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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1answer
31 views

Vector math of applying an X-gate on an $|i\rangle$ basis state

It is well known that the X-gate will apply a rotation about the x-axis on the bloch sphere. Knowing this, the $|i\rangle$ state should be converted to the $|-i\rangle$ state on the application of ...
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0answers
62 views

How to prove that the mutual information is subadditive?

Let $\mathbf x=(x_1,...,x_n)$ and $\mathbf y=(y_1,...,y_n)$ be two vectors of random variables. To make things concrete, assume that Alice sends each component $x_j$ through a noisy channel to Bob, ...
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1answer
100 views

How to generate a quantum circuit from the quantum state $|1000\rangle+|0100\rangle+|0010\rangle+|0001\rangle$?

I am trying to understand the steps of how make a state preparation circuit from a quantum state. For making my question more clearer, for example, for the state is $\frac{|00\rangle+|11\rangle}{\...
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2answers
86 views

What are the constraints for the coefficents of the basis states in quantum computing?

$\newcommand{\ket}[1]{\left|#1\right>}$ It's known that the Kolmogorov axioms characterise a probability distribution: Probability of an event is a non-negative real number. The sum of all ...
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2answers
53 views

Is ($|+⟩$$⟨0|$ + $|-⟩$$⟨1|$ ) similar to ($|0⟩$$⟨+|$ + $|1⟩$$⟨-|$ )?

Is ($|+⟩$$⟨0|$ + $|-⟩$$⟨1|$ ) similar to ($|0⟩$$⟨+|$ + $|1⟩$$⟨-|$ ) ? Can we just reversed it this way when doing Dirac manipulation? I try to calculate HZH = X and i need to reverse the second H
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1answer
77 views

How to normalise when the probability of measurement is zero?

In one of the answers to this question on measuring one qubit it is explained that given a general two-qubit state $$ |\psi\rangle = \begin{bmatrix} \alpha_{00} \\ \alpha_{01} \\ \alpha_{10} \\ \...
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1answer
43 views

Steps to apply Hadamard gate to $n$ qubits

Can someone shows me, step by step, how to apply Hadamard and output the result?
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1answer
46 views

Why probabilites are not 50-50 in this simple circuit?

I am "brand new" to IBM quantum computing. Just created an account and tried my first circuit with a Hadamard applied to $|0 \rangle$ which should lead to $(|0\rangle + | 1 \rangle)/\sqrt{2}$...
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1answer
58 views

Doing $|0\rangle$ then Hadamard gate then measurement

I am starting to use quantum experience and following exactly first example from lecture. After initializing the qubit to $|0\rangle$, then applying a Hadamard gate, the probability for measuring $|1\...
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1answer
61 views

Unitary Transformations for Schmidt Decomposition

$\newcommand{\ket}[1]{|#1\rangle}$ Suppose a pure state $\ket{\psi}$ has a Schmidt decomposition given by $\ket{\psi^{SD}}$, which can be obtained via the diagonalization of the reduced density matrix ...
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1answer
34 views

How to Report an error in the Qiskit Textbook?

In the Qiskit Textbook in Chapter "Introduction to Quantum Computing -> Entangled states -> The CX gate revisited" the following sentence might be wrong: If we do the CX now, we will ...
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1answer
28 views

Write the difference of 2 density operators in terms of a spectral decomposition

An exercise question (9.7) from Quantum computation and Quantum Information by Michael E. Nielson and Isaac L. Chuang says that I can write the difference of any 2 arbitrary density operators $\rho,\...
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1answer
54 views

What are the conditions under which an unknown quantum state is learnable with arbitrary precision?

Assume that we have an unknown quantum state and we need to learn that unknown state with arbitrary precision. Under what conditions can we learn the unknown state with arbitrary precision? One ...
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1answer
53 views

Finding the measurement basis for single qubit with given probability of outcome $0$

I have the general state of a single qubit $|\psi \rangle = \alpha|0\rangle + \beta|1\rangle $. Assume I am given a probability $p$ such that $0 < p <1$. Now I need to find the basis in which ...
3
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1answer
61 views

How do you represent a Hadamard gate as a product of $R_x$ and $R_y$ gates?

I'm looking for a representation of Hadamard gate that uses only $R_x(x)$ and $R_y(y)$ gates. The values $x$ and $y$ may be the same, but they don't necessarily need to be.
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3answers
110 views

Is every single-qubit unitary just a rotation around some unit vector on the Bloch sphere?

I remember reading this somewhere... Is there an elegant proof for this?
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1answer
32 views

In what sense are Pauli matrices measurement operators?

Neilson and Chuang's textbook shows a nice example of measuring in the $Z$ basis on page 89 in section 2.2.5. The Hermitians for measuring in the $Z$ basis, $|0\rangle\langle 0|$ and $|1\rangle\langle ...
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1answer
47 views

Why is it not easy to distinguish $U|\psi\rangle$ and $U'|\psi\rangle$ if $\|U-U'\|<\epsilon$?

So I am currently working on an assignment, which is about the induced Euclidian norm $$ ||A||:= \max_{v\in\mathbb{C}^d\text{ s.t. }||v||_2=1} ||Av||_2 $$ for some $A\in\mathbb{C}^{d\times d}$. For ...
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1answer
102 views

Find the conditions under which the state $|\phi\rangle = \sum_{y=0}^{2^n -1} e^{\frac{2 \pi i a y}{2^n}} |y\rangle$ is unentangled

Show that the state $ |\phi\rangle = \sum_{y=0}^{2^n -1} e^{\frac{2 \pi i a y}{2^n}} |y\rangle $ is unentangled if $a \in \{ 0,1,...,2^n - 1\} $ and $|\phi\rangle$ can be expressed in the form $ \...
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2answers
88 views

How can the state $\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ be entangled if both spins are the same?

I'm reading Qiskit's documentation. https://qiskit.org/textbook/ch-gates/multiple-qubits-entangled-states.html#3.2-Entangled-States- and they show qubits entangled as $|00\rangle$ (both qubits spin ...
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0answers
46 views

Calculating probability that two entangled qubits are the same when measured in different bases

Given the entangled state \begin{equation} |\Phi^+\rangle = \frac{1}{\sqrt 2} |00\rangle + \frac{1}{\sqrt 2} |11\rangle \end{equation} I am trying to calculate the probability that the two qubits end ...
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1answer
52 views

How does applying Z gate to $|0\rangle$ change the phase of other states during reflection about $|s\rangle$ in Grover algorithm in Qiskit textbook

I am trying to understand the Reflection Gate - Us explained for 2 qubits in the qiskit textbook. In the explanation it is mentioned that first Hadamard gate is applied to change the state $|s\rangle$ ...
3
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1answer
81 views

Why is $\langle \psi| \sigma_z |\psi \rangle=\cos(\phi_1)\cos(\phi_2)$ for $|\psi\rangle=R_y(\phi_2)R_x(\phi_1)|0\rangle$?

I'm trying some example with the rotation gates and stuck here: $$\langle \psi| \sigma_z |\psi \rangle = \langle 0 | R_x(\phi_1)^\dagger R_y(\phi_2)^\dagger \sigma_z R_y(\phi_2) R_x(\phi_1) | 0 \...
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1answer
89 views

Distinguishing $\frac{| 0 \rangle + e^{i\theta} |1 \rangle}{\sqrt{2}} $ from $| 0 \rangle/|1 \rangle$ with probability $1/2 + \epsilon$

I am given one copy of one of two quantum states - $\frac{| 0 \rangle + e^{i\theta} | 1 \rangle}{\sqrt{2}} $, for some unknown fixed $\theta$. One of $| 0 \rangle/|1 \rangle$ - don't know which one, ...
3
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1answer
45 views

Exercise 4.41 in N&C book QCQI: how can i implement $R_z(\theta)$ using the circuit shown and $Z$?

I'm studying Nielsen and Chuang's book. I cannot solve one of the questions in the exercise 4.41. The question is the last one that is Explain how repeated use of this circuit and Z gates may be used ...
2
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1answer
122 views

Understanding the definition of entropy in the joint entropy theorem derivation

From section 11.3.2 of Nielsen & Chuang: (4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|...
2
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1answer
74 views

How the single qubit unitary (U) calculates when apply a gate to only one qubit at a time?

Qiskit Textbook, Chapter 2, Section 2.2. Single Qubit Gates on Multi-Qubit Statevectors (here). In here, they have described that: If we want to apply a gate to only one qubit at a time (such as in ...
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1answer
95 views

Creating orthogonal quantum states from a set of given (possibly linearly independent) quantum states

I want to understand how to orthogonalize a system of qubits. Suppose I have $n$ sets of quantum states like $$\{ |1_i\rangle|2_i\rangle|3_i\rangle \cdots|k_i\rangle \mid i=1 \dots n \}$$ where $i=1, \...
3
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1answer
78 views

How can I prove inequality from 4.66 to 4.67 in Nielson and Chuang's book?

I am reading chapter 4 of Nielson and Chuang's QCQI book. I cannot prove the inequality from (4.66) to (4.67) in page 195. That inequality is the following: $$ |\langle\psi|U^\dagger M|\Delta\rangle|+|...
2
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0answers
97 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 $...
4
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1answer
80 views

How to use the Kraus operators to represent the total density matrix instead of the reduced one?

In Nielsen's book, the Kraus operator can be attained by trace out the enviroment: $$\operatorname{Tr}_{\rm env}[\hat{U}(|\psi\rangle\otimes|0\rangle)(\langle\psi|\otimes\langle 0|)\hat{U}^\dagger]. $$...
2
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1answer
55 views

The expectation of a measurement of qubit 2 after qubit 1 has been measured

In section 1.2.4 (page 13) of these lecture notes http://users.cms.caltech.edu/~vidick/teaching/fsmp/fsmp.pdf, it says \begin{aligned}\left\langle\psi\left|X_{1}^{0} Z_{2} X_{1}^{0}\right| \psi\right\...
2
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2answers
99 views

How do we change the basis of a given qubit state?

I'm reading this paper (Link to pdf) about a test of entanglement with three particles. I wanted to ask if there is any mathematical shortcut to express one quantum state on another basis like the ...
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1answer
153 views

Regrouping the terms in expression 1.31 in Quantum Computing and Quantum Information, Nielsen and Chuang

I'm trying to reproduce the passage from expression 1.31 to 1.32 in the book Quantum Computing and Quantum Information, by Michael Nielsen and Isaac Chuang. Expression 1.31 is: $$|\psi_2\rangle = \...
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1answer
69 views

In the amplitude amplification algorithm in Nielsen and Chuang's book, why is the error probability $M/N$?

I was able to follow til the yellow-highlighted sentences, which seems to be important to understand. Why M/N is the probability of error? If so 1 - M/N would be the probability of success?
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2answers
70 views

Can Dirac notation be used with 2 or more gates?

Can Dirac notation be used with 2 or more gates? I've been trying to do the math with the $X$ and $Z$ ($X\otimes Z$) gates but I'm not getting the answer I should. In fact, the answer makes no sense. ...
2
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1answer
148 views

What is the Schmidt decomposition for the 00 + 01 state?

If I try to write the two-qubit state $$ |\psi \rangle = \frac{|0 \rangle |0 \rangle + |0 \rangle |1 \rangle}{\sqrt{2}}$$ as $$ |\psi \rangle = \lambda_0 |\phi_0 \rangle |\phi_0 \rangle + \lambda_1 |\...
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1answer
34 views

How to prove that tensor products of two pure density operators is again pure

I came up with an intuitive guess that tensor product $\phi\otimes\psi$ of two pure states $\phi,~\psi$ (which are density operators) is again pure. However, I tried to use basic linear algebra and ...
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2answers
81 views

Why is there no angle for the $z$ axis in the Bloch sphere?

I see that in Bloch spheres, there is an angle for the $x$ and $y$ axes but not for the $z$ axis. Why?
2
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1answer
54 views

Mathematics Behind Deutsch-Jozsa Algorithm

I am currently learning from Nielsen and Chuang and I am currently learning about Deutsch-Jozsa algorithm. However, I am stumped with the mathematics of the algorithm at the following section: I ...
2
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1answer
48 views

Why are the probabilities $|\alpha|^2$ and $|\beta|^2$ when measuring in the computational basis?

In measurement in the computational basis, I was being told that it is a way to extract information from a qubit, and it outputs a classical bit. For the quantum state $\alpha |0\rangle + \beta |1\...
1
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1answer
52 views

What are the input and output of QFT and IQFT, respectively?

I have read two opposite explanations about QFT and IQFT from 2 books for beginners of Quantum Computing. Which one is correct? The first book said, if we input an n-qubit non-superposition state into ...
2
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1answer
56 views

Does Z gate swap complex amplitudes of $|0\rangle$ and $|1\rangle$?

I am reading Quantum Computing 1st Edition By Parag Lala, this book says It seemed that the Z gate swapped the complex amplitudes $\alpha$ and $\beta$. Can Z gate implement that, or are there any ...
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0answers
66 views

Expectation value of a quantum circuit [closed]

The expectation value of an operator $A$ is defined by this equation $\langle A \rangle_\psi = \sum_j a_j |\langle \psi | \phi_j \rangle|^2 $. My first question is does it mean that the expectation ...
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0answers
24 views

How to trace out qubits from a multipartite density matrix [duplicate]

I have a density made up of 4 qubits. Say system A is made up of the first and second qubits while system B is made up of qubits 3 and 4. I want to trace out 2nd and 3rd qubits. Is there any reference ...
3
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1answer
46 views

What does measurement mean in quantum error correction(syndrome diagnosis)?

In the case of the simple three-qubit repetition code, the encoding consists of the mappings $|0\rangle \rightarrow\left|0_{\mathrm{L}}\right\rangle \equiv|000\rangle$ and $|1\rangle \rightarrow\left|...
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2answers
76 views

How do I apply a matrix to a ket state?

If we have the following matrix: $$\frac{1}{\sqrt{2}}\begin{pmatrix}1&1&0&0\\ 1&-1&0&0\\ 0&0&1&-1\\ 0&0&1&1\end{pmatrix}$$ How do we find the output for ...
3
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1answer
44 views

Knill-Laflamme condition derivation in Nielsen&Chuang: issue to understand a part of the proof

I have some trouble to understand the proof in Nielsen&Chuang about Knill-Laflamme conditions. The conditions: Let $C$ be a quantum code and $P$ the projector onto $C$. Suppose $\mathcal{E}$ is a ...
1
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2answers
89 views

Show when $a_k$ and $b_k$ are correlated when measuring in different bases, in the BB84 protocol

I'm trying to answer the following question about the BB84 protocol from Nielsen and Chuang's Introduction to Quantum Information. As I understand it, the string $b$ is determining whether we are ...
3
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2answers
86 views

In quantum process tomography, how does $\chi$ characterize a quantum process?

I'm working through Nielsen and Chuang and I'm pretty confused by the discussion of quantum process tomography. I'm trying to work through an example of 1-qubit state tomography given by N&C (box ...

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