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.
620
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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 ...
- 53
0
votes
1
answer
50
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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.
5
votes
1
answer
105
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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
...
- 53
3
votes
1
answer
80
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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 ...
- 51
0
votes
2
answers
43
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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 $\hat{n}$.
...
- 510
1
vote
0
answers
55
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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 ...
- 510
3
votes
2
answers
101
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 ...
- 57
0
votes
1
answer
62
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The expectation values for the values of both qubits [closed]
Let’s consider the two-qubit state
|Ψ⟩ =(1/2)|00⟩ + i(√3/4)|01⟩ +(3/4)|10⟩.
a) Find the expectation values for the values of both qubits separately.
1
vote
0
answers
30
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Why does unitary matrix acts only on input qubit state of a vector that is a result of add modulo 2?
Let $|\psi\rangle = \frac{1}{\sqrt{2}}\sum_{k=0}^{1}(-1)^{ka}|k, k \oplus b\rangle $ so that $|\psi'\rangle = \frac{1}{\sqrt{2}}[\sum_{k=2}^{1}(-1)^{ka}(U_{A}|k\rangle \otimes U_{B}|k \oplus b\rangle)]...
- 510
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vote
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What is the expression for $|\psi\rangle\!\langle\psi|$ if $|\psi\rangle=\cos(\theta/2)|0\rangle+\sin(\theta/2)e^{i\phi}|1\rangle$?
Let $|\psi\rangle = \alpha|0\rangle + \beta |1\rangle$. In Bloch sphere representation, this is
$\cos\frac{\theta}{2}|0\rangle + \sin\frac{\theta}{2}e^{i\phi}|1\rangle$.
In matrix representation:
$|\...
- 510
1
vote
1
answer
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views
Why can $(0,0,3/5,0,0,0,4/5,0,0)$ be written as $\frac35|3\rangle+\frac45|7\rangle$?
Context. $\newcommand{\qr}[1]{\left|#1\right\rangle}$ A passage from a lecture by Scott Aaronson: "As an example, instead of writing out a vector like $$(0,0,3/5,0,0,0,4/5,0,0),$$ you can simply ...
- 227
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votes
2
answers
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What we get when measure $|0\rangle$ under computational basis?
It is said if we have been given the state $|0\rangle$, the measurement will yield $0$ with probability $1$ in Nielsen's book.
So here, the measurement will yield $0$ refers to we will get state $|0\...
- 657
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answer
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views
Does the state obtained flipping $a,b$ in the state $(a,b)^T$ have a name?
Suppose we have a qubit with a state vector of $\begin{pmatrix}
a \\
b
\end{pmatrix}
$.
If we flip $a$ with $b$ does the new qubit has a name in relation to the first qubit?
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Define a traceless part of $\rho$ [closed]
I saw in a paper: $|\bar{\rho}\rangle\rangle=|\rho\rangle\rangle-|\hat{I}\rangle\rangle / 2^{n / 2}$ for the $4^n$-dimensional vector representing the traceless part of $\rho$. https://arxiv.org/abs/...
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calculate the reduced density matrix of a 2 qubit state and compare the eigenvalues
So I have the exercise to apply a Cz gate to the following 2 Qubit state
$|a\rangle \otimes |b\rangle = (a_0 |0\rangle + a_1 |1\rangle) \otimes (b_0 |0\rangle + b_1 |1\rangle)\\\\$
Afterwards, I ...
- 31
1
vote
0
answers
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views
Do solutions to the exercises in Preskill's lecture notes exist anywhere?
I'm teaching myself quantum computing and would like to know if solutions to the exercises in Preskill's lecture notes exist anywhere. It's quite hard to see if my approach is correct because I don't ...
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How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = Tr_{2}[\rho_{12}] = \sum_{i}\langle i_{2} | \rho | i_{2} \rangle$?
Let the system be a 2 - qubit system and let $\rho_{12}$ be a density matrix of some state for this 2 - qubit system.
How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = Tr_{2}[\...
- 510
2
votes
1
answer
89
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How to figure out whether a truth table can correspond to a valid quantum gate
I am new to quantum computing and trying to wrap my head around this exercise from Wong's introduction to classical and quantum computer.
I can interpret it mentally that first is a valid quantum ...
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2
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76
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Explanation of the 2.60 equation page 76 in the Nielsen and Chuang [duplicate]
In the Nielsen and Chuang book page 76, equation 2.60 says that we can rewrite the trace $$Tr(A \left|\psi\right>\left<\psi\right|)$$ as follow :
$$Tr(A \left|\psi\right>\left<\psi\right|) ...
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1
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160
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Unital channel which is not mixed unitary
How to prove that for a multi-qubit system a unital channel is not necessarily mixed unitary? This is Problem 8.3 in Nielsen and Chuang. Here's a snippet of the text:
Shall I need to take two ...
4
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1
answer
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How do I find the reduced density matrix of a system where two people share one qubit and have one qubit of their own?
I have the following problem and have attempted to find a solution to it, but to no avail.
Alice and Bob have one qubit each, say $|\psi\rangle$ with Alice and $|\phi\rangle$ with Bob. They also share ...
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vote
3
answers
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Exercise 4.16 in the Nielsen & Chuang book
In the 4.16 exercice in the Quantum Computation and Quantum Information (Michael A. Nielsen & Isaac L. Chuang), I don't understand why the correct answer is not this matrix :
$$
\left[ {\begin{...
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votes
2
answers
479
views
What are "completely positive" and "CPTP" quantum maps?
I am studying quantum computing a little bit by myself, and I have simple questions.
I didn't find a clear definition of what is a completely positive and trace-preserving (CPTP) map. The best I've ...
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2
votes
0
answers
37
views
Mechanics of expanding projector operator (two - qubits) in basis of traceless Hermitian Paul operators
I am currently on a set of lecture notes which says that for a state vector $| \psi \rangle_{AB}$ describing a tensor product state, its density operator $| \psi \rangle \langle \psi |_{AB}$ can be ...
- 510
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votes
5
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257
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How to eliminate the global phase of a state vector?
Say that I have a qubit that began in the $|0\rangle$ state and then the
Hadamard gate is applied, resulting in the following state:
$ \begin{bmatrix}
\frac{1}{\sqrt{2}} \\
\frac{1}{\sqrt{2}}
\end{...
- 143
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votes
0
answers
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views
What is the value of $p(+)$?
I know the formula is $p = \left<\psi\right|M_{m}^{\dagger} M_{m}\left|\psi\right>$, where $\left|\psi\right>= \alpha\left|0\right>+\beta\left|1\right>$ and $M_m=\left|+\right> \left&...
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1
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views
Can a density operator be written equivalently as $\rho=\sum_i p_i|\psi_i〉\!\langle\psi_i|$ and $\rho=\sum_i\lambda_i|\psi_i\rangle\!\langle\psi_i|$?
My doubt arises from page 99, 101 of the book Quantum Computation and Quantum Information by Michael A.Nielson and Issac L.Chung.
Let {${p_{i}, | \psi_{i} \rangle }$} be an ensemble of pure states.
...
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Prove that the eigenvectors of a Hermitian operator form a basis
While I was reading the book Quantum Mechanics The Theoretical Minimum, the author said that if a vector space is $N$ dimensional, an orthonormal basis of $N$ vectors can be constructed from ...
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0
answers
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$T_1$ and $T_2$ time with amplitude damping
Exercise 8.30 of Nielson & Chuang's QCQI says
Equation 7.144, which is mentioned in the text, is
$$\begin{bmatrix}
a & b\\
b^* & 1-a
\end{bmatrix}\rightarrow\begin{bmatrix}
(a-a_0)e^{-t/...
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1
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1
answer
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What is the probability of a state $|0\rangle$ being in another state $\alpha|0\rangle+\beta|1\rangle$?
I am trying to calculate the probability of a state (density matrix) being in a specific other state.
Lets say I have a 2-dimensional state with the states given by the orthonormal basis states $|0\...
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votes
2
answers
200
views
How to prove the matrix identities $HXH = Z$ and $HZH = X$?
As we know Hadamard gates are used to bring quantum bits into superposition states.
I’m trying to understand how identities $HXH = Z$ & $HZH = X$ w.r.t rotation.
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How to go from matrix to bra-ket representation of the CNOT?
I have the following definition for CNOT gate form my notes:
I am trying to derive the bracket notation form the matrix version, can someone help me to see where I am going wrong:
$$U ̂_{CNOT} ...
2
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1
answer
42
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How to show that the trace distance equals the maximal total variation distance?
Let $\rho$ and $\sigma$ be two density operators such that probability of obtaining $a$ is $tr(\rho E_a)$ if the state before measurement was $\rho$ and $tr(\sigma E_a)$ if the state before ...
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1
vote
1
answer
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Decomposing Hadamard gate
I'm following a lesson, and it says that the Hadamard gate can be decomposed to three gates: RZ(pi/2), squared root Z, and RZ(pi/2). However, when I do matrix multiplication of these three matrices, I ...
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Derivation explanation needed [closed]
I'm pretty new to the field. I was reading Preskill Ph219 course notes and came across this. I am a bit confused about the derivation and wondered if someone can write down some skipped steps here.
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Is the quantum state fidelity defined as $|\langle\psi|\phi\rangle|$ or $|\langle\psi|\phi\rangle|^2$? [duplicate]
I am reading Preskill's notes on quantum information as well as Chuang's textbook. I saw that fidelity is defined in two different ways in Preskill's notes and Chuang's book. In Preskill's notes, ...
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0
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Link between Lagrange Multipliers in Quantum Optimal Control and Traditional Method
In the context of quantum optimal control, the term "Lagrange multiplier" has been used, which seems to draw a connection to the traditional Lagrange multiplier method used in problems of ...
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1
vote
1
answer
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How do you write the Hadamard operator on two qubits in braket notation?
I understand how to write the Hadamard operator on one qubit in braket notation using $$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} H=\sum_{i,j} \bra{w_{j}}H\ket{...
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0
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Understanding the Use of Derivatives in Optimal Control for Quantum Computing
I'm currently studying quantum computing and I have a question regarding the application of derivatives in the context of optimal control. I would like to provide a simple and generic example and ...
- 63
0
votes
1
answer
39
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In quantum error correction, what does an "arbitrary error that yields an un-normalized state" mean?
This is from page 434 of Nielsen and Chuang:
. Supposing the state of the encoded qubit is |ψ⟩
before the noise acts, then after the noise has acted the state is
E(|ψ⟩⟨ψ|). To analyze the effects of
...
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answer
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In Shor's algorithm, why do we have ${\rm gcd}(x\pm 1, N) > 1$?
I'm struggling to understand the last part of Shor's algorithm, to be exact the point when we found $x-1$, $x+1$ with $x-1 ≠ 0\mod N$, $x+1 ≠ 0 \mod N$ and $(x+1)(x-1) = 0 \mod N$.
Then, $gcd(x-1, N) &...
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votes
2
answers
498
views
What is the density matrix of a pure state?
By definition of the density matrix for example the density matrix of $|0\rangle$ state (pure state) is:
$$\rho=|0\rangle \langle 0| =
\begin{pmatrix}
1 & 0 \\
...
- 249
1
vote
1
answer
111
views
Why does Sakurai's book use the gyromagnetic ratio of the electron as $\frac{e}{m_e c}$ instead of $\frac{e}{m_e}$?
I'm studying quantum computing, but I had to review some concepts in supplementary readings, where I came across the following question about a simple detail: Why does Sakurai's book (near Eq. 1.1) ...
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0
votes
1
answer
74
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Definition of fault-tolerant measurement in QCQI
In the book "Quantum Computation and Quantum Information," one of the definitions for fault-tolerant measurement is mentioned as "any single component in the procedure results in an ...
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0
votes
1
answer
80
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Clarification regarding application of distributive property in "quantum teleportation" example
For context, this is from Page 27 of Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press:
She then sends the ...
- 147
1
vote
1
answer
109
views
Can we rearrange terms in the tensor product?
Define $o = A \otimes B$. Compute the results of $o^{\otimes N}$. We have
\begin{align}
o^{\otimes N}
&= (A \otimes B) \otimes (A \otimes B) \otimes ... \otimes (A \otimes B).
\end{align}
Can we ...
- 505
-2
votes
1
answer
60
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Unitary matrix knowing input and ouput states
Input: n qubits after H gates.
Output, after measure: the n different bit strings of length 2^n with just one 1, with same probabilities.
Or said differently:
Input state |phi0>=(1,1,...1)/2^n
...
3
votes
2
answers
195
views
Can you distinguish between $|0\rangle, |1\rangle$, and $\frac{1}{\sqrt 2} (|0\rangle + |1\rangle)$?
(A beginner here; possibly a stupid question. Please be gentle. Sorry if I used a wrong tag.)
Suppose that I receive a (classically) random number, which is either $1$ or $2$ or $3$. Depending on this ...
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0
votes
1
answer
52
views
did i calculate the matrix elements correctly?
Suppose the below operator
$$
x\sum_{n=0}^{\infty}\tanh^{2n}(r)|0,n\rangle\langle 0,n|
+y\sum_{n=0}^{\infty}\tanh^{2n}(r)|1,n\rangle\langle 1,n|
+z\sum_{n=0}^{\infty}\tanh^{2n}(r)(n+1)|1,n\rangle\...
- 469
2
votes
1
answer
288
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How would you draw the phase-estimation circuit for the eigenvalues of $U = \mathrm{diag}(1,1,\exp(\pi i/4),\exp(\pi i/8)) $?
How would you draw the phase-estimation circuit for the eigenvalues of:
$U = \mathrm{diag}(1,1,e^{(\pi i)/ 4}, e^{(\pi i)/8}) $
corresponding to the eigenstates $|10\rangle$ and $|11\rangle$?
What is ...
