{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 吉布斯态的制备 （Gibbs State Preparation）\n", "\n", " Copyright (c) 2020 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 概览\n", "- 在这个案例中，我们将展示如何通过Paddle Quantum训练量子神经网络来制备量子吉布斯态。\n", "\n", "- 让我们通过下面几行代码引入必要的library和package。" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "pycharm": { "is_executing": false, "name": "#%%\n" } }, "outputs": [], "source": [ "import scipy\n", "\n", "from numpy import array, concatenate, zeros\n", "from numpy import pi as PI\n", "from numpy import trace as np_trace\n", "\n", "from paddle import fluid\n", "from paddle.complex import matmul, trace\n", "from paddle_quantum.circuit import UAnsatz\n", "from paddle_quantum.state import density_op\n", "from paddle_quantum.utils import state_fidelity, partial_trace, pauli_str_to_matrix" ] }, { "cell_type": "markdown", "metadata": { "pycharm": { "name": "#%% md\n" } }, "source": [ "## 背景\n", "\n", "量子计算中的前沿方向包括量子机器学习和量子优化，在这两个方向中，特定量子态的制备是非常重要的问题。特别的，吉布斯态（Gibbs state）的制备是实现诸多量子算法所必须的一个步骤并且广泛应用于：\n", "- 量子机器学习中受限波尔兹曼机的学习 [1]\n", "- 解决凸优化和半正定规划等优化问题 [2]\n", "- 组合优化问题 [3]\n", "\n", "具体的吉布斯态定义如下：给定一个 $n$ 量子位的哈密顿量 $H$（一般来说这是一个$2^n\\times2^n$的厄米矩阵），其在温度 $T$ 下的吉布斯态为 \n", "$$\n", "\\rho_G = \\frac{{{e^{ - \\beta H}}}}{{tr({e^{ - \\beta H}})}}\n", "$$\n", "\n", "其中 ${e^{ - \\beta H}}$ 是矩阵 $ - \\beta H$ 的矩阵指数，$\\beta = \\frac{1}{{kT}}$ 是系统的逆温度参数，其中 $T$ 是温度参数，$k$ 是玻尔兹曼常数 (这个例子中我们取 $k = 1$)。作为一个上手的例子，这里我们首先考虑一个3量子比特的哈密顿量及其吉布斯态。\n", "\n", "$$\n", "H = -Z \\otimes Z \\otimes I - I \\otimes Z \\otimes Z - Z \\otimes I \\otimes Z,\\,\n", "I=\\left [\n", "\\begin{matrix}\n", "1 & 0 \\\\\n", "0 & 1 \\\\\n", "\\end{matrix} \n", "\\right ],\n", "Z=\\left [\n", "\\begin{matrix}\n", "1 & 0 \\\\\n", "0 & -1 \\\\\n", "\\end{matrix} \n", "\\right ].$$\n", "\n", "\n", "这个例子中，我们将逆温度参数设置为 $\\beta = 1.5$。此外，为了方便测试结果，我们按照定义提前生成好了理想情况的吉布斯态。" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "pycharm": { "is_executing": false, "name": "#%% \n" } }, "outputs": [], "source": [ "N = 4 # 量子神经网络的宽度\n", "N_SYS_B = 3 # 用于生成吉布斯态的子系统B的量子比特数 \n", "SEED = 14 # 固定随机种子" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "pycharm": { "is_executing": false, "name": "#%%\n" } }, "outputs": [], "source": [ "beta = 1.5 # 设置逆温度参数 beta\n", "\n", "# 生成用泡利字符串表示的特定的哈密顿量\n", "H = [[-1.0, 'z0,z1'], [-1.0, 'z1,z2'], [-1.0, 'z0,z2']]\n", "\n", "# 生成哈密顿量的矩阵信息\n", "hamiltonian = pauli_str_to_matrix(H, N_SYS_B)\n", "\n", "# 生成理想情况下的目标吉布斯态 rho\n", "rho_G = scipy.linalg.expm(-1 * beta * hamiltonian) / np_trace(scipy.linalg.expm(-1 * beta * hamiltonian))\n", "\n", "# 设置成 Paddle quantum 所支持的数据类型\n", "hamiltonian = hamiltonian.astype(\"complex128\")\n", "rho_G = rho_G.astype(\"complex128\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 搭建量子神经网络\n", "\n", "- 在这个案例中，我们将通过训练量子神经网络QNN（也可以理解为参数化量子电路）来训练吉布斯态。这里，我们提供一个简单的4量子位的量子电路如下：\n", "![Ugibbs.jpg](https://release-data.cdn.bcebos.com/PIC%2FUgibbs.jpg)\n", "\n", "- 我们需要预设一些电路的参数，比如电路有4量子比特，其中第1个量子位是辅助系统，第2-4个量子位是用以产生吉布斯态的子系统。\n", "\n", "- 初始化其中的变量参数，${\\bf{\\theta }}$ 代表我们量子神经网络中的参数组成的向量。\n", " " ] }, { "cell_type": "markdown", "metadata": { "pycharm": { "name": "#%% md\n" } }, "source": [ "接下来我们根据上图中的电路设计，通过 Paddle Quantum 的 `UAnsatz` 函数和内置的 `real_entangled_layer(theta, D)` 电路模板来高效搭建量子神经网络。 " ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "pycharm": { "is_executing": false, "name": "#%%\n" } }, "outputs": [], "source": [ "def U_theta(initial_state, theta, N, D):\n", " \"\"\"\n", " Quantum Neural Network\n", " \"\"\"\n", " \n", " # 按照量子比特数量/网络宽度初始化量子神经网络\n", " cir = UAnsatz(N)\n", " \n", " # 内置的 {R_y + CNOT} 电路模板\n", " cir.real_entangled_layer(theta[:D], D)\n", " \n", " # 铺上最后一列 R_y 旋转门\n", " for i in range(N):\n", " cir.ry(theta=theta[D][i][0], which_qubit=i)\n", " \n", " # 量子神经网络作用在给定的初始态上\n", " final_state = cir.run_density_matrix(initial_state)\n", "\n", " return final_state" ] }, { "cell_type": "markdown", "metadata": { "pycharm": { "name": "#%% md\n" } }, "source": [ "## 配置训练模型 - 损失函数\n", "- 现在我们已经有了数据和量子神经网络的架构，我们将进一步定义合适的训练参数、模型和损失函数来达到我们的目标。\n", "- 具体的我们参考的是论文[4]中的方法，核心思想是利用吉布斯态达到了最小自由能的性质。\n", "- 通过作用量子神经网络 $U(\\theta)$ 在初始态上，我们可以得到输出态 $\\left| {\\psi \\left( {\\bf{\\theta }} \\right)} \\right\\rangle $, 其在第2-4个量子位的态记为 $\\rho_B(\\theta)$。\n", "- 设置训练模型中的的损失函数。在吉布斯态学习中，我们利用冯诺依曼熵函数的截断来进行自由能的估计，相应的损失函数参考[4]可以设为 \n", "$loss= {L_1} + {L_2} + {L_3}$，其中 ${L_1}= tr(H\\rho_B)$, ${L_2} = 2{\\beta^{-1}}{tr}(\\rho_B^2)$ , $L_3 = - {\\beta ^{ - 1}}\\big(tr(\\rho_B^3) + 3\\big)/2$。" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "pycharm": { "is_executing": false, "name": "#%%\n" } }, "outputs": [], "source": [ "class Net(fluid.dygraph.Layer):\n", " \"\"\"\n", " Construct the model net\n", " \"\"\"\n", "\n", " def __init__(self, shape, param_attr=fluid.initializer.Uniform(low=0.0, high=2*PI, seed=SEED),\n", " dtype='float64'):\n", " super(Net, self).__init__()\n", " \n", " # 初始化 theta 参数列表，并用 [0, 2*pi] 的均匀分布来填充初始值\n", " self.theta = self.create_parameter(shape=shape, attr=param_attr, dtype=dtype, is_bias=False)\n", " \n", " # 初始化 rho = |0..0><0..0| 的密度矩阵\n", " self.initial_state = fluid.dygraph.to_variable(density_op(N))\n", "\n", " # 定义损失函数和前向传播机制\n", " def forward(self, H, N, N_SYS_B, D):\n", "\n", " # 施加量子神经网络\n", " rho_AB = U_theta(self.initial_state, self.theta, N, D)\n", "\n", " # 计算偏迹 partial trace 来获得子系统B所处的量子态 rho_B\n", " rho_B = partial_trace(rho_AB, 2 ** (N - N_SYS_B), 2 ** (N_SYS_B), 1)\n", " \n", " # 计算三个子损失函数\n", " rho_B_squre = matmul(rho_B, rho_B)\n", " loss1 = (trace(matmul(rho_B, H))).real\n", " loss2 = (trace(rho_B_squre)).real * 2 / beta\n", " loss3 = - ((trace(matmul(rho_B_squre, rho_B))).real + 3) / (2 * beta)\n", " \n", " # 最终的损失函数\n", " loss = loss1 + loss2 + loss3 \n", "\n", " return loss, rho_B" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 配置训练模型 - 模型参数\n", "\n", "在进行量子神经网络的训练之前，我们还需要进行一些训练的超参数设置，主要是学习速率 (LR, learning rate)、迭代次数(ITR, iteration)和量子神经网络计算模块的深度 (D, Depth)。这里我们设定学习速率为0.5, 迭代次数为50次。读者不妨自行调整来直观感受下超参数调整对训练效果的影响。" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "pycharm": { "is_executing": false, "name": "#%%\n" } }, "outputs": [], "source": [ "ITR = 50 # 设置训练的总迭代次数\n", "LR = 0.5 # 设置学习速率\n", "D = 1 # 设置量子神经网络中重复计算模块的深度 Depth" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 进行训练\n", "\n", "- 当训练模型的各项参数都设置完成后，我们将数据转化为Paddle动态图中的变量，进而进行量子神经网络的训练。\n", "- 训练过程中我们用的是[Adam Optimizer](https://www.paddlepaddle.org.cn/documentation/docs/zh/api_cn/optimizer_cn/AdagradOptimizer_cn.html)，也可以调用Paddle中提供的其他优化器。\n", "- 我们将训练过程中的结果依次输出。\n", "- 特别的我们依次输出了我们学习到的量子态$\\rho_B(\\theta)$与吉布斯态$\\rho_G$的保真度，保真度越高说明QNN输出的态越接近于吉布斯态。\n" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "pycharm": { "is_executing": false, "name": "#%% \n" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "iter: 5 loss: -2.5615 fid: 0.8631\n", "iter: 10 loss: -3.1189 fid: 0.9504\n", "iter: 15 loss: -3.3290 fid: 0.9732\n", "iter: 20 loss: -3.3502 fid: 0.9846\n", "iter: 25 loss: -3.3446 fid: 0.9868\n", "iter: 30 loss: -3.3630 fid: 0.9873\n", "iter: 35 loss: -3.3937 fid: 0.9916\n", "iter: 40 loss: -3.4087 fid: 0.9948\n", "iter: 45 loss: -3.4108 fid: 0.9954\n", "iter: 50 loss: -3.4110 fid: 0.9953\n" ] } ], "source": [ "# 初始化paddle动态图机制\n", "with fluid.dygraph.guard():\n", " \n", " # 我们需要将 Numpy array 转换成 Paddle 动态图模式中支持的 variable\n", " H = fluid.dygraph.to_variable(hamiltonian)\n", "\n", " # 确定网络的参数维度\n", " net = Net(shape=[D + 1, N, 1])\n", "\n", " # 一般来说，我们利用Adam优化器来获得相对好的收敛，当然你可以改成SGD或者是RMS prop.\n", " opt = fluid.optimizer.AdamOptimizer(learning_rate=LR, parameter_list=net.parameters())\n", "\n", " # 优化循环\n", " for itr in range(1, ITR + 1):\n", " \n", " # 前向传播计算损失函数并返回生成的量子态 rho_B\n", " loss, rho_B = net(H, N, N_SYS_B, D)\n", " \n", " # 在动态图机制下，反向传播极小化损失函数\n", " loss.backward()\n", " opt.minimize(loss)\n", " net.clear_gradients()\n", "\n", " # 转换成 Numpy array 用以计算量子态的保真度 F(rho_B, rho_G)\n", " rho_B = rho_B.numpy()\n", " fid = state_fidelity(rho_B, rho_G)\n", " \n", " # 打印训练结果\n", " if itr % 5 == 0:\n", " print('iter:', itr, 'loss:', '%.4f' % loss.numpy(), 'fid:', '%.4f' % fid)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 总结\n", "根据上面训练得到的结果，通过大概50次迭代，我们就能达到高于99.5%保真度的高精度吉布斯态，高效并精确地完成了吉布斯态的制备。我们可以通过print函数来输出学习到的量子神经网络的参数和它的输出态。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 参考文献\n", "\n", "[1] [Kieferová, M. & Wiebe, N. Tomography and generative training with quantum Boltzmann machines. Phys. Rev. A 96, 062327 (2017).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.96.062327)\n", "\n", "[2] [Brandao, F. G. S. L. & Svore, K. M. Quantum Speed-Ups for Solving Semidefinite Programs. in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) 415–426 (IEEE, 2017). ](https://ieeexplore.ieee.org/abstract/document/8104077)\n", "\n", "[3] [Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum Simulations of Classical Annealing Processes. Phys. Rev. Lett. 101, 130504 (2008).](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.130504)\n", "\n", "[4] [Wang, Y., Li, G. & Wang, X. Variational quantum Gibbs state preparation with a truncated Taylor series. arXiv:2005.08797 (2020).](https://arxiv.org/pdf/2005.08797.pdf)" ] } ], "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": "3.7.7" } }, "nbformat": 4, "nbformat_minor": 1 }