{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "# 变分量子奇异值分解\n", "\n", " Copyright (c) 2021 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 概览\n", "\n", "在本教程中，我们一起学习下经典奇异值分解（singular value decomposition, SVD）的概念以及我们自主研发的量子神经网络版本的量子奇异值分解（variational quantum SVD, VQSVD）[1] 是如何运作的。主体部分包括两个具体案例：\n", "- 分解随机生成的 $8\\times8$ 复数矩阵；\n", "- 应用在图像压缩上的效果。\n", "\n", "## 背景\n", "\n", "奇异值分解有非常多的应用，包括主成分分析（principal component analysis, PCA）、求解线性方程组和推荐系统。其主要任务是给定一个复数矩阵 $M \\in \\mathbb{C}^{m \\times n}$, 找到如下的分解形式：$M = UDV^\\dagger$。其中 $U_{m\\times m}$ 和 $V^\\dagger_{n\\times n}$ 是酉矩阵（Unitary matrix）, 满足性质 $UU^\\dagger = VV^\\dagger = I$。 \n", "\n", "- 矩阵 $U$ 的列向量 $|u_j\\rangle$ 被称为左奇异向量（left singular vectors）, $\\{|u_j\\rangle\\}_{j=1}^{m}$ 组成一组正交向量基。这些列向量本质上是矩阵 $MM^\\dagger$ 的本征向量。\n", "- 类似的，矩阵 $V$ 的列向量 $\\{|v_j\\rangle\\}_{j=1}^{n}$ 是 $M^\\dagger M$ 的本征向量也组成一组正交向量基。\n", "- 中间矩阵 $D_{m\\times n}$ 的对角元素上存储着由大到小排列的奇异值 $d_j$。 \n", "\n", "我们不妨先来看个简单的例子（为了方便讨论，我们假设以下出现的 $M$ 都是方阵）：\n", "\n", "$$\n", "M = 2*X\\otimes Z + 6*Z\\otimes X + 3*I\\otimes I = \n", "\\begin{bmatrix} \n", "3 &6 &2 &0 \\\\\n", "6 &3 &0 &-2 \\\\\n", "2 &0 &3 &-6 \\\\\n", "0 &-2 &-6 &3 \n", "\\end{bmatrix}, \\tag{1}\n", "$$\n", "\n", "那么该矩阵的奇异值分解可表示为:\n", "\n", "$$\n", "M = UDV^\\dagger = \n", "\\frac{1}{2}\n", "\\begin{bmatrix} \n", "-1 &-1 &1 &1 \\\\\n", "-1 &-1 &-1 &-1 \\\\\n", "-1 &1 &-1 &1 \\\\\n", "1 &-1 &-1 &1 \n", "\\end{bmatrix}\n", "\\begin{bmatrix} \n", "11 &0 &0 &0 \\\\\n", "0 &7 &0 &0 \\\\\n", "0 &0 &5 &0 \\\\\n", "0 &0 &0 &1 \n", "\\end{bmatrix}\n", "\\frac{1}{2}\n", "\\begin{bmatrix} \n", "-1 &-1 &-1 &-1 \\\\\n", "-1 &-1 &1 &1 \\\\\n", "-1 &1 &1 &-1 \\\\\n", "1 &-1 &1 &-1 \n", "\\end{bmatrix}. \\tag{2}\n", "$$\n", "\n", "我们通过下面几行代码引入必要的 library和 package。\n", "\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:47:34.433321Z", "start_time": "2021-03-09T03:47:28.730090Z" } }, "outputs": [], "source": [ "import time\n", "import numpy as np\n", "from numpy import pi as PI\n", "from matplotlib import pyplot as plt\n", "from scipy.stats import unitary_group\n", "from scipy.linalg import norm\n", "\n", "import paddle\n", "from paddle import matmul, transpose, trace\n", "from paddle_quantum.circuit import *\n", "from paddle_quantum.utils import *\n", "\n", "\n", "# 画出优化过程中的学习曲线\n", "def loss_plot(loss):\n", " '''\n", " loss is a list, this function plots loss over iteration\n", " '''\n", " plt.plot(list(range(1, len(loss)+1)), loss)\n", " plt.xlabel('iteration')\n", " plt.ylabel('loss')\n", " plt.title('Loss Over Iteration')\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 经典奇异值分解\n", "\n", "那么在了解一些简单的数学背景之后， 我们来学习下如何用 Numpy 完成矩阵的奇异值分解。" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:47:34.469286Z", "start_time": "2021-03-09T03:47:34.440399Z" }, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "我们想要分解的矩阵 M 是：\n", "[[ 3.+0.j 6.+0.j 2.+0.j 0.+0.j]\n", " [ 6.+0.j 3.+0.j 0.+0.j -2.+0.j]\n", " [ 2.+0.j 0.+0.j 3.+0.j -6.+0.j]\n", " [ 0.+0.j -2.+0.j -6.+0.j 3.+0.j]]\n" ] } ], "source": [ "# 生成矩阵 M\n", "def M_generator():\n", " I = np.array([[1, 0], [0, 1]])\n", " Z = np.array([[1, 0], [0, -1]])\n", " X = np.array([[0, 1], [1, 0]])\n", " Y = np.array([[0, -1j], [1j, 0]])\n", " M = 2 *np.kron(X, Z) + 6 * np.kron(Z, X) + 3 * np.kron(I, I)\n", " return M.astype('complex64')\n", "\n", "print('我们想要分解的矩阵 M 是：')\n", "print(M_generator())" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:47:34.541244Z", "start_time": "2021-03-09T03:47:34.489833Z" }, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "矩阵的奇异值从大到小分别是:\n", "[11. 7. 5. 1.]\n", "分解出的酉矩阵 U 是:\n", "[[-0.5+0.j -0.5+0.j 0.5+0.j 0.5+0.j]\n", " [-0.5+0.j -0.5+0.j -0.5+0.j -0.5+0.j]\n", " [-0.5+0.j 0.5+0.j -0.5+0.j 0.5+0.j]\n", " [ 0.5+0.j -0.5+0.j -0.5+0.j 0.5+0.j]]\n", "分解出的酉矩阵 V_dagger 是:\n", "[[-0.5+0.j -0.5+0.j -0.5+0.j 0.5+0.j]\n", " [-0.5+0.j -0.5+0.j 0.5+0.j -0.5+0.j]\n", " [-0.5+0.j 0.5+0.j 0.5+0.j 0.5+0.j]\n", " [-0.5+0.j 0.5+0.j -0.5+0.j -0.5+0.j]]\n" ] } ], "source": [ "# 我们只需要以下一行代码就可以完成 SVD \n", "U, D, V_dagger = np.linalg.svd(M_generator(), full_matrices=True)\n", "\n", "# 打印分解结果\n", "print(\"矩阵的奇异值从大到小分别是:\")\n", "print(D)\n", "print(\"分解出的酉矩阵 U 是:\")\n", "print(U)\n", "print(\"分解出的酉矩阵 V_dagger 是:\")\n", "print(V_dagger)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:47:34.600873Z", "start_time": "2021-03-09T03:47:34.565570Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 3.+0.j 6.+0.j 2.+0.j 0.+0.j]\n", " [ 6.+0.j 3.+0.j 0.+0.j -2.+0.j]\n", " [ 2.+0.j 0.+0.j 3.+0.j -6.+0.j]\n", " [ 0.+0.j -2.+0.j -6.+0.j 3.+0.j]]\n" ] } ], "source": [ "# 再组装回去，能不能复原矩阵？\n", "M_reconst = np.matmul(U, np.matmul(np.diag(D), V_dagger))\n", "print(M_reconst)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "那当然是可以复原成原来的矩阵 $M$ 的！读者也可以自行修改矩阵，试试看不是方阵的情况。\n", "\n", "---\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 量子奇异值分解\n", "\n", "接下来我们来看看量子版本的奇异值分解是怎么一回事。简单的说，我们把矩阵分解这一问题巧妙的转换成了优化问题。通过以下四个步骤：\n", "\n", "- 准备一组正交向量基 $\\{|\\psi_j\\rangle\\}$, 不妨直接取计算基 $\\{ |000\\rangle, |001\\rangle,\\cdots |111\\rangle\\}$ （这是3量子比特的情形）\n", "- 准备两个参数化的量子神经网络 $U(\\theta)$ 和 $V(\\phi)$ 分别用来学习左/右奇异向量\n", "- 利用量子神经网络估算奇异值 $m_j = \\text{Re}\\langle\\psi_j|U(\\theta)^{\\dagger} M V(\\phi)|\\psi_j\\rangle$\n", "- 设计损失函数并且利用飞桨来优化\n", "\n", "$$\n", "L(\\theta,\\phi) = \\sum_{j=1}^T q_j\\times \\text{Re} \\langle\\psi_j|U(\\theta)^{\\dagger} M V(\\phi)|\\psi_j\\rangle, \\tag{3}\n", "$$\n", "\n", "其中 $q_1>\\cdots>q_T>0$ 是可以调节的权重（超参数）, $T$ 表示我们想要学习到的阶数（rank）或者可以解释为总共要学习得到的奇异值个数。\n", "\n", "\n", "\n", "### 案例1：分解随机生成的 $8\\times8$ 复数矩阵\n", "\n", "接着我们来看一个具体的例子，这可以更好的解释整体流程。" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:47:34.656799Z", "start_time": "2021-03-09T03:47:34.620380Z" }, "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "我们想要分解的矩阵 M 是：\n", "[[6.+1.j 3.+9.j 7.+3.j 4.+7.j 6.+6.j 9.+8.j 2.+7.j 6.+4.j]\n", " [7.+1.j 4.+4.j 3.+7.j 7.+9.j 7.+8.j 2.+8.j 5.+0.j 4.+8.j]\n", " [1.+6.j 7.+8.j 5.+7.j 1.+0.j 4.+7.j 0.+7.j 9.+2.j 5.+0.j]\n", " [8.+7.j 0.+2.j 9.+2.j 2.+0.j 6.+4.j 3.+9.j 8.+6.j 2.+9.j]\n", " [4.+8.j 2.+6.j 6.+8.j 4.+7.j 8.+1.j 6.+0.j 1.+6.j 3.+6.j]\n", " [8.+7.j 1.+4.j 9.+2.j 8.+7.j 9.+5.j 4.+2.j 1.+0.j 3.+2.j]\n", " [6.+4.j 7.+2.j 2.+0.j 0.+4.j 3.+9.j 1.+6.j 7.+6.j 3.+8.j]\n", " [1.+9.j 5.+9.j 5.+2.j 9.+6.j 3.+0.j 5.+3.j 1.+3.j 9.+4.j]]\n", "矩阵的奇异值从大到小分别是:\n", "[54.83484985 19.18141073 14.98866247 11.61419557 10.15927045 7.60223249\n", " 5.81040539 3.30116001]\n" ] } ], "source": [ "# 先固定随机种子， 为了能够复现结果\n", "np.random.seed(42)\n", "\n", "# 设置量子比特数量，确定希尔伯特空间的维度\n", "N = 3\n", "\n", "# 制作随机矩阵生成器\n", "def random_M_generator():\n", " M = np.random.randint(10, size = (2**N, 2**N)) + 1j*np.random.randint(10, size = (2**N, 2**N))\n", " M1 = np.random.randint(10, size = (2**N, 2**N)) \n", " return M\n", "\n", "M = random_M_generator()\n", "M_err = np.copy(M)\n", "\n", "# 打印结果\n", "print('我们想要分解的矩阵 M 是：')\n", "print(M)\n", "\n", "U, D, V_dagger = np.linalg.svd(M, full_matrices=True)\n", "print(\"矩阵的奇异值从大到小分别是:\")\n", "print(D)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:47:34.692350Z", "start_time": "2021-03-09T03:47:34.671114Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "选取的等差权重为：\n", "[24.+0.j 21.+0.j 18.+0.j 15.+0.j 12.+0.j 9.+0.j 6.+0.j 3.+0.j]\n" ] } ], "source": [ "# 超参数设置\n", "N = 3 # 量子比特数量\n", "T = 8 # 设置想要学习的阶数\n", "ITR = 100 # 迭代次数\n", "LR = 0.02 # 学习速率\n", "SEED = 14 # 随机数种子\n", "\n", "# 设置等差的学习权重\n", "weight = np.arange(3 * T, 0, -3).astype('complex128')\n", "print('选取的等差权重为：')\n", "print(weight)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "我们搭建如下的量子神经网络结构：" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:47:34.725007Z", "start_time": "2021-03-09T03:47:34.702861Z" } }, "outputs": [], "source": [ "# 设置电路参数\n", "cir_depth = 20 # 电路深度\n", "block_len = 2 # 每个模组的长度\n", "theta_size = N * block_len * cir_depth # 网络参数 theta 的大小\n", "\n", "\n", "# 定义量子神经网络\n", "def U_theta(theta):\n", "\n", " # 用 UAnsatz 初始化网络\n", " cir = UAnsatz(N)\n", " \n", " # 搭建层级结构：\n", " for layer_num in range(cir_depth):\n", " \n", " for which_qubit in range(N):\n", " cir.ry(theta[block_len * layer_num * N + which_qubit], \n", " which_qubit)\n", " \n", " for which_qubit in range(N):\n", " cir.rz(theta[(block_len * layer_num + 1) * N \n", " + which_qubit], which_qubit)\n", "\n", " for which_qubit in range(1, N):\n", " cir.cnot([which_qubit - 1, which_qubit])\n", " cir.cnot([N - 1, 0])\n", "\n", " return cir.U" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "接着我们来完成算法的主体部分：\n", "\n" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:49:24.011232Z", "start_time": "2021-03-09T03:47:40.712257Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "iter: 0 loss: 259.1903\n", "iter: 10 loss: -1672.2268\n", "iter: 20 loss: -2097.8083\n", "iter: 30 loss: -2242.9914\n", "iter: 40 loss: -2310.1948\n", "iter: 50 loss: -2340.2588\n", "iter: 60 loss: -2357.5961\n", "iter: 70 loss: -2369.2625\n", "iter: 80 loss: -2376.9403\n", "iter: 90 loss: -2373.7986\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "class NET(paddle.nn.Layer):\n", " \n", " # 初始化可学习参数列表，并用 [0, 2*pi] 的均匀分布来填充初始值\n", " def __init__(self, shape, dtype='float64'):\n", " super(NET, self).__init__()\n", " \n", " # 创建用来学习 U 的参数 theta\n", " self.theta = self.create_parameter(shape=shape, \n", " default_initializer=paddle.nn.initializer.Uniform(low=0.0, high=2*PI),\n", " dtype=dtype,is_bias=False)\n", " \n", " # 创建用来学习 V_dagger 的参数 phi\n", " self.phi = self.create_parameter(shape=shape,\n", " default_initializer=paddle.nn.initializer.Uniform(low=0.0, high=2*PI),\n", " dtype=dtype, is_bias=False)\n", " \n", " # 将 Numpy array 转换成 Paddle 支持的 Tensor\n", " self.M = paddle.to_tensor(M)\n", " self.weight = paddle.to_tensor(weight)\n", "\n", " # 定义损失函数和前向传播机制\n", " def forward(self):\n", " \n", " # 获取量子神经网络的酉矩阵表示\n", " U = U_theta(self.theta)\n", " U_dagger = dagger(U)\n", " \n", " V = U_theta(self.phi)\n", " V_dagger = dagger(V)\n", " \n", " # 初始化损失函数和奇异值存储器\n", " loss = 0 \n", " singular_values = np.zeros(T)\n", " \n", " # 定义损失函数\n", " for i in range(T):\n", " loss -= paddle.real(self.weight)[i] * paddle.real(matmul(U_dagger,matmul(self.M, V)))[i][i]\n", " singular_values[i] = paddle.real(matmul(U_dagger, matmul(self.M, V)))[i][i].numpy()\n", " \n", " # 函数返回两个矩阵 U 和 V_dagger 学习的奇异值以及损失函数 \n", " return U, V_dagger, loss, singular_values\n", " \n", "# 记录优化中间过程\n", "loss_list, singular_value_list = [], []\n", "U_learned, V_dagger_learned = [], []\n", "\n", "import time\n", "start = time.time()\n", " \n", "# 确定网络的参数维度\n", "net = NET([theta_size])\n", "\n", "# 一般来说，我们利用 Adam 优化器来获得相对好的收敛\n", "# 当然你可以改成 SGD 或者是 RMS prop.\n", "opt = paddle.optimizer.Adam(learning_rate=LR, parameters=net.parameters())\n", "\n", "# 优化循环\n", "for itr in range(ITR):\n", "\n", " # 前向传播计算损失函数\n", " U, V_dagger, loss, singular_values = net()\n", "\n", " # 反向传播极小化损失函数\n", " loss.backward()\n", " opt.minimize(loss)\n", " opt.clear_grad()\n", "\n", " # 记录优化中间结果\n", " loss_list.append(loss[0][0].numpy())\n", " singular_value_list.append(singular_values)\n", " \n", " if itr% 10 == 0:\n", " print('iter:', itr,'loss:','%.4f'% loss.numpy()[0])\n", " \n", "# 绘制学习曲线\n", "loss_plot(loss_list)\n", "\n", "# 记录最后学出的两个酉矩阵 \n", "U_learned = U.numpy()\n", "V_dagger_learned = V_dagger.numpy()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "接着我们来探究下量子版本的奇异值分解的精度问题。在上述部分，我们提到过可以用分解得到的更少的信息来表达原矩阵。具体来说，就是用前 $T$ 个奇异值和前 $T$ 列左右奇异向量重构一个矩阵：\n", "\n", "$$\n", "M_{re}^{(T)} = UDV^{\\dagger}, \\tag{4}\n", "$$\n", "\n", "并且对于一个本身秩 (rank) 为 $r$ 的矩阵 $M$, 误差随着使用奇异值的数量变多会越来越小。经典的奇异值算法可以保证：\n", "\n", "$$\n", "\\lim_{T\\rightarrow r} ||M - M_{re}^{(T)}||^2_2 = 0, \\tag{5}\n", "$$\n", "\n", "其中矩阵间的距离测量由 Frobenius-norm 来计算,\n", "\n", "$$\n", "||M||_2 = \\sqrt{\\sum_{i,j} |M_{ij}|^2}. \\tag{6}\n", "$$\n", "\n", "目前量子版本的奇异值分解还需要很长时间的优化，理论上只能保证上述误差不断减小。" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:49:24.407257Z", "start_time": "2021-03-09T03:49:24.027591Z" } }, "outputs": [ { "data": { "image/png": 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DMmXKqKpqamqqDhw4UL/99ltVVe3Zs6euW7dOVVXnzZunPXr0UFXVwYMH64svvpjxnMTERF2wYIG2bNlSDx8+rElJSdq8eXNdtGiRbt68WYODg3Xx4sWqqjpo0CAdM2aMqqpWr15djx8/rqqqBw4cUFXVxx9/XJ977rmM2IYNG6ZXXHGFpqamZrm/RYsWunnzZl2xYoU2btxY9+zZo6qq+/bty3j+Z599dtr5PvvsMz127JjWqlVL165dq6qqN954Y8Z7qlu3rr7yyiuqqvraa6/pLbfcctZn9sILL+hjjz2mqqo7duzQJk2aqKrqe++9p3fddZeqqu7fv1/T09NVVfWtt97S+++//6zzLF68WKOiorRTp0766KOPZnzeqqpXXHGFTpkyRVVVn376af2///s/VVXdvHmztmjR4rTzHDhwQMPDw886//nK6nsSWKBZfKdaicDTvNcNJFKO5IUfBToUU1wd3AonZ1dRhcSt531Km4Y6cNNQx8bGsmnTJkaOHMn+/fu54IILWL16NXD6AjOZq4WyooVgvjObg9kTElaab2P/y6h5aXy/9wj1qpTJ/UnGZHbT19nvO3EIjidyauC8useNvDrvMpVzfn42bBrq7GMuiGmoTybOAQMGEBQUxDfffEOzZs246qqruO+++1i0aBFHjx6lffv22ca7ePFimjVrdm5v1k+sRJBJz559SA0K4+Nfz/9KzZjTzHoW9IzFaTTdb20FNg11wU9D/fPPP3PgwAEAkpOTWbVqFXXr1gVcgujRowc333xzjqWB+Ph4RowYwT333JOn+P3NEkEm0eXDeaD2Wnr9eivHTyQHOhxTnGz7FdLO+JtKS3bb/cSmofadP6ah3rhxI926dcv4HOLi4rj22msz9g8dOpSlS5eelQg2btyY0X108ODB3Hvvvdx00015it/fCmwaan/yxzTU2Vk9bQzNfrqbrZe+T51O1+TLa5jiwaahNoVZYZ2Gukho2nUIWjaaOpvG5X6wMcYUA5YIzhAUWgppeyO67nuSEjYFOhxjjMl3lgiykBp7IwosmPRSoEMxhVhQUBApKSmBDsOYs6SkpBCUh8k0LRFkIaRyPRbXu43IJtn3nDAmJiaGMWPGWDIwhUpKSgpjxowhJibG5+dYY7Ex52j37t2MGDGCNWvWnNY/3phACgoKIiYmhueff56qVaueti+7xmIbUJaDPds3s3bul3QeeA8iEuhwTCFTtWpVPvzww0CHYcx5s6qhHOyY+RYXr/w7K1csCXQoxhiTbywR5KBx3ztJ1SB2zzx7CLoxxhQXlghyULpKbdZGdqHt3q85cDD7ofPGGFOUWSLIRbkuf6KiJLHkB6sLNsYUT5YIclGn3WVsC67N7+uWkJ5e9HpYGWNMbiwR5CYoiEWXfcljSVczd+O+QEdjjDF+Z4nAB33a1KFi6VAmzT17cQpjjCnqLBH4IDw0mBdqzODxTUNJ2GelAmNM8WKJwEfNOlxCBTlCyKopgQ7FGGP8yhKBj6q16gFRMVRebWsaG2OKlwJNBCISLyLLRWSJiCzwtlUSkR9FZL33s2JBxuQzEYi7GXYsYuPSs1dfMsaYoioQJYIeqhqbaeKjh4BpqtoYmOY9LpTSWg3mOKX4/cfXAx2KMcb4TWGYdO4qoLt3/wNgJvBgoILJSXDpimy65E3aNOkU6FCMMcZvCrpEoMAPIrJQRG73tkWr6k7v/i4gOqsnisjtIrJARBbs2bOnIGLNUuPO11AxqnrAXt8YY/ytoBPBxaraDrgMuEtEumbeqW5xhCyH76rqm6oap6pxUVFRBRBq9tb8NJGfnhvEiZTUgMZhjDH+UKCJQFW3ez93A5OBDkCCiFQH8H7uLsiYzkVw0g66HPmB+T99H+hQjDHmvBVYIhCRMiJS7uR9oA+wAvgCGOYdNgz4vKBiOlcNew7nCBGk//pOoEMxxpjzVpAlgmhgjogsBX4FvlbV74DRwCUish7o7T0u1ILCyxFf80ouPDab9fFbAx2OMcaclwJLBKq6SVXbeLcWqvqUt32fqvZS1caq2ltV9xdUTOejdu+7CJMUNvz4ZqBDMcaY85KnRCAicSIyxKvaOVndUxi6oBa48vXbMq/CFfy4LYQjJ6zR2BhTdPmUCEQkWkTm4ap0PuZUF88XgH/nU2yFXuiAV5l04gK+WLoj0KEYY8w587VE8CKQAFQGjmba/hmu0bdEalenIu2ig1k1exKu56sxxhQ9viaCXsCjqnrgjO0bgTr+DanoEBH+WfEbHkv6ByvWrQ90OMYYc058TQQRQHIW26OA4/4Lp+ip2+dOQiWNmJ1fBDoUY4w5J74mgtnA8EyPVUSCcXMCTfN3UEVJmRrNoF4XQpd8AOnpgQ7HGGPyzNdE8ABwm4j8CIThGohXAZ2Bh/MptiIjvf3NkLiVn38YH+hQjDEmz3xKBKq6CmgFzAV+AMJxDcVtVXVj/oVXNAQ1u5JEieT4+tmBDsUYY/LM5zEAqroLeDwfYym6QkoReu+v9KqY5cSpxhhTqPk6juBuEbkhi+03iMif/R9W0VPGSwJHjx0LcCTGGJM3vrYR/BX4PYvt8cB9/gqmqFvx2T/ZO7oNuxMPBzoUY4zxma+JoBawJYvt27x9Boiq24w6ksBvP3wS6FCMMcZnviaCXUBsFtvbAXv9Fk0RF93+KvYHV6HymrGkpdtIY2NM0eBrIvgYeEVELhGRUO/WB3gJGJtv0RU1wSHsa3odndIXM2/BgkBHY4wxPvE1ETwO/Ax8j5tr6CjwLa476d/zJ7Siqd4ld5JKEIlz3gp0KMYY4xOfuo+qagowVEQeA9ri1hVeoqo2wc4ZQivW4rtGf+O5lZG03n+U2pVKBzokY4zJUZ7WI1DV9ar6qap+Zkkge236/ZmtVOPjX231MmNM4efzgDIRGYKbhbQqZyQQVe3v57iKtOoVIrijXgKV5k8kuffblAopyBVBjTEmb3wdUPYc8BFQD0gE9p1xM2cYGL2DW3Uiq5b9FuhQjDEmR76WCP4IDFXVCfkZTHFSr9ft6LKXiN09BegY6HCMMSZbvtZZBAFL8jGOYieoXFWkeX9YMpb0E0dzf4IxxgSIr4ngTeCsuYZMzrT9cDh+kM8/fi3QoRhjTLZ8rRqKBP4gIpcAy4CUzDtV9V4/x1UsSL0ubCp/AZXKhAY6FGOMyZaviaA5p6qGYs7YZ3MpZEeEBvdPpUGg4zDGmBz4OqCsh79e0FvicgGwXVWvFJH6wDigMrAQuFFVs1ofuchKST7B0sXzaN+hKyIS6HCMMeY0gejg/hdgdabHzwAvqmoj4ABwSwBiylebxv6FmG8Gs2LT9kCHYowxZ/E5EYhIDxF5U0S+E5HpmW95OEct4Argbe+xAD2Bk91SPwCu9jn6IqJW12GUleOsm/puoEMxxpiz+DqgbDhukrlyQHdgD1ARNw31qjy83kvAA0C697gykKiqqd7jbUDNbGK4XUQWiMiCPXv25OElA69Mg07sCG9E8x0TOHikWNV6GWOKAV9LBCOAu1V1KK7H0MOq2hY32tin5bhE5Epgt6ouPJdAVfVNVY1T1bioqKhzOUXgiKDtb6KZbGH2zO8CHY0xxpzG10TQAJjq3T8BlPXuvwoM9/EcnYH+IhKPaxzuCbwMRIrIyUbrWkCxrEiv2XUYxySCo0smomodrYwxhYeviWAfrloI3Bd1S+9+ZSDClxOo6sOqWktV6wHXAdNV9XpgBjDQO2wY8LmPMRUtYeWY2e1THkoayC+bbHomY0zh4Wsi+Ano493/FLda2XvAJ8CP5xnDg8D9IrIBl1jeOc/zFVo9OnemfEQYY+dltfyzMcYEhq8Dyu4Gwr37TwOpuKqeT4F/5vVFVXUmMNO7vwnokNdzFEXhocGMrj2PqLVfsfvQLKqW96kwZYwx+crXAWX7M91Px/X9N+egfaOaVN26lk1rZ1P1gr6BDscYY3zuPpomIlWz2F5ZRNL8H1bxVbXTUDS8Ag22fBroUIwxBvC9jSC7eRHCAOsYnxelSiNthqKrPmfHdlvK0hgTeDlWDYnI/d5dBe4QkcxjBoKBLsCafIqt2NL2w5H5bzBnwisM/svzgQ7HGFPC5dZGcI/3U4BbgczVQMlAPHCH/8Mq3qRqM9a1uI9WDXsGOhRjjMk5EahqfQARmQEMUNUDBRJVCdBk0KhAh2CMMYCPbQSq2uPMJCAijUQkPLvnmNz9vno+X7/7JMmp6bkfbIwx+cTXXkP/EpFh3n0RkanAOmCniNjK7Ocofdln9N3yArMWLA10KMaYEszXXkPXA2u9+5cBbYBOwIfA6HyIq0So1evPhEg6B+YU28HUxpgiwNdEEI2bIhrgcuBTVf0V+A/QNj8CKwmCqzRga8VOXJz0NRt2WfOLMSYw8jLpXF3vfh9gmnc/hOzHGBgfRHb9EzVkP7/9MD7QoRhjSihfE8FE4GMR+RGoBHzvbY8FNuRDXCVG+db92FWqDus3redocmruTzDGGD/zNRHcD7yCW43sElU94m2vDryeH4GVGMGhbBkynXeP9+SrpTsDHY0xpgTyddK5VODfWWx/0e8RlUAdGlShSdUyfDd3AYMvqB3ocIwxJUy2iUBE2gFLVDXdu58tVV3k98hKEBHh5QqfUGXrtyzf0plWdc+a388YY/JNTiWCBUA1YLd3X8m6YVhx8w6Z81C309WU/n0cwQkzoO6QQIdjjClBckoE9YE9me6bfFS6WR+oUIdKqz+CDpYIjDEFJ9vGYlXdot4q6979bG8FF24xFhSMth8Gm2cz77d5gY7GGFOC+DrFRAMRuV9EXhWR/4jIfSJipQQ/k7Y3kkowO2faSGNjTMHJtdeQiPwfbp3iYFx7gQBRwDMi8qD1HPKjctHsu3YClzfuFOhIjDElSI4lAhG5GHgWeA6IUtXqqloNqIrrTvqciHTO/zBLjuhWPQkLL41XK2eMMfkut6qhO4EPVfXRMxaw36eqDwMfAX/OzwBLovXfv8H0p/qzJ+lEoEMxxpQAuSWCTsD7Oex/3zvG+FFFOUKv1Nk89O9Xmf9YR/o9PZEpi7cHOixjTDGVWyKoBmzKYf9G3DQTxo/mV7iUExrKyPR3uUDWMujIxzw8abklA2NMvsgtEUQAOdVPJANhvryQiISLyK8islREVorIE972+iIyX0Q2iMh4ESnlW+jF179mJDA1vS1NZRtBogwKnk3ZlH089/3a3J9sjDF55MtcQ1eIyMFs9kXm4bVOAD1V9bCIhAJzRORb3IR2L6rqOBF5A7iFEj6R3Y7EY0jIqcbiINK5J2QSjyXezP4jyVQqU+JzpTHGj3xJBLl1avepe4s3OO2w9zDUuynQE/iDt/0DYBQlPBG0qnCMnseXIN6EHmGSyuDgWSRoJbo9ncrX911CncqlAxukMabYyLFqSFWDfLj5PM+QiASLyBLceIQfcW0Mid7spuBWQauZzXNvF5EFIrJgz549WR1SbLxU/UfkjPwaQhojQz/ll9IjqL1pHKSl8MHceGas3R2gKI0xxYWv6xH4haqmqWosUAvoAMTk4blvqmqcqsZFRUXlV4iFQoPjKwmT0xepCZF0qFifslXrIV/fh74ax47Z7/Ltsh0ZxxxLTivoUI0xxYBP6xH4m6omisgM4EIgUkRCvFJBLcC6xtwxJ/t9qrD+R2T6kzxYYRGHrvgbAEt/T+SGt+cz+ILaDLuwnlUdGWN8VmAlAhGJEpFI734EcAmwGpgBDPQOGwZ8XlAxFUki0KQP3D6LoMEfElm6FBzcRtOvruHuWhv4YO5muj0/g9s/XMAvG/fZCGVjTK4KskRQHfhARIJxCehTVf1KRFYB40Tkn8Bicm+cNgBBQVC6krt/aCfhyfv5U8Ij3FS3PZMr3sToNaH8sCqBZtXLc1PnevRvU4PwUFs2whhzNimKV4xxcXG6YMGCQIdRuKSlwJKxMOtZOLSdtPrdmBjzEu/M3cbahCQqlynFHd0aclvXBoGO1BgTICKyUFXjztwekDYCkw+CQ6H9cGh9HSx8n+BD2xncsQGDOtRn0ZLFvL4sjcRjyQCkpyurdx2iRY0KgY3ZGFMo+LoeQUUReVlElonILhHZnfmW30GaPAgNh053QJ8nAZCdS2j/eU/eDn+FEW1d6W/mut1c8cocZq0r3t1wjTG+8bVE8CHQAjfgKwEfB5GZQqBSA+j2APzyGrL6S2g9mAsuHMGTV7XgooaVAfh4/lYOHU9h6AV1qFA6NMABG2MKmk9tBCKSBHRT1UX5H1LurI3gHBzZBz+/BL++5UoN96+G0AgA/jpuMVOW7CAiNJgB7WpyU+d6NKpaLrDxGmP8Lrs2Al8TwRLgVlUtFN++lgjOQ9Iu2LnMdUFVhZ9fhtg/sOpQOO/9vJnPl+4gOTWdbk2iuKlzPbo2jiIoSAIdtTHGD843EXQD/gaMAFaoakCHsFoi8JOdS+HNHhASBh3/BBfdy970Mnw8fytj5m1hT9IJGkaV4abO9RnQrialS1nfAmOKsuwSga8DyjbgpqReBCSLSFrmmz8DNQWoehu461doejnMeQlebkOVBS9xb5ea/PxgT14c0obSpUL425QVLNmaCGAD1IwphnwtEcwGKgJvkEVjsapOzJfosmElgnyQsBJm/MtVG92zwJUSVFFg2baDtK5VARHhn1+tYs/hE7w0JBYRqzIypig533EEcUAHVV3h37BMoRHdAq4bC8cPuiSQchzeuwxpM5Q27Ydxck7s8hGhpKSlZySBmWt3szfpBC9OXc+OxGPUiIxgZN+mXN02y0lkjTGFkK+JYBVQPj8DMYVEuDfI7OheCC0N346Eua+4Lqht/sC9vRpnHLp2VxLD3/vttKdvTzzGw5OWA1gyMKaI8LWN4G/ACyLSW0SiRaRS5lt+BmgCpEItGP4V3DgZylaFL+6B1zq4XkeexlXLUjmL1dKOpaTZsprGFCG+lgi+8X7+wOntA+I9ttnMiiMRaNgTGvSAtd/Cmq+hbLTbt2cdQVUas/9IcpZP3Z54jJ837OWihpWtLcGYQs7XRNAjX6MwhZsIxFzubgBH9sKb3aFKY64p149JSTFEkcirpf7D3cn3sodIggSuf3s+HepXYvztnSwZGFOI5ZoIvIXmnwX+qKpW3jcQHgmXPwezRvNCypP8ISyGfWlluEDWck/IJJ6W2/jHVS1IV+XIiTREBFVl7Pyt9Gtdw6axMKaQyTURqGqKiNTH5hcyJwWHQNvrodUgWPQBLab+i4jk/QAMDplNVN+/c1lc7dOesnTbQf42ZQXhocEMbF8rEFEbY7Lha2PxB8Bt+RmIKYJCSkGH24ho2Q+C3DVFeDBctvB2mPoEHIjPODS2diTf3NuFfm2qA/D2T5u4Y8xCFsTvt0FqxgSYr20EZYDrReQSYCFwJPNOVb3X34GZIiJpFywbD+mp7nFaMuxbD3NedLdGvSHuZmjch+Y1TvVAFhF+2bSP71buIrZ2JLd1aUDfFtGEBBfY6qnGGI+vI4tn5LBbVbWn/0LKnY0sLkS+uh8Wj3EJ4KTgUtBiAFSsB4s+gKSd0Otx6HL/aU89mpzKhIXbeGfOZrbsO0qtihHc1Lk+Qy6oTdkwm9fIGH87r0nnChtLBIXIGxfDruVnb6/WCu6YA2mpsO47qNkOytdw3VAXfwQX3AL1u0NQEGnpytTVCbzz02Z+jd9PubAQhnasw82d61OtQnhBvyNjii2/LFUpIuFAI1zD8UZVPe6n+ExRdcecnPcHh0CzK089Proftv4Ca76CivUh7iaCY6+nb4tq9G1RjSW/J/L2T5t4Z85mOjeqQrUK4aSlK8E2FbYx+cbXqqFQ4F/A3UAp3ECyE8B/gEdVNSU/gzyTlQiKuNQTsPpLWPAubPkZqjSFu+ZnzGcEsCPxGNUrhCMi/Oub1azccZAPbupgbQjGnIfzLRE8AwwF7gBOXgJ2AZ7G9Twa4Y8gTQkREgatBrrb7tWuwVnETXQ35hpocTU1Wg8BcSuo1a1cmrR0zUgCU1cl0LlRFSJK2YB2Y/zB1xLBLuBmVf3mjO1XAG+ravV8ii9LViIopvZvggm3wI5FbsK7lte6Hkc122UcsmnPYXr+exYVS4dyQ6e63HhhXaqWs3YEY3xxviuUHQNizxxZLCIxwGJVjfBbpD6wRFDM7VgMC96D5Z9BylG4fSbUaAu4hXF+iz/AWz9tYurqBEKDgri6bQ1uubgBTavZOsvG5OR8E8E8YKGq3nXG9tdxCeJCH85RG/gQiMY1Nr+pqi97s5eOB+oB8cBgVT2Q07ksEZQQxw+6Xkath7iqox8fg+SjEHcTRLdg894jvDtnM58t/J3jKel0bRLFrRfXp0vjKja3kTFZON9E0BU3A+l2YJ63uRNQA7hMVXPpOgIiUh2orqqLRKQcbmDa1cBwYL+qjhaRh4CKqvpgTueyRFBCfXW/63qadgJqd3LVRs2v4kByEGPnb+GDX9w6y+3qRDLxzossGRhzhvMeRyAiNYC7gBhv02rgv6q64xwD+hx41bt1V9WdXrKYqapNc3quJYIS7Mg+WPqx63G0fxN0vBMuGw3AidQ0vliyg4PHUri1S4OMie6uaFWdilmsm2BMSVOoBpSJSD1gNtAS2Kqqkd52AQ6cfHzGc24HbgeoU6dO+y1bthRUuKYwSk+H+NlQoTZUbghb58PMf7lSQtPLITiUFdsPcuV/5vD0gFYM7VAHgCmLt/Pc92ttWU1TIp1TIvB19TFV3Z+HQMoCs4CnVHWSiCRm/uIXkQOqWjGnc1iJwJxlzTfwzUg4tM0tntPuj9BuGGuPR1K3cmnCQ4O5/9MlTFm8nfRMf/IRocE8PaCVJQNTImSXCHIbnbMX2JPLbXcegggFJgJjVXWStznBqxI62Y7g8/mMyRBzOfx1GQwdD9VjYfbz8FYPmkZFEB7qxhtMW737tCQAblnNZ79fU/DxGlOI5DagLKeVyS4F/gKk+vJCXrXPO8BqVX0h064vgGHAaO/n576cz5izBAVD00vdLXEr7FnrprhIT4cxV3NDcjSf0h3Q01ZT25F4nFs/WEDnRpXp2iSKhlFlA/1OjClQeW4jEJG2wHO4kcX/A55U1T0+PO9i4CdgOZDubX4EmA98CtQBtuC6j+ZY1WRVQyZPjuyFCTfD5lmkaDDbtTJ1ZA9j03ry99RbKF0qmCplw9i6/ygD2tXkhcGxqCpfLtvJhQ0qE1UuLNDvwBi/OO9J57xVyp4CBgGTgOaqutHX53tdTLPrz9fL1/MYk2dlqsCwL/jxp5/Z++MLXCdTEYEhwbN4Uwbz0JVtuKJtfX5PSifVqzvatPcI936ymKeuacn1HeuyJ+kEi7ceoFPDypQPt6U2TfHiy5rFlYHHcPMM/QxcpKq/5XdgxvjbJV06s2njh6TEh1CKVBD4oOEMGiSth+feoHaj3hBzJZTtQ/3K5fnm3i4Z02DPWLObByYuI0igda1ILm5Uhc6NqtCubiRhITbnkSnacus19CgwEjfi9yFV/a6A4sqRVQ2Zc5K0C15uA6mZZk8PCYeB78H6H2DtN3A4AYJCoUlfGPJRxoyoyanpLN56gJ837GXOhr0s3XaQtHQlPDSIC+pVykgMzauXJ8imzDaF1Ll2H00HjgEzOFWvfxZV7e+PIH1licCck+xWU2t7I1z5gmtU3vabWyshLSVjoBpT/gxVmrjSQpVGACQdT2H+pv3M2bCXnzfsZf3uw4QGC0sf70PpUiEs33aQChGh1KlcOgBv1JisnWsbwYe4eYGMKfq2/Xp6EgD3eNuv7n5QENTp6G4nJR+F3atgyViY+jhExUDMFZRrfR29mzehd/NoABIOHWf1zkOULuX+pZ74ciUpael8fvfFAMzduJeYauWpZCOcTSFkS1Ua44vE313V0ZqvIP5n6PcytLsRDu+B3SuhbmcIPtWIvHHPYRKPptC+bkWOp6TR5okfOJGaTvPq5bm4satGuqBexYzEYUxBKFRTTJwvSwQmoI7ud1VKYWXh17fgmxEQHglNLoWYK6BRLyhVJuPwtHRl2bZEft6wl5837GPhlgMkp6UTGiy0q1PRtS80rkLrmhVsBTaTrywRGJMfko/Cxumw5mtY9y0cO+AW1fm/NRBewbU7BJ3+5X4sOY3f4vdnNDyv2nkIVfjn1S25oVNdDh5NYc/h4zSMKsvnS3bY3EjGb/yyeL0x5gylSkOzK90tLdWtwbxzqUsCAOP+AMmHXUNzzOUQWYeIUsF0bRJF1yZRAOw/kswvG/fRvq6bYuuHVbsYOWEZD17alFembeBYShoA2xOP8fCk5QCWDIxfWYnAmPw061lYMQn2rHaPq7eBDn+Cttdn+5SEQ8eZvW4PL01dx/bE42ftr1oujPmP9LL1Fkyeneukc8aY89HtAbhrHtyzCC75BwSHubEKAMlH4Ie/w9Z5kJ6W8ZTo8uEMiqvNjiySAMDupBN0eXYGo75Yyc8b9pKcmm3PbmN8YiUCYwqaqhuotmUufNAf0lOgTJRbRyHmSmjQDULC6Dx6OtsTjxHFgdMmyYuMCCWuXkV+Wr+XE6nplAsL4cYL6/LApTG5v7Yp0axEYExhcbJKp+5F8MAmuPYdqHcxrJgIHw+CvesA+HvXilQJPcG9IZO5QNZyT8gkIkKDGdW/BW8Pu4Alj/XhrT/GcXmr6hnjE46npDH8vV/5ecPeQL07UwRZY7ExgRReHloNdLfUE66xObolAJfufps+wZ+gpBOEMiRkFlF9/85lXkNxRKlgLmkezSXeoDaAXQePsyPxGMlprrpo9c5DfLF0B72bRdO2dqRNf2GyZFVDxhRW2xbCl/dCwopT2yo1gHsX5/pUVUVEGP/bVh6ZvIK0dKVK2VL0jKlK72bRXNy4ig1mK4Gs+6gxRU2FmrBvw+nbErdCUoJrU/hkiKteatbfrducyckeRUMuqMOlLaozc91upq7ezbfLd/Hpgm2EhQRxcaMq9G4eTa+YqlQtH15Q78oUQlYiMKawymmSvG4PwCfXwQ6vdBDdEpr1g9g/QGSdbE+ZnJrOb/H7+XFVAlNXJ7DtwDGiyoXxq9cddU/SCaqULWVdU4spKxEYU9TkNEleuWpw+0xXQlj9Faz+AmaOhlpxLhEkbnUrs9Voe6pxGigVEkRnb8rsx/s1Z13CYbYdOIqIoKpc9eocujSO4pmBrQFITUu3aS9KACsRGFNcJO2C0pXd5HdTn4A5L0CF2q6k0Kwf1O7o1nXORmpaOhMWbqNmxQi6NI5i58Fj9HlxNt2aRHFJ82i6N6lKhdK2OltRZnMNGVOSHN0P676DVV+4uZDSTkDF+m5gW5BvV/i/7z/Kq9M3MG1NAnsPJxMcJHSoV4nezaPp3awqdSuXyf0kplCxRGBMSXUiya3AlpQAF/7ZbXv/SldaaN4fGvSA0Owbi9PTlSXbEpm2OoGpq3azNiEJgCbRZenVLJrbuzSgYplSTFm83SbIK+QsERhjnNRk+OIeWPstnDgIpcpC4z7Q4Xaoe2GuT9+67yhTV7vG5qW/J/Lro735cVUCD0xYljF+ASAiNJinB7SyZFCIWCIwxpwuNRniZ8PqL9002r2fcJPhHdoJm2ZC00shomKOpziWnEZEqeCM6TDOVCEilHeGxRFTvTxlw6xvSqBZIjDGZC89zd1CSsGCd+Gr+yAoBOp3deMUYq6AslWzfXr9h77OcU1bEahXuQzNq5eneQ13a1GjPFXL2fiFgmTdR40x2QsKPtWjqN1wqNYGVn/uGpu/+qtbhW3kBldCSE12CSOTGpERWZYIqpUP56lrWrJqxyFW7jjE8u0H+Xr5zoz9N3Wux+P9WpCerny7YhcX1Ktog9sCoMASgYi8C1wJ7FbVlt62SsB4oB4QDwxW1QMFFZMxJgtBQVCrvbv1fgISVsL2haeqiT4ZAscPet1S3ajmkX2b8vCk5ZRN2ZsxU+rh0Mo8dFkMvZpF06vZqfmQDh5LYc1OlxiaRJcDYOv+o9z18SJGD2jFdR3qsGXfEd6Zs5nm1cvTokYFmlQrS1hI9l1fzfkpsKohEekKHAY+zJQIngX2q+poEXkIqKiqD+Z2LqsaMiaA5v7HLbazY5F7XLUFdLqTKdKT9K/u5+rU75kc0pfgK1/wuaE4JS2dtbuSqFYhnCplw5i1bg93jV3E4ROpAIQECY2qlnXVSl5yaF69vI1ryKNC0UYgIvWArzIlgrVAd1XdKSLVgZmq2jS381giMKYQSPwd1nzlqo+aXgqth8DLrd0sqsGl4O6FUDH76S5yk56ubN1/lFU7D7Fyx8GM6qXdSScyjvn8rs60qR3JuoQktuw7SrcmUZQKsZHQ2SmsbQTRqnqywnAXEJ3TwcaYQiSyNnS6091U4ev/O7XSWloyvBIL9btA/W5uDqRy1fJ0+qAgoV6VMtSrUobLW1XP2L4n6QSrdh5i1Y5DNKpaFoDJi7fz1uxNrPxHXwDGzt9C/N4jruRQozwNqpQ5a6oMG/dwSqATQQZVVRHJtngiIrcDtwPUqXPuVxnGmHxwOAGWjIX01NO3H9wB056Appe5RBD/M+xeBQ26Q+VGp82D5KuocmF0KxdFtyZRGdvu7tGIK1pVz2hHWLnjEBMWbMsY1xAWEkRMtXI0r1GBFjXKs/fwCd6YtZHjKW7/9sRjPDxpOUCJTAZWNWSMOX85zZTa/SE3bbYIfPcIzHvN7S9Xwy3LWb+bq1byceoLX6WkpbNxz2FWbj90WvXSoeOp2T6nZmQEPz/U069xFCaFtWroC2AYMNr7+XlgwzHGnJOcZkrNPP6g71NwwS2weRZsmgXrvoet8yB2qNu/+CMIj3RLd0ZEnldIocFBxFQrT0y18lzrbVNVth04RpdnZ2T5nB2Jx1BVbnznV+pULk1MtXI0jS5HTLXi3TBdkN1HPwG6A1VEZBvwOC4BfCoitwBbgMEFFY8xxo/umOPbcSJuEZ3KDSHuZkhPh8O73D5VmPWMm0JbgtwU2vW7ucFstc66iD0nIkLtSqWpmc24hxqRESSdSCU5LZ2vl+3k4/lbM/ZVKx9O02rliKlejphq5YirW4nalUr7Ja5AK7BEoKpDs9nVq6BiMMYUMkFBUL6Guy/iehptX+CmuNg0C+a+4noh1YqDtBT45VWo1xVqxOY4pXZuTo57OJaSlrEtIjSYkX2bUj48lE//dCGqSsKhE6zedYi1u5JYuyuJNbuS+GXjPpLT0nnk8hhu79qQhEPH+efXq/lT1wa0rFmB1LR0goOkSC3uE+iqIWOMOSWklFt+s+5F0OMRN3NqqtddNGElTB3l7odVcNVHDbq7GVTz2CPpZINwTr2GRIRqFcKpViGcHk1PVW+lpKUTv/cIFSJcVVHCoeMs3nqA415S+X5lAg9OXEaT6LLEVC9fJKqXbK4hY0zRcXg3bJ7tSgybZ7lqpGFfuW6qCSth5zLXAH2ylBEAS39PZMLCbV4J4vTG6eoVXPVS02rluL1LAyqXDSvQ2AprY7ExxviubFVoNdDdAPZvhvLeVfzKKTD7WXe/cmOXEBp0hyaXulXbCkib2pG0qR0JuMbpXYeOs+Zk1dLOQ6zZlcTcDfu4s1tDAF6dvp4vl+7k63svJiQ4iA27DxMWEkStihEZ1Uv5PebBEoExpuiqVP/U/e4Pu2qiTbNciWHJJ7D8M3hgs9u/5hsIjYA6ndxPcMt7TrgJBr4P5fw/nlVEqF4hguoVIs6qXgr1BrjVrlSadnUjMwa8PfnVKmat20O5sBCaVCtHqRBhQfwBUtJc7U1+jHmwqiFjTPGUmgwHNkOUNzTpvxe6wWzBYVC7gysx7Fru1mNofxNc+UJg4/Us33aQZdsTMxqnf4vfT1Zf0+cy5sGqhowxJUtIqVNJAOCWH2DLL94Yhpkw/Z+um6qmu1HRFWpDkz4Q1czvg9vyolWtCrSqVSHjcf2Hvs7yuB1ZdH89V5YIjDElQ1g590XfpI97PPlOV3Wk6W6OpGmj3K10FW+OpK7Q5DIoXz2ns+a77NZ6qBEZ4bfXsGn6jDElT9IuWDkJ0lPc4/QUCAmDvk9Do95utPNX98Hv89z+A/GuzeHg9gIPdWTfpkSEnj5m4uSYB3+xEoExpuSZ9awrCWSmCvs2wID/efc3nmpAXvsdfOctlVKpoSstNOjmeiSF+u/KPCu+jHk4X5YIjDElT05zI4Eb5Vyl0al9HW53A9g2z3LjGJZPcJPsPbjF7d8w1Y18rtsZwsv7Pdyr29bM11lRLREYY0oeX+dGOikoCKq1dLcL74K0VNi7DsLcegj8/IpLEhLszZHUFRr1csmjCLBEYIwxeRUcAtHNTz3+w6ew7TdXWtg8282RtHPJqUTw2zsQ3QJqtHO9mQoZSwTGGHO+QsO9nkZdgEfhxGE4utftO34IvhkJmgahZaDuha7EEHOlm4W1ELBEYIwx/hZW9lS1UXh5GLkB4uecKjH8+BiElnaJICkBVn3ukkNU03Nate18WSIwxpj8VrqSm/6ieX/3OGmX664KsOVn+Haku1822iWE+l2h+VUQfmpgWX5Oh2GJwBhjClrmabNbDoCa7U6VFjbPdgPdGvZyiWDzbJcENk534xtmPeP36TAsERhjTKBVrOdu7f54agxDBa+76OKPYNn4U8cuGQvdHvRrqcBGFhtjTGFy5hiGq1+HZle5rqngBsLNesavL2mJwBhjCrMje2D9967XEbiBb0vGukZmP7FEYIwxhVmW02H4t1RgicAYYwqz3KbD8ANrLDbGmMIsr9NhnAMrERhjTAlnicAYY0o4SwTGGFPCWSIwxpgSzhKBMcaUcKKqgY4hz0RkD7DlHJ9eBdjrx3DyW1GK12LNP0Up3qIUKxSteM831rqqGnXmxiKZCM6HiCxQ1bhAx+GrohSvxZp/ilK8RSlWKFrx5lesVjVkjDElnCUCY4wp4UpiIngz0AHkUVGK12LNP0Up3qIUKxStePMl1hLXRmCMMeZ0JbFEYIwxJhNLBMYYU8KVmEQgIu+KyG4RWRHoWHIjIrVFZIaIrBKRlSLyl0DHlBMRCReRX0VkqRfvE4GOKTciEiwii0Xkq0DHkhsRiReR5SKyREQWBDqenIhIpIhMEJE1IrJaRC4MdEzZEZGm3md68nZIRP4a6LiyIyL3ef9fK0TkExEJ99u5S0obgYh0BQ4DH6pqy0DHkxMRqQ5UV9VFIlIOWAhcraqrAhxalkREgDKqelhEQoE5wF9UdV6AQ8uWiNwPxAHlVfXKQMeTExGJB+JUtdAPehKRD4CfVPVtESkFlFbVxACHlSsRCQa2Ax1V9VwHq+YbEamJ+79qrqrHRORT4BtVfd8f5y8xJQJVnQ3sD3QcvlDVnaq6yLufBKwGagY2quypc9h7GOrdCu0VhojUAq4A3g50LMWJiFQAugLvAKhqclFIAp5ewMbCmAQyCQEiRCQEKA3s8NeJS0wiKKpEpB7QFpgf4FBy5FW1LAF2Az+qamGO9yXgASA9l+MKCwV+EJGFInJ7oIPJQX1gD/CeV+32toiUCXRQProO+CTQQWRHVbcDzwNbgZ3AQVX9wV/nt0RQiIlIWWAi8FdVPRToeHKiqmmqGgvUAjqISKGsfhORK4Hdqrow0LHkwcWq2g64DLjLq+YsjEKAdsDrqtoWOAI8FNiQcudVYfUHPgt0LNkRkYrAVbhkWwMoIyI3+Ov8lggKKa+ufSIwVlUnBToeX3lVATOASwMcSnY6A/29evdxQE8R+SiwIeXMuxpEVXcDk4EOgY0oW9uAbZlKgxNwiaGwuwxYpKoJgQ4kB72Bzaq6R1VTgEnARf46uSWCQshrfH0HWK2qLwQ6ntyISJSIRHr3I4BLgDUBDSobqvqwqtZS1Xq46oDpquq3Kyt/E5EyXocBvGqWPkCh7PmmqruA30WkqbepF1AoOzicYSiFuFrIsxXoJCKlve+HXri2Q78oMYlARD4BfgGaisg2Ebkl0DHloDNwI+5q9WTXtssDHVQOqgMzRGQZ8BuujaDQd8ssIqKBOSKyFPgV+FpVvwtwTDm5Bxjr/S3EAv8KbDg585LrJbgr7ELLK2VNABYBy3Hf3X6bbqLEdB81xhiTtRJTIjDGGJM1SwTGGFPCWSIwxpgSzhKBMcaUcJYIjDGmhLNEUIKJyEwReTVAr60iMjAQr+2Lwh6fP3izWI7y4bgZIvLHAggpq9fO8ffgzXp7bUHGVBxZIiimvEFe//WmMD4hIgkiMk1ELsl02ADg4UDF6G8i0s774uiSzf7xIjK3oOPKSXZfdCLyfmGYIltErgBqA2MzbYv34lYROeZNOT3SG+hU0J4ERouIfZedB/vwiq+JuKkIbgGaAFcC3wKVTx6gqvu92U2LHBEJOfOLx5uxdQlwcxbHVwauxmYczau/AO+ratoZ2/+BG0jYDDcZ2r+AQEyI9w1QDjdNhDlHlgiKIW+6hy7AQ6o6TVW3qOpvqvq8qo7LdNxpVUPeld7fROR/3iId20Rk5BnnbiIis0TkuIisFZHLReSwiAz39tfzrhTjznhebkX80d75jnlxPJt54Q0RGeVVZQwXkY3ACSCrmS3fBgZ5E/ZldoP3nPEicqmI/CQiB0Rkv4h8LyLNcojNp/ckIjVFZJx33gMi8rWINM7uvHkhIgNEZJn3+ez3fgfRmfb3Ezc76XER2SwiT4mbTO3k/qoi8rn3/C0iclayzOI1o3Bz3HyZxe4kVd2lqvGq+jawDDf9xcnnNvReb5eIHBGRReIm/Mt8/lz/3rKI6UER2SsincBNdohLBkNzez8me5YIiqfD3q2/5H0Vo/twQ9jbAc8Az4q3ypRX/J4MpAKdgOHA40CYH2I+gruSbwb8GTcP0KNnHFMf+AMwCGgDHM/iPGOBYGDIGdtvAcar6hFcAnkJV2LqDhwEvsz8xZlXIlIaN9necaAbcCFuuuCp3r5zJiLVcBPkfYD7fLoCYzLt74t7368CLXCf40BOn97hfaAR7ov9auCPQL1cXvpiXPLMdm4jcbp7caVk2lUWVwK9BPe7mghMEpGYM06R7d9bFq/zPG4Ki25nLHr0K+4zN+dKVe1WDG/AtbiFeI7j5lh6Hrf6UuZjZgKvZnocD3xyxjHrgb959/vikkDNTPsvws2XP9x7XM97HHfGeRQYmN3jLOK/A9iQ6fEo3BdNtA/v/SNgbqbHF3iv1zGb48sAabjpns+Kz5f3hPvyXY83bYu3LRjYBwzOIdYsPwfcF/dX3v123nF1sznHbODvZ2y7GncxILiqQQU6Z9pf13vPo3KI7a/Aliy2x+MSxGEg2Tv3MeCiXH4v807+Lfny95bp8xkCvAesy+ozwE0hnQ6EFPT/WXG5WYmgmFLVibh5y/vhrswuAuaJyCO5PHXZGY93AFW9+zHADvWmRfb8hh8WeBGRgSIyx6tKOAy8CNQ547Bt6ttUwW8DF2a6+rwZWKHe9MhetcXHIrJRRA4BCbjS8ZmvlxftcSWWJK+q7DCupFERaHge5wVYCkwFVojIRBG506u2yfzaj558Xe+1P8YluGq4q/V03JUzAOpW4spthasIsi51AbyAm1SuG64k9ISqZjTEi5s19Vlx624f8GKK4+zPOKe/t5Oex5XcLtasVxA7hkt4flvDt6SxRFCMqepxVf1RVf+hqhfhprYelUsVSMoZj5W8/Z2cTAoZDbni1lbIllffOw74Hpe42gJ/wy15mdkRH2OYBWwAbhY3LfZQvOUTPV8BUcCfgI7e66UC2X0uvrynIFxDdewZtybA/3KINQmokMX2SFwiQV09eB/vtgxXzbVeRNpkeu0nznjd1kBj3IphJ+V1hsm9uESWlX2qukFVf8GVPkeISI9M+5/HVeH9HZcsYnGJ6MzP2Je/tx9xCS27GXgrAcf11HKpJo9CAh2AKVCrcL/zcFyRPq/WADVEpIaqnryajOP0f9yTXzzVM22LzeW8nYHtqvrkyQ0iUvcc4gPcGsoi8i6ux8sa3JXtGO+8lXElmz+r6gxvWzty/l/w5T0twiWcvZq3dXrX4q7oMxKVuIXU2+CqQzLeE66K7xcR+QewEldlstR77RhV3ZDVC4jIGtzvqAMw19tWB1dizMliIEpEqqjq3uwOUtUD4jodvCgibb1YLwY+9EqmeG1VDXHVO3n1DW6a6M9ERFX1gzP2t8R9BuYcWYmgGBKRyiIyXURuEJHWIlJfRAbh1umdpue+7OWPuC+uD0SkjXcl/wLuatpV6Koew9UFPygiLUTkItzVYU7WATVF5HoRaSAid3L+vUA+AKp4rz1FVfd52w/grnRvE5FGItINeMN7D1ny8T2NxVUxfS4i3bzPvKuI/DuXnkMv4Eoud4nrkRWLm2e+kvcTEenk9a65wPsC74/r239y0Zd/AH8QkX+ISEsRifGq2p714l8LfAf8T0Qu9F7jfVyVSk4W49agvjiX4wD+CzTFlQLA/U6vETe2oxWu3eacq27UrW8xCHhDzh7c1gX3/sw5skRQPB3GfXH9BVdNshLXg+Rjzu5N4zNVTQeuwfUS+hX3ZfsULglkrks+2TXxN1y1yN9yOe+XwHO4njzLcD1NHjvXOL1z7sBdSVYk09gB7z0MwVWdrABew1VfnMjllDm+J1U9iuvNswm39u0a3OdTEZd8sovzE+Am77YA94VWDeiibsUvcFVEnXFVWuuBfwNPqupH3jm+B64AeuB+L7/i1gremumlhgObgem47qAf4xprs+VVSb0LXJ/Tcd6xu3GlrlFe77L7cUnkJ1wb1Tzv/jnzksFgXEL7I7guu7j2r/dyeq7JmS1MY86LV0+9BNejpigtCG98ICJVcSWPC1R1c6DjOZOIPAdUUNVADGYrNqyNwOSJiFyDa7Rdj+tW+QKn6qlNMaOqu8UNPquDK1EUNrvJverR5MJKBCZPvCL533B11AdwYxHu87FbpzGmELJEYIwxJZw1FhtjTAlnicAYY0o4SwTGGFPCWSIwxpgSzhKBMcaUcP8P6f6xra7TpV4AAAAASUVORK5CYII=\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "singular_value = singular_value_list[-1]\n", "err_subfull, err_local, err_SVD = [], [], []\n", "U, D, V_dagger = np.linalg.svd(M, full_matrices=True)\n", "\n", "# 计算 Frobenius-norm 误差\n", "for i in range(T):\n", " lowrank_mat = np.matrix(U[:, :i]) * np.diag(D[:i])* np.matrix(V_dagger[:i, :])\n", " recons_mat = np.matrix(U_learned[:, :i]) * np.diag(singular_value[:i])* np.matrix(V_dagger_learned[:i, :])\n", " err_local.append(norm(lowrank_mat - recons_mat)) \n", " err_subfull.append(norm(M_err - recons_mat))\n", " err_SVD.append(norm(M_err- lowrank_mat))\n", "\n", "\n", "# 画图 \n", "fig, ax = plt.subplots()\n", "ax.plot(list(range(1, T+1)), err_subfull, \"o-.\", \n", " label = 'Reconstruction via VQSVD')\n", "ax.plot(list(range(1, T+1)), err_SVD, \"^--\", \n", " label='Reconstruction via SVD')\n", "plt.xlabel('Singular Value Used (Rank)', fontsize = 14)\n", "plt.ylabel('Norm Distance', fontsize = 14)\n", "leg = plt.legend(frameon=True)\n", "leg.get_frame().set_edgecolor('k')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "---\n", "### 案例2：图像压缩\n", "\n", "为了做图像处理，我们先引入必要的 package。" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:50:17.225100Z", "start_time": "2021-03-09T03:50:17.151352Z" } }, "outputs": [], "source": [ "# 图像处理包 PIL\n", "from PIL import Image\n", "\n", "# 打开提前准备好的图片\n", "img = Image.open('./figures/MNIST_32.png')\n", "imgmat = np.array(list(img.getdata(band=0)), float)\n", "imgmat.shape = (img.size[1], img.size[0])\n", "imgmat = np.matrix(imgmat)/255" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:50:18.337792Z", "start_time": "2021-03-09T03:50:17.231211Z" } }, "outputs": [ { "data": { "image/png": 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\n", 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r3yIySP0mu1UuW94PvOLu3+rTPrXP3T4NbK5/90SkXgZyNf4a4PPAJjPrLtq+CtxmZguplON2AF8YyA6H89I6Ufmn7Lxq73//+8PY3r17w9hTTz0Vxrq7u6u2X3nlleE2d911Vxi77LLLwtixY8fCWGTs2LFhLFVeS8VS8/VFz7fU32w4P0dTBnI1/lmg2m+vmrrIMKJP0IlkQskukgklu0gmlOwimVCyi2RCyz8NQlSSSS0LlZo4MjX54hNPPBHGNm+OP9IwbVr1YQuf/OQnw22uuuqqMFZ2dFg0eWTq8VLltdRxTI0ePHnyZNX21KSS9V7ma6jQmV0kE0p2kUwo2UUyoWQXyYSSXSQTSnaRTKj0NghlSm+pstCbb74ZxtauXRvGtmzZEsY6Ojqqtk+dOrVqO6TLUPUuUaVGm6WOVSpWdj29iEpvIjKsKdlFMqFkF8mEkl0kE0p2kUwo2UUy0fTSW6qUM9RF5Z9UOenw4cNhLDV6bcOGDWHs4MGDYWzBggVV21Olt9Ros2j0GqRLVGUm4Uyt9ZYqoaWeU6n+R8pOIDrU6cwukgklu0gmlOwimVCyi2RCyS6SiX6vxpvZGGAtMLq4/4/d/WtmNhNYBUwE1gOfd/fqE36d51JXilNX43fu3BnGUivetrW1hbFomafZs2eH25S9qp7arkzVpRGDXaLlplL9O3HiRBgbzgZyZj8BXO/uV1NZnnm5mX0Y+AbwbXefDRwE7mxYL0WkZv0mu1ccLX4cWXw5cD3w46L9QeDWRnRQROpjoOuzjyhWcN0PPAlsAw65e29xl11A9TmMRWRIGFCyu/spd18IXAosAf5koDswsxVm1mVmXan3oSLSWIO6Gu/uh4CngaXAB8zszAW+S4HdwTYr3b3T3Tvb29tr6auI1KDfZDezdjP7QHF7LPBx4BUqSf/nxd3uAH7aoD6KSB0MZCDMVOBBMxtB5Z/Dw+7+v2b2W2CVmf0z8CJw/0B2mFr+Z6iL+j5mzJhwm8mTJ4exOXPmhLEpU6aEsdSAkUWLFlVtnzlzZrhNyvHjx0v1IyrL9fb2Vm2HdAkttfxTqox29OjRqu1l5w0czvpNdnffCLznGeTu26m8fxeRYeD8/BcmIu+hZBfJhJJdJBNKdpFMKNlFMmHNnBPOzHqAM0O9JgEHmrbzmPpxNvXjbMOtHx3uXvXTa01N9rN2bNbl7p0t2bn6oX5k2A+9jBfJhJJdJBOtTPaVLdx3X+rH2dSPs503/WjZe3YRaS69jBfJhJJdJBMtSXYzW25mvzOzrWZ2Tyv6UPRjh5ltMrNuM+tq4n4fMLP9Zra5T9vFZvakmW0pvk9oUT/uNbPdxTHpNrObm9CP6Wb2tJn91sxeNrO/LdqbekwS/WjqMTGzMWb2vJm9VPTjH4v2mWa2rsibH5nZ4Bayc/emfgEjqMxhdzkwCngJmNvsfhR92QFMasF+PwosBjb3afsX4J7i9j3AN1rUj3uBv2vy8ZgKLC5utwGvAXObfUwS/WjqMQEMGFfcHgmsAz4MPAx8rmj/T+CLg3ncVpzZlwBb3X27V+aZXwXc0oJ+tIy7rwXePKf5Fiqz9EKTZusN+tF07r7H3TcUt49QmQlpGk0+Jol+NJVX1H1G51Yk+zTgD31+buXMtA48YWbrzWxFi/pwxmR331Pc3gvEU9w03t1mtrF4md/wtxN9mdkMKpOlrKOFx+ScfkCTj0kjZnTO/QLdMndfDNwEfMnMPtrqDkHlPzuVf0StcB8wi8qCIHuAbzZrx2Y2DvgJ8GV3P2spnWYekyr9aPox8RpmdI60Itl3A9P7/BzOTNto7r67+L4feIzWTrO1z8ymAhTf97eiE+6+r3iinQa+S5OOiZmNpJJgP3T3R4vmph+Tav1o1TEp9n2IQc7oHGlFsr8AzCmuLI4CPgc83uxOmNmFZtZ25jZwI7A5vVVDPU5lll5o4Wy9Z5Kr8GmacEysMpPn/cAr7v6tPqGmHpOoH80+Jg2b0blZVxjPudp4M5UrnduAv29RHy6nUgl4CXi5mf0AHqLycvBdKu+97qSyQOYaYAvwFHBxi/rxA2ATsJFKsk1tQj+WUXmJvhHoLr5ubvYxSfSjqccEWEBlxuaNVP6x/EOf5+zzwFbgEWD0YB5XH5cVyUTuF+hEsqFkF8mEkl0kE0p2kUwo2UUyoWQXyYSSXSQT/weYte4bQI3fgAAAAABJRU5ErkJggg==\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# 然后我们看看经典奇异值的分解效果\n", "U, sigma, V = np.linalg.svd(imgmat)\n", "\n", "for i in range(5, 16, 5):\n", " reconstimg = np.matrix(U[:, :i]) * np.diag(sigma[:i]) * np.matrix(V[:i, :])\n", " plt.imshow(reconstimg, cmap='gray')\n", " title = \"n = %s\" % i\n", " plt.title(title)\n", " plt.show()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:50:18.966739Z", "start_time": "2021-03-09T03:50:18.950606Z" } }, "outputs": [], "source": [ "# 然后我们再来看看量子版本的分解效果：\n", "\n", "# 超参数设置\n", "N = 5 # 量子比特数量\n", "T = 8 # 设置想要学习的阶数\n", "ITR = 200 # 迭代次数\n", "LR = 0.02 # 学习速率\n", "SEED = 14 # 随机数种子\n", "\n", "# 设置等差的学习权重\n", "weight = np.arange(2 * T, 0, -2).astype('complex128')\n", "\n", "\n", "def Mat_generator():\n", " imgmat = np.array(list(img.getdata(band=0)), float)\n", " imgmat.shape = (img.size[1], img.size[0])\n", " lenna = np.matrix(imgmat)\n", " return lenna.astype('complex128')\n", "\n", "M_err = Mat_generator()\n", "U, D, V_dagger = np.linalg.svd(Mat_generator(), full_matrices=True)\n", "\n", "# 设置电路参数\n", "cir_depth = 40 # 电路深度\n", "block_len = 1 # 每个模组的长度\n", "theta_size = N * block_len * cir_depth # 网络参数 theta 的大小" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:50:19.553293Z", "start_time": "2021-03-09T03:50:19.531180Z" } }, "outputs": [], "source": [ "# 定义量子神经网络\n", "def U_theta(theta):\n", "\n", " # 用 UAnsatz 初始化网络\n", " cir = UAnsatz(N)\n", " \n", " # 搭建层级结构：\n", " for layer_num in range(cir_depth):\n", " \n", " for which_qubit in range(N):\n", " cir.ry(theta[block_len * layer_num * N + which_qubit],\n", " which_qubit)\n", "\n", " for which_qubit in range(1, N):\n", " cir.cnot([which_qubit - 1, which_qubit])\n", "\n", " return cir.U" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:50:21.636509Z", "start_time": "2021-03-09T03:50:21.593577Z" } }, "outputs": [], "source": [ "class NET(paddle.nn.Layer):\n", " \n", " # 初始化可学习参数列表，并用 [0, 2*pi] 的均匀分布来填充初始值\n", " def __init__(self, shape, dtype='float64'):\n", " super(NET, self).__init__()\n", " \n", " # 创建用来学习 U 的参数 theta\n", " self.theta = self.create_parameter(shape=shape,\n", " default_initializer=paddle.nn.initializer.Uniform(low=0.0, high=2*PI),\n", " dtype=dtype, is_bias=False)\n", " \n", " # 创建用来学习 V_dagger 的参数 phi\n", " self.phi = self.create_parameter(shape=shape,\n", " default_initializer=paddle.nn.initializer.Uniform(low=0.0, high=2*PI),\n", " dtype=dtype, is_bias=False)\n", " \n", " # 将 Numpy array 转换成 Paddle 支持的 Tensor\n", " self.M = paddle.to_tensor(Mat_generator())\n", " self.weight = paddle.to_tensor(weight)\n", "\n", " # 定义损失函数和前向传播机制\n", " def forward(self):\n", " \n", " # 获取量子神经网络的酉矩阵表示\n", " U = U_theta(self.theta)\n", " U_dagger = dagger(U)\n", " \n", " \n", " V = U_theta(self.phi)\n", " V_dagger = dagger(V)\n", " \n", " # 初始化损失函数和奇异值存储器\n", " loss = 0 \n", " singular_values = np.zeros(T)\n", " \n", " # 定义损失函数\n", " for i in range(T):\n", " loss -= paddle.real(self.weight)[i] * paddle.real(matmul(U_dagger,matmul(self.M, V)))[i][i]\n", " singular_values[i] = paddle.real(matmul(U_dagger, matmul(self.M, V)))[i][i].numpy()\n", " \n", " # 函数返回两个矩阵 U 和 V_dagger 学习的奇异值以及损失函数 \n", " return U, V_dagger, loss, singular_values" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "ExecuteTime": { "end_time": "2021-03-09T03:59:58.649381Z", "start_time": "2021-03-09T03:54:30.126561Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "iter: 0 loss: 11569.3661\n", "iter: 10 loss: -84679.3080\n", "iter: 20 loss: -103418.2506\n", "iter: 30 loss: -120854.8849\n", "iter: 40 loss: -130997.4657\n", "iter: 50 loss: -140376.4574\n", "iter: 60 loss: -146738.7417\n", "iter: 70 loss: -149536.6623\n", "iter: 80 loss: -151092.6062\n", "iter: 90 loss: -152296.0673\n", "iter: 100 loss: -153290.4326\n", "iter: 110 loss: -154112.8803\n", "iter: 120 loss: -154812.4464\n", "iter: 130 loss: -155384.0762\n", "iter: 140 loss: -155814.4174\n", "iter: 150 loss: -156132.5890\n", "iter: 160 loss: -156377.1011\n", "iter: 170 loss: -156571.0730\n", "iter: 180 loss: -156728.6039\n", "iter: 190 loss: -156859.1820\n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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x/1ltskgOY5lt27ZtC20ss62zszO0rVy5srSd1aBjWYBMLmUZbNE8srFo5hiZe+YjW1fRGmHrNEW2HTXY3f2PAKJRvznmEYUQLUHfoBMiExTsQmSCgl2ITFCwC5EJCnYhMqHSgpPDw8Nh0UYmaZw6daq0nWWopUorrNhgZEvJaAJ4oUT22Ng3Ebdu3Vravn59/DWIzZs3h7apU6eGtmuvvTa0RRlsUSFKgM89g81VJG8yWYtl0bF+0doejUg6ZNJs0nFSY+4hhPhcomAXIhMU7EJkgoJdiExQsAuRCQp2ITKhUunN3UNZg0llUeYSk0iYHMMykJgEGMEkkqhYJgBMnz49tDFphUk8UYZg1A7wuVq2bFlou/LKK0PbggXlhYtYNt/Ro0dDG3uumfQZrSsmvzJ5LTUTjfkYXXO816le2YXIBAW7EJmgYBciExTsQmSCgl2ITKh0Nx6IdxjZbnyUPJG6+8l2s1k9s8jGdpgZzEeWJLNly5bQtnfv3tJ2VoNu8eLFoe22224LbTfeeGNoi+bqyJEjYR+mQLAjmRjt7e2l7WwNsLWYUqMQ4OsqUhpYYlD0uBo9/kkI8QVAwS5EJijYhcgEBbsQmaBgFyITFOxCZMKo0puZXQbgZ6gdyewA1rn7j8zsKQDfBXD+vKEn3P1Vdq3h4WGcPn261MZqnaVIEymJNUCanMcSSRhM/hkYGAht27dvD2379+8vbWdS3vXXXx/aWJ05liQTPWfsMbPkDnYoKJO8Umq1pfQZjZRkHSbXseuF49Rxn7MAfuDum82sHcCbZvZaYXvO3f9pzKMKISqnnrPe+gH0F7ePm9kuAF3NdkwIMb6M6TO7mS0FcCOATUXTo2a2zcxeMLM54+2cEGL8qDvYzWwGgF8DeMzdjwH4MYDlAFaj9sr/TNBvrZn1mFnP4cOHG/dYCJFEXcFuZpNQC/Sfu/tvAMDdB9z9nLsPA/gJgFvL+rr7OnfvdvfuOXP04i9Eqxg12K32zfrnAexy92dHtC8acbf7AMSZFkKIllPPbvxXATwEYLuZbSnangDwoJmtRk2O2wPge/UMyKStiEiCOHbsWNgnpUYXwGW0SO5IzZI6efJkaPvggw9C23vvvRfahoaGStsXLlwY9rnhhhtC28033xzaOjs7Q1t/f39pO8t6YxlbTGr6+OOPQ1v0fNLsMLJ2mB9M0mW2yBcWKynru57d+D8CKPOGaupCiIsLfYNOiExQsAuRCQp2ITJBwS5EJijYhciESgtOTpgwITzyiEkJKUUqmbSSmvUW+ciOJmIZVCyzbePGjaHtwIEDoS3KDmMS2urVq0Mbkw4jeQ2IJSqWvcYyH1mGY8r8szXAxmLrg2WppR43FZGS9aZXdiEyQcEuRCYo2IXIBAW7EJmgYBciExTsQmRCpdKbmWHKlCnljiRkjjGp49SpU9SPCCa7TJo0qbSdZa+xLC8mvTFZi2XErVixorSdZfPNmzcvtEWPeTRbJIcxCYrJYVGhUoBnvaWsNyblMSmSXTNlfadkyjH0yi5EJijYhcgEBbsQmaBgFyITFOxCZIKCXYhMqFR6A2LphckMY70WkF4YMKXYIJPrmGTEJLve3t7QxmS5qLBkW1tb2IeV+GaPjUleJ06cKG2PpDAAmDlzZmhj/dg6SFlvTNZiNiavpawrJm0q600IEaJgFyITFOxCZIKCXYhMULALkQmj7sab2VQAGwFMKe7/K3d/0syuAPASgEsBvAngIXePMwhq1wp3JdkuZ7RrzWqWMdjOKKsZF8F23FniBOvHkmTYrm9U4+3qq68O+7Cd+r1794Y2tlMf7Z6n1mljCUWsX5TUMmPGjLAPe84YqQlWkTLAkpeOHj1a2k5rKIaW/+c0gDXufgNqxzPfYWZfBvBDAM+5+5UADgN4pI5rCSFaxKjB7jXOi6aTih8HsAbAr4r2FwHc2wwHhRDjQ73ns08sTnAdBPAagD8DOOLu59+b9ALoaoqHQohxoa5gd/dz7r4awGIAtwJYWe8AZrbWzHrMrOfQoUNpXgohGmZMu/HufgTA7wDcBmC2mZ3fKVoMoC/os87du929mx0QIIRoLqMGu5nNM7PZxe1pAL4FYBdqQf+Xxd0eBrC+ST4KIcaBehJhFgF40cwmovbP4WV3/08z2wngJTP7BwBvAXi+ngEj2YtJXpHExiQolijAbEyWiyQSJl0xZs+eHdqWL18e2pgM1dnZWdrOkiqYZBQd1wXwRJ7ommwsJnmlHp8UPWepteTY2mE+MiJf2PyydRoxarC7+zYAN5a0v4/a53chxOcAfYNOiExQsAuRCQp2ITJBwS5EJijYhcgES5ULkgYzOwjgfBrVXAAfVTZ4jPy4EPlxIZ83P5a4e+l5XpUG+wUDm/W4e3dLBpcf8iNDP/Q2XohMULALkQmtDPZ1LRx7JPLjQuTHhXxh/GjZZ3YhRLXobbwQmdCSYDezO8zsXTPbbWaPt8KHwo89ZrbdzLaYWU+F475gZoNmtmNEW4eZvWZm7xW/4zOZmuvHU2bWV8zJFjO7qwI/LjOz35nZTjN728z+umivdE6IH5XOiZlNNbM3zGxr4cffF+1XmNmmIm5+aWZxRcoy3L3SHwATUStrtQzAZABbAayq2o/Clz0A5rZg3K8DuAnAjhFt/wjg8eL24wB+2CI/ngLwNxXPxyIANxW32wH8L4BVVc8J8aPSOQFgAGYUtycB2ATgywBeBvBA0f4vAP5qLNdtxSv7rQB2u/v7Xis9/RKAe1rgR8tw940Ahj7TfA9qhTuBigp4Bn5Ujrv3u/vm4vZx1IqjdKHiOSF+VIrXGPcir60I9i4AH474u5XFKh3Ab83sTTNb2yIfzrPA3c8fz3oAwIIW+vKomW0r3uY3/ePESMxsKWr1EzahhXPyGT+AiuekGUVec9+g+5q73wTgTgDfN7Ovt9ohoPafHbV/RK3gxwCWo3ZGQD+AZ6oa2MxmAPg1gMfc/dhIW5VzUuJH5XPiDRR5jWhFsPcBuGzE32Gxymbj7n3F70EAr6C1lXcGzGwRABS/B1vhhLsPFAttGMBPUNGcmNkk1ALs5+7+m6K58jkp86NVc1KMfQRjLPIa0Ypg/xOAq4qdxckAHgCwoWonzKzNzNrP3wbwbQA7eK+msgG1wp1ACwt4ng+ugvtQwZxYrYjc8wB2ufuzI0yVzknkR9Vz0rQir1XtMH5mt/Eu1HY6/wzgb1vkwzLUlICtAN6u0g8Av0Dt7eAZ1D57PYLamXmvA3gPwP8A6GiRH/8GYDuAbagF26IK/Pgaam/RtwHYUvzcVfWcED8qnRMA16NWxHUbav9Y/m7Emn0DwG4A/wFgyliuq2/QCZEJuW/QCZENCnYhMkHBLkQmKNiFyAQFuxCZoGAXIhMU7EJkgoJdiEz4P3ftN5tVvPS1AAAAAElFTkSuQmCC\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# 记录优化中间过程\n", "loss_list, singular_value_list = [], []\n", "U_learned, V_dagger_learned = [], []\n", " \n", "net = NET([theta_size])\n", "\n", "# 一般来说，我们利用 Adam 优化器来获得相对好的收敛\n", "# 当然你可以改成 SGD 或者是 RMS prop.\n", "opt = paddle.optimizer.Adam(learning_rate=LR, parameters=net.parameters())\n", "\n", "# 优化循环\n", "for itr in range(ITR):\n", "\n", " # 前向传播计算损失函数\n", " U, V_dagger, loss, singular_values = net()\n", "\n", " # 反向传播极小化损失函数\n", " loss.backward()\n", " opt.minimize(loss)\n", " opt.clear_grad()\n", "\n", " # 记录优化中间结果\n", " loss_list.append(loss[0][0].numpy())\n", " singular_value_list.append(singular_values)\n", " \n", " if itr% 10 == 0:\n", " print('iter:', itr,'loss:','%.4f'% loss.numpy()[0])\n", "\n", "# 记录最后学出的两个酉矩阵 \n", "U_learned = U.numpy()\n", "V_dagger_learned = V_dagger.numpy()\n", "\n", "singular_value = singular_value_list[-1]\n", "mat = np.matrix(U_learned.real[:, :T]) * np.diag(singular_value[:T])* np.matrix(V_dagger_learned.real[:T, :])\n", "\n", "reconstimg = mat\n", "plt.imshow(reconstimg, cmap='gray')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "_______\n", "\n", "## 参考文献\n", "\n", "[1] Wang, X., Song, Z., & Wang, Y. Variational Quantum Singular Value Decomposition. [Quantum, 5, 483 (2021).](https://quantum-journal.org/papers/q-2021-06-29-483/)" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.10" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": true } }, "nbformat": 4, "nbformat_minor": 4 }