# Questions tagged [textbook-and-exercises]

Applies to questions of primarily educational value - styled in the format similar to that found in textbook exercises. This tag should be applied to questions that are (1) stated in the form of an exercise and (2) at the level of basic quantum information textbooks.

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### Why is the X-Gate a constant function in the deutsch-josza algorithm?

I don't understand, why the X-Gate represents a constant function. It flips 0 to 1 and 1 to 0, which represents a balanced function. I don't understand this explanation from my professor:
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### Prove a linear map $\mathcal{N}$ is completely positive if its Choi operator is positive semi-definite

I'm doing exercise 4.4.1 in Quantum information theory by Wilde. The exercise asks to prove that a linear map $\mathcal{N}_{A\to B}$ is completely positive if its Choi operator is a positive semi-...
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### Wong's "Introduction to Classical and Quantum Computing" Exercise 7.23

I am currently working my way through "Introduction to Classical and Quantum Computing" by Thomas Wong. I am trying to solve the following problem: Exercise 7.23. Answer the following ...
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### Intro book on classical and quantum computing by Thomas G Wong

Looking at his book, and am obviously new to studying this. Could someone help explain to me how the truth table is valid here? To my understanding, when $C=0$, the circuit behaves like a reversible ...
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### Wong's "Introduction to Classical and Quantum Computing" Exercise 7.20

I am currently working my way through "Introduction to Classical and Quantum Computing" by Thomas Wong. I am trying to solve the following problem on Simon's Algorithm: Exercise 7.20. You ...
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### If eigenvalues of two matrices are equal then the matrices are equal?

Suppose $k_i$ and $f_i$ are eigenvalues of two density matrices A and B, If $k_i=f_i$ then A=B? If the answer is no, under which conditions the statement holds?
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### How to represent a general 3-qubit state as a symmetric ZX-diagram with 14 parameters?

A general pure 1-qubit state can be written as a ZX-diagram like this: Correspondingly, for a general pure 2-qubit state: How can a general pure 3-qubit state be written as a ZX-diagram? Two things ...
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### How does measuring a density matrix give Kraus operators?

I am trying to complete this exercise regarding noisy channels. I need to measure a density matrix to get the Kraus operators. However, if I measure, I only get scalars. Can someone please explain how ...
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### Why do minimal ensemble decompositions for $\rho$ contain $|\psi⟩\in{\rm supp}(\rho)$ with probability $1/\langle\psi|\rho^{-1}|\psi⟩?$

I came across the following exercise (2.73) in Nielsen & Chuang and am trying to understand it intuitively. Here is my reasoning of what is going on: The purpose of this exercise: Let’s say we are ...
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### What is meant with "different ensembles can give rise to the same density matrix?"

I am reading the Nielsen & Chuang section on density matrices and I don't understand the example given to demonstrate a concept. Here is what I am reading: First, they said these two different ...
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### Finding the effect of conjugate transpose on a state $|b\rangle$

Say that I have a unitary gate $U$ such that $U|b\rangle=|b+1$ mod $N\rangle$. How would I go about finding $U^\dagger|b\rangle$?
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### What's the Schmidt decomposition of $|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle)$?

$|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle)$ I absolutely cannot figure out the Schmidt decomposition of this state. I have looked at a ton of ...
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### Solution Nielsen and Chuang exercize 10.71

Exercise 10.71: Verify that when $M = e^{−iπ/4}SX$ the procedure we have described gives a fault-tolerant method for measuring $M$. The book describes a procedure to perform the measurement. Instead ...
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### How to find the eigenvectors and eigenvalues of a hermitian operator?

While reading Theoretical Minimum by Leonard Susskind, I came across the exercise 3.4 where he asked to find the eigenvalues and the eigenvectors of the matrix that represents the $\sigma_{n}$ ...
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