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.
692 questions
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Showing $H(B)_\omega \leq H(B)_\rho$ via concavity of von Neumann entropy
Denoting in the following $\phi$
with a pure state and $\tau$ as the maximally mixed state, furthermore $p\in[0,1]$.
In Wilde's quantum information theory book, according to eq. 20.111,
by the ...
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2
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Why is an operator unitary if and only if its matrix representation is unitary?
I am currently reading Nielsen & Chuang's "Quantum Computation and Quantum Information". In it, for an operator (= a linear function) $f\colon V \to W$, the linear operator $f^\dagger$ ...
- 75
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Computing the expected value of a spin - 1 particle component given density matrix
I have a density matrix $\rho$ where
$$\rho = \frac{1}{4} \cdot \begin{pmatrix} 2 & 1 & 1\\ 1 & 1 & 0\\ 1 & 0 & 1 \end{pmatrix}$$
and the x component of a spin - 1 particle ...
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How does the upper bound on the entropy of the depolarizing channel follow from the concavity of von Neumann entropy?
Let $\sigma_{XA}= \sum_x p_X(x) \Pi_x \otimes \Phi_A $ where $\Pi_x$ is the projector onto $x$ and $\Phi$ is a pure state. The output state after applying the depolarizing channel $D_p$ is
$$\omega_{...
- 155
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What are examples of application of Pauli gates to three-qubit states?
The matrix math from Quantum Computation and Quantum Information is always at most for two qubits, which left the subject vague and unclear to me. So I have a three qubit entangled superposition that ...
- 171
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52
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Reversible Two-Four-Three Swap
I'm trying to solve this. I'm very new to circuit-based reversible computing and have studied about gates. I'm trying to see how can I solve this.
As far as I've come, I see that I might need to put a ...
- 19
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Nielsen and Chuang Exercise 5.9
Let $U$ be a unitary transform with eigenvalues $±1$, which acts on a
state $|ψ〉$. Using the phase estimation procedure, construct a quantum
circuit to collapse $|ψ〉$ into one or the other of the two ...
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Simplifying quantum computing expression
I am working through Lecture Notes by Ronald Wolf: https://arxiv.org/pdf/1907.09415
I am trying to solve exercise 1.5 (page 11 in the pdf) which goes like the following:
Simplify the following: $(\...
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1
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46
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About the isometric extension of the erasure channel
I'm doing exercises 5.2.6 and 5.2.7 in Quantum Information Theory by Wilde. In 5.2.6, it asks us to show that an isometric extension of the erasure channel is
\begin{align}
U &\overset{a}{=} \sqrt{...
- 95
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2
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242
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How to apply a Hadamard gate to the first qubit and then measure that first qubit in the computational basis
I am reading lecture notes from Ronald de Wolf on quantum computing and trying to solve exercise 7 from chapter 1. I am having a bit of trouble understanding exactly what I need to do and how to do it....
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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:
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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-...
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0
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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, ...
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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 ...
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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}$$
Using $\frac{I}{2} = \frac{\rho +X\rho X +Y\rho Y +Z\...
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69
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John Preskill Problem 3.5c
I am trying to solve this problem from John Preskill's lecture notes just for my understanding:
Consider a single-qubit channel with a unitary representation
$$\newcommand\ket[1]{\left|#1\right\...
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2
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117
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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 ...
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58
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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^...
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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 ...
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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 ...
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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 ...
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198
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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?
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108
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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) \...
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1
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157
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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 $|\...
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40
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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 ...
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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\...
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503
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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{\...
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60
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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 ...
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93
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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 ...
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2
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116
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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 ...
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57
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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$?
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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 ...
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2
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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, ...
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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 ...
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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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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{\...
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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,...
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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 ...
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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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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 ...
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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
$$U(1_{H_A} ⊗ |...
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An Introduction to Quantum Computing - Exercise 6.4.1
The Exercise 6.4.1 from Kaye et al. is as follows
Prove that $$\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\...
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2
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83
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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 ...
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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 ...
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3
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285
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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.,
$$ |A\rangle = \frac{1}{...
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1
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How to calculate a density matrix of a given circuit?
I want to find the density matrix of the following quantum circuit, is it correct:
[[0.12499999+0.j 0.12499999+0.j 0.12499999+0.j 0.12499999+0.j
0.12499999+0.j 0.12499999+0.j 0.12499999+0.j 0....
0
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1
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73
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How do I prove the following maps are completely positive?
I am trying to prove that the following superoperators are quantum channels, that is completely positive and trace-perserving linear maps
1 $\Psi[M]=WMW^\dagger$ where $W$ is an isometry
2 $\Psi[M_A]=...
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2
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215
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Prove that $\text{Tr}(M|ψ\rangle\langleϕ|)=\langleϕ|M|ψ\rangle$
Question:
I am studying alone, and I found p.76 of the book quantum computation and quantum information of nielsen &c huang that: $$\text{Tr}(M |\psi\rangle \langle\psi)=\langle\psi| M |\psi\...
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2
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75
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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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1
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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{...
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