From 4088bcbf34afa2ac323afcfc66df7e7495de8ac9 Mon Sep 17 00:00:00 2001 From: Quleaf Date: Sat, 9 Jan 2021 22:29:35 +0800 Subject: [PATCH] update to v1.1.1 --- README.md | 4 +- README_CN.md | 3 - docs/Makefile | 20 - docs/make.bat | 35 - docs/requirements.txt | 10 - docs/source/_static/logo.png | Bin 12406 -> 0 bytes docs/source/conf.py | 76 - docs/source/index.rst | 20 - docs/source/introduction.rst | 121 - docs/source/modules.rst | 7 - docs/source/paddle_quantum.circuit.rst | 7 - docs/source/paddle_quantum.rst | 17 - docs/source/paddle_quantum.state.rst | 7 - docs/source/paddle_quantum.utils.rst | 7 - docs/source/tutorial.rst | 41 - .../PaddleQuantum_GPU_CN.ipynb | 204 +- introduction/PaddleQuantum_GPU_EN.ipynb | 413 ++ introduction/PaddleQuantum_Tutorial_CN.ipynb | 611 ++- introduction/PaddleQuantum_Tutorial_CN.pdf | Bin 1947384 -> 0 bytes introduction/PaddleQuantum_Tutorial_EN.ipynb | 1524 ++++++ introduction/figures/gate.png | Bin 61364 -> 0 bytes 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English | [简体中文](README_CN.md) -# Paddle Quantum (量桨) +# Paddle Quantum (量桨) - [Features](#features) - [Install](#install) @@ -63,7 +63,7 @@ pip install -e . python -m pip install --upgrade -r requirements.txt ``` -### Use OpenFermion to read xyz molecule configuration file +### Use OpenFermion to read .xyz molecule configuration file > Only macOS and linux users can use OpenFermion to read .xyz description files. diff --git a/README_CN.md b/README_CN.md index 28aaba9..a42a094 100644 --- a/README_CN.md +++ b/README_CN.md @@ -168,9 +168,6 @@ Paddle Quantum 使用 setuptools 的 develop 模式进行安装,相关代码 答:量桨是基于百度飞桨开发的量子机器学习工具集。飞桨作为国内首个开源开放的产业级深度学习平台,技术领先且功能完备。拥有飞桨的技术支持,特别是其强大的动态图机制,量桨可以方便地进行机器学习的优化以及 GPU 的加速。同时,基于百度量子计算研究所研发的高性能量子模拟器,量桨在个人笔记本电脑上也能支持20多个量子比特的运算。另外,量桨还有丰富的[量子机器学习案例](./tutorial)供大家参考和学习。 -5. 问:**非常想试用量桨,该怎么入门呢?** - - 答:建议新用户首先阅读量桨的[入门手册](./introduction),它包含量桨详细的安装步骤以及入门教程。另外,量桨提供了丰富的[量子机器学习案例](./tutorial),以 Jupyter Notebook 和 PDF 的方式呈现,方便用户学习和实践。如在学习和使用过程中遇到任何问题,欢迎用户通过 [Github Issues](https://github.com/PaddlePaddle/Quantum/issues) 以及技术交流QQ群(1076223166)与我们交流。 ## Copyright and License diff --git a/docs/Makefile b/docs/Makefile deleted file mode 100644 index d0c3cbf..0000000 --- a/docs/Makefile +++ /dev/null @@ -1,20 +0,0 @@ -# Minimal makefile for Sphinx documentation -# - -# You can set these variables from the command line, and also -# from the environment for the first two. -SPHINXOPTS ?= -SPHINXBUILD ?= sphinx-build -SOURCEDIR = source -BUILDDIR = build - -# Put it first so that "make" without argument is like "make help". -help: - @$(SPHINXBUILD) -M help "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O) - -.PHONY: help Makefile - -# Catch-all target: route all unknown targets to Sphinx using the new -# "make mode" option. $(O) is meant as a shortcut for $(SPHINXOPTS). -%: Makefile - @$(SPHINXBUILD) -M $@ "$(SOURCEDIR)" "$(BUILDDIR)" $(SPHINXOPTS) $(O) diff --git a/docs/make.bat b/docs/make.bat deleted file mode 100644 index 9534b01..0000000 --- a/docs/make.bat +++ /dev/null @@ -1,35 +0,0 @@ -@ECHO OFF - -pushd %~dp0 - -REM Command file for Sphinx documentation - -if "%SPHINXBUILD%" == "" ( - set SPHINXBUILD=sphinx-build -) -set SOURCEDIR=source -set BUILDDIR=build - -if "%1" == "" goto help - -%SPHINXBUILD% >NUL 2>NUL -if errorlevel 9009 ( - echo. - echo.The 'sphinx-build' command was not found. Make sure you have Sphinx - echo.installed, then set the SPHINXBUILD environment variable to point - echo.to the full path of the 'sphinx-build' executable. 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For a full -# list see the documentation: -# https://www.sphinx-doc.org/en/master/usage/configuration.html - -# -- Path setup -------------------------------------------------------------- - -# If extensions (or modules to document with autodoc) are in another directory, -# add these directories to sys.path here. If the directory is relative to the -# documentation root, use os.path.abspath to make it absolute, like shown here. -# -import os -import sys -sys.path.insert(0, os.path.abspath('../..')) - - -# -- Project information ----------------------------------------------------- - -project = 'Paddle Quantum' -copyright = u'2020, Institute for Quantum Computing, Baidu Inc.' -author = u'Institute for Quantum Computing, Baidu Inc.' - -# The full version, including alpha/beta/rc tags -release = '1.1.0' - - -# -- General configuration --------------------------------------------------- - -# Add any Sphinx extension module names here, as strings. They can be -# extensions coming with Sphinx (named 'sphinx.ext.*') or your custom -# ones. -extensions = [ - 'sphinx.ext.autodoc', - 'sphinx.ext.napoleon', - 'sphinx.ext.mathjax', - 'sphinx_search.extension', -] - -# Add any paths that contain templates here, relative to this directory. -templates_path = ['_templates'] - -# The language for content autogenerated by Sphinx. Refer to documentation -# for a list of supported languages. -# -# This is also used if you do content translation via gettext catalogs. -# Usually you set "language" from the command line for these cases. -language = 'zh_CN' - -html_search_language = 'zh' - -# List of patterns, relative to source directory, that match files and -# directories to ignore when looking for source files. -# This pattern also affects html_static_path and html_extra_path. -exclude_patterns = [] - - -# -- Options for HTML output ------------------------------------------------- - -# The theme to use for HTML and HTML Help pages. See the documentation for -# a list of builtin themes. -# -html_theme = 'sphinx_rtd_theme' - -# Add any paths that contain custom static files (such as style sheets) here, -# relative to this directory. They are copied after the builtin static files, -# so a file named "default.css" will overwrite the builtin "default.css". -html_static_path = ['_static'] - -html_logo = '_static/logo.png' - -master_doc = 'index' - -# Autodoc -autodoc_member_order = 'bysource' diff --git a/docs/source/index.rst b/docs/source/index.rst deleted file mode 100644 index 1a39f6f..0000000 --- a/docs/source/index.rst +++ /dev/null @@ -1,20 +0,0 @@ -.. Paddle Quantum documentation master file, created by - sphinx-quickstart on Fri Aug 21 10:54:43 2020. - You can adapt this file completely to your liking, but it should at least - contain the root `toctree` directive. - -Welcome to Paddle Quantum's documentation! -========================================== - -.. toctree:: - :maxdepth: 2 - :caption: Paddle Quantum 入门 - - introduction - tutorial - -.. toctree:: - :maxdepth: 4 - :caption: API 文档 - - modules diff --git a/docs/source/introduction.rst b/docs/source/introduction.rst deleted file mode 100644 index 1b95441..0000000 --- a/docs/source/introduction.rst +++ /dev/null @@ -1,121 +0,0 @@ -.. _header-n0: - -Paddle Quantum (量桨) -======================= - -`Paddle Quantum(量桨) `__\ 是基于百度飞桨开发的量子机器学习工具集,支持量子神经网络的搭建与训练,提供易用的量子机器学习开发套件与量子优化、量子化学等前沿量子应用工具集,使得百度飞桨也因此成为国内首个目前也是唯一一个支持量子机器学习的深度学习框架。 - -.. figure:: https://release-data.cdn.bcebos.com/Paddle%20Quantum.png - :target: https://github.com/PaddlePaddle/Quantum - -量桨建立起了人工智能与量子计算的桥梁,不但可以快速实现量子神经网络的搭建与训练,还提供易用的量子机器学习开发套件与量子优化、量子化学等前沿量子应用工具集,并提供多项自研量子机器学习应用。通过百度飞桨深度学习平台赋能量子计算,量桨为领域内的科研人员以及开发者便捷地开发量子人工智能的应用提供了强有力的支撑,同时也为广大量子计算爱好者提供了一条可行的学习途径。 - - 关于量桨的更多内容可以查看 GitHub 页面:https://github.com/PaddlePaddle/Quantum - -.. _header-n6: - -特色 ----- - -- 易用性 - - - 高效搭建量子神经网络 - - 多种量子神经网络模板 - - 丰富的量子算法教程(10+用例) - -- 可拓展性 - - - 支持通用量子电路模型 - - 高性能模拟器支持20多个量子比特的模拟运算 - - 提供多种优化工具和 GPU 加速 - -- 特色工具集 - - - 提供组合优化和量子化学等前沿领域的计算工具箱 - - 自研多种量子机器学习算法 - -.. _header-n15: - -安装步骤 --------- - -.. _header-n16: - -安装 PaddlePaddle -~~~~~~~~~~~~~~~~~ - -请参考 -`PaddlePaddle `__ -安装配置页面。此项目需求 PaddlePaddle 1.8.3 或更高版本。 - -.. _header-n19: - -下载 Paddle Quantum 并安装 -~~~~~~~~~~~~~~~~~~~~~~~~~~ - -.. code:: shell - - git clone http://github.com/PaddlePaddle/quantum - -.. code:: shell - - cd quantum - pip install -e . - -.. _header-n23: - -或使用 requirements.txt 安装依赖包 -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -.. code:: shell - - python -m pip install --upgrade -r requirements.txt - -.. _header-n25: - -使用 openfermion 读取 xyz 描述文件 -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -.. note:: 仅在 macOS 和 linux 下可以使用 openfermion 读取 xyz 描述文件。 - -VQE中调用 openfermion 读取分子 xyz 文件并计算,因此需要安装 openfermion 和 -openfermionpyscf。 - -.. code:: shell - - pip install openfermion - pip install openfermionpyscf - -.. _header-n29: - -运行 -~~~~ - -现在,可以试着运行一段程序来验证量桨是否已安装成功。这里我们运行量桨提供的量子近似优化算法 -(QAOA) 的例子。 - -.. code:: shell - - cd paddle_quantum/QAOA/example - python main.py - -.. - -.. note:: 关于 QAOA 的介绍可以参考我们的 `QAOA 教程 `__。 - -.. _header-n51: - -交流与反馈 ----------- - -- 我们非常欢迎您通过 `Github - Issues `__ - 来提交问题、报告与建议。 -- 技术交流QQ群:1076223166 - -.. _header-n118: - -Copyright and License ---------------------- - -Paddle Quantum 使用 `Apache-2.0 license `__ 许可证。 diff --git a/docs/source/modules.rst b/docs/source/modules.rst deleted file mode 100644 index a5fd70f..0000000 --- a/docs/source/modules.rst +++ /dev/null @@ -1,7 +0,0 @@ -paddle_quantum -============== - -.. toctree:: - :maxdepth: 4 - - paddle_quantum diff --git a/docs/source/paddle_quantum.circuit.rst b/docs/source/paddle_quantum.circuit.rst deleted file mode 100644 index f370a52..0000000 --- a/docs/source/paddle_quantum.circuit.rst +++ /dev/null @@ -1,7 +0,0 @@ -paddle\_quantum.circuit module -============================== - -.. automodule:: paddle_quantum.circuit - :members: - :undoc-members: - :show-inheritance: diff --git a/docs/source/paddle_quantum.rst b/docs/source/paddle_quantum.rst deleted file mode 100644 index be55387..0000000 --- a/docs/source/paddle_quantum.rst +++ /dev/null @@ -1,17 +0,0 @@ -paddle\_quantum package -======================= - -.. automodule:: paddle_quantum - :members: - :undoc-members: - :show-inheritance: - -Submodules ----------- - -.. toctree:: - :maxdepth: 4 - - paddle_quantum.circuit - paddle_quantum.state - paddle_quantum.utils diff --git a/docs/source/paddle_quantum.state.rst b/docs/source/paddle_quantum.state.rst deleted file mode 100644 index ecb7d63..0000000 --- a/docs/source/paddle_quantum.state.rst +++ /dev/null @@ -1,7 +0,0 @@ -paddle\_quantum.state module -============================ - -.. automodule:: paddle_quantum.state - :members: - :undoc-members: - :show-inheritance: diff --git a/docs/source/paddle_quantum.utils.rst b/docs/source/paddle_quantum.utils.rst deleted file mode 100644 index 7d7fa4a..0000000 --- a/docs/source/paddle_quantum.utils.rst +++ /dev/null @@ -1,7 +0,0 @@ -paddle\_quantum.utils module -============================ - -.. automodule:: paddle_quantum.utils - :members: - :undoc-members: - :show-inheritance: diff --git a/docs/source/tutorial.rst b/docs/source/tutorial.rst deleted file mode 100644 index ea593d6..0000000 --- a/docs/source/tutorial.rst +++ /dev/null @@ -1,41 +0,0 @@ -教程与案例 -======================= - -我们准备了入门教程和入门案例,来帮助用户快速学会如何使用量桨(Paddle Quantum)。 - -.. _header-n33: - -入门教程 --------- - -我们提供了一份 `Paddle Quantum 入门手册 `__\ 来方便用户快速上手 -Paddle Quantum。目前支持 PDF 阅读和运行 Jupyter Notebook -两种方式。内容上,该手册包括以下几个方面: - -- Paddle Quantum 的详细安装教程 -- 量子计算的基础知识介绍 -- Paddle Quantum 的使用介绍 -- PaddlePaddle 飞桨优化器使用教程 -- 具体的量子机器学习案例—VQE - -入门案例 --------- - -我们提供了涵盖量子优化、量子化学、量子机器学习等多个领域的案例供大家学习。与\ `入门手册 `__\ 类似,每个教程目前支持 -PDF 阅读和运行 Jupyter Notebook 两种方式。我们推荐用户下载 Notebook -后,本地运行进行实践。 - -1. `量子近似优化算法 (QAOA) `__ -2. `变分量子特征求解器 (VQE) `__ -3. `量子神经网络的贫瘠高原效应 (Barren Plateaus) `__ -4. `量子分类器 (Quantum Classifier) `__ -5. `量子变分自编码器 (Quantum Autoencoder) `__ -6. `量子生成对抗网络 (Quantum GAN) `__ -7. `子空间搜索 - 量子变分特征求解器 (SSVQE) `__ -8. `变分量子态对角化算法 (VQSD) `__ -9. `吉布斯态的制备 (Gibbs State Preparation) `__ -10. `变分量子奇异值分解 (VQSVD) `__ - -此外,Paddle Quantum 也支持在 GPU -上进行量子机器学习的训练,具体的方法请参考案例:`在 GPU 上使用 Paddle -Quantum `__。 diff --git a/tutorial/GPU/GPU_Tutorial_CN.ipynb b/introduction/PaddleQuantum_GPU_CN.ipynb similarity index 78% rename from tutorial/GPU/GPU_Tutorial_CN.ipynb rename to introduction/PaddleQuantum_GPU_CN.ipynb index 29d5b4b..a991c90 100644 --- a/tutorial/GPU/GPU_Tutorial_CN.ipynb +++ b/introduction/PaddleQuantum_GPU_CN.ipynb @@ -4,28 +4,34 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "# 在 GPU 上使用 Paddle Quantum\n", + "# 在 GPU 上使用量桨\n", "\n", - " Copyright (c) 2020 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " + "\n", + " Copyright (c) 2021 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## 1. 简介\n", + "## 简介\n", "\n", "> 注意,本篇教程具有时效性。同时不同电脑也会有个体差异性,本篇教程不保证所有电脑可以安装成功。\n", "\n", - "在深度学习中,大家通常会使用 GPU 来进行神经网络模型的训练,因为与 CPU 相比,GPU在浮点数运算方面有着显著的优势。因此,使用 GPU 来训练神经网络模型逐渐成为共同的选择。在 Paddle Quantum 中,我们的量子态和量子门也采用基于浮点数的复数表示,因此我们的模型如果能部署到 GPU 上进行训练,也会显著提升训练速度。\n", - "\n", - "## 2. GPU 选择\n", + "在深度学习中,大家通常会使用 GPU 来进行神经网络模型的训练,因为与 CPU 相比,GPU在浮点数运算方面有着显著的优势。因此,使用 GPU 来训练神经网络模型逐渐成为共同的选择。在 Paddle Quantum 中,我们的量子态和量子门也采用基于浮点数的复数表示,因此我们的模型如果能部署到 GPU 上进行训练,也会显著提升训练速度。" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## GPU 选择\n", "\n", - "在这里,我们使用 Nvidia 的硬件设备,其 CUDA(Compute Unified Device Architecture) 对深度学习的框架支持比较好。我们的 PaddlePaddle 也可以比较方便地安装在 CUDA 上。\n", + "在这里,我们选择 Nvidia 的硬件设备,其 CUDA(Compute Unified Device Architecture)对深度学习的框架支持比较好,PaddlePaddle 也可以比较方便地安装在 CUDA 上。\n", "\n", - "## 3. 配置 CUDA 环境\n", + "## 配置 CUDA 环境\n", "\n", - "### 3.1 安装 CUDA\n", + "### 安装 CUDA\n", "\n", "这里,我们介绍如何在 x64 平台上的 Windows10 系统中配置 CUDA 环境。首先,在[CUDA GPUs | NVIDIA Developer](https://developer.nvidia.com/cuda-gpus)上查看你的GPU是否可以安装CUDA环境。然后,在[NVIDIA 驱动程序下载](https://www.nvidia.cn/Download/index.aspx?lang=cn)下载你的显卡的最新版驱动,并安装到电脑上。\n", "\n", @@ -35,39 +41,38 @@ "\n", "安装完成之后,打开 Windows 命令行,输入`nvcc -V`,如果看到版本信息,则说明 CUDA 安装成功。\n", "\n", - "### 3.2 安装 cuDNN\n", + "### 安装 cuDNN\n", "\n", "在[NVIDIA cuDNN | NVIDIA Developer](https://developer.nvidia.com/cudnn)下载 cuDNN,根据[飞桨的安装步骤](https://www.paddlepaddle.org.cn/install/quick)中的要求,我们**需要使用 cuDNN 7.6+** ,因此我们下载支持 CUDA 10.0 的最新版 cuDNN 即可。下载完成 cuDNN 后进行解压缩。然后,假设我们的 CUDA 的安装路径为`C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0`,我们将 cuDNN 解压缩后里面的`bin`、`include`和`lib`中的文件都替换 CUDA 的安装路径下的对应文件(如果文件已存在则进行替换,如果未存在则直接粘贴到对应目录中)。到这里,cuDNN 也就安装完成了。\n", "\n", "\n", - "### 3.3 配置环境变量\n", + "### 配置环境变量\n", "\n", "接下来还需要配置环境变量。右键电脑桌面上的“此电脑”(或“文件资源管理器”左栏的“此电脑”),选择“属性”,然后选择左侧的“高级系统设置”,在“高级”这一栏下选择“环境变量”。\n", "\n", "现在就进入到了环境变量的设置页面,在系统变量中选择`Path`,点击“编辑”。在出现的页面中,查看是否有`C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0\\bin`和`C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0\\libnvvp`这两个地址(其前缀`C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0`应该为你的 CUDA 的安装位置),如果没有,请手动添加。\n", "\n", - "### 3.4 验证是否安装成功\n", + "### 验证是否安装成功\n", "\n", "打开命令行,输入`cd C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0\\extras\\demo_suite`进入到 CUDA 安装路径(这里也应该为你的 CUDA 的安装位置)。然后分别执行`.\\bandwidthTest.exe`和`.\\deviceQuery.exe`,如果都出现`Result = PASS`,则说明安装成功。\n", - "\n", - "\n", - "## 4. 在 CUDA 环境上安装 PaddlePaddle\n", - "\n", - "根据[飞桨的安装步骤](https://www.paddlepaddle.org.cn/install/quick)中的说明,我们首先需要确定自己的python环境,用`python --version`来查看 python 版本,保证**python版本是3.5.1+/3.6+/3.7+**,并且用`python -m ensurepip`和`python -m pip --version`来查看 pip 版本,**确认是 9.0.1+**。然后,使用`python -m pip install paddlepaddle-gpu==1.8.4.post107 -i https://pypi.tuna.tsinghua.edu.cn/simple`来安装 GPU 版本的 PaddlePaddle。\n", - "\n", - "\n", - "## 5. 安装 Paddle Quantum\n", - "\n", - "下载 Paddle Quantum 的安装包,修改`setup.py`,将其中的`paddlepaddle`改为`paddlepaddle-gpu`,然后按照 Paddle Quantum 的安装要求,执行`pip install -e .`即可。\n", - "\n", - "> 如果你是在一个新的 python 环境中安装了 paddlepaddle-gpu 和 paddle_quantum,请在新 python 环境中安装 jupyter,并在新的 jupyter 下重新打开本教程并运行。" + "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## 6. 检测是否安装成功\n", + "## 在 CUDA 环境上安装 PaddlePaddle\n", + "\n", + "根据[飞桨的安装步骤](https://www.paddlepaddle.org.cn/install/quick)中的说明,我们首先需要确定自己的python环境,用`python --version`来查看 python 版本,保证**python版本是3.5.1+/3.6+/3.7+**,并且用`python -m ensurepip`和`python -m pip --version`来查看 pip 版本,**确认是 9.0.1+**。然后,使用`python -m pip install paddlepaddle-gpu==1.8.4.post107 -i https://pypi.tuna.tsinghua.edu.cn/simple`来安装 GPU 版本的 PaddlePaddle。\n", + "\n", + "## 安装 Paddle Quantum\n", + "\n", + "下载 Paddle Quantum 的安装包,修改`setup.py`,将其中的`paddlepaddle`改为`paddlepaddle-gpu`,然后按照 Paddle Quantum 的安装要求,执行`pip install -e .`即可。\n", + "\n", + "> 如果你是在一个新的 python 环境中安装了 paddlepaddle-gpu 和 paddle_quantum,请在新 python 环境中安装 jupyter,并在新的 jupyter 下重新打开本教程并运行。\n", + "\n", + "## 检测是否安装成功\n", "\n", "打开我们 GPU 版本的 PaddlePaddle 环境,执行下面的命令,若输出为`True`则表示当前 PaddlePaddle 框架可以在GPU上运行。" ] @@ -76,7 +81,15 @@ "cell_type": "code", "execution_count": 1, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "True\n" + ] + } + ], "source": [ "import paddle \n", "from paddle import fluid\n", @@ -87,37 +100,53 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "## 7. 使用教程和示例\n", - "\n", - "在 Paddle Quantum 中,我们使用动态图机制来定义和训练我们的参数化量子线路。在这里,我们依然使用动态图机制,只需要定义动态图机制的运行设备即可。方式如下:\n", + "## 使用教程和示例\n", "\n", + "在 Paddle Quantum 中,我们使用动态图机制来定义和训练我们的参数化量子线路。在这里,我们依然使用动态图机制,只需要定义动态图机制的运行设备即可。方式如下:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "```python\n", "# 0 表示使用编号为0的GPU\n", "place = fluid.CUDAPlace(0)\n", "with fluid.dygraph.guard(place):\n", " # build and train your quantum circuit model\n", - "```\n", - "\n", - "当我们想在 CPU 上运行时,也采用类似的方式,定义运行设备为CPU:\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "当我们想在 CPU 上运行时,也采用类似的方式,定义运行设备为CPU:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ "```python\n", "place = fluid.CPUPlace()\n", "with fluid.dygraph.guard(place):\n", " # build and train your quantum circuit model\n", - "```\n", - "\n", - "我们可以在命令行中输入`nvidia-smi`来查看 GPU 的使用情况,包括有哪些程序在哪些 GPU 上运行,以及其显存占用情况。" + "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ + "我们可以在命令行中输入`nvidia-smi`来查看 GPU 的使用情况,包括有哪些程序在哪些 GPU 上运行,以及其显存占用情况。\n", + "\n", "这里,我们以 [VQE](https://github.com/PaddlePaddle/Quantum/blob/master/tutorial/VQE) 为例来说明我们该如何使用 GPU。首先,导入相关的包并定义相关的变量和函数。" ] }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 2, "metadata": {}, "outputs": [], "source": [ @@ -128,6 +157,7 @@ "from paddle import fluid\n", "from paddle.complex import matmul, transpose\n", "from paddle_quantum.circuit import UAnsatz\n", + "\n", "import matplotlib.pyplot as plt\n", "import numpy\n", "from paddle_quantum.VQE.chemistrysub import H2_generator\n", @@ -154,7 +184,7 @@ " # 量子神经网络作用在默认的初始态 |0000>上\n", " cir.run_state_vector()\n", "\n", - " # 计算给定哈密顿量的期望值\n", + " # 计算给定 Hamilton 量的期望值\n", " expectation_val = cir.expecval(Hamiltonian)\n", "\n", " return expectation_val\n", @@ -165,11 +195,13 @@ " Construct the model net\n", " \"\"\"\n", "\n", - " def __init__(self, shape, param_attr=fluid.initializer.Uniform(low=0.0, high=2 * PI), dtype=\"float64\"):\n", + " def __init__(self, shape, param_attr=fluid.initializer.Uniform(\n", + " low=0.0, high=2 * PI), dtype=\"float64\"):\n", " super(StateNet, 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", + " self.theta = self.create_parameter(\n", + " shape=shape, attr=param_attr, dtype=dtype, is_bias=False)\n", "\n", " # 定义损失函数和前向传播机制\n", " def forward(self, Hamiltonian, N, D):\n", @@ -192,18 +224,36 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 3, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "iter: 20 loss: -1.0599\n", + "iter: 20 Ground state energy: -1.0599 Ha\n", + "iter: 40 loss: -1.1229\n", + "iter: 40 Ground state energy: -1.1229 Ha\n", + "iter: 60 loss: -1.1347\n", + "iter: 60 Ground state energy: -1.1347 Ha\n", + "iter: 80 loss: -1.1359\n", + "iter: 80 Ground state energy: -1.1359 Ha\n" + ] + } + ], "source": [ "# 0 表示使用编号为0的GPU\n", "place_gpu = fluid.CUDAPlace(0)\n", - "with fluid.dygraph.guard(palce_gpu):\n", + "with fluid.dygraph.guard(place_gpu):\n", + " \n", " # 确定网络的参数维度\n", " net = StateNet(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", + " # 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 = fluid.optimizer.AdamOptimizer(\n", + " learning_rate=LR, parameter_list=net.parameters())\n", "\n", " # 记录优化结果\n", " summary_iter, summary_loss = [], []\n", @@ -226,11 +276,9 @@ " # 打印结果\n", " if itr % 20 == 0:\n", " print(\"iter:\", itr, \"loss:\", \"%.4f\" % loss.numpy())\n", - " print(\"iter:\", itr, \"Ground state energy:\", \"%.4f Ha\" % loss.numpy())\n", - "\n", - " # 储存训练结果到 output 文件夹\n", - " os.makedirs(\"output\", exist_ok=True)\n", - " savez(\"./output/summary_data\", iter=summary_iter, energy=summary_loss)" + " print(\"iter:\", itr, \"Ground state energy:\", \n", + " \"%.4f Ha\" % loss.numpy())\n", + "\n" ] }, { @@ -249,14 +297,14 @@ "name": "stdout", "output_type": "stream", "text": [ - "iter: 20 loss: -1.0669\n", - "iter: 20 Ground state energy: -1.0669 Ha\n", - "iter: 40 loss: -1.1129\n", - "iter: 40 Ground state energy: -1.1129 Ha\n", - "iter: 60 loss: -1.1163\n", - "iter: 60 Ground state energy: -1.1163 Ha\n", - "iter: 80 loss: -1.1172\n", - "iter: 80 Ground state energy: -1.1172 Ha\n" + "iter: 20 loss: -1.0865\n", + "iter: 20 Ground state energy: -1.0865 Ha\n", + "iter: 40 loss: -1.1153\n", + "iter: 40 Ground state energy: -1.1153 Ha\n", + "iter: 60 loss: -1.1338\n", + "iter: 60 Ground state energy: -1.1338 Ha\n", + "iter: 80 loss: -1.1359\n", + "iter: 80 Ground state energy: -1.1359 Ha\n" ] } ], @@ -264,11 +312,14 @@ "# 表示使用 CPU\n", "place_cpu = fluid.CPUPlace()\n", "with fluid.dygraph.guard(place_cpu):\n", + " \n", " # 确定网络的参数维度\n", " net = StateNet(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", + " # 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 = fluid.optimizer.AdamOptimizer(\n", + " learning_rate=LR, parameter_list=net.parameters())\n", "\n", " # 记录优化结果\n", " summary_iter, summary_loss = [], []\n", @@ -291,26 +342,20 @@ " # 打印结果\n", " if itr % 20 == 0:\n", " print(\"iter:\", itr, \"loss:\", \"%.4f\" % loss.numpy())\n", - " print(\"iter:\", itr, \"Ground state energy:\", \"%.4f Ha\" % loss.numpy())\n", - "\n", - " # 储存训练结果到 output 文件夹\n", - " # os.makedirs(\"output\", exist_ok=True)\n", - " # savez(\"./output/summary_data\", iter=summary_iter, energy=summary_loss)" + " print(\"iter:\", itr, \"Ground state energy:\", \n", + " \"%.4f Ha\" % loss.numpy())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "## 8. 总结\n", + "## 总结\n", + "\n", + "按照我们的测试,现在版本的 paddle_quantum 可以在 GPU 下运行,但是需要比较好的 GPU 资源才能体现出足够的加速效果。在未来的版本中,我们也会不断优化 paddle_quantum 在 GPU 下的性能表现,敬请期待。\n", + "\n", + "_______\n", "\n", - "实践证明,我们现在的 paddle_quantum 可以在 GPU 下运行,目前需要比较好的 GPU 资源才能体现出 GPU 的加速效果。在未来的版本中,我们也会不断优化 paddle_quantum 在 GPU 下的性能表现。" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ "## 参考资料\n", "\n", "[1] [Installation Guide Windows :: CUDA Toolkit Documentation](https://docs.nvidia.com/cuda/cuda-installation-guide-microsoft-windows/index.html)\n", @@ -337,7 +382,20 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.7.7" + "version": "3.7.9" + }, + "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": false } }, "nbformat": 4, diff --git a/introduction/PaddleQuantum_GPU_EN.ipynb b/introduction/PaddleQuantum_GPU_EN.ipynb new file mode 100644 index 0000000..026cbfa --- /dev/null +++ b/introduction/PaddleQuantum_GPU_EN.ipynb @@ -0,0 +1,413 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Use Paddle Quantum on GPU\n", + "\n", + " Copyright (c) 2021 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Introduction\n", + "\n", + "> Note that this tutorial is time-sensitive. And different computers will have individual differences. This tutorial does not guarantee that all computers can install it successfully.\n", + "\n", + "In deep learning, people usually use GPU for neural network model training because GPU has significant advantages in floating-point operations compared with CPU. Therefore, using GPU to train neural network models has gradually become a common choice. In Paddle Quantum, our quantum states and quantum gates are also represented by complex numbers based on floating-point numbers. If our model can be deployed on GPU for training, it will also significantly increase the training speed.\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## GPU selection\n", + "\n", + "Here, we choose Nvidia's hardware devices, and its CUDA (Compute Unified Device Architecture) supports deep learning framework better. PaddlePaddle can also be easily installed on CUDA.\n", + "\n", + "## Configure CUDA environment\n", + "\n", + "### Install CUDA\n", + "\n", + "Here, we introduce how to configure the CUDA environment in Windows 10 on the x64 platform. First, check on [CUDA GPUs | NVIDIA Developer](https://developer.nvidia.com/cuda-gpus) to see if your GPU support the CUDA environment. Then, download the latest version of your graphics card driver from [NVIDIA Driver Download](https://www.nvidia.cn/Download/index.aspx?lang=cn) and install it on your computer.\n", + "\n", + "In [PaddlePaddle Installation Steps](https://www.paddlepaddle.org.cn/install/quick), we found that **Paddle Paddle only supports CUDA 9.0/10.0 single card mode under Windows; it does not support CUDA 9.1/9.2/10.1**, so we need to install CUDA10.0 (CUDA 9.0 is also possible in theory). Find the download link of CUDA 10.0 in [CUDA Toolkit Archive | NVIDIA Developer](https://developer.nvidia.com/cuda-toolkit-archive): [CUDA Toolkit 10.0 Archive | NVIDIA Developer](https://developer.nvidia.com/cuda-10.0-download-archive). After downloading CUDA, run the installation.\n", + "\n", + "During the installation process, select **Custom Installation** in the CUDA options, check all the boxes except for Visual Studio Integration (unless you are familiar with it). Then check CUDA option only. Then select the default location for the installation location (please pay attention to the installation location of your CUDA, you need to set environment variables later), and wait for the installation to complete.\n", + "\n", + "After the installation is complete, open the Windows command line and enter `nvcc -V`. If you see the version information, the CUDA installation is successful.\n", + "\n", + "### Install cuDNN\n", + "\n", + "Download cuDNN in [NVIDIA cuDNN | NVIDIA Developer](https://developer.nvidia.com/cudnn), according to [PaddlePaddle Installation Steps](https://www.paddlepaddle.org.cn/install/quick) requirements, we **need to use cuDNN 7.6+**, so we can download the latest version of cuDNN that supports CUDA 10.0. After downloading cuDNN, unzip it. Assuming the installation path of our CUDA is `C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0`. After decompressing cuDNN, we take the files in `bin`, `include` and `lib` and replace the corresponding original files in the CUDA installation path (if the file already exists, replace it, if it does not exist, paste it directly into the corresponding directory). At this point, cuDNN has been installed.\n", + "\n", + "### Configure environment variables\n", + "\n", + "Next, you need to configure environment variables. Right-click \"This PC\" on the desktop of the computer (or \"This PC\" in the left column of \"File Explorer\"), select \"Properties\", and then select \"Advanced System Settings\" on the left, under the \"Advanced\" column Select \"Environmental Variables\".\n", + "\n", + "Now you enter the setting page of environment variables, select `Path` in the `System variables`, and click `Edit`. In the page that appears, check if there are two addresses `C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0\\bin` and `C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0\\libnvvp` (the prefix `C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0` should be your CUDA installation location), if not, please add them manually.\n", + "\n", + "### Verify that the installation is successful\n", + "\n", + "Open the command line and enter `cd C:\\Program Files\\NVIDIA GPU Computing Toolkit\\CUDA\\v10.0\\extras\\demo_suite` to enter the CUDA installation path (this should also be your CUDA installation location). Then execute `.\\bandwidthTest.exe` and `.\\deviceQuery.exe` respectively. If both `Result = PASS` appear, the installation is successful.\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Install PaddlePaddle on CUDA environment\n", + "\n", + "According to the instructions in [PaddlePaddle Installation Steps](https://www.paddlepaddle.org.cn/install/quick), we first need to make sure our python environment is correct and use `python --version` to check the python version. Ensure that the **python version is 3.5.1+/3.6+/3.7+**, and use `python -m ensurepip` and `python -m pip --version` to check the pip version, **confirm it is 9.0.1+**. Then, use `python -m pip install paddlepaddle-gpu==1.8.4.post107 -i https://pypi.tuna.tsinghua.edu.cn/simple` to install the GPU version of PaddlePaddle.\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Install Paddle Quantum\n", + "\n", + "Download the Paddle Quantum installation package, modify `setup.py`, change paddlepaddle` to `paddlepaddle-gpu`, and then execute `pip install -e .` according to the installation requirements of Paddle Quantum.\n", + "\n", + "> If you have installed paddlepaddle-gpu and paddle_quantum in a new python environment, please also install jupyter in the new python environment, and reopen this tutorial under the new jupyter notebook and run it." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Check if the installation is successful\n", + "\n", + "Open the new environment where we installed the GPU version of PaddlePaddle and execute the following command. If the output is `True`, it means that the current PaddlePaddle framework can run on the GPU.\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "True\n" + ] + } + ], + "source": [ + "import paddle \n", + "from paddle import fluid\n", + "print(fluid.is_compiled_with_cuda())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Use tutorials and examples\n", + "\n", + "In Paddle Quantum, we use the dynamic graph mode to define and train our parameterized quantum circuits. Here, we still use the dynamic graph mode and only need to define the GPU core where we run the dynamic graph mode." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```python\n", + "# 0 means to use GPU number 0\n", + "place = fluid.CUDAPlace(0)\n", + "with fluid.dygraph.guard(place):\n", + " # build and train your quantum circuit model\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If we want to run on CPU, pretty much the same, define the running device as CPU:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "```python\n", + "place = fluid.CPUPlace()\n", + "with fluid.dygraph.guard(place):\n", + " # build and train your quantum circuit model\n", + "```" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can enter `nvidia-smi` in the command line to view the usage of the GPU, including which programs are running on which GPUs, and its memory usage.\n", + "\n", + "Here, we take [VQE](https://github.com/PaddlePaddle/Quantum/blob/master/tutorial/VQE) as an example to illustrate how we should use GPU. First, import the related packages and define some variables and functions." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "import os\n", + "from numpy import concatenate\n", + "from numpy import pi as PI\n", + "from numpy import savez, zeros\n", + "from paddle import fluid\n", + "from paddle.complex import matmul, transpose\n", + "from paddle_quantum.circuit import UAnsatz\n", + "\n", + "import matplotlib.pyplot as plt\n", + "import numpy\n", + "from paddle_quantum.VQE.chemistrysub import H2_generator\n", + "from time import time\n", + "\n", + "Hamiltonian, N = H2_generator()\n", + "\n", + "\n", + "def U_theta(theta, Hamiltonian, N, D):\n", + " \"\"\"\n", + " Quantum Neural Network\n", + " \"\"\"\n", + " \n", + " # Initialize the quantum neural network according to the number of qubits/network width\n", + " cir = UAnsatz(N)\n", + " \n", + " # Built-in {R_y + CNOT} circuit template\n", + " cir.real_entangled_layer(theta[:D], D)\n", + "\n", + " # Add in the last row a layer of R_y rotation gates\n", + " for i in range(N):\n", + " cir.ry(theta=theta[D][i][0], which_qubit=i)\n", + "\n", + " # The quantum neural network acts on the default initial state |0000>\n", + " cir.run_state_vector()\n", + "\n", + " # Calculate the expected value of a given Hamiltonian\n", + " expectation_val = cir.expecval(Hamiltonian)\n", + "\n", + " return expectation_val\n", + "\n", + "\n", + "class StateNet(fluid.dygraph.Layer):\n", + " \"\"\"\n", + " Construct the model net\n", + " \"\"\"\n", + "\n", + " def __init__(self, shape, param_attr=fluid.initializer.Uniform(\n", + " low=0.0, high=2 * PI), dtype=\"float64\"):\n", + " super(StateNet, self).__init__()\n", + "\n", + " # Initialize the theta parameter list and fill the initial value with the uniform distribution of [0, 2*pi]\n", + " self.theta = self.create_parameter(\n", + " shape=shape, attr=param_attr, dtype=dtype, is_bias=False)\n", + "\n", + " # Define loss function and forward propagation mechanism\n", + " def forward(self, Hamiltonian, N, D):\n", + " # Calculate loss function/expected value\n", + " loss = U_theta(self.theta, Hamiltonian, N, D)\n", + "\n", + " return loss\n", + "\n", + "ITR = 80 # Set the total number of training iterations\n", + "LR = 0.2 # Set the learning rate\n", + "D = 2 # Set the depth of the repeated calculation module in the neural network" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If you want to use GPU to train, run the following program:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "iter: 20 loss: -0.8634\n", + "iter: 20 Ground state energy: -0.8634 Ha\n", + "iter: 40 loss: -1.1203\n", + "iter: 40 Ground state energy: -1.1203 Ha\n", + "iter: 60 loss: -1.1350\n", + "iter: 60 Ground state energy: -1.1350 Ha\n", + "iter: 80 loss: -1.1359\n", + "iter: 80 Ground state energy: -1.1359 Ha\n" + ] + } + ], + "source": [ + "# 0 means to use GPU number 0\n", + "place_gpu = fluid.CUDAPlace(0)\n", + "with fluid.dygraph.guard(place_gpu):\n", + " \n", + " # Determine the parameter dimension of the network\n", + " net = StateNet(shape=[D + 1, N, 1])\n", + "\n", + " # Generally speaking, we use Adam optimizer to get relatively good convergence\n", + " # Of course, you can change to SGD or RMS prop.\n", + " opt = fluid.optimizer.AdamOptimizer(\n", + " learning_rate=LR, parameter_list=net.parameters())\n", + "\n", + " # Record optimization results\n", + " summary_iter, summary_loss = [], []\n", + "\n", + " # Optimization cycle\n", + " for itr in range(1, ITR + 1):\n", + "\n", + " # Forward propagation to calculate loss function\n", + " loss = net(Hamiltonian, N, D)\n", + "\n", + " # Under the dynamic graph mechanism, back propagation minimizes the loss function\n", + " loss.backward()\n", + " opt.minimize(loss)\n", + " net.clear_gradients()\n", + "\n", + " # Update optimization results\n", + " summary_loss.append(loss.numpy())\n", + " summary_iter.append(itr)\n", + "\n", + " # Print results\n", + " if itr% 20 == 0:\n", + " print(\"iter:\", itr, \"loss:\", \"%.4f\"% loss.numpy())\n", + " print(\"iter:\", itr, \"Ground state energy:\",\n", + " \"%.4f Ha\"% loss.numpy())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "If you want to use CPU to train, run the following program:" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "iter: 20 loss: -0.7789\n", + "iter: 20 Ground state energy: -0.7789 Ha\n", + "iter: 40 loss: -1.0741\n", + "iter: 40 Ground state energy: -1.0741 Ha\n", + "iter: 60 loss: -1.1271\n", + "iter: 60 Ground state energy: -1.1271 Ha\n", + "iter: 80 loss: -1.1351\n", + "iter: 80 Ground state energy: -1.1351 Ha\n" + ] + } + ], + "source": [ + "# Use CPU\n", + "place_cpu = fluid.CPUPlace()\n", + "with fluid.dygraph.guard(place_cpu):\n", + " \n", + " # Determine the parameter dimension of the network\n", + " net = StateNet(shape=[D + 1, N, 1])\n", + "\n", + " # Generally speaking, we use Adam optimizer to get relatively good convergence\n", + " # Of course you can change to SGD or RMS prop.\n", + " opt = fluid.optimizer.AdamOptimizer(\n", + " learning_rate=LR, parameter_list=net.parameters())\n", + "\n", + " # Record optimization results\n", + " summary_iter, summary_loss = [], []\n", + "\n", + " # Optimization cycle\n", + " for itr in range(1, ITR + 1):\n", + "\n", + " # Forward propagation to calculate loss function\n", + " loss = net(Hamiltonian, N, D)\n", + "\n", + " # Under the dynamic graph mechanism, back propagation minimizes the loss function\n", + " loss.backward()\n", + " opt.minimize(loss)\n", + " net.clear_gradients()\n", + "\n", + " # Update optimization results\n", + " summary_loss.append(loss.numpy())\n", + " summary_iter.append(itr)\n", + "\n", + " # Print results\n", + " if itr% 20 == 0:\n", + " print(\"iter:\", itr, \"loss:\", \"%.4f\"% loss.numpy())\n", + " print(\"iter:\", itr, \"Ground state energy:\",\n", + " \"%.4f Ha\"% loss.numpy())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Summary\n", + "\n", + "According to our test, the current version of paddle_quantum can run under GPU, but it needs better GPU resources to show sufficient acceleration. In future versions, we will continue to optimize the performance of Paddle Quantum under GPU. \n", + "\n", + "_______\n", + "\n", + "## Reference\n", + "\n", + "[1] [Installation Guide Windows :: CUDA Toolkit Documentation](https://docs.nvidia.com/cuda/cuda-installation-guide-microsoft-windows/index.html)\n", + "\n", + "[2] [Installation Guide :: NVIDIA Deep Learning cuDNN Documentation](https://docs.nvidia.com/deeplearning/cudnn/install-guide/index.html#installwindows)\n", + "\n", + "[3] [Getting Started PaddlePaddle](https://www.paddlepaddle.org.cn/install/quick)\n", + "\n" + ] + } + ], + "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": "3.7.9" + }, + "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": false + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/introduction/PaddleQuantum_Tutorial_CN.ipynb b/introduction/PaddleQuantum_Tutorial_CN.ipynb index dba1df2..6c4dc86 100644 --- a/introduction/PaddleQuantum_Tutorial_CN.ipynb +++ b/introduction/PaddleQuantum_Tutorial_CN.ipynb @@ -6,7 +6,7 @@ "source": [ "# Paddle Quantum 入门手册\n", "\n", - " Copyright (c) 2020 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " + " Copyright (c) 2021 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " ] }, { @@ -22,8 +22,17 @@ "source": [ "这是一份简洁、实用的关于量子机器学习(Quantum Machine Learnig,QML)的介绍,面向读者包括但不限于物理、数学和计算机背景。本手册主要采用 Jupyter Notebook 的交互形式 (调用 Numpy, Matplotlib等 Python包以及飞桨Paddlepaddle深度学习框架来实现基于线性代数的量子运算和机器学习优化问题)。我们不仅提供了关于量子计算的一些基础教程同时还能手把手带你完成属于你自己的第一份量子机器学习算法。这并不是一份关于量子计算的百科全书,但我们涉及的案例经常出现在教科书中以及文献中。如果你想深入挖掘一些相关的基础知识,我们也提供了一些外部链接方便用户自己学习。\n", "\n", + "量子计算是由量子力学与计算理论交叉形成的新型学科,本质上是通过量子力学的基本规律去操控信息单元量子比特(quantum bit, qubit)的新型计算模式。与经典计算模型相比,在许多特定的信息处理任务上量子计算被普遍认为具有更强大的信息处理优势。关于量子计算的介绍与入门知识可以参考 [1-2],想要系统性地学习量子计算学科的读者请参阅 Nielsen & Chuang 编写的经典教材 [3]。近期,量子计算领域中一个热门的课题是如何有效地结合量子计算和人工智能两者的潜能。量子机器学习便是这样一门结合了量子计算与机器学习的交叉学科,一方面研究者们希望利用量子计算的信息处理优势去促进人工智能的发展,另一方面也存在可能性去利用现有的人工智能的技术突破量子计算的研发瓶颈。关于量子机器学习的入门资料可以参考 [4-6]。\n", "\n", - "最后修改于: 2020年9月9日 由量桨 Paddle Quantum开发小组共同完成。\n", + "内容上,这份快速入门包括以下几个方面:\n", + "\n", + "- 量子计算和量子神经网络的基础知识介绍\n", + "- 量桨(Paddle Quantum)的使用介绍\n", + "- 飞桨(PaddlePaddle)优化器的使用教程\n", + "- 具体的量子机器学习案例—— 变分量子特征求解器(VQE)\n", + "\n", + "\n", + "最后修改于: 2021年1月9日 由量桨 Paddle Quantum 开发小组共同完成。\n", "\n", "
" ] @@ -83,12 +92,12 @@ "source": [ "我们推荐使用 [Anaconda](https://www.anaconda.com/download) 作为 Python3的开发环境,支持多种主流操作系统(Windows, MacOS, 以及 Linux)。Anaconda本身提供 Scipy, Numpy, Matplotlib等科学计算、作图包,最主要的是其自带 Python开发环境的管理器 conda,可以用来安装或者更新主流 Python包。这里我们提供一个例子来学习使用 conda创建和管理环境:\n", "\n", - "1. 首先进入命令行 (Terminal) 界面:Windows用户可以使用组合键 `win + R` 打开运行程序再输入 `cmd`/ Mac用户可以使用组合键 `command⌘ + 空格` 再输入 `Terminal`。\n", + "1. 首先进入命令行 (Terminal) 界面:Windows用户可以通过 `Anaconda Prompt`/ Mac用户可以使用组合键 `command⌘ + 空格` 再输入 `Terminal`。\n", "1. 进入 Terminal 后输入 `conda create --name paddle_quantum_env python=3.6` 创建名为 `paddle_quantum_env` 的 Python3.6 环境。\n", "1. 在 Terminal 内通过 `conda env list` 查看已有的环境,然后通过 `conda activate paddle_quantum_env ` 进入我们刚建立的环境。\n", - "1. 为了能正确运行 Jupyter Notebook 我们还需要安装 `conda install jupyter notebook`。安装完成之后,如果你想开启 Jupyter 只需要在Terminal内激活正确的环境然后输入 `jupyter notebook` 即可。\n", + "1. 为了能正确运行 Jupyter Notebook 我们还需要安装 `conda install jupyter notebook` 或者 `pip install jupyter notebook`。安装完成之后,如果你想开启 Jupyter 只需要在Terminal内激活正确的环境然后输入 `jupyter notebook` 即可。\n", "\n", - "\n", + "\n", "\n", "关于 conda 更多的本地指令请参考 [官方教程](https://docs.conda.io/projects/conda/en/latest/user-guide/getting-started.html)。\n", " 此外,你也可以通过使用
Anaconda Navigator 开启 jupyter notebook。\n", @@ -96,7 +105,7 @@ "以下是这个教程中你需要使用的包:\n", "
    \n", "
  • Numpy\n", - "
  • Paddlepaddle 1.8.3 以上\n", + "
  • Paddlepaddle 1.8.5\n", "
  • Paddle Quantum \n", "
" ] @@ -112,67 +121,18 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "接着我们先讲解下如何安装飞桨Paddlepaddle深度学习框架。用户需要再次打开 Terminal 界面,输入 `conda activate paddle_quantum_env `进入我们新建的Python 环境。接着输入 `conda install paddlepaddle` 安装最新的飞桨Paddle。如果你在按照以上方案安装 paddle-cpu 或者 paddle-gpu 遇到问题们可以参考 [官方链接](https://www.paddlepaddle.org.cn/documentation/docs/zh/install/install_Conda.html)。或者考虑以下备选方案:在 Terminal 界面输入 `pip install paddlepaddle` 或者开启 jupyter notebook 后运行以下的命令。" + "接着我们安装 Paddle Quantum 包,用户可以直接通过 `pip install paddle-quantum` 完成安装。关于本地安装方式,用户可以通过 Terminal 界面使用 git指令 `git clone http://github.com/PaddlePaddle/quantum` 或者直接下载 `zip` 压缩包,然后找到对应本地文件的路径输入 `cd quantum` 和 `pip install -e .` 完成安装。接着在 Terminal 界面输入`pip list`查看是否在正确的环境中安装完成。关于 git的使用和安装,请参考这篇 [教程](https://git-scm.com/book/zh/v2/%E8%B5%B7%E6%AD%A5-%E5%AE%89%E8%A3%85-Git)。此外,如果你需要更多的关于安装 Paddle Quantum 的帮助,可以参考我们的 [Github链接](https://github.com/PaddlePaddle/Quantum) 或者通过 Github Issues联系我们。" ] }, { "cell_type": "code", "execution_count": 1, - "metadata": {}, - "outputs": [], - "source": [ - "from IPython.display import clear_output\n", - "\n", - "# 备用方法: 通过PyPI安装最新的飞桨Paddle\n", - "!pip install paddlepaddle\n", - "clear_output()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "在安装完成后,请通过以下代码段来检验是否成功安装 Paddle。" - ] - }, - { - "cell_type": "code", - "execution_count": 2, "metadata": { - "colab": {}, - "colab_type": "code", - "id": "HPV-UyAqsKFp" - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Running Verify Fluid Program ... \n", - "Your Paddle Fluid works well on SINGLE GPU or CPU.\n", - "Your Paddle Fluid works well on MUTIPLE GPU or CPU.\n", - "Your Paddle Fluid is installed successfully! Let's start deep Learning with Paddle Fluid now\n" - ] + "ExecuteTime": { + "end_time": "2021-01-09T12:47:03.252187Z", + "start_time": "2021-01-09T12:46:59.940104Z" } - ], - "source": [ - "from paddle import fluid\n", - "\n", - "# 检查飞桨Paddle是否成功安装\n", - "fluid.install_check.run_check()" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "接着我们安装 Paddle Quantum包,很遗憾目前我们还不支持通过 pip安装。需要用户在 Terminal 界面通过 git指令 `git clone http://github.com/PaddlePaddle/quantum` 下载文件,然后输入 `cd quantum` 和 `pip install -e .` 完成安装。接着在 Terminal 界面输入`pip list`查看是否在正确的环境中安装完成。关于 git的使用和安装,请参考这篇 [教程](https://git-scm.com/book/zh/v2/%E8%B5%B7%E6%AD%A5-%E5%AE%89%E8%A3%85-Git)。此外,如果你需要更多的关于安装Paddle-Quantum 的帮助,可以参考我们的 [Github链接](https://github.com/PaddlePaddle/Quantum) 或者通过 Github Issues联系我们。" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": {}, + }, "outputs": [], "source": [ "import numpy as np\n", @@ -210,7 +170,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "量子计算 (Quantum Computing QC )是利用量子物理中特有的现象 (量子叠加态,量子纠缠)来设计相应的量子算法以解决 (物理、化学、计算机等领域)特定的任务。现有的量子计算有好几种模型,比如有基于绝热定理的绝热计算模型以及基于测量的MBQC模型等等。在本手册中,我们主要讨论目前影响力最大、使用最广泛的量子电路(Quantum circuit)模型。在量子电路的框架下,运算最基本的组成单元是量子比特 (qubit)。这与经典计算机中比特 (bit) 的概念很相似。经典比特只能处于0和1两种状态中的某一种 (物理图景上可以对应高低电位)。与之不同的是,量子比特不仅可以处于两个状态$\\lvert {0}\\rangle$ 还有 $\\lvert {1}\\rangle$还可以处于两者的叠加态 (稍后我们来具体讲解下这一概念)。而常见的量子计算模型,就是利用量子逻辑门操控这些量子比特。逻辑门运算的基本理论是线性代数,在此我们假定读者已经具备一定的线性代数基础。" + "量子计算(Quantum Computing, QC)是利用量子物理中特有的现象(量子叠加态、量子相干性和量子纠缠等)来设计相应的量子算法以解决 (物理、化学、计算机等领域)特定的任务。现有的量子计算有存在几种模型,例如基于绝热定理的绝热量子计算模型(Adiabatic Quantum Computation, AQC)以及基于测量的量子计算模型(Measurement-Based Quantum Computation, MBQC)等等。在本介绍中,我们主要讨论目前影响力最大、使用最广泛的量子电路(Quantum Circuit)模型。在量子电路的框架下,运算最基本的组成单元是量子比特(qubit)。这与经典计算机中比特(bit)的概念很相似。经典比特只能处于0和1两种状态中的某一种(物理图景上可以对应晶体管的高低电位)。与之不同的是,量子比特不仅可以处于两个状态 $|0\\rangle$ 还有 $|1\\rangle$ 还可以处于两者的叠加态(稍后我们来具体讲解下这一概念)。在量子电路模型中,我们通过由一系列量子逻辑门构成的量子电路来操控这些量子比特的状态从而完成计算任务。逻辑门运算的基本理论是线性代数,在此我们假定读者已经具备一定的线性代数基础。" ] }, { @@ -226,22 +186,26 @@ "source": [ "### 数学表示\n", "\n", - "在量子力学中,一个微观粒子的量子态可以表示为由两个正规正交基线性组合得到的向量,这些基向量一般可以写为:\n", + "在量子力学中,一个二能级系统微观粒子(qubit)的量子态(quantum state)可以表示为由两个正规正交基线性组合得到的向量,这些基向量一般可以写为\n", "\n", - "$$ \\lvert {0}\\rangle := \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \\quad \\lvert {1}\\rangle := \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} $$\n", + "$$\n", + "|0\\rangle := \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \\quad |1\\rangle := \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}.\n", + "\\tag{1}\n", + "$$\n", "\n", - "我们这里向量的表示方法采用了量子物理上传统的狄拉克表示 (bra-ket)。这两个单位正交向量$\\{\\lvert {0}\\rangle, \\lvert {1}\\rangle \\}$,它俩一般被称为**计算基** (computational basis)。物理图景中我们可以认为 $\\lvert {0}\\rangle$ 和 $\\lvert {1}\\rangle$ 分别对应一个原子的能量基态和激发态或者其他一些二分类状态。 一个量子比特所有可能的态可以看作是二维希尔伯特空间中所有的归一化向量,这个希尔伯特空间的一组正规正交基正是$\\{\\lvert {0}\\rangle, \\lvert {1}\\rangle \\}$。而更多的量子比特系统也同样可以由高维度的希尔伯特空间中的的单位向量表示,而这个高维希尔伯特空间的正交基就是$\\{\\lvert {0}\\rangle, \\lvert {1}\\rangle \\}$的张量积。比如说,一个两量子比特 (2-qubit)系统可以被一个4维的希尔伯特空间里的单位复数向量表示,而这个希尔伯特空间的正规正交基是:\n", + "这里向量的表示方法采用了量子物理上传统的狄拉克表示(bra-ket)。这两个单位正交向量 $\\{|0\\rangle, |1\\rangle \\}$ 一般被称为**计算基**(computational basis)。物理图景中我们可以认为 $|0\\rangle$ 和 $|1\\rangle$ 分别对应一个原子的能量基态和激发态或者其他一些二分类状态。 一个量子比特所有可能的态可以看作是二维希尔伯特空间中所有的归一化向量,这个希尔伯特空间的一组正规正交基正是 $\\{|0\\rangle, |1\\rangle \\}$。更多的量子比特系统也同样可以由高维度的希尔伯特空间中的的单位向量表示,而这个高维希尔伯特空间的正交基就是 $\\{|0\\rangle, |1\\rangle \\}$ 的张量积。比如说,一个两量子比特(2-qubit)系统可以被一个4维的希尔伯特空间里的单位复数向量表示,而这个希尔伯特空间的正规正交基是\n", "\n", - "$$ \n", - "\\lvert {00}\\rangle = \\lvert {0}\\rangle\\otimes \\lvert {0}\\rangle := \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\quad \n", - "\\lvert {01}\\rangle = \\lvert {0}\\rangle\\otimes \\lvert {1}\\rangle := \\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\quad\n", - "\\lvert {10}\\rangle = \\lvert {1}\\rangle\\otimes \\lvert {0}\\rangle := \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{bmatrix}, \\quad\n", - "\\lvert {11}\\rangle = \\lvert {1}\\rangle\\otimes \\lvert {0}\\rangle := \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}\n", + "$$\n", + "\\left\\{\n", + "|00\\rangle = |0\\rangle\\otimes |0\\rangle := \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\quad \n", + "|01\\rangle = |0\\rangle\\otimes |1\\rangle := \\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\quad\n", + "|10\\rangle = |1\\rangle\\otimes |0\\rangle := \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{bmatrix}, \\quad\n", + "|11\\rangle = |1\\rangle\\otimes |0\\rangle := \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}\n", + "\\right\\}.\n", + "\\tag{2}\n", "$$\n", "\n", - "我们默认最左边的位置代表第一个量子比特,依此类推。其中符号 $\\otimes$ 是张量积运算。其工作原理大概如下:\n", - "给定两个矩阵$A_{m\\times n}$ 还有 $B_{p \\times q}$,那么$A,B$的张量积为\n", - "\n", + "我们默认最左边的位置代表第一个量子比特,依此类推。其中符号 $\\otimes$ 是张量积运算。其工作原理大概如下:给定两个矩阵 $A_{m\\times n}$ 还有 $B_{p \\times q}$,那么 $A,B$ 的张量积为\n", "\n", "$$\n", "A \\otimes B = \n", @@ -250,17 +214,18 @@ "\\vdots & \\ddots & \\vdots \\\\\n", "a_{m1}B & \\cdots & a_{m n}B\n", "\\end{bmatrix}_{(mp)\\times (nq)}\n", + "\\tag{3}\n", "$$\n", "\n", - "一个单量子比特所处的任意量子态 $\\lvert {\\psi}\\rangle$可以写成基向量 $\\lvert {0}\\rangle$ 和 $\\lvert {1}\\rangle$ 的线性叠加,也就是说,它可以被描述成一个$\\lvert {0}\\rangle$ 和 $\\lvert {1}\\rangle$的线性组合\n", + "一个单量子比特所处的任意量子态 $|\\psi\\rangle$ 可以写成基向量 $|0\\rangle$ 和 $|1\\rangle$ 的线性叠加,也就是说,它可以被描述成一个 $|0\\rangle$ 和 $|1\\rangle$ 的线性组合:\n", "\n", + "$$\n", + "|\\psi\\rangle = \\alpha |0\\rangle + \\beta |1\\rangle\n", + ":= \\begin{bmatrix} \\alpha \\\\ \\beta \\end{bmatrix}.\n", + "\\tag{4}\n", + "$$\n", "\n", - "$$\\lvert {\\psi}\\rangle = \\alpha \\lvert {0}\\rangle + \\beta \\lvert {1}\\rangle\n", - ":= \\begin{bmatrix} \\alpha \\\\ \\beta \\end{bmatrix},\\,\\alpha, \\beta \\in \\mathbb{C}$$\n", - "\n", - "其中$\\alpha$ 和 $\\beta$ 可以是**复数**,他们表示概率振幅。这意味着当我们测量这个量子比特时,根据波恩法则,测量得到量子比特处于$\\lvert {0}\\rangle$ 状态的概率是$|\\alpha|^2$;而测量得到$\\lvert {1}\\rangle$的概率是$|\\beta|^2$。由于概率相加等于1,我们必须要加入如下的限制条件:\n", - "\n", - "$$|\\alpha|^2 + |\\beta|^2 = 1$$" + "其中 $\\alpha$ 和 $\\beta$ 可以是**复数**,他们表示概率振幅。这意味着当我们测量这个量子比特时,根据波恩法则,测量得到量子比特处于 $|0\\rangle$ 状态的概率是 $|\\alpha|^2$;而测量得到 $|1\\rangle$ 的概率是 $|\\beta|^2$。由于概率相加等于 1,我们必须要加入如下的限制条件:$|\\alpha|^2 + |\\beta|^2 = 1$。" ] }, { @@ -269,27 +234,18 @@ "source": [ "### 布洛赫球面 (Bloch Sphere) 表示\n", "\n", - "我们用一个球面上的点来表示一个量子比特可能处于的量子态,这个球面被称为**布洛赫球面**,(见图 1)\n", + "我们用一个球面上的点来表示一个量子比特可能处于的量子态,这个球面被称为**布洛赫球面**(Bloch Sphere),(见图1)\n", "\n", - "$$ \n", - "\\lvert {\\psi}\\rangle = \\alpha \\lvert {0}\\rangle + \\beta \\lvert {1}\\rangle \n", - "= \\cos\\bigg(\\frac{\\theta}{2}\\bigg) \\lvert {0}\\rangle + e^{i\\varphi}\\sin\\bigg(\\frac{\\theta}{2}\\bigg) \\lvert {1}\\rangle\n", + "$$\n", + "|\\psi\\rangle = \\alpha |0\\rangle + \\beta |1\\rangle \n", + "= \\cos\\bigg(\\frac{\\theta}{2}\\bigg) |0\\rangle + e^{i\\varphi}\\sin\\bigg(\\frac{\\theta}{2}\\bigg) |1\\rangle.\n", + "\\tag{5}\n", "$$\n", "\n", - "注意:多个量子系统的状态就无法用布洛赫球面来表示。如果是一个经典比特的话,那么它只有两个状态0和1,也就是布洛赫球面的北极和南极。这两个位置恰好对应着$\\lvert {0}\\rangle$ 和 $\\lvert {1}\\rangle$。**而一个量子比特不光可以处于两极,它可以在球面上任意一点,这样一种叠加的状态是经典比特做不到的**。举例来说,量子态 $\\frac{1}{\\sqrt{2}}\\big(\\lvert {0}\\rangle + i\\lvert {1}\\rangle\\big)$就处于球面赤道和 y-正半轴的交界处。" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ + "**注意**:多个量子系统的状态就无法用布洛赫球面来表示。如果是一个经典比特的话,那么它只有两个状态0和1,也就是布洛赫球面的北极和南极。这两个位置恰好对应着 $|0\\rangle$ 和 $|1\\rangle$。**而一个量子比特不光可以处于两极,它可以在球面上任意一点,这样一种叠加的状态是经典比特做不到的**。举例来说,量子态 $\\frac{1}{\\sqrt{2}}\\big(|0\\rangle + i|1\\rangle\\big)$就处于球面赤道和 $y$-正半轴的交界处。\n", + "\n", + "\n", + "\n", "          \n", "         \n", "         \n", @@ -298,25 +254,17 @@ "**图 1.** \n", "单量子比特的布洛赫球面表示. [[图片来源]](https://en.wikipedia.org/wiki/Qubit)\n", "\n", - "下面的内容面向对量子计算更熟悉的读者。如果你阅读这段感到困难,不用担心,您可以选择略过这一节,这不会对理解接下的内容产生影响。由于量子比特之间的交互以及去相干问题 (Decoherence),因此,对于一个具有多量子比特的系统来说,它的单量子比特子系统将不再处于纯态 (pure state),而是演变成混合态 (mixed state)。混合态可以看成不同纯态的按照一定概率的混合。\n", - " **单比特的混合态可以看成是布洛赫球内部的点,而不是存在于球表面**。通常来说,混合态需要用到量子力学的密度矩阵形式来描述,比如\n", "\n", - "$$ \n", - "\\rho_{\\text{mixed}} \n", - "= \\sum_i P_i \\lvert {\\psi_i}\\rangle\\langle{\\psi_i} \\lvert\n", - "= \\frac{1}{2} \\lvert {0}\\rangle \\langle{0} \\lvert + \\frac{1}{2} \\lvert {1}\\rangle\\langle{1} \\lvert\n", - ":= \\frac{1}{2} \\begin{bmatrix} 1 \\\\ 0\\end{bmatrix} \\begin{bmatrix} 1 & 0 \\end{bmatrix} + \\frac{1}{2} \\begin{bmatrix} 0 \\\\ 1\\end{bmatrix} \\begin{bmatrix} 0 & 1 \\end{bmatrix} \n", - "= \\frac{1}{2} \\begin{bmatrix} 1 & 0\\\\ 0 & 1 \\end{bmatrix}\n", + "下面的内容面向对量子计算更熟悉的读者。如果你阅读这段感到困难,不用担心,您可以选择略过这一节,这不会对理解接下的内容产生影响。由于量子比特之间的交互以及去相干问题 (Decoherence),因此,对于一个具有多量子比特的系统来说,它的单量子比特子系统将不再处于纯态 (pure state),而是演变成混合态 (mixed state)。混合态可以看成不同纯态的按照一定概率的混合。 **单比特的混合态可以看成是布洛赫球内部的点,而不是存在于球表面**。通常来说,混合态需要用到量子力学的密度矩阵形式来描述,比如以下量子态分别有 $1/2$ 的概率处于 $|0\\rangle$ 或 $|1\\rangle$ 态,\n", + "\n", + "$$\n", + "\\rho_{\\text{mixed}} = \\sum_i P_i |\\psi_i\\rangle\\langle\\psi_i| = \\frac{1}{2} |0\\rangle\\langle0| + \\frac{1}{2} |1\\rangle\\langle1| := \\frac{1}{2} \\begin{bmatrix} 1 \\\\ 0\\end{bmatrix} \\begin{bmatrix} 1 & 0 \\end{bmatrix} + \\frac{1}{2} \\begin{bmatrix} 0 \\\\ 1\\end{bmatrix} \\begin{bmatrix} 0 & 1 \\end{bmatrix} = \\frac{1}{2} \\begin{bmatrix} 1 & 0\\\\ 0 & 1 \\end{bmatrix}.\n", + "\\tag{6}\n", "$$\n", "\n", - "其中行向量 (bra) $ \\langle{0} \\lvert $ 是列向量 (ket) $\\lvert {0}\\rangle$ 的复共轭转置。" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "**注:**如需更多信息,可参考维基百科 [链接](https://en.wikipedia.org/wiki/Qubit)" + "其中行向量(bra)$\\langle0|$ 是列向量 (ket)$|0\\rangle$ 的复共轭转置。\n", + "\n", + "**注:** 如需更多信息,可参考维基百科 [链接](https://en.wikipedia.org/wiki/Qubit)\n" ] }, { @@ -325,16 +273,16 @@ "source": [ "### 什么是量子逻辑门?\n", "\n", - "在经典计算机中,我们可以在经典比特上施加基本的逻辑运算( 非门 NOT, 与非门 NAND, 异或门 XOR, 与门 AND, 或门 OR)并组合成更复杂的运算。而量子计算则有完全不同的一套逻辑运算,它们被称为量子门 (quantum gate)。我们并不能在一个量子计算机上编译现有的C++程序。因为**经典计算机和量子计算机有不同的逻辑门构造,所以量子算法是需要利用这些量子门的特殊性来构造的**。\n", + "在经典计算机中,我们可以在经典比特上施加基本的逻辑运算(非门 NOT, 与非门 NAND, 异或门 XOR, 与门 AND, 或门 OR)并组合成更复杂的运算。而量子计算则有完全不同的一套逻辑运算,它们被称为量子门 (quantum gate)。我们并不能在一个量子计算机上编译现有的C++程序。因为**经典计算机和量子计算机有不同的逻辑门构造,所以量子算法是需要利用这些量子门的特殊性来构造的**。量子门在数学上可以被表示成酉矩阵(unitary matrix)。酉矩阵操作可以保证向量的长度不变,这是个很好的性质。不然我们对一个纯态量子比特进行操作,会让它劣化成混合态导致其无法接着很好地使用。酉矩阵定义为:\n", "\n", - "量子门在数学上可以被表示成酉矩阵 (unitary matrix)。酉矩阵操作可以保证向量的长度不变,这是个很好的性质。不然我们对一个纯态量子比特进行操作,会让它劣化成混合态导致其无法接着使用。那么什么是酉矩阵呢?\n", - "\n", - "$$ \n", - "U^{\\dagger}U = UU^{\\dagger} = I,\\,\n", - "\\Vert \\lvert {\\psi}\\rangle \\Vert = \\Vert U\\lvert {\\psi}\\rangle\\Vert = 1\n", + "$$\n", + "U^{\\dagger}U = UU^{\\dagger} = I, \n", + "\\quad \\text{并且} \\quad \n", + "\\Vert |\\psi\\rangle \\Vert = \\Vert U|\\psi\\rangle\\Vert = 1.\n", + "\\tag{7}\n", "$$\n", "\n", - "其中$U^{\\dagger}$是$U$的埃尔米特转置,$I$ 表示单位矩阵。但是酉矩阵作为量子门的物理意义是什么?这意味着**所有的量子门都必须是可逆的**。对于任何一个量子门运算,都可以找到一个与其对应的反向运算。除此之外,酉矩阵必须是个方阵。因为量子门的输入和输出要求有同样数量的量子比特。一个作用在 $n$ 量子比特的量子门可以写成一个 $2^n \\times 2^n$ 的酉矩阵。最常见的(也是物理上最容易实现的)量子门作用在一个或两个量子比特上,就像经典逻辑门那样。" + "其中 $U^{\\dagger}$ 是 $U$ 的 Hermite 转置,$I$ 表示单位矩阵。但是酉矩阵作为量子门的物理意义是什么?这意味着**所有的量子门都必须是可逆的**。对于任何一个量子门运算,都可以找到一个与其对应的反向运算。除此之外,酉矩阵必须是个方阵。因为量子门的输入和输出要求有同样数量的量子比特。一个作用在 $n$ 量子比特的量子门可以写成一个 $2^n \\times 2^n$ 的酉矩阵。最常见的(也是物理上最容易实现的)量子门作用在一个或两个量子比特上,就像经典逻辑门那样。" ] }, { @@ -343,35 +291,36 @@ "source": [ "### 单量子比特门\n", "\n", - "接下来,我们介绍在量子计算中非常重要的单量子比特门,包括泡利矩阵 $\\{X, Y, Z\\}$ 或 $\\{\\sigma_x, \\sigma_y, \\sigma_z\\}$、单比特旋转门 $\\{R_x, R_y, R_z\\}$ 和哈达玛门 $H$。有一个对于经典或量子计算都很重要的门是 **非门 (NOT gate)**,其可以表示成如下酉矩阵:\n", - "\n", + "接下来,我们介绍在量子计算中非常重要的单量子比特门,包括泡利矩阵 $\\{X, Y, Z\\}$、单比特旋转门 $\\{R_x, R_y, R_z\\}$ 和 Hadamard 门 $H$。其中 **非门(NOT gate)** 对于经典或量子计算都很重要,酉矩阵表示为:\n", "\n", - "$$ X := \\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix}$$\n", - "\n", - "这个量子门(酉矩阵)作用在单量子比特(一个复向量)上本质上的运算是**矩阵乘以向量**\n", + "$$\n", + "X := \\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix},\n", + "\\tag{8}\n", + "$$\n", "\n", + "这个量子门(酉矩阵)作用在单量子比特(一个复向量)上本质上的运算是**矩阵乘以向量**:\n", "\n", - "$$ \n", - "X \\lvert {0}\\rangle := \\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix} \\begin{bmatrix} 1 \\\\0 \\end{bmatrix} \n", - "=\\begin{bmatrix} 0 \\\\1 \\end{bmatrix} = \\lvert {1}\\rangle \n", - "\\quad \\text{and} \\quad \n", - "X \\lvert {1}\\rangle := \\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix} \\begin{bmatrix} 0 \\\\1 \\end{bmatrix} \n", - "=\\begin{bmatrix} 1 \\\\0 \\end{bmatrix}=\\lvert {0}\\rangle\n", + "$$\n", + "X |0\\rangle := \\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix} \\begin{bmatrix} 1 \\\\0 \\end{bmatrix} \n", + "=\\begin{bmatrix} 0 \\\\1 \\end{bmatrix} = |1\\rangle, \n", + "\\quad \n", + "X |1\\rangle := \\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix} \\begin{bmatrix} 0 \\\\1 \\end{bmatrix} \n", + "=\\begin{bmatrix} 1 \\\\0 \\end{bmatrix}=|0\\rangle.\n", + "\\tag{9}\n", "$$\n", "\n", - "回忆起前面的布洛赫球面表示,这个矩阵 $X$ 作用在一个量子比特(布洛赫球面上的一点)就相当于**关于布洛赫球的X轴旋转角度 $\\pi$**。这就是为什么\n", - " $X$ 可以表示成 $R_x(\\pi)$(只相差了一个无关紧要的全局相位 $e^{-i\\pi/2} = -i$ )。其他两个泡利矩阵 $Y$ 和 $Z$ 在这一点上也非常相似 (代表绕 Y和 Z轴旋转 $\\pi$运算 ):\n", - "\n", + "回忆起前面的布洛赫球面表示,这个矩阵 $X$ 作用在一个量子比特(布洛赫球面上的一点)就相当于**关于布洛赫球的 $x$ 轴旋转角度 $\\pi$**。这就是为什么 $X$ 可以表示成 $R_x(\\pi)$(相差全局相位 $e^{-i\\pi/2} = -i$)。其他两个泡利矩阵 $Y$ 和 $Z$ 在这一点上也非常相似(代表绕 $y$ 和 $z$ 轴旋转 $\\pi$ 角度 ):\n", "\n", - "$$ \n", - "Y := \\begin{bmatrix} 0 &-i \\\\ i &0 \\end{bmatrix}\n", - "\\quad \\text{and} \\quad \n", - "Z := \\begin{bmatrix} 1 &0 \\\\ 0 &-1 \\end{bmatrix}\n", + "$$\n", + "Y := \\begin{bmatrix} 0 &-i \\\\ i &0 \\end{bmatrix},\n", + "\\quad \n", + "Z := \\begin{bmatrix} 1 &0 \\\\ 0 &-1 \\end{bmatrix}.\n", + "\\tag{10}\n", "$$\n", "\n", "一般来说,任何一个在布洛赫球关于相应的轴旋转 $\\theta$ 角度的量子门可以表示为:\n", "\n", - "$$ \n", + "$$\n", "R_x(\\theta) := \n", "\\begin{bmatrix} \n", "\\cos \\frac{\\theta}{2} &-i\\sin \\frac{\\theta}{2} \\\\ \n", @@ -388,13 +337,16 @@ "\\begin{bmatrix}\n", "e^{-i\\frac{\\theta}{2}} & 0 \\\\ \n", "0 & e^{i\\frac{\\theta}{2}}\n", - "\\end{bmatrix}\n", + "\\end{bmatrix}.\n", + "\\tag{11}\n", "$$\n", "\n", - "除了旋转门之外,最重要的单比特门就是哈达玛门了。对应的布洛赫球面解释是两个旋转组成的,先是按Z轴旋转$\\pi$,然后按Y轴旋转$\\pi/2$。它的矩阵表示是:\n", + "除了旋转门之外,最重要的单比特门就是 Hadamard 门了。对应的布洛赫球面解释是两个旋转组成的,先是按 $z$ 轴旋转 $\\pi$,然后按 $y$ 轴旋转 $\\pi/2$。它的矩阵表示是\n", "\n", - "$$H := \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1 &1 \\\\ 1 &-1 \\end{bmatrix} $$\n", - "\n" + "$$\n", + "H := \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1 &1 \\\\ 1 &-1 \\end{bmatrix}.\n", + "\\tag{12}\n", + "$$" ] }, { @@ -410,7 +362,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "\n", + "\n", "\n", "          \n", "         \n", @@ -424,7 +376,8 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "作用在两量子比特上的量子门可以表示成一个$4\\times4$酉矩阵\n", + "作用在两量子比特上的量子门可以表示成一个 $4\\times4$ 酉矩阵:\n", + "\n", "$$\n", "U = H \\otimes I \n", "= \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 &1 \\\\ 1 &-1 \\end{bmatrix} \n", @@ -436,56 +389,36 @@ "1 &0 &-1 &0 \\\\\n", "0 &1 &0 &-1 \n", "\\end{bmatrix}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "另一种拓展方式是作用在全部两个量子比特上。比如CNOT,这个门会使得一个量子比特的状态影响到另一个量子比特的状态\n", + "\\tag{13}\n", + "$$\n", + "\n", + "另一种拓展方式是将逻辑门直接作用在两个量子比特上。比如受控非门 $\\text{CNOT}$,这个门会使得一个量子比特的状态影响到另一个量子比特的状态\n", "\n", "$$\n", - "CNOT := \n", + "\\text{CNOT} := \n", "\\begin{bmatrix} \n", "1 &0 &0 &0 \\\\ \n", "0 &1 &0 &0 \\\\\n", "0 &0 &0 &1 \\\\\n", "0 &0 &1 &0 \n", - "\\end{bmatrix}\n", + "\\end{bmatrix}.\n", + "\\tag{14}\n", "$$\n", "\n", - "我们近距离观察一下它作用在不同的初始量子态上\n", + "我们观察一下它作用在不同的初始量子态上:\n", "\n", "$$\n", - "CNOT \\lvert {00}\\rangle = \\lvert {00}\\rangle, \\quad\n", - "CNOT \\lvert {01}\\rangle = \\lvert {01}\\rangle, \\quad\n", - "CNOT \\lvert {10}\\rangle = \\lvert {11}\\rangle, \\quad\n", - "CNOT \\lvert {11}\\rangle = \\lvert {10}\\rangle\n", + "\\text{CNOT} |00\\rangle = |00\\rangle, \\quad\n", + "\\text{CNOT} |01\\rangle = |01\\rangle, \\quad\n", + "\\text{CNOT} |10\\rangle = |11\\rangle, \\quad\n", + "\\text{CNOT} |11\\rangle = |10\\rangle.\n", + "\\tag{15}\n", "$$\n", "\n", - "也就是说,当第一个量子比特处于 $\\lvert {1}\\rangle$状态时,CNOT会在第二个量子比特上施加$X$门,如果第一个量子比特处于 $\\lvert {0}\\rangle$状态,那么第二个量子比特则不受任何影响。这也是为什么 CNOT 会被称为受控非门。下面是一些常见的量子门及它们的矩阵表示" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", + "也就是说,当第一个量子比特处于 $|1\\rangle$ 状态时,$\\text{CNOT}$ 会在第二个量子比特上施加 $X$ 门,如果第一个量子比特处于 $|0\\rangle$ 状态,那么第二个量子比特则不受任何影响。这也是为什么 $\\text{CNOT}$ 会被称为受控非门。下面是一些常见的量子门及它们的矩阵表示,**这些量子门都可以在量桨内被调用**。\n", + "\n", + "![intro-fig-gates](./figures/intro-fig-gates.png \"**图 3.** 常见的量子门. [[图片来源]](https://en.wikipedia.org/wiki/Quantum_logic_gate)\")\n", "\n", - "          \n", - "         \n", - "         \n", - "         \n", - "  \n", - "**图 3.** \n", - "常见的量子门. [[图片来源]](https://en.wikipedia.org/wiki/Quantum_logic_gate)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ "**注**:更多信息可见如下维基百科 [链接](https://en.wikipedia.org/wiki/Quantum_logic_gate)" ] }, @@ -510,7 +443,7 @@ "\n", "**注意:** 所有的单比特旋转门都按如下规定建立:\n", "\n", - "$$ \n", + "$$\n", "R_x(\\theta) := \n", "\\begin{bmatrix} \n", "\\cos \\frac{\\theta}{2} &-i\\sin \\frac{\\theta}{2} \\\\ \n", @@ -527,16 +460,22 @@ "\\begin{bmatrix}\n", "e^{-i\\frac{\\theta}{2}} & 0 \\\\ \n", "0 & e^{i\\frac{\\theta}{2}}\n", - "\\end{bmatrix}\n", + "\\end{bmatrix}.\n", + "\\tag{16}\n", "$$\n", "\n", - "因此,我们不难看出 $X$ 门可以表示为$R_x(\\pi)$。以下是代码展示:" + "因此,我们不难看出 $X$ 门可以表示为 $R_x(\\pi)$。以下是代码展示:" ] }, { "cell_type": "code", - "execution_count": 4, - "metadata": {}, + "execution_count": 2, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:47:08.582194Z", + "start_time": "2021-01-09T12:47:08.337302Z" + } + }, "outputs": [ { "name": "stdout", @@ -578,14 +517,15 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "结果和$X (NOT)$门只相差一个全局相位$-i$\n", + "结果和 $X$(NOT)门只相差一个全局相位$-i$\n", "\n", - "\n", - "$$ \\text{output} = \\begin{bmatrix} 0 &-i \\\\ -i &0 \\end{bmatrix}\n", - "= -i\\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix} = -i X\n", + "$$\n", + "\\text{output} = \\begin{bmatrix} 0 &-i \\\\ -i &0 \\end{bmatrix}\n", + "= -i\\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix} = -i X.\n", + "\\tag{17}\n", "$$\n", "\n", - "有兴趣的话,你可以仔细思考一下为什么在量子计算中,全局相位并不重要。\n" + "有兴趣的话,你可以仔细思考一下为什么在量子计算中,全局相位并不重要。" ] }, { @@ -619,10 +559,11 @@ "source": [ "和原来一样,我们还是多了一个全局相位\n", "\n", - "\n", - "$$ \\text{output} = \\begin{bmatrix} 0 &-1 \\\\ 1 &0 \\end{bmatrix}\n", - "= -i\\begin{bmatrix} 0 &-i \\\\ i &0 \\end{bmatrix} = -i Y\n", - "$$\n" + "$$\n", + "\\text{output} = \\begin{bmatrix} 0 &-1 \\\\ 1 &0 \\end{bmatrix}\n", + "= -i\\begin{bmatrix} 0 &-i \\\\ i &0 \\end{bmatrix} = -i Y.\n", + "\\tag{18}\n", + "$$" ] }, { @@ -645,7 +586,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "经过上面的准备,你现在有一定的知识基础可以了解量子机器学习了。简单来说我们要做的就是利用量子电路来替代传统的神经网络来完成机器学习的任务。我们一般会准备一个可调节参数的量子电路(量子神经网络 QNN),很多时候也被称为模板 (Ansatz),里面的参数是人为可调节的(这些参数其实就是旋转门的角度$\\theta$)。如果再加上一个精心设计的损失函数,就可以将一个量子计算问题转化为经典的寻找损失函数最小值问题。然后不断调节电路中的参数直到损失函数下降至收敛 (此时损失函数达到最优值或次优值), 我们就完成优化了。\n" + "经过上面的准备,你现在有一定的知识基础可以了解量子机器学习了。简单来说,我们要做的就是利用参数化量子电路(Parametrized Quantum Circuit, PQC)来替代传统的神经网络来完成机器学习的任务。处理的对象可以是经典数据也可以是量子数据。我们一般会准备一个可调节参数的量子电路(PQC),也被称作量子神经网络(Quantum Neural Network, QNN)或者电路模板(ansatz),里面的参数是人为可调节的(这些参数大多数情况下就是旋转门的角度 $\\theta$)。例如上一节中看到的用参数 $\\pi$ 构造 $X$ 门,这其实就是最简单的量子神经网络。如果再加上一个精心设计的损失函数,就可以将一个计算问题转化为寻找损失函数的最值问题。然后不断调节电路中的参数直到损失函数下降至收敛(此时损失函数达到最优值或次优值),我们就完成了优化。这样的一种在量子设备上估值损失函数然后在经典设备上进行优化的框架被称为量子-经典混合优化,或者变分量子算法(Variational Quantum Algorithms, VQA)。" ] }, { @@ -654,27 +595,25 @@ "source": [ "### 示例: 如何创建量子神经网络 QNN?\n", "\n", - "QNN通常可以表示为一些单比特量子旋转门和双比特门的组合。其中一个可以高效利用硬件的架构是只包含 $\\{R_x, R_y, R_z, \\text{CNOT} \\}$这四种量子门的模板。它们很容易在NISQ设备(通常是超导量子比特)上实现,因为CNOT只需要实施在相邻量子比特上。一个例子可见下图:\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "通常来说,每条线代表一个量子比特。我们把图最上端的认为是第一个量子比特,依次往下。从左到右代表我们施加门的时间顺序,先施加最左边的量子门。接下来,我们来看看如何在量桨上建造这个简单的两比特量子神经网络\n" + "QNN 通常可以表示为一些单比特量子旋转门和双比特门的组合。其中一个可以高效利用硬件的架构是只包含 $\\{R_x, R_y, R_z, \\text{CNOT}\\}$ 这四种量子门的模板。它们很容易在 NISQ (Noisy-Intermidiate-Scale-Quantum)设备(通常是超导量子比特)上实现,因为 $\\text{CNOT}$ 只需要实施在相邻量子比特上。一个例子可见下图:\n", + "\n", + "\n", + "![intro-fig-gate1](./figures/intro-fig-gate1.png)\n", + "\n", + "通常来说,每条线代表一个量子比特。我们把图最上端的认为是第一个量子比特 $q_0$,依次往下。从左到右代表我们施加门的时间顺序,先施加最左边的量子门。接下来,我们来看看如何在量桨上建造这个简单的两比特量子神经网络\n", + "\n", + "\n" ] }, { "cell_type": "code", - "execution_count": 5, - "metadata": {}, + "execution_count": 3, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:47:11.924347Z", + "start_time": "2021-01-09T12:47:11.900263Z" + } + }, "outputs": [ { "name": "stdout", @@ -727,7 +666,8 @@ "-1 &0 &0 &0 \\\\\n", "0 &0 &1 &0 \\\\\n", "0 &0 &0 &1 \n", - "\\end{bmatrix}\n", + "\\end{bmatrix}.\n", + "\\tag{19}\n", "$$" ] }, @@ -778,7 +718,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "\n" + "\n" ] }, { @@ -797,12 +737,17 @@ }, { "cell_type": "code", - "execution_count": 7, - "metadata": {}, + "execution_count": 4, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:47:17.393379Z", + "start_time": "2021-01-09T12:47:16.964739Z" + } + }, "outputs": [ { "data": { - "image/png": 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\n", 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\n", 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" ] @@ -835,12 +780,17 @@ }, { "cell_type": "code", - "execution_count": 8, - "metadata": {}, + "execution_count": 5, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:47:21.610944Z", + "start_time": "2021-01-09T12:47:21.228136Z" + } + }, "outputs": [ { "data": { - "image/png": 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\n", 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\n", "text/plain": [ "
" ] @@ -882,7 +832,7 @@ "\\begin{bmatrix} \n", "\\cos \\frac{\\theta}{2} & -e^{i \\varphi}\\sin \\frac{\\theta}{2} \\\\ \n", "e^{i \\phi}\\sin \\frac{\\theta}{2} &e^{i (\\phi+\\varphi)} \\cos \\frac{\\theta}{2} \n", - "\\end{bmatrix}\n", + "\\end{bmatrix},\\tag{20}\n", "$$\n", "\n", "$U_3$ 旋转门在效果上是等价于以下组合旋转门的,\n", @@ -902,26 +852,30 @@ "\\begin{bmatrix}\n", "e^{-i\\frac{\\varphi}{2}} & 0 \\\\ \n", "0 & e^{i\\frac{\\varphi}{2}}\n", - "\\end{bmatrix}\n", + "\\end{bmatrix}.\\tag{21}\n", "$$\n", "\n", "感兴趣的读者不妨自行验证一下。\n", "\n", - "\n", + "\n", "\n", "特别地,当我们处理的任务不涉及虚数时,使用电路模板 `real_entangled_layer(theta, DEPTH)` 会更加高效 ($R_y$旋转门替代$U_3$)。" ] }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 6, "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:47:40.125099Z", + "start_time": "2021-01-09T12:47:39.631740Z" + }, "scrolled": true }, "outputs": [ { "data": { - "image/png": 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+ "image/png": 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\n", "text/plain": [ "
" ] @@ -981,22 +935,27 @@ "\n", "所谓的波函数模式也就是用复数向量表示和储存量子态。向量模式只能处理纯态,但这种模式在家用电脑硬件高效支持**20+量子比特**的运算。用户可以测试下自己电脑的极限在哪里。在这种表示下,量子门 (酉矩阵)作用在量子比特 (一个复向量)上本质上的运算是**矩阵乘以向量**:\n", "\n", - "$$\\lvert {\\psi}\\rangle = U \\lvert {\\psi_0}\\rangle$$\n", + "$$\\lvert {\\psi}\\rangle = U \\lvert {\\psi_0}\\rangle. \\tag{22}$$\n", "\n", "代码中,具体体现在 UAnsatz的调用 `cir.run_state_vector(input_state = None)`。如果我们不输入任何初始量子态,就会默认所有的量子比特都处于$\\lvert {0}\\rangle$态。接着来看个具体的例子:" ] }, { "cell_type": "code", - "execution_count": 10, - "metadata": {}, + "execution_count": 7, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:41.024881Z", + "start_time": "2021-01-09T12:47:50.905119Z" + } + }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "[[ 1.36742103e-03+0.j -1.15621578e-03+0.j 1.59433644e-03+0.j ...\n", - " 2.36662139e-04+0.j -2.47895067e-05+0.j -9.47071583e-04+0.j]]\n" + "[[-0.00172454+0.j -0.00029211+0.j 0.00071415+0.j ... 0.00040607+0.j\n", + " -0.00104915+0.j 0.00085908+0.j]]\n" ] } ], @@ -1039,24 +998,29 @@ "\n", "同时 Paddle quantum也支持了密度矩阵运算模式,也就是用一个密度矩阵 $\\rho = \\sum_i P_i \\lvert {\\psi_i}\\rangle\\langle{\\psi_i} \\lvert$表示和储存量子态。该模式下可以根据算法需要支持**混合态模拟**。但是在密度矩阵模式下,家用电脑硬件只能运行10个左右的量子比特。请用户注意这方面的限制,我们也在不断优化这个模式下的模拟器性能。在这种表示下,量子门 (酉矩阵)作用在量子态(一个迹为1的厄尔米特矩阵)上本质上的运算是**矩阵乘法**:\n", "\n", - "$$\\rho = U \\rho_0 U^\\dagger$$\n", + "$$\\rho = U \\rho_0 U^\\dagger. \\tag{23}$$\n", "\n", "代码中,具体体现在 UAnsatz的调用 `cir.run_density_matrix()`。接着来看个具体的例子:" ] }, { "cell_type": "code", - "execution_count": 11, - "metadata": {}, + "execution_count": 8, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:41.307338Z", + "start_time": "2021-01-09T12:48:41.028222Z" + } + }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "[[0.62679516+0.j 0.38101066+0.j 0.13601748+0.j 0.26505304+0.j]\n", - " [0.38101066+0.j 0.23160536+0.j 0.08268109+0.j 0.16111808+0.j]\n", - " [0.13601748+0.j 0.08268109+0.j 0.02951643+0.j 0.05751775+0.j]\n", - " [0.26505304+0.j 0.16111808+0.j 0.05751775+0.j 0.11208305+0.j]]\n" + "[[ 0.20642449+0.j -0.24205478+0.j -0.27795534+0.j 0.1672236 +0.j]\n", + " [-0.24205478+0.j 0.2838351 +0.j 0.32593235+0.j -0.19608755+0.j]\n", + " [-0.27795534+0.j 0.32593235+0.j 0.37427328+0.j -0.22517044+0.j]\n", + " [ 0.1672236 +0.j -0.19608755+0.j -0.22517044+0.j 0.13546713+0.j]]\n" ] } ], @@ -1100,15 +1064,6 @@ "如果你对源代码有兴趣,可以利用如下 magic command即时查看。或者关注我们的 [[Github]](https://github.com/PaddlePaddle/Quantum)!" ] }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": {}, - "outputs": [], - "source": [ - "%psource density_op_random" - ] - }, { "cell_type": "markdown", "metadata": {}, @@ -1125,7 +1080,7 @@ "0 \\\\\n", "0 \\\\\n", "1\n", - "\\end{bmatrix}\n", + "\\end{bmatrix}.\\tag{24}\n", "$$\n", "\n", "那么我们如何用量桨来制备一个贝尔态呢? 只需要如下的量子电路:\n" @@ -1135,17 +1090,22 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "" + "" ] }, { "cell_type": "code", - "execution_count": 13, - "metadata": {}, + "execution_count": 9, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:42.231220Z", + "start_time": "2021-01-09T12:48:41.331283Z" + } + }, "outputs": [ { "data": { - "image/png": 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" ] @@ -1215,7 +1175,7 @@ "\n", "$$\n", "\\min_{\\boldsymbol{\\theta}}\\mathcal{L}(\\theta_1, \\theta_2, \\theta_3)\n", - "= (\\theta_1)^2 + (\\theta_2)^2 + (\\theta_3)^2 + 10\n", + "= (\\theta_1)^2 + (\\theta_2)^2 + (\\theta_3)^2 + 10 \\tag{25}\n", "$$\n", "\n", "可以看出,只有当 $\\theta_1 = \\theta_2 = \\theta_3 = 0$ 的时候,$\\mathcal{L}$ 取最小值10。\n" @@ -1223,8 +1183,13 @@ }, { "cell_type": "code", - "execution_count": 14, - "metadata": {}, + "execution_count": 10, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:42.863949Z", + "start_time": "2021-01-09T12:48:42.262829Z" + } + }, "outputs": [ { "name": "stdout", @@ -1295,7 +1260,7 @@ "\n", "接下来,我们试一个更复杂的损失函数。 首先我们介绍一个随机的埃尔米特矩阵 $H$ 其**特征值**为矩阵 $D$ 的对角元素。 \n", "\n", - "$$ D = \\begin{bmatrix} 0.2 &0 \\\\ 0 &0.8 \\end{bmatrix} $$\n", + "$$ D = \\begin{bmatrix} 0.2 &0 \\\\ 0 &0.8 \\end{bmatrix}, \\tag{26} $$\n", "\n", "不用担心,我们会帮你生成这个埃尔米特矩阵$H$. \n", "\n", @@ -1315,14 +1280,14 @@ "\\begin{bmatrix}\n", "e^{-i\\frac{\\theta_3}{2}} & 0 \\\\ \n", "0 & e^{i\\frac{\\theta_3}{2}}\n", - "\\end{bmatrix}\n", + "\\end{bmatrix}, \\tag{27}\n", "$$\n", "\n", "我们让这个矩阵(模板)乘以 $\\lvert {0}\\rangle$,得到一个新的2维复向量$\\lvert {\\phi}\\rangle$\n", "\n", "\n", "$$ \n", - "\\lvert {\\phi}\\rangle = U(\\theta_1, \\theta_2, \\theta_3)\\lvert {0}\\rangle\n", + "\\lvert {\\phi}\\rangle = U(\\theta_1, \\theta_2, \\theta_3)\\lvert {0}\\rangle, \\tag{28}\n", "$$\n", "\n", "然后,我们定义损失函数为:\n", @@ -1330,7 +1295,7 @@ "$$\n", "\\min_{\\boldsymbol{\\theta}}\\mathcal{L}(\\theta_1, \\theta_2, \\theta_3) \n", "= \\langle{\\phi} \\lvert H \\lvert {\\phi}\\rangle \n", - "= \\langle{0} \\lvert U^{\\dagger}H U \\lvert {0}\\rangle\n", + "= \\langle{0} \\lvert U^{\\dagger}H U \\lvert {0}\\rangle. \\tag{29}\n", "$$\n", "\n", "来看看优化后我们得到了什么!" @@ -1338,16 +1303,21 @@ }, { "cell_type": "code", - "execution_count": 15, - "metadata": {}, + "execution_count": 11, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:43.137909Z", + "start_time": "2021-01-09T12:48:42.896539Z" + } + }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "随机生成的矩阵 H 是:\n", - "[[ 0.55386934+2.77555756e-17j -0.282673 -8.48178487e-02j]\n", - " [-0.282673 +8.48178487e-02j 0.44613066+0.00000000e+00j]] \n", + "[[0.76063331+0.j 0.06546144+0.13336071j]\n", + " [0.06546144-0.13336071j 0.23936669+0.j ]] \n", "\n", "不出所料,H 的特征值是:\n", "[0.2 0.8]\n" @@ -1377,8 +1347,13 @@ }, { "cell_type": "code", - "execution_count": 16, - "metadata": {}, + "execution_count": 12, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:43.155166Z", + "start_time": "2021-01-09T12:48:43.146544Z" + } + }, "outputs": [], "source": [ "# 超参数设置\n", @@ -1404,8 +1379,13 @@ }, { "cell_type": "code", - "execution_count": 17, - "metadata": {}, + "execution_count": 13, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:43.179953Z", + "start_time": "2021-01-09T12:48:43.161434Z" + } + }, "outputs": [], "source": [ "class Optimization_ex2(fluid.dygraph.Layer):\n", @@ -1435,24 +1415,29 @@ }, { "cell_type": "code", - "execution_count": 18, - "metadata": {}, + "execution_count": 14, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:43.581225Z", + "start_time": "2021-01-09T12:48:43.201135Z" + } + }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ - "iter: 0 loss: 0.5581\n", - "iter: 1 loss: 0.3127\n", - "iter: 2 loss: 0.2455\n", - "iter: 3 loss: 0.2072\n", - "iter: 4 loss: 0.2010\n", - "iter: 5 loss: 0.2002\n", - "iter: 6 loss: 0.2000\n", - "iter: 7 loss: 0.2000\n", - "iter: 8 loss: 0.2000\n", - "iter: 9 loss: 0.2000\n", - "损失函数的最小值是: 0.20000008859899268\n" + "iter: 0 loss: 0.7653\n", + "iter: 1 loss: 0.5138\n", + "iter: 2 loss: 0.2556\n", + "iter: 3 loss: 0.2068\n", + "iter: 4 loss: 0.2030\n", + "iter: 5 loss: 0.2015\n", + "iter: 6 loss: 0.2007\n", + "iter: 7 loss: 0.2004\n", + "iter: 8 loss: 0.2002\n", + "iter: 9 loss: 0.2001\n", + "损失函数的最小值是: 0.20007871016474926\n" ] } ], @@ -1494,7 +1479,7 @@ "source": [ "我们可以改变一下$H$的特征值。如果将它对角化后的的对角矩阵改变为\n", "\n", - "$$ D = \\begin{bmatrix} 0.8 &0 \\\\ 0 &1.2 \\end{bmatrix} $$\n", + "$$ D = \\begin{bmatrix} 0.8 &0 \\\\ 0 &1.2 \\end{bmatrix}. \\tag{30} $$\n", "\n", "你会发现我们仍然得到了$H$的最小特征值0.8, 你能找到背后的原因吗?还是说这背后隐藏着什么理论?\n" ] @@ -1525,7 +1510,7 @@ "\n", "假设我们想找到如下哈密顿量的基态:\n", "\n", - "$$ H = 0.4 \\, Z \\otimes I + 0.4 \\, I \\otimes Z + 0.2 \\, X \\otimes X$$\n", + "$$ H = 0.4 \\, Z \\otimes I + 0.4 \\, I \\otimes Z + 0.2 \\, X \\otimes X. \\tag{31}$$\n", "\n", "给定一种常见的量子神经网络架构" ] @@ -1534,7 +1519,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "" + "" ] }, { @@ -1546,8 +1531,13 @@ }, { "cell_type": "code", - "execution_count": 19, - "metadata": {}, + "execution_count": 15, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:43.603696Z", + "start_time": "2021-01-09T12:48:43.592558Z" + } + }, "outputs": [], "source": [ "from paddle_quantum.utils import pauli_str_to_matrix\n", @@ -1570,8 +1560,13 @@ }, { "cell_type": "code", - "execution_count": 20, - "metadata": {}, + "execution_count": 16, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:43.719137Z", + "start_time": "2021-01-09T12:48:43.608167Z" + } + }, "outputs": [], "source": [ "class vqe_demo(fluid.dygraph.Layer):\n", @@ -1609,8 +1604,12 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 17, "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:48:46.185285Z", + "start_time": "2021-01-09T12:48:43.831438Z" + }, "colab": { "base_uri": "https://localhost:8080/", "height": 1000 @@ -1689,13 +1688,23 @@ "source": [ "## 参考文献\n", "\n", - "[1] [Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).](https://www.nature.com/articles/ncomms5213)\n", + "[1] Nielsen, M. A. & Chuang, I. L. Quantum computation and quantum information. (Cambridge university press, 2010).\n", + "\n", + "[2] Phillip Kaye, Laflamme, R. & Mosca, M. An Introduction to Quantum Computing. (2007).\n", + "\n", + "[3] Biamonte, J. et al. Quantum machine learning. [Nature 549, 195–202 (2017).](https://www.nature.com/articles/nature23474)\n", + "\n", + "[4] Schuld, M., Sinayskiy, I. & Petruccione, F. An introduction to quantum machine learning. [Contemp. Phys. 56, 172–185 (2015).](https://www.tandfonline.com/doi/abs/10.1080/00107514.2014.964942)\n", "\n", - "[2] [McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016).](https://iopscience.iop.org/article/10.1088/1367-2630/18/2/023023)\n", + "[5] Benedetti, M., Lloyd, E., Sack, S. & Fiorentini, M. Parameterized quantum circuits as machine learning models. [Quantum Sci. Technol. 4, 043001 (2019).](https://iopscience.iop.org/article/10.1088/2058-9565/ab4eb5)\n", "\n", - "[3] [Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).](https://www.nature.com/articles/nature23879)\n", + "[6] [Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).](https://www.nature.com/articles/ncomms5213)\n", "\n", - "[4] [Mitarai, K., Negoro, M., Kitagawa, M. & Fujii, K. Quantum circuit learning. Phys. Rev. A 98, 032309 (2018).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.032309)" + "[7] [McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016).](https://iopscience.iop.org/article/10.1088/1367-2630/18/2/023023)\n", + "\n", + "[8] [Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).](https://www.nature.com/articles/nature23879)\n", + "\n", + "[9] [Mitarai, K., Negoro, M., Kitagawa, M. & Fujii, K. Quantum circuit learning. Phys. Rev. A 98, 032309 (2018).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.032309)" ] }, { @@ -1729,6 +1738,24 @@ "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.10" + }, + "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": { + "height": "calc(100% - 180px)", + "left": "10px", + "top": "150px", + "width": "426.667px" + }, + "toc_section_display": true, + "toc_window_display": true } }, "nbformat": 4, diff --git a/introduction/PaddleQuantum_Tutorial_CN.pdf b/introduction/PaddleQuantum_Tutorial_CN.pdf deleted file mode 100644 index 2f9314e7239de496fb449d367dc7f75b58afcd82..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 1947384 zcmcG#Wk4O}K@n8dWt$+1CZJjDQd&Efqa1ao_g&_F>UQ_K&^+SVjO1z)IH?mV*O8En#43 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b/introduction/PaddleQuantum_Tutorial_EN.ipynb new file mode 100644 index 0000000..028c48a --- /dev/null +++ b/introduction/PaddleQuantum_Tutorial_EN.ipynb @@ -0,0 +1,1524 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Paddle Quantum Quick Start Manual\n", + "\n", + " Copyright (c) 2021 Institute for Quantum Computing, Baidu Inc. All Rights Reserved. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "# Overview\n", + "\n", + "Quantum computing (QC) is a new computing model laying in the intersection of quantum mechanics and theory of computation. QC follows the fundamental laws of quantum theory to manipulate quantum bits (qubits). For many specific computational tasks, it is widely believed that quantum algorithms exhibit advantages over classical algorithms at least in theory. For a systematic introduction to the subject of QC, we refer to [1-2].\n", + "\n", + "In recent years, one of the popular research topics in QC is combining the potential of quantum computing and artificial intelligence. Quantum machine learning (QML) is such an interdisciplinary subject. Researchers want to utilize the information processing advantages of quantum computing to promote the development of artificial intelligence. On the other side, it is also worth exploring the possibility of using artificial intelligence technology to break through the bottleneck of quantum computing research and development. For introductory materials about quantum machine learning, please refer to [3-5].\n", + "\n", + "Here, we provide a quick start for users to get started with Paddle Quantum. Currently, you can read all the content online or download the Jupyter Notebook from our [GitHub](https://github.com/PaddlePaddle/Quantum/tree/master/introduction). In terms of content, the quick start includes the following sections:\n", + "\n", + "- Introduction to quantum computing and quantum neural network (QNN)\n", + "- Introduction to Paddle Quantum\n", + "- PaddlePaddle optimizer tutorial\n", + "- A case study on quantum machine learning - Variational Quantum Eigensolver (VQE)\n", + "\n", + "**Latest version updated on:** Jan. 9th, 2021 by Paddle Quantum developers.\n", + "\n", + "---\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Quantum Computing Fundamentals\n", + "\n", + "Quantum Computing (QC) uses unique phenomena in quantum physics (quantum superposition, quantum interference, and quantum entanglement) to design algorithms and help solve specific tasks in physics, chemistry, and optimization theory. There are several existing quantum computation models including the Adiabatic Quantum Computation (AQC) based on the adiabatic theorem and Measurement-Based Quantum Computation (MBQC). This introduction will focus on the most widely used Quantum Circuit model. In quantum circuits, the basic computation unit is the quantum bit (qubit), which is similar to the concept of bit in classical computers. Classical bits can only be in one of the two states, 0 or 1. By comparison, qubits can not only be in states $|0\\rangle$ and $|1\\rangle$ but also in a superposition state (we will explain this concept later). Quantum circuit model utilize quantum logic gates to manipulate the states of these qubits. The mathematics behind this process is linear algebra in the complex domain. Here we assume the readers are already familiar with linear algebra." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## What is a qubit?\n", + "\n", + "### Mathematical representation\n", + "\n", + "In quantum mechanics, the state of a two-level quantum system (e.g. electron spin) can be expressed as a state vector obtained through linear combinations of the following orthonormal basis,\n", + "\n", + "$$\n", + "|0\\rangle := \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \\quad |1\\rangle := \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix}.\n", + "\\tag{1}\n", + "$$\n", + "\n", + "The vector representation here follows the Dirac notation (bra-ket) in quantum physics. This orthonormal basis $\\{|0\\rangle, |1\\rangle \\}$ is known as the **computational basis**. Physically, one could consider $|0\\rangle$ and $|1\\rangle$ as the energy ground state and excited state of an atom, respectively. All possible pure states of a qubit can be regarded as normalized vectors in the two-dimensional Hilbert space. Moreover, multi-qubit states can be represented by unit vectors in high-dimensional Hilbert space where the basis is the tensor product of $\\{|0\\rangle, |1\\rangle\\}$. For example, a 2-qubit quantum state can be represented by a unit complex vector in a 4-dimensional Hilbert space with the orthonormal basis,\n", + "\n", + "$$\n", + "\\left\\{\n", + "|00\\rangle = |0\\rangle\\otimes |0\\rangle := \\begin{bmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\quad \n", + "|01\\rangle = |0\\rangle\\otimes |1\\rangle := \\begin{bmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{bmatrix}, \\quad\n", + "|10\\rangle = |1\\rangle\\otimes |0\\rangle := \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{bmatrix}, \\quad\n", + "|11\\rangle = |1\\rangle\\otimes |0\\rangle := \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix}\n", + "\\right\\}.\n", + "\\tag{2}\n", + "$$\n", + "\n", + "By convention, the leftmost position in ket notation represents the first qubit $q_0$, the second position represents the second qubit $q_1$, and so on. The symbol $\\otimes$ denotes the tensor product operation. It works as follows: Given two matrices $A_{m\\times n}$ and $B_{p \\times q}$, then the tensor product of $A, B$ is\n", + "\n", + "$$\n", + "A \\otimes B =\n", + "\\begin{bmatrix}\n", + "a_{11}B & \\cdots & a_{1 n}B\\\\\n", + "\\vdots & \\ddots & \\vdots \\\\\n", + "a_{m1}B & \\cdots & a_{m n}B\n", + "\\end{bmatrix}_{(mp)\\times (nq)}.\n", + "\\tag{3}\n", + "$$\n", + "\n", + "Any single qubit quantum state $|\\psi\\rangle$ can be written as a linear combination (superposition) of the basis vectors $|0\\rangle$ and $|1\\rangle$. \n", + "\n", + "$$\n", + "|\\psi\\rangle = \\alpha |0\\rangle + \\beta |1\\rangle\n", + ":= \\begin{bmatrix} \\alpha \\\\ \\beta \\end{bmatrix}.\n", + "\\tag{4}\n", + "$$\n", + "\n", + "where $\\alpha$ and $\\beta$ are **complex numbers** referred as the probability amplitudes. According to Born Rule, the probability to find the qubit in $|0\\rangle$ state is $|\\alpha|^2$; and the probability of $|1\\rangle$ is $|\\beta|^2$. Since the sum of probabilities equals to 1, one should introduce the constraint: $|\\alpha|^2 + |\\beta|^2 = 1$.\n", + "\n", + "\n", + "### Bloch sphere representation\n", + "\n", + "Geometrically, one can use a point on the unit sphere to represent the pure quantum state of a qubit. This sphere is called the **Bloch sphere**, (see Figure 1)\n", + "\n", + "$$\n", + "|\\psi\\rangle = \\alpha |0\\rangle + \\beta |1\\rangle\n", + "= \\cos\\bigg(\\frac{\\theta}{2}\\bigg) |0\\rangle + e^{i\\varphi}\\sin\\bigg(\\frac{\\theta}{2}\\bigg) |1\\rangle.\n", + "\\tag{5}\n", + "$$\n", + "\n", + "**Note**: The states of multi-qubit quantum systems cannot be directly represented by Bloch spheres. For a classical bit in state 0 or 1, it corresponds to the north or south pole of the Bloch sphere. **A qubit state can be at any point on the sphere. Such a superposition state is impossible for classical bits**. For example, the quantum state $\\frac{1}{\\sqrt{2}}\\big(|0\\rangle + i|1\\rangle\\big)$ is at the intersection of the equator and $+y$ axis.\n", + "\n", + "![intro-fig-bloch](./figures/intro-fig-bloch.png \"**Figure 1.** Bloch sphere representation of a single qubit. [[Image source]](https://en.wikipedia.org/wiki/Qubit)\")\n", + "\n", + "The following content is for readers who are more familiar with quantum computing. If you find it difficult to follow, don’t worry, you can skip it. Due to the interaction between qubits and decoherence, for a system with multiple qubits, its single-qubit subsystem will no longer stay in pure states, but a mixed state in general. The concept of mixed state can be regarded as a statistical mixture of multiple pure states. **The single-bit mixed state can be seen as a point inside the Bloch sphere**. The mixed state needs to be described in the density matrix formulation of quantum mechanics, for example the following mixed state has $1/2$ probability to be in $|0\\rangle$ or $|1\\rangle$,\n", + "\n", + "$$\n", + "\\rho_{\\text{mixed}} = \\sum_i P_i |\\psi_i\\rangle\\langle\\psi_i| = \\frac{1}{2} |0\\rangle\\langle0| + \\frac{1}{2} |1\\rangle\\langle1| := \\frac{1}{2} \\begin{bmatrix} 1 \\\\ 0\\end{bmatrix} \\begin{bmatrix} 1 & 0 \\end{bmatrix} + \\frac{1}{2} \\begin{bmatrix} 0 \\\\ 1\\end{bmatrix} \\begin{bmatrix} 0 & 1 \\end{bmatrix} = \\frac{1}{2} \\begin{bmatrix} 1 & 0\\\\ 0 & 1 \\end{bmatrix}.\n", + "\\tag{6}\n", + "$$\n", + "\n", + "The row vector (bra) $\\langle0| = |0\\rangle^\\dagger$ is the conjugate transpose of the column vector (ket) $|0\\rangle$.\n", + "\n", + "**Note:** For more information, please refer to [Wikipedia](https://en.wikipedia.org/wiki/Qubit).\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## What is a quantum logic gate?\n", + "\n", + "In classical computers, we can apply basic logical operations (NOT gates, NAND gates, XOR gates, AND gates, and OR gates) on classical bits and combine them into more complicated operations. Quantum computing has a completely different set of logical operations, which are called quantum gates. We cannot compile existing C++ programs on a quantum computer. Because **classical computers and quantum computers have different logic gate structures, quantum algorithms need to be constructed using quantum gates**. Mathematically, a quantum gates can be expressed as a unitary matrix. Unitary operations could preserve vector length, which is a desirable property. Otherwise, if we operate on a pure state, it will be degraded into a mixed state, making it unreliable for the following running time. The unitary matrix is defined as:\n", + "\n", + "$$\n", + "U^{\\dagger}U = UU^{\\dagger} = I,\n", + "\\quad \\text{and} \\quad\n", + "\\Vert |\\psi\\rangle \\Vert = \\Vert U|\\psi\\rangle\\Vert = 1.\n", + "\\tag{7}\n", + "$$\n", + "\n", + "where $U^{\\dagger}$ is the conjugate transpose of $U$, and $I$ represents the identity matrix. But what is the physical meaning of representing quantum gates as unitary matrices? This implies that all quantum gates must be reversible. For any gate logic $U$, one can always find the corresponding reverse operation $U^\\dagger$. In addition, the unitary matrix must be a square matrix, because the input and output of the quantum operation require the same number of qubits. A quantum gate acting on $n$ qubits can be written as a $2^n \\times 2^n$ unitary matrix. The most common quantum gates act on one or two qubits, just like classical logic gates.\n", + "\n", + "### Single-qubit gate\n", + "\n", + "Next, we introduce single-qubit gates in quantum computing, including the Pauli matrices $\\{X, Y, Z\\}$, single-bit rotation gates $\\{R_x, R_y, R_z\\}$ and the Hadamard gate $H$. Firstly, **NOT gate** is important for both classical and quantum computing,\n", + "\n", + "$$\n", + "X := \\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix}.\n", + "\\tag{8}\n", + "$$\n", + "\n", + "This quantum gate (unitary matrix) acts on the state of a single qubit (a complex vector). The operation is essentially **multiplication between a matrix and a column vector**:\n", + "\n", + "$$\n", + "X |0\\rangle := \\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix} \\begin{bmatrix} 1 \\\\0 \\end{bmatrix}\n", + "=\\begin{bmatrix} 0 \\\\1 \\end{bmatrix} = |1\\rangle,\n", + "\\quad \\text{and} \\quad\n", + "X |1\\rangle := \\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix} \\begin{bmatrix} 0 \\\\1 \\end{bmatrix}\n", + "=\\begin{bmatrix} 1 \\\\0 \\end{bmatrix}=|0\\rangle.\n", + "\\tag{9}\n", + "$$\n", + "\n", + "Recall the Bloch sphere representation, this operation $X$ acting on a qubit state (a point on the Bloch sphere) is equivalent to **a rotation about the $x$ axis in the Bloch sphere with angle $\\pi$** . This is why $X$ can be expressed as $R_x(\\pi)$ (differ by a global phase $e^{-i\\pi/2} = -i$ ). The other two Pauli matrices $Y$ and $Z$ are very similar in this sense (representing rotation around the $y$ and $z$ axes with angle $\\pi$):\n", + "\n", + "$$\n", + "Y := \\begin{bmatrix} 0 &-i \\\\ i &0 \\end{bmatrix},\n", + "\\quad \\text{and} \\quad \n", + "Z := \\begin{bmatrix} 1 &0 \\\\ 0 &-1 \\end{bmatrix}.\n", + "\\tag{10}\n", + "$$\n", + "\n", + "Generally speaking, any quantum gate that rotates $\\theta$ on the corresponding axis on the Bloch sphere can be expressed as\n", + "\n", + "$$\n", + "R_x(\\theta) := \n", + "\\begin{bmatrix} \n", + "\\cos \\frac{\\theta}{2} &-i\\sin \\frac{\\theta}{2} \\\\ \n", + "-i\\sin \\frac{\\theta}{2} &\\cos \\frac{\\theta}{2} \n", + "\\end{bmatrix}\n", + ",\\quad \n", + "R_y(\\theta) := \n", + "\\begin{bmatrix}\n", + "\\cos \\frac{\\theta}{2} &-\\sin \\frac{\\theta}{2} \\\\ \n", + "\\sin \\frac{\\theta}{2} &\\cos \\frac{\\theta}{2} \n", + "\\end{bmatrix}\n", + ",\\quad \n", + "R_z(\\theta) := \n", + "\\begin{bmatrix}\n", + "e^{-i\\frac{\\theta}{2}} & 0 \\\\ \n", + "0 & e^{i\\frac{\\theta}{2}}\n", + "\\end{bmatrix}.\n", + "\\tag{11}\n", + "$$\n", + "\n", + "In addition to the rotation gates, the most important single-qubit gate is the Hadamard gate. The corresponding Bloch spherical interpretation consists of two separate rotations, first rotating $\\pi$ around the $z$-axis, and then rotating $\\pi/2$ around the $y$-axis. Its matrix representation is\n", + "\n", + "$$\n", + "H := \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1 &1 \\\\ 1 &-1 \\end{bmatrix}.\n", + "\\tag{12}\n", + "$$\n", + "\n", + "### Two-bit quantum gate\n", + "\n", + "We can expand the idea of single-qubit gates to multi-qubit. There are two ways to realize this expansion. The first is to apply single-qubit gates on selected qubits, while the other qubits are not operated. The figure below gives a concrete example:\n", + "\n", + "![intro-fig-hadamard](./figures/intro-fig-hadamard.png \"**Figure 2.** Circuit representation and interpretation of two-qubit logic operations. [[Picture source]](https://en.wikipedia.org/wiki/Quantum_logic_gate)\")\n", + "\n", + "The quantum gate acting on two-qubit system can be expressed as a $4\\times4$ unitary matrix\n", + "\n", + "$$\n", + "U = H \\otimes I \n", + "= \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 &1 \\\\ 1 &-1 \\end{bmatrix} \n", + "\\otimes \\begin{bmatrix} 1 &0 \\\\ 0 &1 \\end{bmatrix} \n", + "= \\frac{1}{\\sqrt{2}} \\,\n", + "\\begin{bmatrix}\n", + "1 &0 &1 &0 \\\\ \n", + "0 &1 &0 &1 \\\\\n", + "1 &0 &-1 &0 \\\\\n", + "0 &1 &0 &-1 \n", + "\\end{bmatrix}.\n", + "\\tag{13}\n", + "$$\n", + "\n", + "Another way is to apply two-qubit gates directly. For example, $\\text{CNOT}$ gate will make the state of one qubit affect another qubit state.\n", + "\n", + "$$\n", + "\\text{CNOT} :=\n", + "\\begin{bmatrix}\n", + "1 &0 &0 &0 \\\\\n", + "0 &1 &0 &0 \\\\\n", + "0 &0 &0 &1 \\\\\n", + "0 &0 &1 &0\n", + "\\end{bmatrix},\n", + "\\tag{14}\n", + "$$\n", + "\n", + "When $\\text{CNOT}$ acts on the computational basis, we have\n", + "\n", + "$$\n", + "\\text{CNOT} |00\\rangle = |00\\rangle, \\quad\n", + "\\text{CNOT} |01\\rangle = |01\\rangle, \\quad\n", + "\\text{CNOT} |10\\rangle = |11\\rangle, \\quad\n", + "\\text{CNOT} |11\\rangle = |10\\rangle.\n", + "\\tag{15}\n", + "$$\n", + "\n", + "We can conclude that when the first qubit is in the $|1\\rangle$ state, $\\text{CNOT}$ will act $X$ gate on the second qubit. If the first qubit is in the $|0\\rangle$ state, then the second qubit is not affected in any way. This is why $\\text{CNOT}$ stands for the controlled-$\\text{NOT}$ gate. The following list contains frequently used quantum gates and their matrix representations. **All of these quantum gates can be called in Paddle Quantum**.\n", + "\n", + "![intro-fig-gates](./figures/intro-fig-gates.png \"**Figure 3.** List of common quantum gates. [[Image source]](https://en.wikipedia.org/wiki/Quantum_logic_gate)\")\n", + "\n", + "**Note**: For more information, please see the following Wikipedia [link](https://en.wikipedia.org/wiki/Quantum_logic_gate)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## What is measurement in quantum mechanics?\n", + "\n", + "For a two-level quantum system, such as the spin of an electron, it can be spin up $\\uparrow$ or spin down $\\downarrow$, corresponding to state $|0\\rangle$ and state $|1\\rangle$. As mentioned before, the electron can be in a superposition state of spin up and down, which is $|\\psi\\rangle =\\alpha |0\\rangle + \\beta |1\\rangle$. The measurement will help us further understand what is a superposition state. It is worth noting that the measurement in quantum mechanics usually refers to a statistical result rather than a single measurement. This is due to the nature of measurements in quantum physics, which collapses the observed quantum state. For example, if we measure an electron in state $|\\psi\\rangle =\\alpha |0\\rangle + \\beta |1\\rangle$, we will have a probability of $|\\alpha|^2$ to obtain the measurement results of spin up, and after measurement, the quantum state collapses to the post-measurement state $ |0\\rangle$. Similarly, we also have a probability of $|\\beta|^2$ to get the spin down post-measurement state $|1\\rangle$. So if we want to get the value of $\\alpha$ accurately, one experiment is obviously not enough. We need to prepare a lot of electrons in the superposition state $\\alpha |0\\rangle + \\beta |1\\rangle$, measure the spin of each electron, and then count the frequency. Measurement has a special place in quantum mechanics. If the reader finds it difficult to understand, we refer to [Measurement in Quantum Mechanics](https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics) for more information." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example and exercise\n", + "\n", + "### Example: Using Paddle Quantum to create a $X$ gate\n", + "\n", + "**Note:** All single-bit rotation gates are established as follows:\n", + "\n", + "$$\n", + "R_x(\\theta) :=\n", + "\\begin{bmatrix}\n", + "\\cos \\frac{\\theta}{2} &-i\\sin \\frac{\\theta}{2} \\\\\n", + "-i\\sin \\frac{\\theta}{2} &\\cos \\frac{\\theta}{2}\n", + "\\end{bmatrix}\n", + ",\\quad\n", + "R_y(\\theta) :=\n", + "\\begin{bmatrix}\n", + "\\cos \\frac{\\theta}{2} &-\\sin \\frac{\\theta}{2} \\\\\n", + "\\sin \\frac{\\theta}{2} &\\cos \\frac{\\theta}{2}\n", + "\\end{bmatrix}\n", + ",\\quad\n", + "R_z(\\theta) :=\n", + "\\begin{bmatrix}\n", + "e^{-i\\frac{\\theta}{2}} & 0 \\\\\n", + "0 & e^{i\\frac{\\theta}{2}}\n", + "\\end{bmatrix}.\n", + "\\tag{16}\n", + "$$\n", + "\n", + "Therefore, it is not difficult to see that the $X$ gate can be expressed as $R_x(\\pi)$. The following code will generate the $X$ gate:\n", + "\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:57:05.944723Z", + "start_time": "2021-01-09T12:57:02.038199Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The matrix representation of quantum gate is:\n", + "[[ 6.123234e-17+0.j -6.123234e-17-1.j]\n", + " [ 6.123234e-17-1.j 6.123234e-17+0.j]]\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "from paddle import fluid\n", + "from paddle_quantum.circuit import UAnsatz\n", + "\n", + "# Set the angle parameter theta = pi\n", + "theta = np.array([np.pi])\n", + "\n", + "# Initiate Paddle dynamic graph mode\n", + "with fluid.dygraph.guard():\n", + " \n", + " # We need to convert numpy.ndarray to variable supported in Paddle dynamic graph mode\n", + " theta = fluid.dygraph.to_variable(theta)\n", + " \n", + " # Set the number of qubits required for calculation\n", + " num_qubits = 1\n", + " \n", + " # Initialize our single-bit quantum circuit\n", + " cir = UAnsatz(num_qubits)\n", + " \n", + " # Apply an Rx rotation gate to the first qubit (q0), the angle is pi\n", + " which_qubit = 0\n", + " cir.rx(theta, which_qubit)\n", + " \n", + " # Convert to numpy.ndarray\n", + " # Print out this quantum gate\n", + " print('The matrix representation of quantum gate is:')\n", + " print(cir.U.numpy())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "There is a global phase $-i$ in front between the output and the $X$ (NOT) gate:\n", + "\n", + "$$\n", + "\\text{output} = \\begin{bmatrix} 0 &-i \\\\ -i &0 \\end{bmatrix}\n", + "= -i\\begin{bmatrix} 0 &1 \\\\ 1 &0 \\end{bmatrix} = -i X.\n", + "\\tag{17}\n", + "$$\n", + "\n", + "Can you figure out why such a global phase is not important in quantum computing? And not important in what sense?" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Exercise: Create a $Y$ gate\n", + "\n", + "Similar to the $X$ gate, try to create a $Y$ gate by filling the following code. " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "theta = \"your code\"\n", + "\n", + "with fluid.dygraph.guard(): \n", + " theta = fluid.dygraph.to_variable(theta)\n", + " num_qubits = 1\n", + " cir = UAnsatz(\"your code\")\n", + " cir.ry(\"your code\")\n", + " print(cir.U.numpy())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "As mentioned before, we have a global phase $-i$ in front:\n", + "\n", + "$$\n", + "\\text{output} = \\begin{bmatrix} 0 &-1 \\\\ 1 &0 \\end{bmatrix}\n", + "= -i\\begin{bmatrix} 0 &-i \\\\ i &0 \\end{bmatrix} = -i Y.\n", + "\\tag{18}\n", + "$$\n", + "\n", + "---" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Quantum Neural Network\n", + "\n", + "After the preparations above, we now have the necessary knowledge to have a taste on quantum machine learning (QML). QML uses quantum circuits to replace classical neural networks to complete machine learning tasks. So we need to prepare a parameterized quantum circuit (PQC), which is also called Quantum neural network (QNN) or Ansatz. The quantum circuit parameters are adjustable (usually these parameters are the angles $\\theta$ of rotation gates). As we have seen in the last section, using angle $\\pi$ to create a $X$ gate is probably the simplest QNN. If we design an appropriate loss function, then certain computational tasks can be transformed into optimization problems. Keep adjusting the parameters in PQC until the loss function convergences. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example: How to create a quantum neural network?\n", + "\n", + "QNN can usually be expressed as a combination of single-qubit gates and two-qubit gates. One of the widely used circuit architectures is the hardware-efficient ansatz consists of $\\{R_x, R_y, R_z, \\text{CNOT}_{j,j+1} \\}$. They are easy to implement on near-term devices (usually superconducting qubits) because $\\text{CNOT}_{j,j+1} $ only works on adjacent qubits and hence mitigate the topological connectivity issues. The figure below gives us a concrete example:\n", + "\n", + "![intro-fig-gate1](./figures/intro-fig-gate1.png)\n", + "\n", + "Each horizontal line here represents a qubit. We define the upper qubit to be the first qubit $q_0$; the lower one is the second qubit $q_1$. From left to right, it represents the order that we apply quantum gates. The leftmost quantum gate will be applied first. Next, let’s take a look on how to build this simple two-qubit quantum neural network on Paddle Quantum." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:58:40.113133Z", + "start_time": "2021-01-09T12:58:40.081753Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The matrix representation of the quantum neural network U(theta=pi) in the figure is:\n", + "[[ 0.0000000e+00 -1.0000000e+00 6.1232340e-17 -6.1232340e-17]\n", + " [-1.0000000e+00 0.0000000e+00 -6.1232340e-17 6.1232340e-17]\n", + " [-6.1232340e-17 6.1232340e-17 1.0000000e+00 1.2246468e-16]\n", + " [ 6.1232340e-17 -6.1232340e-17 -1.2246468e-16 1.0000000e+00]]\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "from paddle import fluid\n", + "from paddle_quantum.circuit import UAnsatz\n", + "\n", + "# Set the angle parameter theta\n", + "theta = np.full([4], np.pi)\n", + "\n", + "# Start Paddle dynamic graph mode\n", + "with fluid.dygraph.guard():\n", + " \n", + " # We need to convert numpy.ndarray to variable supported in Paddle dynamic graph mode\n", + " theta = fluid.dygraph.to_variable(theta)\n", + " \n", + " # Initialize the quantum circuit\n", + " num_qubits = 2\n", + " cir = UAnsatz(num_qubits)\n", + " \n", + " # Add single-qubit rotation gates\n", + " cir.ry(theta[0], 0)\n", + " cir.ry(theta[1], 1)\n", + "\n", + " # Add two-qubit gate\n", + " cir.cnot([0, 1])\n", + "\n", + " # Add single-qubit rotation gates\n", + " cir.ry(theta[2], 0)\n", + " cir.ry(theta[3], 1)\n", + " \n", + " print('The matrix representation of the quantum neural network U(theta=pi) in the figure is:')\n", + " print(cir.U.numpy().real)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\n", + "\\text{output} = \n", + "\\begin{bmatrix} \n", + "0 &-1 &0 &0 \\\\ \n", + "-1 &0 &0 &0 \\\\\n", + "0 &0 &1 &0 \\\\\n", + "0 &0 &0 &1 \n", + "\\end{bmatrix}.\n", + "\\tag{19}\n", + "$$\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exercise\n", + "\n", + "Given the following code, can you work out the corresponding circuit?" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "theta = np.full([6], np.pi)\n", + "with fluid.dygraph.guard():\n", + " \n", + " theta = fluid.dygraph.to_variable(theta)\n", + " \n", + " num_qubits = 3\n", + " cir = UAnsatz(num_qubits)\n", + " \n", + " cir.ry(theta[0], 0)\n", + " cir.ry(theta[1], 1)\n", + " cir.ry(theta[2], 2)\n", + " \n", + " cir.cnot([0, 1])\n", + " cir.cnot([1, 2])\n", + "\n", + " cir.ry(theta[3], 0)\n", + " cir.ry(theta[4], 1)\n", + " cir.ry(theta[5], 2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Answer:\n", + "\n", + "![intro-fig-gate2](./figures/intro-fig-gate2.png)\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Built-in circuit templates\n", + "\n", + "In the latest version of Paddle Quantum, we provide some built-in circuit templates to make users' life easier." + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:59:37.481620Z", + "start_time": "2021-01-09T12:59:37.040582Z" + }, + "scrolled": true + }, + "outputs": [ + { + "data": { + "image/png": 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2+VpgjyS7DzGrJKnPMAviAODWnvcb2mWd61TVJuAeYL++dV4JXF5VP+//BklOSrImyZqNGzdus+CSpDGfpE5yMM1hpzd2jVfViqpaUlVL5s2bN7fhJGk7N8yCuA1Y0PN+frusc50kuwB7AXe07+cDnwdeV1XfH2JOSVKHYRbEpcCiJAcm2Q1YBqzqW2cVzSQ0wDHARVVVSfYGLgCWV9W3hphRkjSJoRVEO6dwMnAhcC1wXlWtTXJakpe3q50F7JdkPfAWYPOpsCcDBwGnJrmi/Xj8sLJKkibaZZhfvKpWA6v7lp3a8/p+4NiO7d4LvHeY2SRJUxvrSWpJ0uhYEJKkThaEJKmTBSFJ6mRBSJI6WRCSpE4WhCSpkwUhSepkQUiSOlkQkqROFoQkqZMFIUnqZEFIkjpZEJKkThaEJKmTBSFJ6mRBSJI6WRCSpE4WhCSpkwUhSepkQUiSOlkQkqROFoQkqZMFIUnqNNSCSHJkknVJ1idZ3jG+e5Jz2/FLkizsGXtHu3xdkpcMM6ckaaKhFUSSnYEzgaOAxcBxSRb3rXYicFdVHQScAZzebrsYWAYcDBwJfLT9epKkOTLMPYjDgPVVdUNVPQCsBJb2rbMUOLt9fT5weJK0y1dW1c+r6kZgffv1JElzZJchfu0DgFt73m8AnjPZOlW1Kck9wH7t8ov7tj2g/xskOQk4qX17b5J12yb6jD0OuH22G+f0bZhkIrPNjtlmx2yzM8psT55sYJgFMXRVtQJYMeocSdZU1ZJR5+hittkx2+yYbXbGNdswDzHdBizoeT+/Xda5TpJdgL2AOwbcVpI0RMMsiEuBRUkOTLIbzaTzqr51VgHHt6+PAS6qqmqXL2vPcjoQWAR8Z4hZJUl9hnaIqZ1TOBm4ENgZ+HhVrU1yGrCmqlYBZwHnJFkP3ElTIrTrnQdcA2wC3lxVvxxW1m1g5Ie5pmC22THb7JhtdsYyW5p/2CVJ2ppXUkuSOlkQkqROFoQkqZMF8TAk2TfJvqPOIUnDYEHMUJInJVmZZCNwCfCdJD9uly0ccbyxl2T/JM9sP/YfdZ7pJNlz1BmkUfEsphlK8m3gQ8D5m0+9bW8keCxwSlU9d4TxJpXk6qo6ZITf/1Dg72guhtx80eN84G7gTVV1+WiSTS3JLVX1pDHIsT9bbjdzW1X96yjzDCLJnlV174gzhOY+bg/97oDv1Bj/4UvytKq6btQ5wIKYsSTXV9WimY7NhSS/P9kQ8HdVNW8u82wVILkCeGNVXdK3/LnAx6rqGSMJ1mR4y2RDwJ9X1cgOIz5SixVGX65JjgA+ClzP1r+7g2h+d18ZVbapjPr31usRfS+mEbksyUdp7kK7+WaEC2iuCP/uyFI1zgX+B9DV+o+a4yz9HtNfDgBVdXGSx4wiUI//AvwNzUWZ/UZ9GPaTTF6snwBGVqxtjqnKddSH5/4WeFFV3dS7sL07w2rgN0YRqs3w4cmGgL3nMMqULIiZex3Ncyzew9a7rZuvDB+lq4D3V9W/9A8kedEI8vT6UpILgE+xdbG+DvjyyFI1Lge+UFWX9Q8kecMI8vQa52KF8S7XXWjuBN3vNmDXOc7S7/XAW4Gfd4wdN8dZJuUhpu1Ikt8Fbq6qWzrGllTVmhHE6s1wFM2zPrYq1qpaPbpUkOSpwB1VNeF2y0n2H+Xx/vY/zV+nu1hvrKqTR5UNIMk/A/9xknK9taoWdGw2J5K8A3gVzbNoen93y4DzquqvRpjtIuCdVfXPHWM3VtWBI4g1gQUxQ+1dZ08EjmbrP3RfBM6qql+MKJq2U+NarPBQud5ZVRs7xkZarm2G36D7d3fN6FI1p8gD91fVfaPMMR0LYoaSfJZmgvBstuy+zqeZg9i3ql49omi95fUK4Int4rEvryQrquqk6dece+OcTRo2C2KGknyvqp4y07G5MOblNdmZQAGurKr5c5lnqwDjnW0v4B00/wXvT3MCwo9pSv99VXX3qLLBVvmOBh7PmOWbTJIvVdVRo87RZZyyOUk9c3cmORb4h6p6ECDJTjTXQdw10mTwrI6C2gBcnOR7owjUYyNwM80f3c2qff/4kSTaYpyznQdcBLygqn4EkORXgRPasSNGFw3Yku/5ffmOZ8T5kjxzsiHg0DmMMjHAGGfr5R7EDLVXS58OvIDmv3VoTkv7OrC8qm4cSTAgycXAB+gur7dUVf8zwecy2/XA4ZNMoI96MnOcs62rqqfOdGyujHO+JL8E/omti3+z51bVHnMc6SHjnK2XexAzVFU3JXk3zTUPW01Sj7IcWstoyuvMJHe3y/amKa9lI8q02YeAfYAJf4SBv57bKBN8iPHNdnOStwNnb57wba+qPoEtZ+aM0jjnu5bmGpLr+weSmG0A7kHMUJI/o/lju5Ktr85cBqysqveNKhtMetbGF6vq2tGlaiR5Gt1nlJhtEkn2AZbTZNt8uOtfaa67eV9VjfSw5jjnS3IMcHVVresYO7qqvjD3qR76/mObrZcFMUPtsfyD+88Iap+7vXbEt9oY2/Jq/8t8TZutdwLdbLOU5PVV9YlR55jMOOcz22AsiBlKch3wkqq6uW/5k4GvjPiY6ziXl9m2sXG6Z0+Xcc5ntsE4BzFzpwBfayc2Nx8rfBLNDcBGelUr8CDN9Q839y1/Qjs2SmabhSRXTTZEc9rrSI1zPrM9fBbEDFXVl5M8hYm3EL508+2/R+gUxre8TsFss7E/8BImnkIdYMJtGkZgnPOZ7WGyIGahPYX04lHn6DfO5WW2WftHYM+quqJ/IMk35jzNROOcz2wPk3MQkqROo74dryRpTFkQkqROFoR2WEnmJ/likuuT3JDkI0l2H2C7zucsJzlt84OZkpyS5NGTrPfSJN9NcmWSa5K8sV1+dJLFA3z/gdaTHi4LQjukJAE+R/MkuUXAImAPHsatNarq1Kr63+3bU4AJBZFkV2AF8LL2Ody/BXyjHT4aGOQP/6DrSQ+Lk9TaISU5HHhXVT2vZ9mv0FwLsQA4Bliy+YltSf6R5nGu32j3IP6e5k6lPwKWVdXGJJ+kOTvlicD7gXXA7VX1gp7vsS9wHfDkqvp/Pct/u932nvbjlcALgZOA3YD1wB/Q3Omzfz2AM4F5wH3AH1bVddvkF6UdmnsQ2lEdDGz1mMyq+glwE831D1N5DLCmqg6muSPnu/q+zoeBH9DcovsFfWN30tyn6OYkn03y2iQ7tY+eXAX8aVUdWlXfBz5XVc9u9zSuBU6cZL0VNI/9fBbwNuCjM/5tSB28DkKauQeBc9vXn6Y5VDWwqnpDkkOAF9H8QX8xzd1P+/1mkvfS3JF3T+DC/hWS7An8NvA/m6NmAEw7jyINwoLQjuoamsNID2kPMf0qzaGh32TrPexHTfG1ZnyctqquBq5Ocg5wI90F8Ung6Kq6MskJwPM71tkJuLuqDp1pBmk6HmLSjuprwKOTvA4gyc40D1v6SDs3cBNwaJKdkiygucp6s53YUi6vAf5vx9f/KfDY/oVJ9kzy/J5Fh7LlHlD92zwW+GE7sf3arq/dHha7sX3KIWk8Y6ofXBqUBaEdUjVnZ7wCOKa9B9MdwINV9Z/bVb5F85/9NcCHgct7Nv8ZcFiSf6GZSD6t41usAL6c5Ot9ywO8Pcm6JFcA72HL3sNK4E/bU2B/HfgL4JI2S++kc/96rwVOTHIlsJbm2QzSw+ZZTBIPnUX0WeAVVXX5dOtLOwILQpLUyUNMkqROFoQkqZMFIUnqZEFIkjpZEJKkThaEJKnT/weRuZs3hsDuNQAAAABJRU5ErkJggg==\n", + "text/plain": [ + "

" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "N = 3 # Set the number of qubits\n", + "\n", + "# Start PaddlePaddle dynamic graph mode\n", + "with fluid.dygraph.guard():\n", + " \n", + " # Initialize the quantum circuit\n", + " cir = UAnsatz(N)\n", + " \n", + " # Apply Hadamard gate on each qubit\n", + " cir.superposition_layer()\n", + " \n", + " # Prepare output state\n", + " # If the user does not enter the initial quantum state, the default initial is |00..0>\n", + " final_state = cir.run_state_vector()\n", + " \n", + " # Get the theoretical value of the probability distribution, set shots = 0\n", + " cir.measure(shots = 0, plot = True)" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T12:59:50.443828Z", + "start_time": "2021-01-09T12:59:50.101308Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "N = 3 # Set the number of qubits\n", + "\n", + "# Start PaddlePaddle dynamic graph mode\n", + "with fluid.dygraph.guard():\n", + " \n", + " # Initialize the quantum circuit\n", + " cir = UAnsatz(N)\n", + " \n", + " # Apply Ry(pi/4) rotation gate on each qubit\n", + " cir.weak_superposition_layer()\n", + " \n", + " # Prepare output state\n", + " # If the user does not enter the initial quantum state, the default initial state is |00..0>\n", + " final_state = cir.run_state_vector()\n", + " \n", + " # Get the theoretical value of the probability distribution, set shots = 0\n", + " cir.measure(shots = 0, plot = True)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The following figure depicts a handy circuit template `complex_entangled_layer(theta, DEPTH)` . Users can extend the circuit architecture by changing the circuit depth parameter `DEPTH`. Define generalized rotation gate $U_3$ as\n", + "\n", + "$$\n", + "U_3(\\theta, \\phi, \\varphi) :=\n", + "\\begin{bmatrix}\n", + "\\cos \\frac{\\theta}{2} & -e^{i \\varphi}\\sin \\frac{\\theta}{2} \\\\\n", + "e^{i \\phi}\\sin \\frac{\\theta}{2} &e^{i (\\phi+\\varphi)} \\cos \\frac{\\theta}{2}\n", + "\\end{bmatrix}.\n", + "\\tag{20}\n", + "$$\n", + "\n", + "The $U_3$ rotation gate is equivalent to the combination of three different rotation gates:\n", + "\n", + "$$\n", + "U_3(\\theta, \\phi, \\varphi) \n", + "= R_z(\\phi)*R_y(\\theta)*R_z(\\varphi)\n", + ":=\n", + "\\begin{bmatrix}\n", + "e^{-i\\frac{\\phi}{2}} & 0 \\\\ \n", + "0 & e^{i\\frac{\\phi}{2}}\n", + "\\end{bmatrix}\n", + "\\begin{bmatrix}\n", + "\\cos \\frac{\\theta}{2} &-\\sin \\frac{\\theta}{2} \\\\ \n", + "\\sin \\frac{\\theta}{2} &\\cos \\frac{\\theta}{2} \n", + "\\end{bmatrix}\n", + "\\begin{bmatrix}\n", + "e^{-i\\frac{\\varphi}{2}} & 0 \\\\ \n", + "0 & e^{i\\frac{\\varphi}{2}}\n", + "\\end{bmatrix}.\n", + "\\tag{21}\n", + "$$\n", + "\n", + "![intro-fig-complex_entangled_layer2](./figures/intro-fig-complex_entangled_layer2.png)\n", + "\n", + "When our task does not involve imaginary numbers, it is more efficient to use the circuit template `real_entangled_layer(theta, DEPTH)` ($R_y$ instead of $U_3$).\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:00:30.035538Z", + "start_time": "2021-01-09T13:00:29.585215Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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\n", 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" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "N = 4 # Set the number of qubits\n", + "DEPTH = 6 # Set the quantum circuit depth\n", + "theta = np.random.randn(DEPTH, N, 3)\n", + "\n", + "# Start Paddle dynamic graph mode\n", + "with fluid.dygraph.guard():\n", + " \n", + " # We need to convert numpy.ndarray to variable supported in Paddle dynamic graph mode\n", + " theta = fluid.dygraph.to_variable(theta)\n", + " \n", + " # Initialize the quantum circuit\n", + " cir = UAnsatz(N)\n", + " \n", + " # Add a complex strong entanglement structure QNN with depth D = 6 {Rz+Ry+Rz/U3 + CNOT's}\n", + " cir.complex_entangled_layer(theta, DEPTH)\n", + " \n", + " # Prepare output state\n", + " # If the user does not enter the initial quantum state, the default initial is |00..0>\n", + " final_state = cir.run_state_vector()\n", + " \n", + " # Measure the output state [0, 1, 2] qubits 2048 times, and count the frequency of the measurement results\n", + " cir.measure(shots = 2048, which_qubits = [0, 1, 2], plot = True)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "---\n", + "\n", + "# Operating Mode of Paddle Quantum\n", + "\n", + "## Wave function vector mode\n", + "\n", + "The so-called wave function mode is to use complex vectors to represent and store the quantum states. Vector mode can only handle pure states, but this mode efficiently supports **20+ qubit** operations on personal computer hardware. Users can test the limits of their computers. Under this representation, the essential operation of the quantum gate (unitary matrix) acting on qubits (a complex vector to describe the state) is **multiplying a matrix by a vector**:\n", + "\n", + "$$\n", + "|\\psi\\rangle = U |\\psi_0\\rangle.\n", + "\\tag{22}\n", + "$$\n", + "\n", + "A function `cir.run_state_vector(input_state = None)` will be used in the following code. If we don't enter any initial quantum state, all qubits will be set into the $\\lvert {0}\\rangle$ state by default. Let's take a specific example:" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:02:13.332713Z", + "start_time": "2021-01-09T13:01:13.592149Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[ 0.00072194+0.j 0.00044133+0.j 0.00064311+0.j ... -0.00045617+0.j\n", + " 0.00025925+0.j 0.00080118+0.j]]\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "from paddle import fluid\n", + "from paddle_quantum.circuit import UAnsatz\n", + "from paddle_quantum.state import vec, vec_random\n", + "\n", + "N = 20 # Set the number of qubits\n", + "DEPTH = 6 # Set the quantum circuit depth\n", + "theta = np.random.randn(DEPTH, N, 1)\n", + "\n", + "# Call the built-in |00..0> initial state\n", + "initial_state1 = vec(N)\n", + "# Call the built-in random quantum state |psi>\n", + "initial_state2 = vec_random(N)\n", + "\n", + "# Start Paddle dynamic graph mode\n", + "with fluid.dygraph.guard():\n", + " \n", + " # We need to convert numpy.ndarray to variable supported in PaddlePaddle dynamic graph mode\n", + " theta = fluid.dygraph.to_variable(theta)\n", + " initial_state = fluid.dygraph.to_variable(initial_state1)\n", + " \n", + " # Initialize the quantum circuit\n", + " cir = UAnsatz(N)\n", + " \n", + " # Add a real entanglement structure QNN {Ry+CNOT's} with depth of DEPTH\n", + " cir.real_entangled_layer(theta, DEPTH)\n", + " \n", + " # Prepare output state\n", + " # If the user does not enter the initial quantum state, the default initial state is |00..0>\n", + " final_state = cir.run_state_vector(initial_state)\n", + " print(final_state.numpy())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "## Density matrix mode\n", + "\n", + "Paddle Quantum also supports the density matrix mode, which is to use density matrices $\\rho = \\sum_i P_i |\\psi_i\\rangle\\langle\\psi_i|$ to represent and store quantum states. This mode can supports **mixed state simulation** . But in density matrix mode, personal computer hardware can only support around 10 qubits. Please pay attention to this limitation. We are constantly optimizing the performance of the simulator in this mode. Under this representation, quantum gates (unitary matrices) acting on the quantum states (Hermitian matrix with a trace of 1) can be viewed as **matrix multiplication**:\n", + "\n", + "$$\n", + "\\rho = U \\rho_0 U^\\dagger.\n", + "\\tag{23}\n", + "$$\n", + "\n", + "Function `cir.run_density_matrix()` will be used in the following code. Here is an example:\n", + "\n" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:02:13.431007Z", + "start_time": "2021-01-09T13:02:13.337255Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[[ 2.58020326e-01+0.j -3.54982418e-01+0.j 2.55776343e-01+0.j\n", + " 3.43259769e-03+0.j]\n", + " [-3.54982418e-01+0.j 4.88382132e-01+0.j -3.51895163e-01+0.j\n", + " -4.72254201e-03+0.j]\n", + " [ 2.55776343e-01+0.j -3.51895163e-01+0.j 2.53551876e-01+0.j\n", + " 3.40274466e-03+0.j]\n", + " [ 3.43259769e-03+0.j -4.72254201e-03+0.j 3.40274466e-03+0.j\n", + " 4.56658865e-05+0.j]]\n" + ] + } + ], + "source": [ + "from paddle_quantum.state import density_op, density_op_random, completely_mixed_computational\n", + "\n", + "N = 2 # Set the number of qubits\n", + "DEPTH = 6 # Set the quantum circuit depth\n", + "theta = np.random.randn(DEPTH, N, 1)\n", + "\n", + "# Call the built-in |00..0><00..0| initial state\n", + "initial_state1 = density_op(N)\n", + "# Call the built-in random quantum state, you can specify whether to allow complex number elements and matrix rank\n", + "initial_state2 = density_op_random(N, real_or_complex=2, rank=4)\n", + "# Call the complete mixed state under the built-in calculation base\n", + "initial_state3 = completely_mixed_computational(N)\n", + "\n", + "# Start Paddle dynamic graph mode\n", + "with fluid.dygraph.guard():\n", + " \n", + " # We need to convert numpy.ndarray to variable supported in PaddlePaddle dynamic graph mode\n", + " theta = fluid.dygraph.to_variable(theta)\n", + " initial_state = fluid.dygraph.to_variable(initial_state1)\n", + " \n", + " # Initialize the quantum circuit\n", + " cir = UAnsatz(N)\n", + " \n", + " # Add a real number strong entanglement structure QNN {Ry+CNOT's} with depth of DEPTH\n", + " cir.real_entangled_layer(theta, DEPTH)\n", + " \n", + " # Prepare output state\n", + " # If the user does not enter the initial quantum state, the default initial is |00..0><00..0|\n", + " final_state = cir.run_density_matrix(initial_state)\n", + " print(final_state.numpy())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Exercise: How to prepare Bell states from computational basis\n", + "\n", + "Bell state is a widely used quantum entangled state, which can be expressed as\n", + "\n", + "$$\n", + "|\\Phi^+\\rangle = \\frac{1}{\\sqrt{2}} \\big(|00\\rangle + |11\\rangle\\big)\n", + "= \\frac{1}{\\sqrt{2}} \\,\n", + "\\begin{bmatrix}\n", + "1 \\\\\n", + "0 \\\\\n", + "0 \\\\\n", + "1\n", + "\\end{bmatrix}.\n", + "\\tag{24}\n", + "$$\n", + "\n", + "So how do we use Paddle Quantum to prepare a Bell state? We can use the following quantum circuit :\n", + "\n", + "![intro-fig-bell2](./figures/intro-fig-bell2.png)" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:02:13.775333Z", + "start_time": "2021-01-09T13:02:13.434980Z" + } + }, + "outputs": [ + { + "data": { + "image/png": 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" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The Bell state is:\n", + " [0.70710678+0.j 0. +0.j 0. +0.j 0.70710678+0.j]\n" + ] + } + ], + "source": [ + "# Start Paddle dynamic graph mode\n", + "with fluid.dygraph.guard():\n", + " \n", + " # Initialize the quantum circuit\n", + " cir = UAnsatz(2)\n", + "\n", + " # Add quantum gates\n", + " cir.h(0)\n", + " cir.cnot([0, 1])\n", + " \n", + " # If the user does not enter the initial quantum state, the default initial is |00..0>\n", + " output_state = cir.run_state_vector()\n", + " \n", + " # We measure the output state 2048 times and obtain the frequency distribution of the measurement results\n", + " cir.measure(shots = 2048, plot = True)\n", + " \n", + " print('The Bell state is:\\n', output_state.numpy())" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "---" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# PaddlePaddle Optimizer Tutorial\n", + "\n", + "## Example: Using gradient descent in PaddlePaddle to optimize multivariable functions\n", + "\n", + "\n", + "In this section, we will learn how to use the PaddlePaddle dynamic computational graph to find the minimum value of a multivariable function, for example,\n", + "\n", + "$$\n", + "\\min_{\\boldsymbol{\\theta}}\\mathcal{L}(\\theta_1, \\theta_2, \\theta_3)\n", + "= (\\theta_1)^2 + (\\theta_2)^2 + (\\theta_3)^2 + 10.\n", + "\\tag{25}\n", + "$$\n", + "\n", + "It is clear when $\\theta_1 = \\theta_2 = \\theta_3 = 0$, $\\mathcal{L}$ takes the minimum value of 10." + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:03:57.626603Z", + "start_time": "2021-01-09T13:03:57.398028Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The minimum value of the loss function is: 10.000000020981819\n" + ] + } + ], + "source": [ + "import numpy as np\n", + "from paddle import fluid\n", + "from paddle.complex import matmul\n", + "from paddle_quantum.utils import dagger\n", + "\n", + "# Set hyper parameter\n", + "theta_size = 3\n", + "ITR = 200 # Set the number of iterations\n", + "LR = 0.5 # Set the learning rate\n", + "SEED = 1 # Fix random number seed\n", + "\n", + "class Optimization_ex1(fluid.dygraph.Layer):\n", + " def __init__(self, shape, param_attr=fluid.initializer.Uniform(\n", + " low=-5., high=5., seed=SEED), dtype='float64'):\n", + " super(Optimization_ex1, self).__init__()\n", + " \n", + " # Initialize a list of learnable parameters with length theta_size\n", + " # Use the uniform distribution of [-5, 5] to fill the initial value\n", + " self.theta = self.create_parameter(\n", + " shape=shape, attr=param_attr, dtype=dtype, is_bias=False)\n", + "\n", + " # Define loss function and forward propagation mechanism\n", + " def forward(self):\n", + " loss = self.theta[0] ** 2 + self.theta[1] ** 2 + self.theta[2] ** 2 + 10\n", + " return loss\n", + " \n", + "# Record intermediate optimization results\n", + "loss_list = []\n", + "parameter_list = []\n", + "\n", + "# Initialize paddle dynamic computational graph \n", + "with fluid.dygraph.guard():\n", + " \n", + " # Define network dimensions\n", + " myLayer = Optimization_ex1([theta_size])\n", + " \n", + " # Use Adam optimizer to get relatively good convergence\n", + " # Of course you can change to SGD or RMSprop.\n", + " optimizer = fluid.optimizer.AdamOptimizer(\n", + " learning_rate = LR, parameter_list = myLayer.parameters())\n", + " \n", + " # Iteration of optimization\n", + " for itr in range(ITR):\n", + " \n", + " # Forward propagation calculates the loss function\n", + " loss = myLayer()[0]\n", + " \n", + " # Under the dynamic graph mode, backpropagation optimizes the loss function\n", + " loss.backward()\n", + " optimizer.minimize(loss)\n", + " myLayer.clear_gradients()\n", + " \n", + " # Record the learning process\n", + " loss_list.append(loss.numpy()[0])\n", + " parameter_list.append(myLayer.parameters()[0].numpy())\n", + " \n", + " print('The minimum value of the loss function is:', loss_list[-1])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Exercise: Finding Eigenvalues\n", + "\n", + "Next, let's try a more complicated loss function. First, we introduce a random Hermitian matrix $H$ whose **eigenvalues** are the diagonal elements of matrix $D$, where\n", + "\n", + "$$\n", + "D = \\begin{bmatrix} 0.2 &0 \\\\ 0 &0.8 \\end{bmatrix}.\n", + "\\tag{26}\n", + "$$\n", + "\n", + "Don't worry, we will help you generate this Hermitian matrix $H$.\n", + "\n", + "Initialize the parameter vector $\\boldsymbol{\\theta}$ to construct a simple linear operation $U(\\boldsymbol{\\theta}) = R_z(\\theta_1)*R_y(\\theta_2)*R_z(\\theta_3)$:\n", + "\n", + "$$\n", + "U(\\theta_1, \\theta_2, \\theta_3) =\n", + "\\begin{bmatrix}\n", + "e^{-i\\frac{\\theta_1}{2}} & 0 \\\\\n", + "0 & e^{i\\frac{\\theta_1}{2}}\n", + "\\end{bmatrix}\n", + "\\begin{bmatrix}\n", + "\\cos \\frac{\\theta_2}{2} &-\\sin \\frac{\\theta_2}{2} \\\\\n", + "\\sin \\frac{\\theta_2}{2} &\\cos \\frac{\\theta_2}{2}\n", + "\\end{bmatrix}\n", + "\\begin{bmatrix}\n", + "e^{-i\\frac{\\theta_3}{2}} & 0 \\\\\n", + "0 & e^{i\\frac{\\theta_3}{2}}\n", + "\\end{bmatrix}.\n", + "\\tag{27}\n", + "$$\n", + "\n", + "Multiply this matrix (ansatz) by $|0\\rangle$ to get a new 2-dimensional complex vector\n", + "\n", + "$$\n", + "|\\phi\\rangle = U(\\theta_1, \\theta_2, \\theta_3)|0\\rangle.\n", + "\\tag{28}\n", + "$$\n", + "\n", + "Then, we define the loss function as\n", + "\n", + "$$\n", + "\\min_{\\boldsymbol{\\theta}}\\mathcal{L}(\\theta_1, \\theta_2, \\theta_3)\n", + "= \\langle\\phi| H |\\phi\\rangle\n", + "= \\langle0| U^{\\dagger}H U |0\\rangle.\n", + "\\tag{29}\n", + "$$\n", + "\n", + "Let's see what we got after optimization!" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:04:24.823063Z", + "start_time": "2021-01-09T13:04:24.807113Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The randomly generated matrix H according to the spectral decomposition is:\n", + "[[0.30704576+0.j 0.06071073-0.22154654j]\n", + " [0.06071073+0.22154654j 0.69295424+0.j ]] \n", + "\n", + "The eigenvalues of H are:\n", + "[0.2 0.8]\n" + ] + } + ], + "source": [ + "from scipy.stats import unitary_group\n", + "\n", + "# V is a 2x2 random unitary matrix\n", + "V = unitary_group.rvs(2)\n", + "\n", + "# The diagonal elements in D are the eigenvalue of H\n", + "# You can change the diagonal element value here\n", + "D = np.diag([0.2, 0.8])\n", + "\n", + "# V_dagger is the Hermitian transpose of V\n", + "V_dagger = V.conj().T\n", + "\n", + "# @: Represents matrix multiplication\n", + "H = (V @ D @ V_dagger)\n", + "print('The randomly generated matrix H according to the spectral decomposition is:')\n", + "print(H,'\\n')\n", + "print('The eigenvalues of H are:')\n", + "print(np.linalg.eigh(H)[0])" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:05:28.870563Z", + "start_time": "2021-01-09T13:05:28.864497Z" + } + }, + "outputs": [], + "source": [ + "# Hyper parameter settings\n", + "theta_size = 3 # set theta dimension\n", + "num_qubits = 1 # Set the number of qubits\n", + "ITR = 10 # Set the number of iterations\n", + "LR = 0.2 # Set the learning rate\n", + "SEED = 1 # Fix random seed for initializing theta parameter\n", + "\n", + "\n", + "# Set the circuit module separately\n", + "def U_theta(theta):\n", + " \n", + " # Initialize the circuit and add the quantum gates\n", + " cir = UAnsatz(num_qubits)\n", + " cir.rz(theta[0], 0)\n", + " cir.ry(theta[1], 0)\n", + " cir.rz(theta[2], 0)\n", + " \n", + " # Return parameterized matrix\n", + " return cir.U" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:05:29.349729Z", + "start_time": "2021-01-09T13:05:29.340787Z" + } + }, + "outputs": [], + "source": [ + "class Optimization_ex2(fluid.dygraph.Layer):\n", + " def __init__(self, shape, param_attr=fluid.initializer.Uniform(\n", + " low=0., high=2*np.pi, seed=SEED), dtype='float64'):\n", + " super(Optimization_ex2, self).__init__()\n", + " \n", + " # Initialize a list of trainable parameters with length theta_size\n", + " # And use the uniform distribution of [0, 2*pi] to fill the initial value\n", + " self.theta = self.create_parameter(\n", + " shape=shape, attr=param_attr, dtype=dtype, is_bias=False)\n", + " self.H = fluid.dygraph.to_variable(H)\n", + " \n", + " # Define loss function and forward propagation mechanism\n", + " def forward(self):\n", + " \n", + " # Get the unitary matrix representation of the quantum neural network\n", + " U = U_theta(self.theta)\n", + " \n", + " # Conjugate transpose operation\n", + " U_dagger = dagger(U)\n", + " \n", + " # Calculate the loss function\n", + " loss = matmul(U_dagger, matmul(self.H, U)).real[0][0]\n", + " \n", + " return loss" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:05:30.086892Z", + "start_time": "2021-01-09T13:05:29.956476Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "iter: 0 loss: 0.2064\n", + "iter: 1 loss: 0.2007\n", + "iter: 2 loss: 0.2033\n", + "iter: 3 loss: 0.2028\n", + "iter: 4 loss: 0.2013\n", + "iter: 5 loss: 0.2004\n", + "iter: 6 loss: 0.2008\n", + "iter: 7 loss: 0.2015\n", + "iter: 8 loss: 0.2016\n", + "iter: 9 loss: 0.2009\n", + "The minimum value of the loss function is: 0.20090539960637277\n" + ] + } + ], + "source": [ + "loss_list = []\n", + "parameter_list = []\n", + "\n", + "# Initialize paddle dynamic graph mechanism\n", + "with fluid.dygraph.guard():\n", + " \n", + " myLayer = Optimization_ex2([theta_size])\n", + " \n", + " # Generally speaking, we use Adam optimizer to get relatively good convergence\n", + " # Of course you can change to SGD or RMS prop.\n", + " optimizer = fluid.optimizer.AdamOptimizer(\n", + " learning_rate = LR, parameter_list = myLayer.parameters())\n", + " \n", + " # Optimization cycle\n", + " for itr in range(ITR):\n", + " \n", + " # Forward propagation calculates loss function\n", + " loss = myLayer()[0]\n", + " \n", + " # Under the dynamic graph mechanism, back propagation minimizes the loss function\n", + " loss.backward()\n", + " optimizer.minimize(loss)\n", + " myLayer.clear_gradients()\n", + " \n", + " # Record the learning curve\n", + " loss_list.append(loss.numpy()[0])\n", + " parameter_list.append(myLayer.parameters()[0].numpy())\n", + " print('iter:', itr, 'loss: %.4f'% loss.numpy())\n", + " \n", + " print('The minimum value of the loss function is:', loss_list[-1])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can change the eigenvalues of $H$. If the diagonal matrix after diagonalization is changed to\n", + "\n", + "$$\n", + "D = \\begin{bmatrix} 0.8 &0 \\\\ 0 &1.2 \\end{bmatrix}.\n", + "\\tag{30}\n", + "$$\n", + "\n", + "We still get the minimum eigenvalue of $\\lambda_{\\text{min}}(H)=0.8$. Can you find the reason behind it? Or is there any theory behind this?\n", + "\n", + "---" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Quantum Machine Learning Case Study\n", + "\n", + "## Variational Quantum Eigensolver - Unsupervised Learning\n", + "\n", + "At this stage, large-scale fault-tolerant quantum computers are still far from us. We can only build noisy intermediate-scale quantum (NISQ) computing devices. A promising type of algorithm suitable for NISQ devices is the quantum-classical hybrid algorithm, or variational quantum algorithms. People expect that this approach could surpass the performance of classical computers in certain applications. Among which, the Variational Quantum Eigensolver (VQE) is such an important application. It uses a parameterized circuit to search the vast Hilbert space and uses the gradient descent or other classical optimization methods to find the optimal parameters, which yield a state that is close to the ground state of a given Hamiltonian (find the minimum eigenvalue of a Hermitian matrix). Let's go through the following two-qubit example.\n", + "\n", + "Suppose we want to find the ground state of the following Hamiltonian:\n", + "\n", + "$$\n", + "H = 0.4 \\, Z \\otimes I + 0.4 \\, I \\otimes Z + 0.2 \\, X \\otimes X.\n", + "\\tag{31}\n", + "$$\n", + "\n", + "Given the following quantum neural network architecture\n", + "\n", + "![intro-fig-vqeAnsatz](./figures/intro-fig-vqeAnsatz.png)\n", + "\n", + "We have learned how to build this circuit. If you forget, please go to section **Quantum Neural Network**." + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:08:40.727331Z", + "start_time": "2021-01-09T13:08:40.721736Z" + } + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "from paddle import fluid\n", + "from paddle_quantum.circuit import UAnsatz\n", + "from paddle_quantum.utils import pauli_str_to_matrix\n", + "\n", + "# First generate the Hamiltonian under Pauli string representation\n", + "# H_info is equivalent to 0.4*kron(I, Z) + 0.4*kron(Z, I) + 0.2*kron(X, X)\n", + "# Among them, X, Y, Z are the Pauli matrix and I is the identity matrix\n", + "H_info = [[0.4,'z0'], [0.4,'z1'], [0.2,'x0,x1']]\n", + "\n", + "# Set hyper parameter\n", + "num_qubits = 2\n", + "theta_size = 4\n", + "ITR = 10\n", + "LR = 0.6\n", + "SEED = 1\n", + "\n", + "# Convert the Hamiltonian into matrix representation\n", + "H_matrix = pauli_str_to_matrix(H_info, num_qubits)" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:08:41.137003Z", + "start_time": "2021-01-09T13:08:41.125057Z" + } + }, + "outputs": [], + "source": [ + "class vqe_demo(fluid.dygraph.Layer):\n", + " def __init__(self, shape, param_attr=fluid.initializer.Uniform(\n", + " low=0., high=2*np.pi, seed=SEED), dtype='float64'):\n", + " super(vqe_demo, self).__init__()\n", + " \n", + " # Initialize a list of learnable parameters with length theta_size\n", + " # Use the uniform distribution of [0, 2*pi] to fill the initial value\n", + " self.theta = self.create_parameter(\n", + " shape=shape, attr=param_attr, dtype=dtype, is_bias=False)\n", + " \n", + " # Define loss function and forward propagation mechanism\n", + " def forward(self):\n", + " \n", + " # Initial quantum circuit\n", + " cir = UAnsatz(num_qubits)\n", + " \n", + " # Add quantum gates\n", + " cir.ry(self.theta[0], 0)\n", + " cir.ry(self.theta[1], 1)\n", + " cir.cnot([0, 1])\n", + " cir.ry(self.theta[2], 0)\n", + " cir.ry(self.theta[3], 1)\n", + " \n", + " # Choose state vector operation mode\n", + " cir.run_state_vector()\n", + " \n", + " # Calculate the expected value of H_info in the current quantum state\n", + " # The formula is given by \n", + " loss = cir.expecval(H_info)\n", + " \n", + " return loss" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "ExecuteTime": { + "end_time": "2021-01-09T13:08:44.573571Z", + "start_time": "2021-01-09T13:08:41.546193Z" + } + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "iter: 0 loss: 0.3195\n", + "iter: 1 loss: -0.4650\n", + "iter: 2 loss: -0.7123\n", + "iter: 3 loss: -0.7690\n", + "iter: 4 loss: -0.7925\n", + "iter: 5 loss: -0.8096\n", + "iter: 6 loss: -0.8186\n", + "iter: 7 loss: -0.8225\n", + "iter: 8 loss: -0.8239\n", + "iter: 9 loss: -0.8244\n", + "The calculated ground state energy is: -0.8243582047336566\n", + "The real ground state energy is: -0.8246211251235321\n" + ] + } + ], + "source": [ + "loss_list = []\n", + "parameter_list = []\n", + "\n", + "# Initialize paddle dynamic computational graph\n", + "with fluid.dygraph.guard():\n", + " \n", + " # Define network dimensions\n", + " vqe = vqe_demo([theta_size])\n", + " \n", + " # We usually use Adam optimizer to get relatively good convergence\n", + " # Of course you can change it to SGD, Adagrad, or RMS prop as we did here.\n", + " optimizer = fluid.optimizer.AdagradOptimizer(\n", + " learning_rate = LR, parameter_list = vqe.parameters())\n", + " \n", + " # Optimization cycle\n", + " for itr in range(ITR):\n", + " \n", + " # Forward propagation calculates loss function\n", + " loss = vqe()\n", + " \n", + " # Under the dynamic graph mechanism, back propagation minimizes the loss function\n", + " loss.backward()\n", + " optimizer.minimize(loss)\n", + " vqe.clear_gradients()\n", + " \n", + " # Record the learning curve\n", + " loss_list.append(loss.numpy()[0])\n", + " parameter_list.append(vqe.parameters()[0].numpy())\n", + " print('iter:', itr, 'loss: %.4f'% loss.numpy())\n", + " \n", + " \n", + " print('The calculated ground state energy is:', loss_list[-1])\n", + " print('The real ground state energy is:', np.linalg.eigh(H_matrix)[0][0])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "_______\n", + "\n", + "## References\n", + "\n", + "[1] Nielsen, M. A. & Chuang, I. L. Quantum computation and quantum information. (Cambridge university press, 2010).\n", + "\n", + "[2] Phillip Kaye, Laflamme, R. & Mosca, M. An Introduction to Quantum Computing. (2007).\n", + "\n", + "[3] Biamonte, J. et al. Quantum machine learning. [Nature 549, 195–202 (2017).](https://www.nature.com/articles/nature23474)\n", + "\n", + "[4] Schuld, M., Sinayskiy, I. & Petruccione, F. An introduction to quantum machine learning. [Contemp. Phys. 56, 172–185 (2015).](https://www.tandfonline.com/doi/abs/10.1080/00107514.2014.964942)\n", + "\n", + "[5] Benedetti, M., Lloyd, E., Sack, S. & Fiorentini, M. Parameterized quantum circuits as machine learning models. [Quantum Sci. Technol. 4, 043001 (2019).](https://iopscience.iop.org/article/10.1088/2058-9565/ab4eb5)\n", + "\n", + "[6] [Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 4213 (2014).](https://www.nature.com/articles/ncomms5213)\n", + "\n", + "[7] [McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New J. Phys. 18, 023023 (2016).](https://iopscience.iop.org/article/10.1088/1367-2630/18/2/023023)\n", + "\n", + "[8] [Kandala, A. et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature 549, 242–246 (2017).](https://www.nature.com/articles/nature23879)\n", + "\n", + "[9] [Mitarai, K., Negoro, M., Kitagawa, M. & Fujii, K. Quantum circuit learning. Phys. Rev. A 98, 032309 (2018).](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.98.032309)" + ] + } + ], + "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": "3.6.10" + }, + "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 +} diff --git a/introduction/figures/gate.png b/introduction/figures/gate.png deleted file mode 100644 index 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