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.
682
questions
0
votes
1
answer
34
views
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:
1
vote
1
answer
52
views
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-...
- 53
1
vote
1
answer
44
views
Build a query circuit that set the first bit to 0
This is a question from Chapter 2, 8(c) in Quantum Computing: Lecture notes by Ronald de Wolf.
Suppose we can make queries of the type $|i, b\rangle→ |i, b\oplus x_i\rangle$ to input $x\in\{0, 1\}^N, ...
- 11
4
votes
2
answers
149
views
Prove ${\rm tr}(\rho^2)={\rm tr}((\rho \otimes \rho')S)$ with $S$ the swap operator
I'm trying to prove $P(\rho)={\rm tr}((\rho \otimes \rho')S)$, where $P(\rho)={\rm tr}(\rho^2)$ is the purity and $S$ is the swap operator. My question is that since ${\rm tr}(AB)={\rm tr}(BA)$, do we ...
- 53
1
vote
1
answer
138
views
What is the operator-sum representation of the two-qubit depolarizing channel?
I want to get the operator-sum representation of the two-qubit depolarizing channel
$$ \mathcal{E}(\rho) = (1-\lambda)\rho + \frac{\lambda I}{4}$$
Using $\frac{I}{2} = \frac{\rho +X\rho X +Y\rho Y +Z\...
- 135
2
votes
0
answers
61
views
John Preskill Problem 3.5c
I am trying to solve this problem from John Preskill's lecture notes just for my understanding:
Consider a single-qubit channel with a unitary representation
$$\newcommand\ket[1]{\left|#1\right\...
- 391
-1
votes
2
answers
88
views
Trouble understanding operator sum representation [duplicate]
I am having lot of trouble trying to understand the operator sum representation in Nielsen and Chuang:
I get the very first equation in above but how does that translate to 2 and how does the 3rd ...
- 391
0
votes
1
answer
49
views
Quantum Cryptography without Bell's Theorem -- Brassard - Bennett - Mermin
It is an old paper but I'm trying to understand one of their argument. They say that if
$$U|u\rangle |a\rangle = |u\rangle |a^\prime\rangle \ \ \ \mathrm{and} \ \ \ U|v\rangle |a\rangle = |v\rangle |a^...
- 173
3
votes
1
answer
355
views
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 ...
- 91
1
vote
3
answers
77
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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 ...
- 11
2
votes
1
answer
195
views
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 ...
- 91
1
vote
2
answers
173
views
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?
- 791
3
votes
2
answers
99
views
Show that transformation $U_f: \left| x, y \right\rangle \to \left| x, y \oplus f(x) \right\rangle$ is unitary [duplicate]
I am reading Quantum Computation and Quantum Information by Chuang and Nielsen and they claim that it is easy to show that transformation $U_f: \left| x, y \right\rangle \to \left| x, y \oplus f(x) \...
- 131
-1
votes
1
answer
127
views
Hong Ou Mandel interference and bell basis measurment
It is well known that using Hong Ou Mandel interference in polarization one can only detect 2 out of the 4 bell states($|\psi^+\rangle$ and $|\psi^-\rangle$ can be detected but $|\phi^+\rangle$ and $|\...
0
votes
0
answers
38
views
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 ...
- 894
1
vote
1
answer
68
views
Clarification about the Alberti's Theorem proof given by Watrous in his condensed lecture notes
In the John Watrous condensed TQI lecture notes, an alternative proof of the Alberti's Theorem is given. He use an auxiliary lemma that states;
Lemma 4.9. Let $P \in Pos(X)$. It holds that $${inf}_{R\...
- 63
5
votes
2
answers
432
views
Represent Hadamard gate in terms of rotations and reflections in Bloch sphere
I read in a book that any single qubit operation can be decomposed as
$$
\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{\...
- 51
0
votes
1
answer
58
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 ...
- 173
4
votes
2
answers
85
views
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 ...
- 173
1
vote
2
answers
78
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 ...
- 173
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 $U^\dagger|b\rangle$?
- 63
2
votes
1
answer
91
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 ...
- 21
1
vote
2
answers
64
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, ...
- 13
0
votes
0
answers
121
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 ...
- 63
0
votes
1
answer
31
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
$$\rho_{AB'}=\sum_{i=1}^{k}p_{i}...
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3
votes
1
answer
60
views
How large does the isometry in Naimark's theorem need to be for a 3-outcome POVM?
I am interested in the POVM example Nielsen and Chuang give in the discussion about indistinguishability. They define the POVM
$E_1 = \frac{\sqrt{2}}{1+\sqrt{2}} |1\rangle \langle 1|$,
$E_2 = \frac{\...
- 31
1
vote
0
answers
67
views
How to express a traceless matrix in Pauli basis
This question is probably too obvious, so sorry beforehand.
We know that the generalized Pauli elements $P\in \mathcal{P}_d \setminus {\mathrm{Id}_d}$ in Sylvesters representation, hence not Hermitian,...
- 135
2
votes
2
answers
221
views
Exercise 11.7 in Nielsen & Chuang and basic properties of Shannon entropy
I apologize in advance if this question is trivial, I'm aware I'm a total beginner in this field. This is the exercise I would like to solve:
As to the first point, what I get is that one should ...
- 61
4
votes
1
answer
79
views
Bound on success Probability for Regev's factoring algorithm
Theorem 4.1 in Regev's paper talks about a theorem due to Pomerance as follows:
Theorem 4.1: Suppose G is a finite abelian group with minimal number of generators $r$. Then, when choosing elements ...
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1
vote
1
answer
52
views
Why can't the eigenvalues of a unitary matrix have the form $e^{i\theta}$?
The textbook says that since $U$ is a unitary matrix, its eigenvalue should be of the form $e^{2 \pi i \theta}$. The thing I don't understand is why it's not $e^{i \theta}$ because it also lies on the ...
- 23
1
vote
1
answer
71
views
Can any isometry $V$ be written as $U(I\otimes |\psi\rangle)=V$ for some unitary $U$ and vector $|\psi\rangle$?
I have the following exercise:
Let $V : H_A → H_A ⊗ H_E$ denote an isometry and $|ψ_E⟩ ∈ H_E$ a normalized
vector. Show that there exists a unitary $U : H_A ⊗ H_E → H_A ⊗ H_E$ such
that
$$U(1_{H_A} ⊗ |...
- 163
0
votes
2
answers
120
views
An Introduction to Quantum Computing - Exercise 6.4.1
The Exercise 6.4.1 from Kaye et al. is as follows
Prove that $$\bigg({|0\rangle +(-1)^{x_1}|1\rangle
\over\sqrt{2}}\bigg)\cdot\bigg({|0\rangle +(-1)^{x_2}|1\rangle
\over\sqrt{2}}\bigg)\cdots\...
- 63
0
votes
2
answers
75
views
Help finding mistake when modifying $T$ injection protocols
I am a little confused about where I am going wrong when computing the action of the following circuit:
My understanding is that the CNOT gate acts on the second qubit as a control and the first ...
- 705
1
vote
0
answers
68
views
How does Chernoff's bound help to solve Exercise 6.4.2 in Kaye et al.'s textbook? [duplicate]
I was wondering if anyone could help me with this question, I'm kind of new to quantum computing in general. I understand the Deutsch Josza Algorithm, but I'm not really sure where to even begin with ...
- 19
2
votes
3
answers
197
views
Is $|A\rangle = \frac{1}{\sqrt2} |00\rangle + \frac{1}{\sqrt2} |01\rangle$ a valid quantum state?
Is $|A\rangle = \frac{1}{\sqrt2} |00\rangle + \frac{1}{\sqrt2} |01\rangle$ a valid quantum state? Or does a quantum state need to be a superposition of the entire basis, i.e.,
$$ |A\rangle = \frac{1}{...
- 31
-1
votes
1
answer
162
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....
0
votes
1
answer
63
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]=...
- 137
2
votes
2
answers
210
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: $$\text{Tr}(M |\psi\rangle \langle\psi)=\langle\psi| M |\psi\...
- 193
0
votes
2
answers
73
views
Performing a projective measurement, is the resulting expectation value $\langle \Psi|M|\Psi\rangle$ bounded between $+1$ and $-1$?
Suppose we have a quantum state $|\Psi\rangle = \alpha|0\rangle + \beta|1\rangle$.According to a measurement operator M, the projective measurement of $|\Psi\rangle$ is given by $\langle\Psi|M|\Psi\...
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-2
votes
1
answer
61
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help understanding gate to hamiltonian and representation
So I have this question: Given an operator, find some Hamiltonian implementing this operator/gate. I have realized that this is a swap gate and I know the matrix for it. I also know that $U = \text{...
- 1
2
votes
1
answer
63
views
When are two Hermitian operators unitarily similar?
Let $A$ and $B$ $2^n \times 2^n$ Hermitian matrices. What are sufficient and necessary conditions that they are equal up to some unitary, i.e. there exists $U$ such that $A = U B U^\dagger$?
The first ...
- 525
1
vote
1
answer
73
views
How to take partial trace of a $n - 1$ qubit subsystem from a $n$ qubit system
I would like to calculate the expression
$$
\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\,,$$
...
- 45
1
vote
1
answer
86
views
Quantum teleportation of unknown qubit when the entangled state is not a Bell state
Assume Bob and Alice have two particles with a prior entanglement: $A$ and $B$. The entangled state $|Ψ⟩$ is maximally entangled, and $$|Ψ⟩ = \frac{1}{\sqrt{2}}(|00⟩ + j|11⟩)\,,$$ where $j$ is a ...
- 21
2
votes
1
answer
269
views
Trace Distance in Bloch sphere, what is the vector of Pauli matrices?
While reading Chapter 9.2.1 Trace distance in "Quantum Computation and Quantum Information," I encountered a question. What is the vector of Pauli matrices referring to?
$$
\vec{\sigma} = (\...
0
votes
2
answers
102
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 the state
$|\psi\rangle:=A|00\rangle+B|11\rangle$.
If I have another qubit
in the state $|\phi\rangle:=a|0\rangle+b|1\...
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 $(...
- 705
2
votes
0
answers
149
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 ...
- 21
2
votes
1
answer
258
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 $\sigma_{n}$ ...
- 93
2
votes
1
answer
86
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 ...
0
votes
1
answer
69
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?
- 93