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東京大学 新領域創成科学研究科 メディカル情報生命専攻 2024年1月実施 問題11

Author​

zephyr

Description​

A quantum state of a 1-qubit quantum computer can be represented by a complex matrix called a density matrix. Here, is the imaginary unit that satisfies and are real numbers that satisfy . By measuring the qubit represented by matrix in the computational basis (also called the Z-basis), we observe state 0 or state 1 with probabilities and , which are the diagonal elements of matrix , respectively. Answer the following questions with mathematical derivation.

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

  2. Answer the probability that state 0 is observed by measurement in the computational basis after applying quantum gate operation to quantum state .

  3. Answer the probability that state 0 is observed by measurement in the computational basis after applying quantum gate operation to quantum state .

  4. Let be the quantum state after applying quantum gate operation to quantum state . Compute .

1个量子比特量子计算机的量子态可以表示为一个 的复数矩阵 ，称为密度矩阵。这里， 是满足 的虚数单位， 是满足 的实数。通过在计算基（也称为Z基）中测量由矩阵 表示的量子比特，我们以概率 和 观察到状态0或状态1，分别对应密度矩阵 的对角元素。用数学推导回答以下问题。

  1. 证明矩阵 的所有特征值都是非负实数。

  2. 在对量子态 施加量子门操作 后，在计算基中测量观察到状态0的概率是多少。

  3. 在对量子态 施加量子门操作 后，在计算基中测量观察到状态0的概率是多少。

  4. 设 是施加量子门操作 后的量子态。计算 。

Kai​

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。

1. Show that all the eigenvalues of matrix are non-negative real numbers​

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

The solutions to this quadratic equation are:

Since , we have . Therefore, both eigenvalues are real and non-negative.

2. Probability of observing state 0 after applying H gate​

The Hadamard gate operation transforms to . Let's calculate this:

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

3. Probability of observing state 0 after applying Y gate​

Similarly, for the Y gate:

The probability of observing state 0 is:

4. Compute after applying U gate​

Let and

We need to calculate . This is a complex calculation, so let's break it down:

Now, multiplying this by and focusing on the diagonal elements:

Where:

Now, we need to compute . This is a very long and complex calculation. However, we can use a property of unitary matrices: . This implies that the transformation preserves the trace and the purity of the density matrix.

The purity of a density matrix is defined as .

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

Therefore:

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

Knowledge​

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 ，其中 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 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 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 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.