.16831387 -7.17873365 -7.21905991 -7.17714641 -7.21910053 -7.17729778\n",
      " -7.23594046 -7.1978075  -7.2311691  -7.18566164 -7.15141273 -7.1760751\n",
      " -7.20727055 -7.22174427 -7.15227955 -7.15343225]\n",
      "[-7.24047827 -7.23486717 -7.26382185 -7.25267406 -7.23938877 -7.24135079\n",
      " -7.28655961 -7.24413064 -7.28070556 -7.24825735 -7.23400189 -7.25234153\n",
      " -7.25756263 -7.2505181  -7.22647645 -7.2589444 ]\n",
      "[-7.28642159 -7.23707926 -7.28988032 -7.28627451 -7.28716418 -7.25068739\n",
      " -7.29122589 -7.2510777  -7.2906953  -7.25976327 -7.23891735 -7.29227009\n",
      " -7.28973637 -7.26238069 -7.245065   -7.29155041]\n",
      "[-7.29198674 -7.24196434 -7.29188725 -7.29243688 -7.2926968  -7.26254168\n",
      " -7.29233808 -7.26729904 -7.29277165 -7.28066403 -7.24315235 -7.29344766\n",
      " -7.2920645  -7.26717433 -7.26959622 -7.29307748]\n",
      "[-7.29320541 -7.27162341 -7.29245991 -7.2934821  -7.29360574 -7.27103573\n",
      " -7.29311302 -7.29213505 -7.29356392 -7.29162927 -7.24981922 -7.29384423\n",
      " -7.2930642  -7.27323089 -7.29156252 -7.293684  ]\n",
      "[-7.29384132 -7.29130123 -7.29333191 -7.29442748 -7.29396942 -7.29011734\n",
      " -7.2936549  -7.2929216  -7.29430937 -7.29288465 -7.27758063 -7.29446428\n",
      " -7.2939575  -7.27958774 -7.29260011 -7.29408982]\n",
      "[-7.29433271 -7.29273603 -7.29467579 -7.29507754 -7.29438517 -7.29298445\n",
      " -7.2940647  -7.29297896 -7.29494014 -7.29348052 -7.29142477 -7.29524322\n",
      " -7.29455267 -7.28904449 -7.29319333 -7.29437335]\n",
      "[-7.29480095 -7.29294578 -7.29507043 -7.29530148 -7.29479745 -7.29390014\n",
      " -7.29439439 -7.29304737 -7.29520739 -7.29397645 -7.29276852 -7.2955135\n",
      " -7.29492853 -7.29145529 -7.29390729 -7.29462868]\n",
      "[-7.29530498 -7.29299558 -7.29520643 -7.29537379 -7.29505127 -7.2945667\n",
      " -7.29466477 -7.29312996 -7.29529573 -7.29447499 -7.29327567 -7.29560796\n",
      " -7.29520605 -7.29179062 -7.29484071 -7.29489978]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "DeviceArray([-7.29575181, -7.29302602, -7.29529976, -7.29541094,\n",
       "             -7.29517778, -7.29488194, -7.29487651, -7.29323608,\n",
       "             -7.29532772, -7.29494698, -7.29369784, -7.29567791,\n",
       "             -7.29534388, -7.29187906, -7.29536221, -7.29516005],            dtype=float64)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vqe_tfim_vvag = tc.backend.jit(\n",
    "    tc.backend.vectorized_value_and_grad(vqe_tfim), static_argnums=(1, 2)\n",
    ")\n",
    "\n",
    "\n",
    "def batched_train_step_jax(batch, n, nlayers, maxiter=10000):\n",
    "\n",
    "    key = jax.random.PRNGKey(42)\n",
    "    param = jax.random.normal(key, shape=[batch, nlayers * 2, n]) * 0.1\n",
    "    opt_init, opt_update, get_params = optimizers.adam(step_size=1e-2)\n",
    "    opt_state = opt_init(param)\n",
    "\n",
    "    def update(i, opt_state):\n",
    "        param = get_params(opt_state)\n",
    "        (value, gradient) = vqe_tfim_vvag(param, n, nlayers)\n",
    "        return value, opt_update(i, gradient, opt_state)\n",
    "\n",
    "    for i in range(maxiter):\n",
    "        value, opt_state = update(i, opt_state)\n",
    "        param = get_params(opt_state)\n",
    "        if i % 200 == 0:\n",
    "            print(value)\n",
    "    return value\n",
    "\n",
    "\n",
    "batched_train_step_jax(16, 6, 3, 2000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 更低层的 API\n",
    "\n",
    "`TensorCircuit` 命名空间下的更高级别 API 提供了一个统一的框架来进行线性代数和自动微分，这与后端无关。\n",
    "也可以使用 TensorFlow 或 Jax 直接提供的相关 API（ops、自动微分相关、可即时编译相关），只要坚持一个固定后端即可。 请参阅下面的 TensorFlow 后端示例。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "tc.set_backend(\"tensorflow\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "def tfi_energy(c: tc.Circuit, j: float = 1.0, h: float = -1.0):\n",
    "    e = 0.0\n",
    "    n = c._nqubits\n",
    "    for i in range(n):\n",
    "        e += h * c.expectation((tc.gates.x(), [i]))  # <X_i>\n",
    "    for i in range(n - 1):  # OBC\n",
    "        e += j * c.expectation(\n",
    "            (tc.gates.z(), [i]), (tc.gates.z(), [(i + 1) % n])\n",
    "        )  # <Z_iZ_{i+1}>\n",
    "    return tf.math.real(e)\n",
    "\n",
    "\n",
    "def vqe_tfim(param, n, nlayers):\n",
    "    c = tc.Circuit(n)\n",
    "    paramc = tf.cast(param, tf.complex128)\n",
    "    for i in range(n):\n",
    "        c.H(i)\n",
    "    for j in range(nlayers):\n",
    "        for i in range(n - 1):\n",
    "            c.exp1(i, i + 1, unitary=zz, theta=paramc[2 * j, i])\n",
    "        for i in range(n):\n",
    "            c.rx(i, theta=paramc[2 * j + 1, i])\n",
    "    e = tfi_energy(c)\n",
    "    return e\n",
    "\n",
    "\n",
    "@tf.function\n",
    "def vqe_tfim_vag(param, n, nlayers):\n",
    "    with tf.GradientTape() as tape:\n",
    "        tape.watch(param)\n",
    "        v = vqe_tfim(param, n, nlayers)\n",
    "    grad = tape.gradient(v, param)\n",
    "    return v, grad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tf.Tensor(-5.5454151788179376, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.167693061786028, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.254761404891117, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.290050014550046, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.29133881232428, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.2918048286324915, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.292590929769901, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.294195015132205, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.295013538531699, shape=(), dtype=float64)\n",
      "tf.Tensor(-7.2951174084838835, shape=(), dtype=float64)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<tf.Tensor: shape=(), dtype=float64, numpy=-7.295170441938537>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "train_step_tf(6, 3, 2000)"
   ]
  }
 ],
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