# 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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### In Nielsen and Chuang, how can $\frac{1}{2(e-1)}$ result from $\frac12\int_{e-1}^{2^{t-1}-1}dl\frac{1}{l^2}$?

From Nielsen and Chuang's book: $\textit{Quantum computation and quantum information}$, how can (5.34) equal (5.33)? I.e. $$\dfrac{1}{2} \int_{e-1}^{2^{t-1}-1} dl \dfrac{1}{l^2} = \dfrac{1}{2(e-1)}.$$...
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### IF statement in OpenQASM on IBM Quantum Experience

This is a simple circuit introduced in Moran's book "Mastering Quantum Computing with IBM QX" to demonstrate how if works in OpenQASM: ...
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### Show that for any measurement operator $M_m$ there exists unitary $U_m$ such that $M_m=U_m\sqrt{E_m}$ with $E_m$ POVM

Exercise 2.63 of Nielsen & Chuang asks one to show that if a measurement is described by measurement operators $M_m$, there exists unitary $U_m$ such that $M_m = U_m \sqrt{E_m}$ where $E_m$ are ...
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### Calculating outcomes of projective measurements

I'm pretty new to quantum computing, and I'm wondering how I can compute the outcome of a projective measurement of a spin along the +Z axis followed by a projective measurement along the -Z axis. I ...
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### What is the matrix representation of the Hadamard gate in the computational basis?

I read about Hadamard gate H and found it's matrix representation as follows: $$H_1=\frac{1}{\sqrt 2}\begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix}$$ I wanted to know what will be the matrix ...
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### What is the thought process for circuit making after seeing input and output of a matrix?

Here is an exercise (4.27) from Nielsen and Chuang and I found the answer (given in the figure below) online without any explanation. The question was to construct a circuit by seeing a matrix (given ...
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### What is meaning of 'up to a global phase'? [duplicate]

What is the importance of global phase? How it affect a unit vector if we see it on a Bloch sphere? What is the meaning of 'up to global phase' in exercise 4.3 of Nielsen and Chuang?
I am doing the 5th exercise on https://qiskit.org/textbook/ch-states/representing-qubit-states.html#Quick-Exercise (all the way at the bottom). Which states find the angle for the vector $\frac{1}{\... 1answer 98 views ### What is the output of applying the Hadamard matrix to$\sum_{y\in\{0,1\}^n} (-1)^{xy}|y\rangle$? If, for some$x$, I have the$n$-qubit state $$\sum_{y\in\{0,1\}^n} (-1)^{xy}|y\rangle,$$ and I would like to apply to that the$n$-qubit Hadamard transform, with the aim of calculating the final ... 1answer 149 views ### Nielsen & Chuang Exercise 2.55: Prove that$\exp \left[ -\frac{iH(t_2 - t_1)}{\hbar} \right]$is unitary$\newcommand{\expterm}{\frac{-iH(t_2 - t_1)}{\hbar}} \newcommand{\exptermp}{\frac{iH(t_2 - t_1)}{\hbar}}$Nielsen & Chuang (10th edition, page 82) states that$H$is a fixed Hermitian ... 1answer 96 views ### How to find the unitary operation of a depolarizing channel? Suppose we have a depolarizing channel operation $$E(\rho)=\frac{p}{2}\textbf{1}+(1-p)\rho$$ acting on a Spin$\frac{1}{2}$density matrix of the form$\rho=\frac{1}{2}(\textbf{1}+\textbf{s}\cdot\...
Let $|\phi\rangle,|\psi\rangle$ be two state vectors, and let $d=\frac{1}{2}\mathrm{Tr}(\sqrt{(|\phi\rangle\langle\phi|-|\psi\rangle\langle\psi|)^2})$ be their trace distance. Then it will always hold ...