用量子计算模拟器ProjectQ生成随机数，并用pytest进行单元测试与覆盖率测试

技术背景

本文中主要包含有三个领域的知识点：随机数的应用、量子计算模拟产生随机数与基于pytest框架的单元测试与覆盖率测试，这里先简单分别介绍一下背景知识。

随机数的应用

在上一篇介绍量子态模拟采样的算法中，我们就使用到了随机数，随机数在各种蒙特卡洛方法与数值计算中，扮演着非常重要的角色。在金融领域，随机数则是在加密算法中扮演重要角色，其风险在于，如果随机数可被预测，那么恶意用户就可以利用这一特点，破解对称加密甚至是非对称加密的算法(非对称加密如RSA算法的私钥的挑选其实也有可能使用到随机数)。在python代码中，可以通过调用random库或者numpy.random库进行随机数的生成，仅需要通过如下的代码即可实现：

import random
random_number = random.random() # 产生[0,1)之间的均匀随机数
import numpy as np
random_array = random.randn(10) # 产生10个随机数并存储到np格式的一维数组中

这些库所产生的随机数，往往被认为是不安全的随机数，其主要特点是通过随机数生成算法，结合硬件的一些独有编号来产生一系列数串。但是随机数生成算法一般都是已知的，因此还是有一定的潜在风险。有一个python库函数叫secrets可以产生"安全随机数"，其特点在于不仅利用了一些硬件系统独有信息，如ip和MAC等，主要还利用了系统熵源来生成随机数，增强了所产生随机数的不可预测性。关于secrets的使用方法， 可以参考其官方文档。但是，即使是这里所谓的安全随机数，也并不是真随机数，真随机数的产生不依赖于算法。

量子计算与随机数

前面写过一篇博客介绍两比特量子系统的模拟，读者可以从中了解到量子计算的一些基本原理与模拟实现。而另一篇关于量子线路模拟的博客，则介绍了关于开源量子计算框架ProjectQ的一些基本使用方法，在这篇文章中我们也会使用到。通过量子硬件，我们可以直接读取一系列的随机串，由于这些随机串是由硬件直接给出来的结果，不是通过算法来生成的，因此是完成不可预测的，被称为"真随机数"。

基于pytest的python单元测试框架

在使用python编程实现算法时，需要对算法函数或者类进行功能验证，这就需要写测试用例。python第三方库pytest提供了非常好的自动化测试的功能，配合html插件和覆盖率测试插件，还可以进行更多的测试以及更好的展示测试的结果。

量子计算产生随机数

量子计算是基于量子叠加与量子纠缠两种特性进行计算的方案，其结合量子算法的复杂性优势，在量子处理器上可以实现经典计算机无法在有效时间内完成的任务，一般称之为量子优越性。量子叠加是对量子比特重要特性的一个总结，如下图所示(图片来自于参考链接1)，区分于经典比特，量子比特不仅可以表示0和1，还可以表示处于0和1之间的状态，比如0.5，0.2243等等。这些状态我们不能直接读取，在硬件上我们只能够读取到0或者1两种状态，这跟经典比特是一样的。区别在于，即使我们不改变量子比特的状态，不同的测试却会给出不同的结果，而0.5等中间状态，则表示的是出现0或者1状态分别的概率。



由于量子测量会导致量子态的坍缩，也就是说，每一次测量之后，都需要重新制备量子态。而制备一个产生随机数的量子态，我们可以用如下所示的\(H\)量子门操作来实现。其具体的矩阵表述可以参考这篇博客，量子计算本质上就是用特殊的硬件方法，快速的实现大规模的矩阵运算，这也是其展现量子优势的来源。



这里我们直接使用开源量子计算框架ProjectQ来实现这个门操作，就省去了自己去模拟实习的繁琐操作：

# random_number.py
from projectq import MainEngine
from projectq.ops import H, Measure

def random_number_generator():
    eng = MainEngine()
    qubit = eng.allocate_qubit()
    H | qubit
    eng.flush()
    Measure | qubit
    random_number = int(qubit)
    eng.flush(deallocate_qubits=True)
    return random_number

if __name__ == '__main__':
    random_number = random_number_generator()
    print ('Random number generated by quantum simulator is: {}'.format(random_number))

这里顺带的介绍下ProjectQ的一些操作实现的方法，在ProjectQ中将众多的量子比特作为寄存器来进行调用，这些寄存器都在给定的engine下赋予了各种操作，如量子门操作、打印线路、编译优化等。编程框架与实际的量子比特并不在架构上的同一个层次，每次传送指令都是通过eng.flush()打包进行编译优化之后，再传送到后端去逐一执行。因此如果要测量量子比特(在ProjectQ模拟器中必须只能全部执行测量操作，不能只测量其中的某一个部分)，或者是读取某一个量子态的概率时，都需要先执行eng.flush()才行。接下来简单看下上述代码的执行效果：

[dechin@dechin-manjaro pytest]$ python3 random_number.py
Random number generated by quantum simulator is: 1
[dechin@dechin-manjaro pytest]$ python3 random_number.py
Random number generated by quantum simulator is: 1
[dechin@dechin-manjaro pytest]$ python3 random_number.py
Random number generated by quantum simulator is: 0
[dechin@dechin-manjaro pytest]$ python3 random_number.py
Random number generated by quantum simulator is: 0
[dechin@dechin-manjaro pytest]$ python3 random_number.py
Random number generated by quantum simulator is: 1

在这个案例中，每一次都会产生一个新的随机数0或者1，并且在量子比特使用结束之后，通过eng.flush(deallocate_qubits=True)可以初始化该寄存器，所有的振幅和测量结果都会被丢弃。

注意！！！

需要注意的是，由于这里我们是使用了模拟器来模拟量子计算机的行为，因此得到的随机数结果还是伪随机数。只有当这一串代码执行在量子处理器上时，我们才能得到真随机数。

pytest测试

安装pytest库

首先我们可以通过pip来更新安装pytest：

[dechin@dechin-manjaro pytest]$ python3 -m pip install -U pytest
Collecting pytest
  Downloading pytest-6.2.2-py3-none-any.whl (280 kB)
     |████████████████████████████████| 280 kB 464 kB/s
Requirement already satisfied, skipping upgrade: iniconfig in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest) (1.1.1)
Requirement already satisfied, skipping upgrade: pluggy<1.0.0a1,>=0.12 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest) (0.13.1)
Requirement already satisfied, skipping upgrade: attrs>=19.2.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest) (20.3.0)
Requirement already satisfied, skipping upgrade: toml in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest) (0.10.1)
Requirement already satisfied, skipping upgrade: packaging in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest) (20.4)
Requirement already satisfied, skipping upgrade: py>=1.8.2 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest) (1.9.0)
Requirement already satisfied, skipping upgrade: pyparsing>=2.0.2 in /home/dechin/anaconda3/lib/python3.8/site-packages (from packaging->pytest) (2.4.7)
Requirement already satisfied, skipping upgrade: six in /home/dechin/anaconda3/lib/python3.8/site-packages (from packaging->pytest) (1.15.0)
Installing collected packages: pytest
  Attempting uninstall: pytest
    Found existing installation: pytest 6.2.1
    Uninstalling pytest-6.2.1:
      Successfully uninstalled pytest-6.2.1
Successfully installed pytest-6.2.2

安装完成后，可以通过如下指令来查看安装的pytest版本：

[dechin@dechin-manjaro pytest]$ pytest --version
pytest 6.2.2

pytest单元测试用例撰写

根据前面一个章节中的random_number.py文件，我们可以对照的写一个简单测试用例：

# test_random_number.py
import pytest
from random_number import random_number_generator as rng

def test_random_number_generator():
    for i in range(10):
        random_number = rng()
        assert random_number == 0 or random_number == 1

该测试用例的含义为：导入rng函数之后，测试10次该函数的返回值，所返回的值必须是0或者1的随机数，如果输出了这两个数字以外的返回结果，那么说明这个随机数产生器功能上存在问题。基于pytest的测试代码可以通过如下的指令来运行：

[dechin@dechin-20n2s01200 pytest]$ py.test
=========================================== test session starts ============================================
platform linux -- Python 3.8.5, pytest-6.2.2, py-1.9.0, pluggy-0.13.1
rootdir: /home/dechin/projects/2021-python/pytest
plugins: cov-2.11.1, metadata-1.11.0, html-3.1.1
collected 1 item                                                                                           

test_random_number.py .                                                                              [100%]

============================================= warnings summary =============================================
../../../anaconda3/lib/python3.8/site-packages/projectq/ops/_gates.py:118
  /home/dechin/anaconda3/lib/python3.8/site-packages/projectq/ops/_gates.py:118: PendingDeprecationWarning: the matrix subclass is not the recommended way to represent matrices or deal with linear algebra (see https://docs.scipy.org/doc/numpy/user/numpy-for-matlab-users.html). Please adjust your code to use regular ndarray.
    return np.matrix([[1, 0], [0, 1j]])

../../../anaconda3/lib/python3.8/site-packages/projectq/ops/_gates.py:133
  /home/dechin/anaconda3/lib/python3.8/site-packages/projectq/ops/_gates.py:133: PendingDeprecationWarning: the matrix subclass is not the recommended way to represent matrices or deal with linear algebra (see https://docs.scipy.org/doc/numpy/user/numpy-for-matlab-users.html). Please adjust your code to use regular ndarray.
    return np.matrix([[1, 0], [0, cmath.exp(1j * cmath.pi / 4)]])

test_random_number.py: 40 warnings
  /home/dechin/anaconda3/lib/python3.8/site-packages/projectq/ops/_gates.py:69: PendingDeprecationWarning: the matrix subclass is not the recommended way to represent matrices or deal with linear algebra (see https://docs.scipy.org/doc/numpy/user/numpy-for-matlab-users.html). Please adjust your code to use regular ndarray.
    return 1. / cmath.sqrt(2.) * np.matrix([[1, 1], [1, -1]])

-- Docs: https://docs.pytest.org/en/stable/warnings.html
====================================== 1 passed, 42 warnings in 0.50s ======================================

从返回的结果来看，出现了1 passed而没有failed，说明所有的测试用例都已经执行成功了，但是这里存在不少的告警warnings信息。

pytest初始化配置文件

在上一节的测试结果中，我们发现有非常多的测试告警。假如我们确认这些告警信息可以忽略，那么我们可以通过在指令中配置忽略告警信息，或者直接使用这里介绍的pytest.ini来忽略相应的告警信息：

# pytest.ini
[pytest]
filterwarnings =
        ignore::PendingDeprecationWarning

在当前目录下的ini配置文件中，我们添加了PendingDeprecationWarning作为忽略项，然后我们再回头看一下上述用例的测试结果：

[dechin@dechin-20n2s01200 pytest]$ py.test
=========================================== test session starts ============================================
platform linux -- Python 3.8.5, pytest-6.2.2, py-1.9.0, pluggy-0.13.1
rootdir: /home/dechin/projects/2021-python/pytest, configfile: pytest.ini
plugins: cov-2.11.1, metadata-1.11.0, html-3.1.1
collected 1 item                                                                                           

test_random_number.py .                                                                              [100%]

============================================ 1 passed in 0.50s =============================================

这里返回的结果中就没有告警信息了。

pytest生成html格式报告

为了更好的展现测试的结果，这里我们需要先安装一个组件pytest-html：

[dechin@dechin-manjaro pytest]$ python3 -m pip install pytest-html
Collecting pytest-html
  Downloading pytest_html-3.1.1-py3-none-any.whl (14 kB)
Collecting pytest-metadata
  Downloading pytest_metadata-1.11.0-py2.py3-none-any.whl (10 kB)
Requirement already satisfied: pytest!=6.0.0,>=5.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest-html) (6.2.1)
Requirement already satisfied: attrs>=19.2.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest!=6.0.0,>=5.0->pytest-html) (20.3.0)
Requirement already satisfied: py>=1.8.2 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest!=6.0.0,>=5.0->pytest-html) (1.9.0)
Requirement already satisfied: toml in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest!=6.0.0,>=5.0->pytest-html) (0.10.1)
Requirement already satisfied: packaging in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest!=6.0.0,>=5.0->pytest-html) (20.4)
Requirement already satisfied: pluggy<1.0.0a1,>=0.12 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest!=6.0.0,>=5.0->pytest-html) (0.13.1)
Requirement already satisfied: iniconfig in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest!=6.0.0,>=5.0->pytest-html) (1.1.1)
Requirement already satisfied: six in /home/dechin/anaconda3/lib/python3.8/site-packages (from packaging->pytest!=6.0.0,>=5.0->pytest-html) (1.15.0)
Requirement already satisfied: pyparsing>=2.0.2 in /home/dechin/anaconda3/lib/python3.8/site-packages (from packaging->pytest!=6.0.0,>=5.0->pytest-html) (2.4.7)
Installing collected packages: pytest-metadata, pytest-html
Successfully installed pytest-html-3.1.1 pytest-metadata-1.11.0

安装成功后，执行如下指令，可以在当前目录下生成一个指定文件名的html文件，如下图所示，就比较全面且美观的展示了测试中的信息：

[dechin@dechin-manjaro pytest]$ py.test --html=pytest_report.html
=========================================== test session starts ============================================
platform linux -- Python 3.8.5, pytest-6.2.2, py-1.9.0, pluggy-0.13.1
rootdir: /home/dechin/projects/2021-python/pytest, configfile: pytest.ini
plugins: cov-2.11.1, metadata-1.11.0, html-3.1.1
collected 1 item                                                                                           

test_random_number.py .                                                                              [100%]

--------- generated html file: file:///home/dechin/projects/2021-python/pytest/pytest_report.html ----------
============================================ 1 passed in 0.51s =============================================

不过在问题数量并不是很多的情况下，直接看命令行输出也是比较方便的。

pytest覆盖率测试

在一个python仓库中我们有可能有非常多的函数、类和文件等，为了保障结果的准确性，我们需要能够给出一个可信的覆盖率测试的结论，只有当覆盖率达到100%时，我们才能认为测试工作已经比较全面的考虑。当然，覆盖率100%的测试，其实并不能保障100%的不出问题，这就是另外的话题了。首先我们又需要补充安装一个组件pytest-cov：

[dechin@dechin-manjaro pytest]$ python3 -m pip install pytest-cov
Collecting pytest-cov
  Downloading pytest_cov-2.11.1-py2.py3-none-any.whl (20 kB)
Collecting coverage>=5.2.1
  Downloading coverage-5.4-cp38-cp38-manylinux2010_x86_64.whl (245 kB)
     |████████████████████████████████| 245 kB 15 kB/s
Requirement already satisfied: pytest>=4.6 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest-cov) (6.2.1)
Requirement already satisfied: attrs>=19.2.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest>=4.6->pytest-cov) (20.3.0)
Requirement already satisfied: packaging in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest>=4.6->pytest-cov) (20.4)
Requirement already satisfied: iniconfig in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest>=4.6->pytest-cov) (1.1.1)
Requirement already satisfied: toml in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest>=4.6->pytest-cov) (0.10.1)
Requirement already satisfied: py>=1.8.2 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest>=4.6->pytest-cov) (1.9.0)
Requirement already satisfied: pluggy<1.0.0a1,>=0.12 in /home/dechin/anaconda3/lib/python3.8/site-packages (from pytest>=4.6->pytest-cov) (0.13.1)
Requirement already satisfied: six in /home/dechin/anaconda3/lib/python3.8/site-packages (from packaging->pytest>=4.6->pytest-cov) (1.15.0)
Requirement already satisfied: pyparsing>=2.0.2 in /home/dechin/anaconda3/lib/python3.8/site-packages (from packaging->pytest>=4.6->pytest-cov) (2.4.7)
Installing collected packages: coverage, pytest-cov
Successfully installed coverage-5.4 pytest-cov-2.11.1

然后在当前目录下执行以下指令：

[dechin@dechin-manjaro pytest]$ py.test --cov=random_number ./ --cov-report=html
=========================================== test session starts ============================================
platform linux -- Python 3.8.5, pytest-6.2.1, py-1.9.0, pluggy-0.13.1
rootdir: /home/dechin/projects/2021-python/pytest, configfile: pytest.ini
plugins: cov-2.11.1, metadata-1.11.0, html-3.1.1
collected 1 item                                                                                           

test_random_number.py .                                                                              [100%]

----------- coverage: platform linux, python 3.8.5-final-0 -----------
Coverage HTML written to dir htmlcov

============================================ 1 passed in 0.76s =============================================

这里会提示我们去htmlcov目录下找测试报告，我们打开相应的index.html文件，效果如下：



这里我们看到random_number.py文件的测试覆盖率为86%，我们可以点开链接查看剩下未覆盖的测试是什么内容：



原来是main函数中的几行指令没有被测试到，一般我们在正式仓库中，是需要去掉main函数的，如果不是测试文件的话。

使用flake8进行python编码规范检查

首先我们还是使用pip来直接安装flake8组件，其集成了PEP8的python编码规范：

[dechin@dechin-manjaro pytest]$ python3 -m pip install -U flake8
Requirement already up-to-date: flake8 in /home/dechin/anaconda3/lib/python3.8/site-packages (3.8.4)
Requirement already satisfied, skipping upgrade: mccabe<0.7.0,>=0.6.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8) (0.6.1)
Requirement already satisfied, skipping upgrade: pyflakes<2.3.0,>=2.2.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8) (2.2.0)
Requirement already satisfied, skipping upgrade: pycodestyle<2.7.0,>=2.6.0a1 in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8) (2.6.0)

查看安装的flake8版本号：

[dechin@dechin-manjaro pytest]$ flake8 --version
3.8.4 (mccabe: 0.6.1, pycodestyle: 2.6.0, pyflakes: 2.2.0) CPython 3.8.5 on Linux

使用方式较为简单，在当前目录下直接执行flake8即可，检查项会以当前路径为rootdir，递归的进行检索：

[dechin@dechin-manjaro pytest]$ flake8
./test_random_number.py:2:1: F401 'pytest' imported but unused
./test_random_number.py:5:1: E302 expected 2 blank lines, found 1
./random_number.py:5:1: E302 expected 2 blank lines, found 1
./random_number.py:15:1: E305 expected 2 blank lines after class or function definition, found 1
./random_number.py:17:10: E211 whitespace before '('
./random_number.py:17:80: E501 line too long (87 > 79 characters)

假如我们不想包含测试用例的风格检查，我们可以在命令行中屏蔽相关文件，或者创建配置文件，在配置文件中屏蔽相关文件，这里介绍后者的使用方法：

# .flake8
[flake8]
exclude = ./test*

这个配置的意义在于，将当前目录下，所有以test开头的文件，都将会忽略python编码规范的检查，配置文件支持通配符的表达形式。接下来看下最新的执行结果：

[dechin@dechin-manjaro pytest]$ flake8
./random_number.py:5:1: E302 expected 2 blank lines, found 1
./random_number.py:15:1: E305 expected 2 blank lines after class or function definition, found 1
./random_number.py:17:10: E211 whitespace before '('
./random_number.py:17:80: E501 line too long (87 > 79 characters)

我们发现前面出现的关于测试用例中的编码规范，已经不在这个列表的范围了。这个列表中的问题我们一般最好是处理下，但是如果遇到一些判断不需要处理的规范，则同样可以在配置文件中添加相应的规范ID，这里仅作示例使用，前面显示的编码规范问题后面都会改。如下所示就是两个忽略：

# .flake8
[flake8]
exclude = ./test*
ignore = E302, E305

对应的执行结果如下所示：

[dechin@dechin-manjaro pytest]$ flake8
./random_number.py:17:10: E211 whitespace before '('
./random_number.py:17:80: E501 line too long (87 > 79 characters)

我们发现被忽略的问题已经没有显示了。flake8还有一个特点，是可以配置max-complexity，其采用的基本复杂性计算方法为McCabe度量法，可以参考这篇博客中的介绍内容，这里就不过多的赘述。为了更加美观简明的看到输出的结果，我们可以补充安装一个组件flake8-html：

[dechin@dechin-manjaro pytest]$ python3 -m pip install flake8-html
Collecting flake8-html
  Downloading flake8_html-0.4.1-py2.py3-none-any.whl (13 kB)
Requirement already satisfied: pygments>=2.2.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8-html) (2.7.2)
Requirement already satisfied: flake8>=3.3.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8-html) (3.8.4)
Requirement already satisfied: jinja2>=2.9.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8-html) (2.11.2)
Requirement already satisfied: importlib-metadata in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8-html) (2.0.0)
Requirement already satisfied: pycodestyle<2.7.0,>=2.6.0a1 in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8>=3.3.0->flake8-html) (2.6.0)
Requirement already satisfied: mccabe<0.7.0,>=0.6.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8>=3.3.0->flake8-html) (0.6.1)
Requirement already satisfied: pyflakes<2.3.0,>=2.2.0 in /home/dechin/anaconda3/lib/python3.8/site-packages (from flake8>=3.3.0->flake8-html) (2.2.0)
Requirement already satisfied: MarkupSafe>=0.23 in /home/dechin/anaconda3/lib/python3.8/site-packages (from jinja2>=2.9.0->flake8-html) (1.1.1)
Requirement already satisfied: zipp>=0.5 in /home/dechin/anaconda3/lib/python3.8/site-packages (from importlib-metadata->flake8-html) (3.4.0)
Installing collected packages: flake8-html
Successfully installed flake8-html-0.4.1

使用方法如下：

[dechin@dechin-20n2s01200 pytest]$ flake8 --format=html --htmldir=flake-report
./random_number.py has issues: medium: 4

在当前目录执行后，相关的规范的issue就不会在命令行里面逐一显示，都在flake-report目录下的index.html中可以查看：



我们可以点击进入相应文件的issue清单中去查看：



可以点击每一个issue，展开内容中包含了issue所对应行的代码内容：



经过一番修改之后，我们得到的flake8配置文件和源代码文件如下：

# .flake8
[flake8]
exclude = ./test*
ignore = W391

# random_number.py
from projectq import MainEngine
from projectq.ops import H, Measure

def random_number_generator():
    eng = MainEngine()
    qubit = eng.allocate_qubit()
    H | qubit
    eng.flush()
    Measure | qubit
    random_number = int(qubit)
    eng.flush(deallocate_qubits=True)
    return random_number

if __name__ == '__main__':
    random_number = random_number_generator()
    print('Random number generated by quantum simulator is:\
            {}'.format(random_number))

最终我们的目的是使得flake8执行的issue清零：

[dechin@dechin-manjaro pytest]$ flake8 --format=html --htmldir=flake-report
[dechin@dechin-manjaro pytest]$

总结概要

本文通过引入一个随机数生成器的案例，介绍了在量子计算中产生真随机数的方案，同时给出了量子计算模拟实现。借此机会也同时介绍了python的单元测试库pytest的一些常规和扩展使用方法，以及python的编码规范测试库flake8的基本使用方法，希望能够对大家有所启发以及帮助。

版权声明

本文首发链接为：https://www.cnblogs.com/dechinphy/p/random_number_test.html

作者ID：DechinPhy

更多原著文章请参考：https://www.cnblogs.com/dechinphy/

参考链接

1. https://cn.bing.com/images/search?view=detailV2&ccid=X1pIP1mm&id=0CF2CDDD1C700A051F3FEB4701669068CE05B9FA&thid=OIP.X1pIP1mmVKohO5UC1CvbuQHaED&mediaurl=https%3A%2F%2Fth.bing.com%2Fth%2Fid%2FR5f5a483f59a654aa213b9502d42bdbb9%3Frik%3D%2brkFzmiQZgFH6w%26riu%3Dhttp%3a%2f%2fwww.supernovapcrepair.co.uk%2fimages%2fquantum%2Bsuperposition.png%26ehk%3DEt3zyrD9NqYgaMFwpPb%2bcZAohbR0ERpeRfCEkgKy6i0%3d%26risl%3D%26pid%3DImgRaw&exph=199&expw=363&q=quantum+superposition&simid=608026555008484117&ck=E74E3E03947D89427ED1A5352402AADB&selectedIndex=3&FORM=IRPRST&ajaxhist=0