diff --git a/tutorial/GIBBS.ipynb b/tutorial/GIBBS.ipynb
index 783b1f835dcf92462eb7cebd3a6b82cd416176c0..25ba86e111f1c14e42d402276ae3c17a8f6d9ceb 100644
--- a/tutorial/GIBBS.ipynb
+++ b/tutorial/GIBBS.ipynb
@@ -178,7 +178,7 @@
     "- 具体的我们参考的是[4]中的方法（核心思想是利用吉布斯态达到了最小自由能的性质）。\n",
     "- 通过作用量子神经网络$U(\\theta)$在初始态上，我们可以得到输出态$\\left| {\\psi \\left( {\\bf{\\theta }} \\right)} \\right\\rangle $, 其在第2-4个量子位的态记为$\\rho_B(\\theta)$.\n",
     "- 设置训练模型中的的损失函数，在吉布斯态学习中，我们利用冯诺依曼熵函数的截断来进行自由能的估计，相应的损失函数参考[4]可以设为 \n",
-    "$loss= {L_1} + {L_2} + {L_3}$，其中 ${L_1}= tr(H\\rho_B)$, ${L_2} = 2{\\beta^{-1}}{Tr}(\\rho_B^2)$ , $L_3 = - {\\beta ^{ - 1}}\\frac{{Tr(\\rho_B^3) + 3}}{2}$."
+    "$loss= {L_1} + {L_2} + {L_3}$，其中 ${L_1}= tr(H\\rho_B)$, ${L_2} = 2{\\beta^{-1}}{tr}(\\rho_B^2)$ , $L_3 = - {\\beta ^{ - 1}}\\frac{{tr(\\rho_B^3) + 3}}{2}$."
    ]
   },
   {
diff --git a/tutorial/QAOA.ipynb b/tutorial/QAOA.ipynb
index 5fd93bb7d546ceab441ddc0a74b20833765d36c2..c3e9aead34d159e113aee4ce55eeb3c22e2f5f78 100644
--- a/tutorial/QAOA.ipynb
+++ b/tutorial/QAOA.ipynb
@@ -57,7 +57,7 @@
     "量子近似优化算法（QAOA，Quantum Approximate Optimization Algorithm）是可以在近期有噪中等规模（NISQ，Noisy Intermediate-Scale Quantum）量子计算机上运行且具有广泛应用前景的量子算法。例如，QAOA 可以用来处理压缩图信号和二次优化等领域常见的离散组合优化问题。这类优化问题通常可以归结为下面的数学模型：\n",
     "\n",
     "\n",
-    " $$F=\\max_{z_i\\in\\{-1,1\\}} \\sum_{（i,j）} q_{ij}(1-z_iz_j)=-\\min_{z_i\\in\\{-1,1\\}} \\sum_{（i,j）} q_{ij}z_iz_j+ \\sum_{（i,j）}q_{ij}. $$\n",
+    " $$F=\\max_{z_i\\in\\{-1,1\\}} \\sum q_{ij}(1-z_iz_j)=-\\min_{z_i\\in\\{-1,1\\}} \\sum q_{ij}z_iz_j+ \\sum q_{ij}. $$\n",
     "\n",
     "\n",
     "其中, $z_i \\in \\{-1 ,1\\} $ 是待求的二元参数，系数 $q_{ij}$ 是 $z_i z_j$ 的权重 (weight)。一般地，精确求解该问题对于经典计算机是 NP-hard 的，而 QAOA 被认为对近似求解这类困难问题具有潜在速度优势。\n",
@@ -772,7 +772,7 @@
    "source": [
     "# 参考文献\n",
     "\n",
-    "[1] Farhi, E., Goldstone, J., & Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv:1411.4028."
+    "[1] E. Farhi, J. Goldstone, and S. Gutman. 2014. A quantum approximate optimization algorithm. arXiv:1411.4028 "
    ]
   }
  ],
diff --git a/tutorial/QAOA_En.ipynb b/tutorial/QAOA_En.ipynb
index 99755079cf40c25759a06e05b67e898489876255..f40247c7426bf11ccc3f67ec66248c68dcd14bb6 100644
--- a/tutorial/QAOA_En.ipynb
+++ b/tutorial/QAOA_En.ipynb
@@ -49,7 +49,7 @@
     "\n",
     "QAOA is one of quantum algorithms which can be implemented on near-term quantum processors, also called as noisy intermediate-scale quantum (NISQ) processors, and may have wide applications in solving hard computational problems.  For example, it could be applied to tackle a large family of optimization problems, named as the quadratic unconstrained binary optimization (QUBO) which is ubiquitous in the computer science and operation research. Basically, this class can be modeled with the form of\n",
     "\n",
-    "$$F=\\max_{z_i\\in\\{-1,1\\}} \\sum_{（i,j）} q_{ij}(1-z_iz_j)=-\\min_{z_i\\in\\{-1,1\\}} \\sum_{（i,j）} q_{ij}z_iz_j+ \\sum_{（i,j）}q_{ij} $$\n",
+    "$$F=\\max_{z_i\\in\\{-1,1\\}} \\sum q_{ij}(1-z_iz_j)=-\\min_{z_i\\in\\{-1,1\\}} \\sum q_{ij}z_iz_j+ \\sum q_{ij} $$\n",
     "\n",
     "\n",
     "where $z_i$s are  binary parameters and coefficients $q_{ij}$ refer to the weight associated to $x_i x_j$. Indeed, it is usually extremely difficult for classical computers to give the exact optimal solution, while QAOA provides an alternative approach which may have a speedup advantage over classical ones to solve these hard problems.\n",
@@ -758,7 +758,7 @@
    "source": [
     "# References\n",
     "\n",
-    "[1] Farhi, E., Goldstone, J., & Gutmann, S. (2014). A quantum approximate optimization algorithm. arXiv:1411.4028."
+    "[1] E. Farhi, J. Goldstone, and S. Gutman. 2014. A quantum approximate optimization algorithm. arXiv:1411.4028 "
    ]
   }
  ],
diff --git a/tutorial/VQE.ipynb b/tutorial/VQE.ipynb
index 39772f730d346f3c14b88a5ddbb9f6b3f9fc3899..9acffb26c5a9fb9ff5ecacb64d9644540b234000 100644
--- a/tutorial/VQE.ipynb
+++ b/tutorial/VQE.ipynb
@@ -38,7 +38,7 @@
    "metadata": {},
    "source": [
     "## 背景\n",
-    "- 量子计算中在近期非常有前途的一个量子算法是变分量子特征求解器(VQE, variational quantum eigensolver (VQE)) [1-3].\n",
+    "- 量子计算中在近期非常有前途的一个量子算法是变分量子特征求解器(VQE, variational quantum eigensolver) [1-3].\n",
     "- VQE是量子化学在近期有噪量子设备（NISQ device）上的核心应用之一。其核心是去求解一个物理系统的哈密顿量的基态及其对应的能量。数学上，可以理解为求解一个厄米矩阵(Hermitian matrix)的最小特征值及其对应的特征向量。\n",
     "- 接下来我们将通过一个简单的例子学习如何通过训练量子神经网络解决这个问题，我们的目标是通过训练量子神经网络去找到量子态 $\\left| \\phi  \\right\\rangle $ (可以理解为一个归一化的复数向量), 使得 $$\\left\\langle \\phi  \\right|H\\left| \\phi  \\right\\rangle =\\lambda_{\\min}(H)$$, 其中$\\left\\langle \\phi  \\right|$是$\\left| \\phi  \\right\\rangle$的共轭转置，$\\lambda_{\\min}(H)$是矩阵$H$的最小特征值。\n",
     "\n",