第 1 章 量子计算前奏：量子力学. .............................. 1
1.1 用线性代数打开量子力学 ..................................................................2
1.1.1 量子力学与经典力学 ....................................................................3
1.1.2 “薛定谔的猫”与叠加态 ..............................................................5
1.1.3 波函数 ..........................................................................................10
1.1.4 小结 ..............................................................................................14
1.2 测量、概率与“投影” ......................................................................14
1.2.1 量子力学中的测量 ......................................................................15
1.2.2 测量结果出现的概率 ..................................................................17
1.2.3 波函数的坍缩 ..............................................................................19
1.2.4 总概率的归一化 ..........................................................................20
1.2.5 在量子态上测量其他力学量 ......................................................22
1.2.6 小结 ..............................................................................................22
1.3 直观认识不确定性 ............................................................................23
1.3.1 光的单缝衍射实验与不确定性原理 ..........................................24
1.3.2 不确定性原理的“几何”解释 ..................................................25
1.3.3 位置- 动量不确定性 ..................................................................28
1.3.4 小结 ..............................................................................................31
1.4 自旋：通向量子计算的桥梁 ............................................................32
1.4.1 什么是自旋 ..................................................................................33
1.4.2 自旋的测量性质 ..........................................................................33
1.4.3 特殊方向自旋的量子态 ..............................................................37
1.4.4 布洛赫球与任意自旋态 ..............................................................40
*1.4.5 自旋态和波函数的关系 ............................................................47
1.4.6 小结 ..............................................................................................48
1.5 让量子态“动”起来 ........................................................................48
1.5.1 量子态的演化与薛定谔方程 ......................................................49
1.5.2 自旋态的时间演化 ......................................................................51
1.5.3 薛定谔方程的物理意义与解的结构 ..........................................52
1.5.4 小结 ..............................................................................................56
第 2 章 量子计算掠影. ..........................................58
2.1 经典计算机与通用计算 ....................................................................59
2.1.1 经典计算的通用性 ......................................................................60
2.1.2 有限位二进制函数的穷举 ..........................................................62
2.1.3 基本逻辑运算与逻辑门 ..............................................................64
2.1.4 通用计算与完全集 ......................................................................67
2.1.5 最小完全集 ..................................................................................71
2.1.6 小结 ..............................................................................................72
2.2 初识量子比特 ....................................................................................73
2.2.1 单量子比特 ..................................................................................73
2.2.2 多量子比特系统 ..........................................................................75
2.2.3 直积态与纠缠态 ..........................................................................76
2.2.4 多量子比特系统的测量 ..............................................................80
2.2.5 从物理角度认识纠缠态 ..............................................................83
2.2.6 小结 ..............................................................................................87
2.3 量子优越性如何实现 ........................................................................87
2.3.1 典型的量子计算过程 ..................................................................87
2.3.2 量子计算结果的输出方式 ..........................................................89
2.3.3 量子优越性的实现途径 ..............................................................91
2.3.4 初识量子电路 ..............................................................................96
2.3.5 小结 ..............................................................................................99
2.4 量子算法初体验 ................................................................................99
2.4.1 多伊奇算法 ................................................................................100
2.4.2 格罗弗算法 ................................................................................105
2.4.3 肖尔算法 .................................................................................... 111
2.4.4 小结 ............................................................................................ 113
第 3 章 数学准备.............................................. 114
3.1 向量与矩阵的快速回顾 .................................................................. 115
3.1.1 向量与向量空间 ........................................................................ 116
3.1.2 基底、投影与内积 .................................................................... 118
3.1.3 矩阵与变换 ................................................................................ 119
3.1.4 复合变换与矩阵乘法 ................................................................122
3.1.5 矩阵的可逆性 ............................................................................126
3.1.6 相似矩阵 ....................................................................................128
3.1.7 特征值与特征向量 ....................................................................131
3.1.8 正交变换与正交矩阵 ................................................................133
3.1.9 小结 ............................................................................................135
3.2 当线性代数遇上复数 ......................................................................135
3.2.1 复线性空间中的向量及其内积 ................................................136
3.2.2 复线性空间中的线性变换及矩阵 ............................................140
3.2.3 酉变换与酉矩阵 ........................................................................141
3.2.4 单量子比特的酉变换与泡利矩阵 ............................................142
3.2.5 小结 ............................................................................................146
3.3 向量与矩阵的张量积 ......................................................................146
3.3.1 向量的张量积 ............................................................................147
3.3.2 多量子比特系统状态的表示 ....................................................149
3.3.3 矩阵的张量积 ............................................................................152
*3.3.4 波函数与自旋态的张量积 ......................................................156
3.3.5 小结 ............................................................................................157
第 4 章 量子门与量子电路. .................................. 158
4.1 单量子比特门：量子比特的“独舞” ............................................159
4.1.1 单量子比特门的一般形式 ........................................................160
4.1.2 X 门 ............................................................................................161
4.1.3 相位门和 Z 门 ............................................................................162
4.1.4 阿达玛门 ....................................................................................164
4.1.5 单量子比特门的组合 ................................................................167
4.1.6 小结 ............................................................................................172
4.2 受控门：量子比特的“互动” ........................................................173
4.2.1 什么是受控门？ ........................................................................173
4.2.2 常用的双量子比特受控门 ........................................................176
4.2.3 常用的三量子比特受控门 ........................................................180
4.2.4 受控门的矩阵形式 ....................................................................182
4.2.5 受控门矩阵的一般构造规则 ....................................................185
4.2.6 小结 ............................................................................................188
4.3 量子电路的“看图计算” ................................................................188
4.3.1 利用变换规则计算 ....................................................................189
4.3.2 利用变换矩阵计算 ....................................................................191
4.3.3 等效量子电路 ............................................................................195
4.3.4 量子电路中的“信息转移”
......................................................198
4.3.5 量子电路的仿真 ........................................................................201
4.3.6 小结 ............................................................................................202
4.4 量子门的通用性 ..............................................................................203
4.4.1 量子门为什么不同于经典逻辑门？ ........................................204
4.4.2 经典计算的通用性能否实现？ ................................................207
4.4.3 量子计算的通用性能否实现？ ................................................209
4.4.4 小结 ............................................................................................ 211
*4.5 量子门的物理实现 ........................................................................212
4.5.1 拉莫尔进动与自旋态变换 ........................................................212
4.5.2 量子门的核磁共振方案 ............................................................214
4.5.3 多量子比特门的实现案例 ........................................................217
4.5.4 小结 ............................................................................................218
第 5 章 基础量子算法. ........................................220
5.1 格罗弗算法 ......................................................................................222
5.1.1 背景与思路的回顾 ....................................................................222
5.1.2 旋转变换的分解 ........................................................................226
5.1.3 格罗弗算法的量子电路 ............................................................230
5.1.4 多解情形的格罗弗算法 ............................................................234
5.1.5 小结 ............................................................................................239
5.2 量子相位估计算法 ..........................................................................239
5.2.1 背景介绍 ....................................................................................240
5.2.2 量子相位估计算法的思路 ........................................................241
5.2.3 量子相位估计算法的步骤 ........................................................242
5.2.4 量子相位估计算法的量子电路 ................................................248
5.2.5 小结 ............................................................................................251
5.3 基于概率输出的量子算法 ..............................................................251
5.3.1 背景介绍 ....................................................................................251
5.3.2 SWAP 测试 ................................................................................252
5.3.3 阿达玛测试 ................................................................................256
5.3.4 小结 ............................................................................................259
第 6 章 量子算法的应用. ..................................... 261
6.1 更快的搜索与优化 ..........................................................................262
6.1.1 格罗弗算法与数据搜索 ............................................................262
6.1.2 格罗弗算法与全局寻优 ............................................................268
6.1.3 小结 ............................................................................................270
6.2 破解密码的“神器” ........................................................................270
6.2.1 RSA 加密算法简介 ...................................................................271
6.2.2 数论知识补遗 ............................................................................279
6.2.3 肖尔算法的思路 ........................................................................281
6.2.4 肖尔算法的后处理 ....................................................................286
6.2.5 小结 ............................................................................................286
6.3 线性方程组的快速求解 ..................................................................286
6.3.1 HHL 算法的理论基础 ..............................................................287
6.3.2 HHL 算法的主要步骤 ..............................................................289
6.3.3 小结 ............................................................................................294
6.4 量子机器学习案例 ..........................................................................295
6.4.1 机器学习中的量子计算 ............................................................296
6.4.2 支持向量机 ................................................................................297
6.4.3 量子支持向量机的训练过程 ....................................................299
6.4.4 量子支持向量机的预测过程 ....................................................302
6.4.5 小结 ............................................................................................306
6.5 量子化学模拟 ..................................................................................307
6.5.1 量子化学要计算什么 ................................................................307
6.5.2 怎样计算基态能量：经典算法 ................................................310
6.5.3 怎样计算基态能量：量子算法 ................................................317
6.5.4 小结 ............................................................................................321
6.6 NISQ 与变分量子算法简介 ............................................................322
6.6.1 背景介绍 ....................................................................................322
6.6.2 变分简介 ....................................................................................323
6.6.3 变分量子算法的基本思路 ........................................................326
6.6.4 变分量子特征值求解器 ............................................................328
6.6.5 能量期望值的求解 ....................................................................329
6.6.6 小结 ............................................................................................331
