# 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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### How to find the operator sum representation of the depolarizing channel?

In Nielsen and Chuang (page:379), it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity ...
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### Procedures and intuition for designing simple quantum circuits?

I'm working my way through one of the quantum circuits sections in Nielsen and Chuang and I'm struggling to get a feel for the basics of circuit construction. For example, one of the exercises is as ...
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### What is the unitary operator realizing a given CPTP operator

Complete Positive Trace Preserving Map (CPTP) operator is the most general operation that can be performed on a quantum system. This post mentioned that a CPTP operator is nothing but a unitary ...
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### Is the tensor product of two states commutative?

I'm reading "Quantum Computing Expained" of David McMahon, and encountered a confusing concept. In the beginning of Chapter 4, author described the tensor product as below: To construct a ...
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### How is it possible to guess what state the qubit was in by measuring it?

Let's say that the qubit is in the state $\psi = \alpha|0\rangle+\beta|1\rangle$. We want to find out the values $\alpha$ and $\beta$. If we measure it in, say, the standard basis, then the outcome we ...
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### How to decompose a unitary single qubit gate by universal quantum gate set?

How to decompose a unitary single qubit gate? I have read some paper or books, which told me a unitary single qubit gate could be decomposed by universal quantum gates set. For example {phase gate, ...
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### Bipartite states whose coefficients are entries of a unitary matrix

I've been trying to solve this question It seems that in order to show it has unit length, we must show that $$\frac{1}{d} \sum_{m, n=0}^{d=1} \lvert U_{m, n}\rvert ^2 = 1$$ I've tried searching ...
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### Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates

Show that a $CZ$ gate can be implemented using a $CNOT$ gate and Hadamard gates and write down the corresponding circuit. Recall from Quantum Information Theory that $Z=HXH$. As $CNOT$ is a ...
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### What happens when you send a Bell state through depolarizing channel?

For noise parameter $Q$ and a density matrix $\rho$, we know that the depolarization channel $\mathcal{E}$ would act like: $$\mathcal{E}(\rho) = (1 - Q)\rho +Q\frac{I}{2},$$ where $I$ is the ...
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### Find the probability of a measurement outcome in terms of the coefficients of the state

Suppose we have a quantum state $|\psi \rangle$ of $n$ qubits, where $|\psi\rangle=\sum_{x∈\{0,1\}^n}\alpha_x |x\rangle$,and we measure the first qubit of $|\psi\rangle$ in the computational basis. ...
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### Can Eve perform this operation?

I am a beginner in quantum computing. Please consider the following scenario: Suppose Alice wants to send $\frac{1}{\sqrt{N}}\sum_{j=0,1,2,..N-1} |j\rangle$ to Bob. Eve has intercepted the state ...
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### How to compute the tensor product of the depolarizing channel with the identity?

Consider two quantum systems A and B, B goes through a depolarizing noise channel, while A is not changed, i.e., they go through the channel $\mathbb{I}_A \otimes \mathcal{E_{\text{depol}}}$. If the ...
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### Representing a von Neumann measurement as $[\mathcal{I} \otimes P_i] U(\rho_s \otimes \rho_a)U^{-1} [\mathcal{I} \otimes P_i]$, how do we choose $U$?

Given the state of a system as $\rho_s$ and that of the ancilla (pointer) as $\rho_a$, the Von-Neumann measurement involves entangling a system with ancilla and then performing a projective ...
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### Generalized set of Pauli elements for a basis for the linear transformations on the vector space [duplicate]

I have been doing some practice problems from "Gentle introduction to Quantum Computing". I am a little bit lost with this one: The generalized Pauli group $\mathcal G_n$ is defined by all elements ...
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### Proof of quantum data processing inequality in N&C on pg 566

On page 566, it states that using $S(\rho^{'})-S(\rho,\varepsilon) \ge S(\rho)$ and combining this with $S(\rho) \ge S(\rho^{'})-S(\rho,\varepsilon))$, we get $S(\rho^{'})=S(\rho)-S(\rho,\varepsilon)$....
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### How does the graphical notation used to denote doubly-controlled gates work?

$\qquad$ $\qquad$ What is the difference between solid and hollow? How to express the corresponding matrix of these figures? In addition, if they are not adjacent, what should be done in the middle of ...
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### Prove that $A\preceq B$ implies $A=\Psi(B)$ for some channel $\Psi$

Define $\newcommand{\PP}{\mathbb{P}}\newcommand{\ket}{\lvert #1\rangle}\newcommand{\tr}{\operatorname{tr}}\newcommand{\ketbra}{\lvert #1\rangle\!\langle #1\rvert}\PP_\psi\equiv\ketbra\psi$, and ...
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### How to find the matrix representation of an operator from its action on a basis?

First, I apologize if something is poorly written but English is not my first language. I know that these exercises have been solved in this question. But I do not agree. Inner product and concrete ...
I understand that every Hilbert space $H$ can be decomposed into two mutually orthogonal subspaces $H_1$ and $H_2$ whose direct sum is $H$. Therefore, every vector $v\in H$ can be decomposed into \$...