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.

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3
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0answers
47 views

In Nielsen and Chuang, how can $\frac{1}{2(e-1)}$ result from $\frac12\int_{e-1}^{2^{t-1}-1}dl\frac{1}{l^2}$?

From Nielsen and Chuang's book: $\textit{Quantum computation and quantum information}$, how can (5.34) equal (5.33)? I.e. $$\dfrac{1}{2} \int_{e-1}^{2^{t-1}-1} dl \dfrac{1}{l^2} = \dfrac{1}{2(e-1)}.$$...
2
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1answer
59 views

Discrepancy in inner product between tensor products

I have noticed one identity in case of tensor product from this post. But I can't understand why it is true. $\langle v_i| \otimes \langle w_j| \cdot |w_k\rangle \otimes |v_m\rangle = \langle v_i|v_m\...
1
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1answer
45 views

Inequality in overlap of quantum states

For quantum states $\vert\psi_1\rangle, \vert\psi_2\rangle, \vert\phi\rangle$, is it true that $$\tag{1}\langle \phi\vert\psi_1\rangle\langle\psi_1\vert\phi\rangle\langle \phi\vert\psi_2\rangle\langle\...
2
votes
1answer
41 views

In Uhlmann's theorem, should the polar decomposition be written as $A=|A|V$ or $A=V|A|$?

In the proof of Uhlmann's theorem, the book writes the polar decomposition: $A = |A|V$, with $|A| = \sqrt{A^\dagger A}$. Shouldn't it be $V|A|$ instead? The former case is $A^\dagger A = V^\dagger|A||...
3
votes
1answer
55 views

How are the eigenvalues of $\rho=\frac12(|a\rangle\!\langle a| +|b\rangle\!\langle b|)$ derived?

Let's say I have a density matrix of the following form: $$ \rho := \frac{1}{2} (|a \rangle \langle a| + |b \rangle \langle b|), $$ where $|a\rangle$ and $|b\rangle$ are quantum states. I saw that ...
1
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1answer
59 views

CSS Code and Fault-tolerant Problem in Nielsen and Isaac Chuang‘s book

Could someone help me with $Problem-10.52$ in Nielsen and Isaac Chuang‘s book? The screenshot is shown below. I have no idea about that. Hope someone can give me some suggestions. By the way, if I ...
1
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1answer
60 views

Pauli Matrix tolerant and encoding

I'm stumped by these questions in Chuang's book. If I have a state $|ψ\rangle=1/2\sqrt2(1+M_0)(1+M_1)(1+M_2)|0\rangle_7$, where $$M_0=X_0X_4X_5X_6; M_1=X_1X_3X_5X_6; M_2=X_2X_3X_4X_6$$ I rewrite its ...
1
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2answers
74 views

What is $\sum_{i}\langle i \vert U \vert j\rangle$ for unitary $U$?

The question is basically the title but given a unitary operator $U$ and a computational basis, can we say anything about the complex number below? $$c = \sum_{i}\langle i \vert U \vert j\rangle$$ I ...
-2
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0answers
37 views

Quantum Principal component analysis [closed]

Please explain me how the repeated application of (1) creates the line i marked in qpca paper.
1
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1answer
50 views

How does feedback work in simple Grovers algorithm where $n=4$?

In this example implementation of Grovers Algorithm from the Qiskit Textbook which solves a $2\times 2$ sudoku puzzle: https://qiskit.org/textbook/ch-algorithms/grover.html The circuit iterates twice (...
0
votes
1answer
68 views

What are the possible results of measuring $X$ and $Z$ on the state $|01\rangle+|10\rangle$?

When calculating the probability of getting +1 on X-basis on the first qubit of Bell's state $|01\rangle+|10\rangle$, the result is 1/2 with the state after measurement |++⟩ while the probability of ...
2
votes
1answer
48 views

Nielsen & Chuang Exercise 6.13: Standard deviation of classical counting algorithm

$\newcommand{\expectation}[1]{\mathop{\mathbb{E}} \left[ #1 \right] } \newcommand{\Var}{\mathrm{Var}}$ From Nielsen & Chuang 10th edition page 261: Consider a classical algorithm for the counting ...
2
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1answer
71 views

What is the probability of finding the second qubit as $0$ in the state $|\psi\rangle=\frac1{\sqrt2}|00\rangle+\frac12|10\rangle-\frac12|11\rangle $?

Assuming two qubits start in the state: $|\psi\rangle = \frac{1}{\sqrt 2}|00\rangle + \frac{1}{2}|10\rangle- \frac{1}{2}|11\rangle $ What is the probability of measuring the second qubit as 0? And ...
1
vote
2answers
157 views

What is the result of measuring $\sigma_x$ on the state $|01\rangle+|10\rangle$?

I confused about how to calculate the probabilities and getting a certain result of measuring Bell's states with Pauli matrices as the operator. When you measure something, the state involved would be ...
1
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2answers
62 views

What are the properties of the matrices representing quantum gates?

Quantum gates are basically matrices belonging to $C_{2\times2}$. Now, what are the properties of these matrices? We know that to preserve the normalization factor of the qubits these are unitary. All ...
2
votes
1answer
27 views

Hybrid lower bound proof Kaye Laflamme Mosca (lemma 9.3.6)

I am confused about one point in the proof on the lower bounds in Kaye, Laflamme Mosca's lemma 9.3.6. Context: $|\psi_T\rangle$ is the final state of the search algorithm that started on the all-zero ...
1
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0answers
20 views

Non-ideal coin tossing

Can someone please check if the following makes sense? We have a non-ideal coin-tossing scheme as follows. Alice and Bob know what $|0\rangle,|1\rangle$ are. Bob wins when the coin is 1. Honest Alice ...
2
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1answer
50 views

Non-ideal bit commitment and coin tossing

Can someone please explain rigorously the reasoning behing non-ideal ($\epsilon$-concealing and $\delta$-binding) bit commitment scheme and the impossibility of coin tossing. What is bias in coin ...
2
votes
1answer
43 views

Is it wrong to say that $a$ and $b$ are the square roots of the detection probabilities in a qubit state $|\psi \rangle = a|0 \rangle +b|1 \rangle $?

Is it wrong to say in $a$ and $b$ are the square roots of the probability of the qubit being in the state 0 and 1 when measured for a qubit in the state $|\psi \rangle = a|0 \rangle +b|1 \rangle $? ...
1
vote
1answer
76 views

What is the difference between the states $i|1\rangle$ and $|+i\rangle$?

I am new to Quantum computing. I see $|\mbox{+}i\rangle$ state maps to y-axis on bloch sphere ($\theta = 90$ degree and $\phi = 90$ degree) while $i|1\rangle$ maps on x-axis, $i|1\rangle$ is stated as ...
1
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2answers
37 views

How to represent the state vector form of a qubit in density matrix representation? [duplicate]

While I'm studying state vector and density matrix. I wonder how to write qubit state as density matrix. qubit state can be represented with state vector form. But how about density matrix?
2
votes
2answers
113 views

Prove that the trace norm is dual to the spectral norm

Suppose $A\in L(X,Y)$. $||\cdot||$ denotes spectral norm and denotes the largest singular value of a matrix, i.e. the largest eigenvalue of $\sqrt{A^*A}$. $||\cdot||_{tr}$ denotes trace norm. We have ...
2
votes
1answer
70 views

von Neumann entropy in a limiting case

I am stuck with a question from the book Quantum theory by Asher Peres. Excercise (9.11): Three different preparation procedures of a spin 1/2 particle are represented by the vectors $\begin{pmatrix} ...
1
vote
1answer
35 views

What does a negative ket state represent?

I'm studying - well trudging through - Quantum Computing for Everyone by Bernhardt. Towards the bottom of the book's physical page 41 he states, "spins are given by either," and then shows ...
3
votes
2answers
78 views

Optimizations in quantum circuits

In a paper called On quantum circuits employing roots of the Pauli matrices, I found this figure, where I couldn't understand the equality in the circled circuits. I need an explanation of how the ...
0
votes
0answers
55 views

Is the tensor product with the multiplication distributive or associative?

Hello is the tensor product with the multiplication distributive or associative? When having the formula $$X_{1} \prod_{i\in (2,3)}(Z_{i})$$ is the then $$X_{1} \prod_{i\in (2,3)}(Z_{i}) = (X_{1}\...
5
votes
3answers
284 views

Nielsen&Chuang 5-qubit quantum error-correction encoding gate

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ In ...
6
votes
2answers
208 views

How to compute the measurement probability in swap test?

The figure of a circuit and the state are as follows. The final state before the measurement is $|O_{out}\rangle=\frac{1}{2}|0\rangle(|\phi\rangle|\psi\rangle+|\psi\rangle|\phi\rangle)+\frac{1}{2}|1\...
0
votes
1answer
33 views

Basis Change Substitution

question: "A spin right $\frac{1}{\sqrt 2}(|0\rangle + |1\rangle)$ is sent through a Hadamard gate, creating the superposition of $|+\rangle$ and $|-\rangle$, given by $\frac{1}{\sqrt 2}(|+\...
-1
votes
1answer
49 views

Outer Product Intution [duplicate]

Please, help me understand this statement. The outer product notation for matrices also gives an intuitive input-output relation for them. For instance, the matrix |0⟩ ⟨1| + |1⟩ ⟨0| can be read as &...
3
votes
3answers
45 views

Prove that different purifications of a state can be mapped into one another via local unitaries

Let $\rho \in \mathfrak{D}(A)$ be a density matrix. Show that $\left|\psi^{A B}\right\rangle \in A B$ and $\left|\phi^{A C}\right\rangle \in A C$ (assuming $\left.|B| \leqslant|C|\right)$ are two ...
3
votes
2answers
81 views

What is a basis (not necessarily orthogonal) of Herm(A) consisting of pure density matrices in D(A)?($A \cong \mathbb{C}^{n}$)

Let $A \cong \mathbb{C}^{n}$ be a Hilbert space $A,$ and let $\operatorname{Herm}(A)$ be the Hilbert space consisting of all Hermitian matrices on $A$. Give an example of a basis (not necessarily ...
2
votes
1answer
137 views

How are the Pauli $X$ and $Z$ matrices expressed in bra-ket notation? [duplicate]

For example: $$\rm{X=\sigma_x=NOT=|0\rangle\langle 1|+|1\rangle\langle 0|=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}}$$ $$\rm{Z=\sigma_Z=signflip=|0\rangle\langle 0|-|1\rangle\langle 1|=\...
1
vote
1answer
59 views

How does a projective measurement distinguish between two states in a $d$-dimensional Hilbert space?

Let $\mathcal{H}$ be a $d$ -dimensional Hilbert space, and let $|\psi\rangle,|\phi\rangle \in \mathcal{H}$ be two quantum states. Show that if $|\psi\rangle$ and $|\phi\rangle$ are orthogonal, then ...
3
votes
1answer
312 views

How to translate the Hadamard gate matrix into Dirac notation?

Hadamard gate matrix is: $$\frac{1}{\sqrt 2}\begin{bmatrix}1 && 1 \\ 1 && -1\end{bmatrix}$$ The Dirac notation for it is: $$\frac{|0\rangle+|1\rangle}{\sqrt 2}\langle0|+\frac{|0\...
1
vote
0answers
72 views

Wrong Expectation value when implementing a VQE for the Heisenberg Hamiltonian

I tried to implement an extended Heisenberg-Hamiltonian as an extra exercise further than my homework. My Hamiltonian is the following: $H = \sum_{NN} \sigma_x\sigma_x + \sigma_z\sigma_z$ I try to ...
2
votes
1answer
50 views

What linear map is needed for acting on a maximally entangled state?

I was reading a textbook and I encountered this question. I was wondering why we don't consider $M^\dagger$ instead of $M^{T}$, so I didn't show this relation, could you please help me to show below ...
1
vote
1answer
57 views

Why is local qasm simulation taking so long for VQE?

I've successfully run the $LiH$ VQE simulation detailed in Simulating Molecules using VQE using the statevector_simulator, however when attempting to run the same ...
1
vote
1answer
76 views

Example of Quantum Error Correction [closed]

Shor's 9 Qubit code. Imagine that we encode the state $| \psi \rangle = \alpha | 0 \rangle + \beta | 1 \rangle$ using Shor's 9 qubit code, then an X error occurs on the 8th qubit of the encoded state $...
3
votes
1answer
77 views

IF statement in OpenQASM on IBM Quantum Experience

This is a simple circuit introduced in Moran's book "Mastering Quantum Computing with IBM QX" to demonstrate how if works in OpenQASM: ...
3
votes
1answer
48 views

Show that for any measurement operator $M_m$ there exists unitary $U_m$ such that $M_m=U_m\sqrt{E_m}$ with $E_m$ POVM

Exercise 2.63 of Nielsen & Chuang asks one to show that if a measurement is described by measurement operators $M_m$, there exists unitary $U_m$ such that $M_m = U_m \sqrt{E_m}$ where $E_m$ are ...
3
votes
0answers
44 views

Calculating outcomes of projective measurements

I'm pretty new to quantum computing, and I'm wondering how I can compute the outcome of a projective measurement of a spin along the +Z axis followed by a projective measurement along the -Z axis. I ...
0
votes
1answer
92 views

What is the matrix representation of the Hadamard gate in the computational basis?

I read about Hadamard gate H and found it's matrix representation as follows: $$H_1=\frac{1}{\sqrt 2}\begin{pmatrix}1 & 1 \\1 & -1\end{pmatrix}$$ I wanted to know what will be the matrix ...
3
votes
0answers
66 views

What is the thought process for circuit making after seeing input and output of a matrix?

Here is an exercise (4.27) from Nielsen and Chuang and I found the answer (given in the figure below) online without any explanation. The question was to construct a circuit by seeing a matrix (given ...
1
vote
1answer
49 views

What is meaning of 'up to a global phase'? [duplicate]

What is the importance of global phase? How it affect a unit vector if we see it on a Bloch sphere? What is the meaning of 'up to global phase' in exercise 4.3 of Nielsen and Chuang?
2
votes
2answers
75 views

Having trouble finding angles for Bloch vector

I am doing the 5th exercise on https://qiskit.org/textbook/ch-states/representing-qubit-states.html#Quick-Exercise (all the way at the bottom). Which states find the angle for the vector $\frac{1}{\...
0
votes
1answer
98 views

What is the output of applying the Hadamard matrix to $\sum_{y\in\{0,1\}^n} (-1)^{xy}|y\rangle$?

If, for some $x$, I have the $n$-qubit state $$\sum_{y\in\{0,1\}^n} (-1)^{xy}|y\rangle,$$ and I would like to apply to that the $n$-qubit Hadamard transform, with the aim of calculating the final ...
3
votes
1answer
149 views

Nielsen & Chuang Exercise 2.55: Prove that $\exp \left[ -\frac{iH(t_2 - t_1)}{\hbar} \right]$ is unitary

$\newcommand{\expterm}[0]{\frac{-iH(t_2 - t_1)}{\hbar}} \newcommand{\exptermp}[0]{\frac{iH(t_2 - t_1)}{\hbar}}$Nielsen & Chuang (10th edition, page 82) states that $H$ is a fixed Hermitian ...
3
votes
1answer
96 views

How to find the unitary operation of a depolarizing channel?

Suppose we have a depolarizing channel operation $$E(\rho)=\frac{p}{2}\textbf{1}+(1-p)\rho$$ acting on a Spin$\frac{1}{2}$ density matrix of the form $\rho=\frac{1}{2}(\textbf{1}+\textbf{s}\cdot\...
1
vote
2answers
49 views

Relation between trace distance and inner product between pure states

Let $|\phi\rangle,|\psi\rangle$ be two state vectors, and let $d=\frac{1}{2}\mathrm{Tr}(\sqrt{(|\phi\rangle\langle\phi|-|\psi\rangle\langle\psi|)^2})$ be their trace distance. Then it will always hold ...