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.
44
questions with no upvoted or accepted answers
5
votes
1
answer
361
views
Regarding the inductive proof that any Clifford gate can be made of Hadamard, phase and c-not
In Exercise 10.40 of Nielsen and Chunang's textbook, the reader is supposed to construct an inductive proof of Theorem 10.6 that any Clifford gate can be made of Hadamard, phase and c-not. There it is ...
- 51
5
votes
0
answers
379
views
Prove a circuit with controlled $iR_x(\pi \alpha)$ is universal for quantum computation whenever $\alpha$ is irrational
Show that the three qubit gate $G$ defined by the circuit is universal for quantum computation whenever $\alpha$ is irrational.
My Observations
The unitary gate on the third qubit is activated only ...
- 831
5
votes
0
answers
95
views
Pure state ensembles achieving the Holevo $\chi$-quantity with at most $d^2$ states
Theorem 8.10 in Chapter 8 of Theory of Quantum Information asserts that the Holevo capacity of a quantum channel (between density operators on $\mathbb{C}^d$) can be achieved by an ensemble consisting ...
- 2,111
5
votes
0
answers
123
views
Complexity of controlled operations in a two-level unitary operation
In Neilsen and Chuang, chapter 4.5.2 (~p.193), why did the authors come to the conclusion that complexity of operations $C^n(X)$ and $C^n(\tilde{U})$ is $O(n)$?
Did they assume using work qubits? If ...
- 166
4
votes
0
answers
368
views
Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy
The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is
$$
S(A,B)\ge|S(A)-S(B)|
$$
this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
- 831
4
votes
0
answers
66
views
What does the $I$ mean when measuring ${\rm Tr}(\rho (I\otimes\sigma\otimes\cdots))$ in quantum tomography?
In Nielsen and Chuang's QCQI, I learned that the quantum tomography for n qubit can be described easily in math as we need to measure $Tr(\rho W_k),\forall k$ where $W_k\in\{I,\sigma_x,\sigma_y,\...
- 3,009
4
votes
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80
views
A question in classical and quantum information
Let $\rho, \sigma \in \mathfrak{D}(A)$ with $\operatorname{supp}(\rho) \subseteq \operatorname{supp}(\sigma),$ and spectral decomposition
$$
\rho=\sum_{x} p_{x}\left|\psi_{x}\right\rangle\left\langle\...
- 133
3
votes
2
answers
897
views
How to choose a suitable number of iterations for Grover's algorithm?
In Nielsen and Chuang (2010), section 6.1.1. it is written:
"For an N item search problem with M solutions, it turns out that we need only apply the search oracle ...
- 51
3
votes
0
answers
118
views
Upper bound on the distance between two distinct orthonormal vectors
I need to prove that if $\phi$ and $\psi$ are distinct vectors of an orthonormal set then $|| \phi - \psi|| \leq \sqrt{2} $.
Going by the definition of norm, $|| \phi - \psi||^2$ is the inner product $...
- 199
2
votes
2
answers
161
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
2
votes
0
answers
140
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
0
answers
53
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 ...
- 29
2
votes
0
answers
41
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 ...
- 518
2
votes
0
answers
102
views
$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/...
- 63
2
votes
0
answers
33
views
How to obtain an expression of the complexity of the period-finding algorithm with respect to the period length?
Intro. Nielsen and Chuang in Quantum computation and quantum information on section 5.4.1 state that the period-finding algorithm has a runtime of $U + O(L^2)$ operations where $L$ is the size of the ...
