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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0answers
38 views

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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1answer
87 views

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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58 views

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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1answer
30 views

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?
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35 views

Secret string in least significant bit with Simon algorithm [closed]

The function $f:\{0,1\}^2\to\{0,1\}$ , $f(x_1x_0)=x_0$ returns the least significant bit of its argument. Solve the Simon's problem for this function and write down the number "a" (secret ...
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1answer
50 views

Least significant bit with Simon algorithm [closed]

would you kindly help me solve this problem: The function $f:\{0,1\}^2\to\{0,1\}$ , $f(x_1x_0)=x_0$ returns the least significant bit of its argument. What is the $U_f$ operator for this function?
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2answers
54 views

Having trouble finding angles for Bloch vector

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}{\...
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1answer
90 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 ...
3
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1answer
134 views

Nielsen & Chuang Exercise 2.55: Prove that $\exp \left[ -\frac{iH(t_2 - t_1)}{\hbar} \right]$ is unitary

$\newcommand{\expterm}[0]{\frac{-iH(t_2 - t_1)}{\hbar}} \newcommand{\exptermp}[0]{\frac{iH(t_2 - t_1)}{\hbar}}$Nielsen & Chuang (10th edition, page 82) states that $H$ is a fixed Hermitian ...
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1answer
86 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\...
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2answers
37 views

Relation between trace distance and inner product between pure states

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 ...
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1answer
64 views

Nielsen & Chuang Exercise 2.32: Show that the tensor product of two projectors is a projector

$\newcommand{\bra}[1]{\left<#1\right|} \newcommand{\ket}[1]{\left|#1\right>}$Here is what I tried: Given that we have two projectors: $$ A = \sum_i \ket{i} \bra{i}, \hspace{2em} B = \sum_j \ket{...
3
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1answer
62 views

Eigenvectors and eigenvalues of the gate $U_a:|s\rangle\mapsto|sa \bmod N\rangle$

I'm studying Shor algorithm. This is a demostration about the eigenvectors and eigenvalues of $U_a$ gate: Can somebody explain me from the third step to the last?
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1answer
44 views

Nielsen & Chuang Theorem 2.6 Proof

I got a problem in understanding the proof of the Theorem 2.6 (Unitary freedom in the ensenble for density matrices), 2.168 and 2.169 in the Nielsen and Chuang book Equation 2.168 Suppose $|{\tilde\...
2
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1answer
26 views

Asymmetry in distributing phase change across components

The quantum computing text books and theory in general seems to have added an asymmetry in the distribution of change in phase across the components in the context of a qubit. Is there any reason for ...
3
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1answer
46 views

Prove entanglement in the final state of the Deutsch-Jozsa circuit

I am asked to prove the following: Consider the Deutsch-Jozsa circuit. The output of the circuit is of the form $|\psi\rangle \otimes \frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$. Prove that the state$|\...
2
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1answer
57 views

How is $S(\rho)=H(p_{i})+\sum_{i}p_{i}S(\rho_{i})\le \log(d)$ possible if $\rho_{i}$ are not pure states?

I know how this can be proved using the quantum relative entropy. However, even with this proof, and am still confused about how this emerges. Say I have a source that produces two states $\rho_1$ and ...
3
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1answer
71 views

Deriving $\left( A | v \rangle \right)^\dagger = \langle v | A^\dagger$ without using $A^\dagger=\left(A^* \right)^T$

From Nielsen & Chuang (10th edition), page 69: Suppose $A$ is any linear operator on a Hilbert space, $V$. It turns out that there exists a unique linear operator $A^\dagger$ on $V$ such that for ...
2
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1answer
50 views

How to construct a non-trivial (non-projective) POVM measurement example?

We know that generalized (POVM) measurement is defined by matrices $M_i$ which are Positive semidefinite Add up to a unit matrix, $\sum_i M_i = \mathbb{I}$ and the probability of obtaining outcome $...
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1answer
40 views

Confused about associativity of outer product notation

Consider this expression where $A$ and $B$ are matrices, $|i \rangle$ is a ket (column vector) and $\langle j |$ is a bra (row vector) : $$ A | i \rangle \langle j | B \tag1\label1 $$ Due to the ...
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1answer
57 views

Hadamard direct mapping of input to output in $\theta$ and $\varphi$ form

I was wondering what would be an equation for Hadamard operation for a single qubit, given the input as the current $\theta$ (0 to $+\pi/2$) and $\varphi$ ($-\pi$ to $+\pi$) and output expected in $\...
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3answers
67 views

Why can every $|X\rangle\in H_1\otimes H_0$ be written as $|X\rangle=(X\otimes I_{H_0})|\Omega \rangle$ for some $X\in\mathcal L(H_0,H_1)$?

In A theoretical framework for quantum networks is proven that a linear map $\mathcal{M} \in \mathcal{L}(\mathcal{H_0},\mathcal{H_1})$ is CP (completely positive) iff its Choi operator $M$ is semi ...
3
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1answer
64 views

Why is the action of controlled-Z unaltered by exchanging target control qubits?

In the book "Quantum Computer Science", when explaining the error correction code, it uses this picture and says "the action of controlled-z is unaltered by exchanging the target and ...
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2answers
93 views

Why does the Hadamard gate satisfy $H|x\rangle=\frac{1}{\sqrt2}\sum_{z\in\{0,1\}}(-1)^{xz}\lvert z\rangle$?

I'm studying Deutsch–Jozsa algorithm and I don't understand this passage about Hadamard gate result: $$\newcommand{\ket}[1]{\lvert #1\rangle}H\ket x=\frac{1}{\sqrt2}(\ket0+(-1)^x\ket1)=\frac{1}{\sqrt2}...
4
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1answer
45 views

What to do when the amount of solutions is not known before applying Grovers Algorithm?

When running Grovers Algorithm one has to know how many solutions there are right? When the number of solutions are not known is then what do you do then?
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2answers
44 views

How are eigenvectors and eigenvalues expressed in the Bloch sphere?

I'm relatively new to the subject of quantum computing, and I recently came across the idea of eigenvalues and eigenvectors. I believe I understand the relationship between the two, where eigenvalues ...
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1answer
51 views

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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1answer
68 views

What does the unitary $[|0\rangle\langle 0|\otimes I+|1\rangle\langle1|\otimes(|1\rangle\langle 0|+|0\rangle\langle1|)]\otimes I$ represent?

Consider the following unitary defined for a system $A$ interacting with a bipartite system $BB^\prime$ $$U_{AB} = \Big[|0\rangle \langle 0|_{A} \otimes \mathbf{I}_{B} + |1\rangle \langle 1|_{A} \...
3
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3answers
140 views

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 ...
2
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2answers
56 views

Calculate probability of a state after depolarization

Let's say I have a particle in the quantum state $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, represented as a density operator (1st matrix) that went through a depolarizing chanel (2nd ...
2
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1answer
95 views

Prove that $A\preceq B$ implies $A=\Psi(B)$ for some channel $\Psi$

Define $\newcommand{\PP}{\mathbb{P}}\newcommand{\ket}[1]{\lvert #1\rangle}\newcommand{\tr}{\operatorname{tr}}\newcommand{\ketbra}[1]{\lvert #1\rangle\!\langle #1\rvert}\PP_\psi\equiv\ketbra\psi$, and ...
1
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1answer
41 views

How do I measure a single qubit in a two-qubit state?

Let us suppose that I have the state \begin{equation} \frac{1}{\sqrt2}(\alpha|0\rangle|+\rangle+\beta|1\rangle|-\rangle) \end{equation} and I choose to measure the first qubit in the basis $\{(1/\...
2
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2answers
50 views

How does Bell measurement work in the teleportation?

I'm a complete beginner and one of the first things I was taught was the teleportation protocol. In the protocol, the party sending its state (which we call say $|\phi\rangle$) makes a Bell ...
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2answers
70 views

Meaning behind obtaining a hermitian operator for measurement in another basis?

If $$P_{+} = |+\rangle\langle+|=\frac{1}{2}(|0\rangle\langle0|+|0\rangle\langle1|+|1\rangle\langle0| +|1\rangle\langle1|)$$ and $$P_{-} = |-\rangle\langle-|=\frac{1}{2}(|0\rangle\langle0|-|0\rangle\...
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1answer
41 views

What is the Kraus representation of quantum-to-classical channels?

As discussed in Watrous' book, quantum-to-classical channels are CPTP maps whose output is always fully depolarised. These can always be written as $$\Phi_\mu(X) = \sum_a \langle X,\mu(a)\rangle E_{a,...
2
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2answers
105 views

Why do we divide by $\sqrt2$ in the qubit states $\lvert\pm\rangle=\frac{1}{\sqrt2}(\lvert0\rangle\pm\lvert1\rangle)$?

I have a very basic question. I have found qubits are represented as complex vectors. I get it totally. I understand bracket notation and vector\matrix algebra. However, I cannot move further from ...
2
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0answers
70 views

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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1answer
25 views

In quantum teleportation what is the need for the extra X/Z gate after classical information is sent from Alice to Bob?

In the Qiskit textbook, at step 4, Bob who has received the classical bits from Alice then needs to apply a X and/or Z gate depending on what the classical bits received are, why is that?
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3answers
92 views

Can a Kraus representation act as the identity on any operator?

In the textbook “Quantum Computation and Quantum Information” by Nielsen and Chuang, it is stated that there exists a set of unitaries $U_i$ and a probability distribution $p_i$ for any matrix A, $$\...
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1answer
88 views

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 ...
2
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2answers
71 views

Find the Kraus operators of a combined amplitude and phase damping channel

I am going through the paper Surface code with decoherence: An analysis of three superconducting architectures and I have a doubt about how the authors get what they refer to as the combined channel ...
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2answers
95 views

Nielsen and Chuang: Demonstration of equation 2.12

Reproduced from Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition) in page 64: We've seen that matrices can be regarded as linear operators. [...] Suppose $...
2
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1answer
51 views

Calcuate $\langle x | D | y \rangle$ for arbitrary $x,y \in \{0,1\}^n$

We are considering Grover's algorithm with a search space of size $2^n$ for an arbitrary integer $n$ for arbitrary $n$, and a unique marked element $x_0$. Question: Calculate $\langle x | D | y \...
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1answer
36 views

What is the representative matrix for a measurement in the Bell-state basis?

I have a few questions about measurement in Bell-state basis. In particular, if $Z = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}$ is for a measurement on the computational basis, then what ...
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0answers
44 views

Step-by-step passages in calculation

I would like to better understand some passages in a paper (Appendix A): Properties of Tensor Product Bilinearity: $A\otimes(B+ C) = A \otimes B + A \otimes C $ Mixed-product property: $(A\otimes B)(...
3
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1answer
67 views

How do you embed a POVM matrix in a Unitary?

In QuantumKatas Measurement Task 2.3 - Peres-Wooter's Game, we are given 3 states A,B and C. We construct a POVM of these states. But how do we convert that POVM into a Unitary that we can apply. ...
3
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1answer
53 views

Problem about entanglement swapping

I have been learning about the concept of entanglement swapping and found an equation mentioned in the textbook, Mathematics Of Quantum Computing: An Introduction written by Wolfgang Scherer. At ...
0
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1answer
22 views

Compute the negativity of maximally entangled bipartite states

The entanglement negativity $\mathcal N(\rho)$ of a (bipartite) state $\rho$ is defined as the absolute value of the sum of the negative eigenvalues of the partial transpose of a state, or ...
0
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1answer
23 views

Why does the entanglement negativity equal (in magnitude) the sum of the negative eigenvalues?

The entanglement negativity, introduced in (Vidal and Werner 2002), is defined as $$\mathcal N(\rho) \equiv \frac{\|\rho^{T_B}\|_1-1}{2}.$$ It is mentioned there that this equals the sum of the ...
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2answers
143 views

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 ...