Skip to main content
Share Your Experience: Take the 2024 Developer Survey

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.

Filter by
Sorted by
Tagged with
5 votes
2 answers
354 views

Represent Hadamard gate in terms of rotations and reflections in Bloch sphere

I read in a book that any single qubit operation can be decomposed as $$ \bf{U} =e^{i\gamma}\begin{pmatrix}e^{-i\phi/2}&0\\ 0&e^{i\phi/2}\end{pmatrix}\begin{pmatrix}\cos{\theta/2}&-\sin{\...
A1Y's user avatar
  • 51
0 votes
1 answer
45 views

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 ...
researcher101's user avatar
1 vote
0 answers
25 views

Decomposing density matrix into arbitrary minimal ensemble

I came across this exercise 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 given a density ...
researcher101's user avatar
1 vote
2 answers
36 views

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 ...
researcher101's user avatar
1 vote
1 answer
53 views

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 $U^\dagger|b\rangle$?
afebs's user avatar
  • 57
-3 votes
0 answers
53 views

Eigenvalue Estimation Question

Consider the eigenvalue estimation algorithm. Suppose that instead of being given a single eigenstate $|\psi\rangle$ with eigenvalue $e^{2\pi i\omega}$ you are given the state $$|\phi\rangle=\frac{1}{\...
afebs's user avatar
  • 57
2 votes
1 answer
66 views

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 ...
qityhd's user avatar
  • 21
1 vote
2 answers
53 views

In the QECC condition $\langle\psi|E_a^\dagger E_b|\phi\rangle=C_{ab}\langle\psi|\phi\rangle$, what is $C_{ab}$?

In this book, Theorem 2.7 has the QECC conditions. I attach a snippet here Theorem 2.7 (QECC Conditions). $(Q, \mathcal{E})$ is a $Q E C C$ iff $\forall|\psi\rangle,|\phi\rangle \in Q, \forall E_a, ...
Polya's user avatar
  • 13
0 votes
0 answers
116 views

An Introduction to Quantum Computing Exercise 7.1.6

This question is from An Introduction to Quantum Computing Kaye et al. I'm having a difficult time coming up with a solution for this question. It is in relation to period finding however I cannot ...
afebs's user avatar
  • 57
-2 votes
0 answers
48 views

How to prove that the function is continuous?

Let $A_0, A_1 \in \operatorname{Pos}(\mathcal{X})$ and $B_0, B_1 \in \operatorname{Pos}(\mathcal{Y})$ be positive semidefinite operators, for complex Euclidean spaces $\mathcal{X}$ and $\mathcal{Y}$, ...
Aimin Xu's user avatar
  • 129
0 votes
1 answer
28 views

Can any separable $\rho=\sum_i p_i\sigma_i\otimes\tau_i$ be written as $\rho=(I\otimes T)(\sum_ip_i\sigma_i\otimes|i⟩\!⟨i|)$ for some channel $T$?

I am struggling with the following exercise, and was wondering if anybody had any good tips on how to attack the problem/where to begin: Given a separable quantum state $$\rho_{AB'}=\sum_{i=1}^{k}p_{i}...
Pink Elephants's user avatar
3 votes
1 answer
52 views

How large does the isometry in Naimark's theorem need to be for a 3-outcome POVM?

I am interested in the POVM example Nielsen and Chuang give in the discussion about indistinguishability. They define the POVM $E_1 = \frac{\sqrt{2}}{1+\sqrt{2}} |1\rangle \langle 1|$, $E_2 = \frac{\...
BenPhys's user avatar
  • 31
1 vote
0 answers
60 views

How to express a traceless matrix in Pauli basis

This question is probably too obvious, so sorry beforehand. We know that the generalized Pauli elements $P\in \mathcal{P}_d \setminus {\mathrm{Id}_d}$ in Sylvesters representation, hence not Hermitian,...
relativeentropy's user avatar
2 votes
2 answers
142 views

Exercise 11.7 in Nielsen & Chuang and basic properties of Shannon entropy

I apologize in advance if this question is trivial, I'm aware I'm a total beginner in this field. This is the exercise I would like to solve: As to the first point, what I get is that one should ...
atlantropa's user avatar
3 votes
1 answer
62 views

Bound on success Probability for Regev's factoring algorithm

Theorem 4.1 in Regev's paper talks about a theorem due to Pomerance as follows: Theorem 4.1: Suppose G is a finite abelian group with minimal number of generators $r$. Then, when choosing elements ...
Manish Kumar's user avatar
1 vote
1 answer
43 views

Why can't the eigenvalues of a unitary matrix have the form $e^{i\theta}$?

The textbook says that since $U$ is a unitary matrix, its eigenvalue should be of the form $e^{2 \pi i \theta}$. The thing I don't understand is why it's not $e^{i \theta}$ because it also lies on the ...
Nir Sharma's user avatar
1 vote
1 answer
60 views

Can any isometry $V$ be written as $U(I\otimes |\psi\rangle)=V$ for some unitary $U$ and vector $|\psi\rangle$?

I have the following exercise: Let $V : H_A → H_A ⊗ H_E$ denote an isometry and $|ψ_E⟩ ∈ H_E$ a normalized vector. Show that there exists a unitary $U : H_A ⊗ H_E → H_A ⊗ H_E$ such that $$U(1_{H_A} ⊗ |...
Pink Elephants's user avatar
0 votes
2 answers
108 views

An Introduction to Quantum Computing - Exercise 6.4.1

The Exercise 6.4.1 from Kaye et al. is as follows Prove that $$\bigg({|0\rangle +(-1)^{x_1}|1\rangle \over\sqrt{2}}\bigg)\cdot\bigg({|0\rangle +(-1)^{x_2}|1\rangle \over\sqrt{2}}\bigg)\cdots\...
afebs's user avatar
  • 57
0 votes
2 answers
71 views

Help finding mistake when modifying $T$ injection protocols

I am a little confused about where I am going wrong when computing the action of the following circuit: My understanding is that the CNOT gate acts on the second qubit as a control and the first ...
am567's user avatar
  • 585
1 vote
0 answers
67 views

How does Chernoff's bound help to solve Exercise 6.4.2 in Kaye et al.'s textbook? [duplicate]

I was wondering if anyone could help me with this question, I'm kind of new to quantum computing in general. I understand the Deutsch Josza Algorithm, but I'm not really sure where to even begin with ...
sdfsdfsdf555's user avatar
2 votes
3 answers
135 views

Is $|A\rangle = \frac{1}{\sqrt2} |00\rangle + \frac{1}{\sqrt2} |01\rangle$ a valid quantum state?

Is $|A\rangle = \frac{1}{\sqrt2} |00\rangle + \frac{1}{\sqrt2} |01\rangle$ a valid quantum state? Or does a quantum state need to be a superposition of the entire basis, i.e., $$ |A\rangle = \frac{1}{...
user29495's user avatar
-1 votes
1 answer
103 views

How to calculate a density matrix of a given circuit?

I want to find the density matrix of the following quantum circuit, is it correct: [[0.12499999+0.j 0.12499999+0.j 0.12499999+0.j 0.12499999+0.j 0.12499999+0.j 0.12499999+0.j 0.12499999+0.j 0....
Frank Hansen's user avatar
0 votes
1 answer
56 views

How do I prove the following maps are completely positive?

I am trying to prove that the following superoperators are quantum channels, that is completely positive and trace-perserving linear maps 1 $\Psi[M]=WMW^\dagger$ where $W$ is an isometry 2 $\Psi[M_A]=...
darkside's user avatar
  • 137
2 votes
2 answers
206 views

Prove that $\text{Tr}(M|ψ\rangle\langleϕ|)=\langleϕ|M|ψ\rangle$

Question: I am studying alone, and I found p.76 of the book quantum computation and quantum information of nielsen &c huang that: $$\text{Tr}(M |\psi\rangle \langle\psi)=\langle\psi| M |\psi\...
OffHakhol's user avatar
  • 155
0 votes
2 answers
73 views

Performing a projective measurement, is the resulting expectation value $\langle \Psi|M|\Psi\rangle$ bounded between $+1$ and $-1$?

Suppose we have a quantum state $|\Psi\rangle = \alpha|0\rangle + \beta|1\rangle$.According to a measurement operator M, the projective measurement of $|\Psi\rangle$ is given by $\langle\Psi|M|\Psi\...
aghin's user avatar
  • 129
-2 votes
1 answer
55 views

help understanding gate to hamiltonian and representation

So I have this question: Given an operator, find some Hamiltonian implementing this operator/gate. I have realized that this is a swap gate and I know the matrix for it. I also know that $U = \text{...
George's user avatar
  • 1
1 vote
1 answer
46 views

conditions for two hermitians operators same up to unitary

Let $A$ and $B$ $2^n \times 2^n$ Hermitian matrices. What are sufficient and necessary conditions that they are equal up to some unitary, i.e. there exists $U$ such that $A = U B U^\dagger$? The first ...
Jon Megan's user avatar
  • 497
1 vote
1 answer
60 views

How to take partial trace of a $n - 1$ qubit subsystem from a $n$ qubit system

I would like to calculate the expression $$ \text{Tr}_2\left\{R^z \sigma\right\}\,, $$ where $$ \sigma = \rho \otimes |0\rangle \langle0|^{{\otimes}(n-1)}\,. $$ Here $$ R = \sum{\theta_m}G_m\,,$$ ...
Sudhir Kumar's user avatar
1 vote
1 answer
79 views

Quantum teleportation of unknown qubit when the entangled state is not a Bell state

Assume Bob and Alice have two particles with a prior entanglement: $A$ and $B$. The entangled state $|Ψ⟩$ is maximally entangled, and $$|Ψ⟩ = \frac{1}{\sqrt{2}}(|00⟩ + j|11⟩)\,,$$ where $j$ is a ...
Dogukan's user avatar
  • 21
2 votes
1 answer
218 views

Trace Distance in Bloch sphere, what is the vector of Pauli matrices?

While reading Chapter 9.2.1 Trace distance in "Quantum Computation and Quantum Information," I encountered a question. What is the vector of Pauli matrices referring to? $$ \vec{\sigma} = (\...
Wang Sheffield's user avatar
0 votes
1 answer
63 views

How does a three-qubit state evolve through a CNOT gate?

Suppose I have a qubit which is entangled with another; let's say they are in $A|00\rangle+B|11\rangle$. If I have another qubit in the state $a|0\rangle+b|1\rangle$ then the combined state is $Aa|000\...
Tamilaruvi Saravanan's user avatar
0 votes
1 answer
41 views

Understanding the error operator representation $E = i^{\lambda}X(a)Z(b)$

Question regarding exercise $27.3.2$ in "Concise Encyclopedia of Coding Theory". The exercise states: We write $E = X((0,1))Z((0,0))$ and $E' = iX((0,1))Z((1,1))$. We choose the ordering $(...
am567's user avatar
  • 585
2 votes
0 answers
126 views

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 ...
Maria 's user avatar
  • 21
2 votes
1 answer
215 views

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 $\sigma_{n}$ ...
zizaaooo's user avatar
2 votes
1 answer
81 views

Error 'LocalSimulator' with Googlecolab

I have an error when I want to run the 'LocalSimulator'. I am not inside AWS, its mean I runnig from Google Colab. The code is the same on the notebooks from ...
Sandra Rosales's user avatar
0 votes
1 answer
62 views

What does "the eigenvectors of a Hermitian operator are a complete set" mean?

I read in my book that the eigenvectors of a Hermitian operator are a complete set. What does the author mean by that?
zizaaooo's user avatar
1 vote
0 answers
65 views

Distinguish two states with their priors probability

EDIT: This is a computer programming / coding exercise The states $\left|\psi\right>$ and $\left|\phi\right>$ are defined as $∣ϕ⟩=\cos(θ_ϕ)\left|0\right>+\sin(θ_ϕ) \left|1\right>$ with ...
Minh Triet's user avatar
-2 votes
1 answer
76 views

Measuring probabilities of 0 or 1 in a two qubit state

I'm preparing for my upcoming exam, I need to determine the probabilities with which Bob measures 0 or 1 and in both cases describe the state of Alice, this is my state: $$ |\psi\rangle =\frac{1}{\...
embe99's user avatar
  • 1
1 vote
1 answer
127 views

Is a linear combination of unitaries unitary?

Suppose you have a pure state $\vert\psi\rangle$. Consider the following operation. For unitaries $U_1$ and $U_2$, one can take complex numbers $\alpha, \beta$ where $|\alpha|^2 + |\beta|^2 = 1$ and ...
user1936752's user avatar
  • 2,997
0 votes
1 answer
44 views

Calculate of theoretical probabilities for the outcomes

I have a $|+\rangle$ state qubit and I measure it in a random basis. The random basis is made with random $\theta$, $\varphi$ and $\lambda$ of $U3$ gate. How can I calculate the theoretical ...
hdsa's user avatar
  • 9
2 votes
0 answers
52 views

Measuring an entangled quantum state

I have this exercise to solve, but I can't figure out how to proceed. First, I don't think the state proposed is valid (the probabilities don't sum up to 1). 'Secondly, given that it should be an ...
Giulia's user avatar
  • 29
0 votes
0 answers
30 views

Deriving a equation in "Data re-uploading for a universal quantum classifier" paper: U(phi1,phi2,phi3) U(x1,x2,x3) = U(theta + w *x)

In "Data re-uploading for a universal quantum classifier" paper the U(phi1,phi2,phi3) U(x1,x2,x3) is derived as U(theta + w *x) in compact state. How to derive the above equation ?
Aakash's user avatar
  • 1
1 vote
1 answer
32 views

Is qiskit.pulse.SamplePulse deprecated?

I'm currently following along the book "Learn Quantum Computing with Python and IBM Quantum Experience" by Rodert Loredo and I am at chapter 8 - Generating pulse schedules on hardware. The ...
Johannes Jyrgenson's user avatar
6 votes
2 answers
177 views

Decomposition of a $4 \times 4$ unitary matrix

I am currently studying the paper "Decomposition of unitary matrices and quantum gates (2012)" and referring to the textbook Quantum Computation and Quantum Information. Among the topics, I ...
junghyunHa's user avatar
0 votes
1 answer
151 views

How to show that the GHZ state is absolutely maximally entangled?

A multipartite state is called absolutely maximally entangled if for its any bipartition the reduced density matrix of smaller part is maximally mixed. Show that GHZ state has this property.
user27383's user avatar
5 votes
1 answer
119 views

Question about Nielson & Chuang Problem 9.2

I am working on the following problem from the book "Quantum Computation and Quantum Information" by Nielsen and Chuang. Problem 9.2: Let $\mathcal{E}$ be a trace-preserving quantum ...
DJD's user avatar
  • 53
3 votes
1 answer
104 views

Why is the operator $M_a |x\rangle= |a \cdot x \pmod{N} \rangle $ unitary?

If $N\geq 2$, $a\in \mathbb{Z}_N$, and $a^r= 1$ for some $r$. Consider the operator $M_a$, which is related to order finding : $M_a |x\rangle= |a \cdot x \pmod{N} \rangle $ if $x\in \mathbb{Z}_N$ What ...
metaUser's user avatar
0 votes
2 answers
69 views

Why is a density matrix an orthogonal projector?

Suppose I have a density matrix like $\rho = \frac{1}{2}[I + \hat{n}\vec{\sigma}]$. The claim is that $\rho$ is an orthogonal projector for the state $|+\rangle$ in an arbitrary direction $\hat{n}$. ...
Physkid's user avatar
  • 520
1 vote
0 answers
65 views

Possible post - measurement states for Bell state $\frac{1}{\sqrt{2}}[|00\rangle + |11\rangle]$

This is in reference to page 241 of Introduction to classical and quantum computing by Thomas.G Wong. The author starts off with a Bell state $\frac{1}{\sqrt{2}}[|00\rangle + |11\rangle]$. In trying ...
Physkid's user avatar
  • 520
3 votes
2 answers
121 views

How to know what eigenvalue corresponds to measurements of individual qubits in a multiqubit system?

I'm working through the book "Introduction to the Theory of Quantum Information Processing" by Bergou and Hillary, and I've encountered a scenario that I'm not sure how to approach. In ...
YaGoi Root's user avatar

1
2 3 4 5
14