CBMS-2024B-11

题目来源：Question 11 日期：2024-08-04 题目主题：Math-量子计算-密度矩阵与量子门操作

解题思路

这道题目涉及量子计算中的密度矩阵和量子门操作，需要运用线性代数和复数运算的知识。我们将逐步解答每个小问：

1. 求密度矩阵的特征值，证明它们是非负实数。
2. 计算 Hadamard 门（H 门）操作后的测量概率。
3. 计算 Y 门操作后的测量概率。
4. 计算一般量子门 U 操作后的密度矩阵参数。

每个小问都需要详细的数学推导。

Solution

1. Show that all the eigenvalues of matrix $\rho$ are non-negative real numbers

To find the eigenvalues of $\rho$, we need to solve the characteristic equation:

$\det (\rho -\lambda I)=0$

$\det \left(\left(\begin{array}{cc}\frac{1+a}{2}-\lambda & \frac{b-ic}{2}\\ \frac{b+ic}{2}& \frac{1-a}{2}-\lambda \end{array}\right)\right)=0$

$(\frac{1+a}{2}-\lambda )(\frac{1-a}{2}-\lambda )-(\frac{b-ic}{2})(\frac{b+ic}{2})=0$

${\lambda }^2-\lambda +\frac{1-a^2-b^2-c^2}{4}=0$

The solutions to this quadratic equation are:

$\lambda =\frac{1\pm \sqrt{1-(1-a^2-b^2-c^2)}}{2}=\frac{1\pm \sqrt{a^2+b^2+c^2}}{2}$

Since $a^2+b^2+c^2\le 1$, we have $0\le \sqrt{a^2+b^2+c^2}\le 1$. Therefore, both eigenvalues are real and non-negative.

2. Probability of observing state 0 after applying H gate

The Hadamard gate operation transforms $\rho$ to $H\rho H^{†}$. Let’s calculate this:

$H\rho H^{†}=\frac{1}{2}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)\left(\begin{array}{cc}1+a& b-ic\\ b+ic& 1-a\end{array}\right)\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$

$=\frac{1}{4}\left(\begin{array}{cc}(1+a)+(b+ic)+(b-ic)+(1-a)& (1+a)-(b+ic)+(b-ic)-(1-a)\\ (1+a)+(b+ic)-(b-ic)-(1-a)& (1+a)-(b+ic)-(b-ic)+(1-a)\end{array}\right)$

$=\frac{1}{2}\left(\begin{array}{cc}1+b& a-ic\\ a+ic& 1-b\end{array}\right)$

The probability of observing state 0 is the top-left element of this matrix:

$p_0=\frac{1+b}{2}$

3. Probability of observing state 0 after applying Y gate

Similarly, for the Y gate:

$Y\rho Y^{†}=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\left(\begin{array}{cc}1+a& b-ic\\ b+ic& 1-a\end{array}\right)\left(\begin{array}{cc}0& i\\ -i& 0\end{array}\right)$

$=\left(\begin{array}{cc}1-a& -b-ic\\ -b+ic& 1+a\end{array}\right)$

The probability of observing state 0 is:

$p_0=\frac{1-a}{2}$

4. Compute $a^{\prime 2}+b^{\prime 2}+c^{\prime 2}$ after applying U gate

Let $U=\left(\begin{array}{cc}{u}_{00}^{\prime }+i{u}_{00}^{\prime \prime }& u_{01}^{\prime }+i{u}_{01}^{\prime \prime }\\ u_{10}^{\prime }+i{u}_{10}^{\prime \prime }& u_{11}^{\prime }+i{u}_{11}^{\prime \prime }\end{array}\right)$ and $U^{†}=\left(\begin{array}{cc}{u}_{00}^{\prime }-i{u}_{00}^{\prime \prime }& u_{10}^{\prime }-i{u}_{10}^{\prime \prime }\\ u_{01}^{\prime }-i{u}_{01}^{\prime \prime }& u_{11}^{\prime }-i{u}_{11}^{\prime \prime }\end{array}\right)$

We need to calculate $U\rho U^{†}$. This is a complex calculation, so let’s break it down:

$U\rho =\left(\begin{array}{cc}(u_{00}^{\prime }+i{u}_{00}^{\prime \prime })(1+a)+(u_{01}^{\prime }+i{u}_{01}^{\prime \prime })(b+ic)& (u_{00}^{\prime }+i{u}_{00}^{\prime \prime })(b-ic)+(u_{01}^{\prime }+i{u}_{01}^{\prime \prime })(1-a)\\ (u_{10}^{\prime }+i{u}_{10}^{\prime \prime })(1+a)+(u_{11}^{\prime }+i{u}_{11}^{\prime \prime })(b+ic)& (u_{10}^{\prime }+i{u}_{10}^{\prime \prime })(b-ic)+(u_{11}^{\prime }+i{u}_{11}^{\prime \prime })(1-a)\end{array}\right)$

Now, multiplying this by $U^{†}$ and focusing on the diagonal elements:

${\rho }^{\prime }=U\rho U^{†}=\left(\begin{array}{cc}\frac{1+a^{\prime }}{2}& \frac{b^{\prime }-i{c}^{\prime }}{2}\\ \frac{b^{\prime }+i{c}^{\prime }}{2}& \frac{1-a^{\prime }}{2}\end{array}\right)$

Where:

$a^{\prime }=(|u_{00}{|}^2-|u_{01}{|}^2)(1+a)+(u_{00}^{\prime }{u}_{01}^{\prime }+u_{00}^{\prime \prime }{u}_{01}^{\prime \prime })2b+(u_{00}^{\prime }{u}_{01}^{\prime \prime }-u_{00}^{\prime \prime }{u}_{01}^{\prime })2c+(|u_{10}{|}^2-|u_{11}{|}^2)(1-a)$

$b^{\prime }=(u_{00}^{\prime }{u}_{10}^{\prime }+u_{00}^{\prime \prime }{u}_{10}^{\prime \prime })(1+a)+(u_{01}^{\prime }{u}_{11}^{\prime }+u_{01}^{\prime \prime }{u}_{11}^{\prime \prime })(1-a)+(u_{00}^{\prime }{u}_{11}^{\prime }+u_{00}^{\prime \prime }{u}_{11}^{\prime \prime }+u_{01}^{\prime }{u}_{10}^{\prime }+u_{01}^{\prime \prime }{u}_{10}^{\prime \prime })b-(u_{00}^{\prime }{u}_{11}^{\prime \prime }-u_{00}^{\prime \prime }{u}_{11}^{\prime }-u_{01}^{\prime }{u}_{10}^{\prime \prime }+u_{01}^{\prime \prime }{u}_{10}^{\prime })c$

$c^{\prime }=(u_{00}^{\prime }{u}_{10}^{\prime \prime }-u_{00}^{\prime \prime }{u}_{10}^{\prime })(1+a)+(u_{01}^{\prime }{u}_{11}^{\prime \prime }-u_{01}^{\prime \prime }{u}_{11}^{\prime })(1-a)+(u_{00}^{\prime }{u}_{11}^{\prime \prime }-u_{00}^{\prime \prime }{u}_{11}^{\prime }-u_{01}^{\prime }{u}_{10}^{\prime \prime }+u_{01}^{\prime \prime }{u}_{10}^{\prime })b+(u_{00}^{\prime }{u}_{11}^{\prime }+u_{00}^{\prime \prime }{u}_{11}^{\prime \prime }-u_{01}^{\prime }{u}_{10}^{\prime }-u_{01}^{\prime \prime }{u}_{10}^{\prime \prime })c$

Now, we need to compute $a^{\prime 2}+b^{\prime 2}+c^{\prime 2}$. This is a very long and complex calculation. However, we can use a property of unitary matrices: $U{U}^{†}=I$. This implies that the transformation $\rho \to U\rho U^{†}$ preserves the trace and the purity of the density matrix.

The purity of a density matrix is defined as $\text{Tr}({\rho }^2)=\frac{1+a^2+b^2+c^2}{2}$.

Since this quantity is preserved under unitary transformations, we have:

$\frac{1+a^2+b^2+c^2}{2}=\frac{1+a^{\prime 2}+b^{\prime 2}+c^{\prime 2}}{2}$

Therefore:

$a^{\prime 2}+b^{\prime 2}+c^{\prime 2}=a^2+b^2+c^2$

This result shows that the sum of squares of the parameters in the density matrix is invariant under unitary transformations.

知识点

量子计算密度矩阵量子门操作线性代数复数厄米矩阵

难点思路

第 4 小问的计算过程非常复杂，直接计算会非常繁琐。关键是要认识到酉变换的性质，即它保持密度矩阵的纯度不变。这样可以大大简化计算。

解题技巧和信息

1. 在处理密度矩阵时，要注意其特殊性质：Hermitian（自伴）、半正定、迹为 1。
2. 量子门操作可以表示为 $U\rho U^{†}$，其中 $U$ 是酉矩阵。
3. 酉变换保持密度矩阵的迹和纯度不变，意味着新态的 $a^{\prime 2}+b^{\prime 2}+c^{\prime 2}$ 保持不变。这是解决复杂问题的关键。
4. 在计算复杂的矩阵乘法时，可以先关注最终需要的元素，而不必计算整个矩阵。
5. Hadamard 门 $H$ 将计算基的状态均匀地混合到对角线基。测量概率可以通过变换后的密度矩阵来计算。
6. Pauli-Y 门 $Y$ 交换计算基的状态并引入相位因子。

重点词汇

• density matrix 密度矩阵
• eigenvalue 特征值
• quantum gate 量子门
• Hadamard gate H 门
• unitary transformation 酉变换
• purity 纯度
• trace 迹

参考资料

1. Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press. Chapter 2 and 4.
2. Wilde, M. M. (2017). Quantum Information Theory. Cambridge University Press. Chapter 3.