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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25 views

Evaluation of Wigner function representation of a Bloch Sphere

Consider Wigner function representation of a qubit in the basis labeled by $\sigma_z$ and $\sigma_x$ eigenvalues. A general single qubit mixed state has the Bloch representation,$\rho = 1/2 (I + r.\...
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1answer
51 views

Implication of SWAP being not positive in terms of quantum channel

I am going over chapter 3 of Preskill's lecture notes regarding complete positivity. Specifically, on page 19, it is mentioned that since SWAP has eigenstates with eigenvalue -1, it is not positive, ...
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2answers
51 views

For two-qubit systems, do we have $\langle 01|01\rangle = \langle 0|0\rangle\langle 1|1\rangle$?

I am new to quantum computing and I want to know the following: If I have a 2 qubit system in state e.g. $\left|01\right>$ and I want to calculate the probability of measuring e.g. $\left<01\...
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1answer
63 views

Why does $H^2=X^2 =I$ not imply $H=X$?

if $HH = I$ and $XX =I$, then is $H=X$? $HH = I = XX$ or, $HH = XX$ then, taking under root, is $H = X$? This is absurd but how to disprove it?
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1answer
49 views

Factoring Decision Problem - why not in P? [closed]

Nielsen and Chuang, 10th Anniversary Edition, page 142, refers to the following (classical) computation problem: Given a composite integer m and L <m, does m have a non-trivial factor less than L? ...
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1answer
58 views

When discussing error correction, what are the objects in the expression $PE_i^\dagger E_j P=\alpha_{ij} P$?

I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm ...
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1answer
35 views

Making sense of the terms Polynomial and Exponential Precision in a Quantum Circuit

The quantum circuit construction of the quantum Fourier transform apparently requires gates of exponential precision in the number of qubits used. However, such precision is never required in any ...
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1answer
39 views

What is meant by a "projection operator" in the book "Quantum Computation and Quantum Information"?

I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm ...
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2answers
68 views

Compute the output of this Approximate Quantum Cloning three-qubit circuit

Above is a circuit for "Approximate Quantum Cloning" The preparation state is given by $$|\psi\rangle =\alpha|0\rangle+\beta|1\rangle$$ The gates labelled $\theta_i$ denote single qubit ...
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1answer
45 views

Show that the trace of squared density matrix gives ${\rm tr}(\rho^2)=\frac12(1+\|\mathbf n\|^2)$ [duplicate]

Equation 7.7 is given below: $$\hat\rho = \frac12(I +n_x(\hat X)+n_y(\hat Y)+n_z(\hat Z)) $$ Where $I$ is the identity matrix and $\hat X,\hat Y,\hat Z$ are Pauli matrices. Now my attempt of this was ...
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2answers
123 views

Are the first and second qubits of the state $| 111 \rangle + | 010 \rangle + | 101 \rangle + | 000 \rangle$ entangled with each another?

State of qubits: $\frac{1}{2} (| 111 \rangle + | 010 \rangle + | 101 \rangle + | 000 \rangle)$ Are the first and second qubits of this register entangled with each another?
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1answer
36 views

How to calculate the coefficients of a qubit from the angles of its Bloch representation?

A quantum bit $|\psi\rangle=a|0\rangle+b|1\rangle$ is represented on the Bloch sphere as a point on the spherical surface with $\theta = 40^°$ and $\phi = 245^°$. Calculate the (complex) coefficients $...
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1answer
65 views

How to apply the Hadamard gate to a given qubit state?

I have this qubit state: $$ H \left[ \frac{1}{\sqrt{2}} |0\rangle + \left( \sqrt{\frac{2}{7}}+\frac{1}{\sqrt{7}}i \right) |1\rangle \right] $$ How to solve this given Hadamard gate on qubit? Hadamard ...
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1answer
52 views

Is it possible to find a 2x2 Hermitian matrix whose eigenvalues have 1:2 ratio? [closed]

Is it possible to find 2x2 Hermitian matrix whose eigenvalues have 1:2 ratio and if it is how is it done?
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2answers
251 views

Is the SWAP gate a Clifford Gate? How would I express it using the Clifford Gate generators?

By my calculations, it looks like the SWAP gate is a Clifford Gate. See the following table: I follow the same method as in this paper for showing a gate is a Clifford Gate. I got the above table by ...
2
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1answer
55 views

What is the physical significance of the general rotation operator $R(\hat n,d\theta)=I-id\theta (\hat n\cdot\vec J)$?

John Preskills lecture notes (here) contain an equation for a general rotation operator (eqn 2.25, page 11): $$R(\hat{n},d\theta)=I-id\theta \hat{n}\cdot \vec{J},$$ where $\vec{J}$ is angular momentum,...
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1answer
43 views

What is the physical process causing a Bell state to be shared? [duplicate]

Nielsen and Chuang, QCQI, page 57, last paragraph, says "suppose Alice and Bob share between them a Bell state." I know how to prepare a Bell state, but what would be the physical process ...
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27 views

prove that $E(U^m_{\Delta t},e^{-2miH\Delta t})\leq m\alpha \Delta t^3$ where $U_{\Delta t}=e^{-2iH\Delta t}+O(\Delta t^3)$

Let $H=\sum_{k=1}^LH_k$ and define $U_{\Delta t}=[e^{-iH_1\Delta t}e^{-iH_2\Delta t}\cdots e^{-iH_L\Delta t}][e^{-iH_L\Delta t}e^{-iH_{L-1}\Delta t}\cdots e^{-iH_1\Delta t}]=e^{-2iH\Delta t}+O(\Delta ...
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1answer
51 views

What conditions on the coefficients of a bipartite pure state imply it being entangled?

With $\{ |e\rangle_j \}_{j=1}^{dim. \mathcal{H}_A}$ for $\mathcal{H}_A$ and $\{|f\rangle_j \}_{j=1}^{dim. \mathcal{H}_B}$ for $\mathcal{H}_B$, the product state reads \begin{equation} |u\rangle \...
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1answer
35 views

Show that $\langle v,O(v)\rangle= \mathrm{tr}(O|v\rangle\langle v|)$ for $v \in V$

I have a question regarding this exercise: Let O be an observable on V. Show that $\langle v,O(v)\rangle= \mathrm{tr}(O|v\rangle\langle v|)$ for $v \in V$. I thought that this exercise is quite easy ...
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32 views

Solving Particle in Box problem using Qiskit

I am new to quantum computing, I would like to ask how can I solve simple particle in box problem using qiskit which allows to compute the bound states in finite quantum well. Finite quantum well
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0answers
84 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 ...
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45 views

Circuit to show that Hadamard, phase, controlled- and Toffoli gates are universal

Part 1 The final output state is, $|\psi_{out}\rangle=\frac{1}{4}[|00\rangle(3S+XSX)|\psi\rangle+|01\rangle(S-XSX)|\psi\rangle+|10\rangle(S-XSX)|\psi\rangle+|11\rangle(-S+XSX)|\psi\rangle]$ When the ...
2
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2answers
57 views

Show that there are unitaries $U_m$ such that $M_m=U_m \sqrt{E_m}$, for any measurement $M_m$ and associated POVM $E_m$

Nielsen and Chuang's QCQI, section 2.2.6, page 92, asks Suppose a measurement is described by measurement operators $M_m$. Show that there exist unitary operators $U_m$ such that $M_m=U_m\sqrt{E_m}$, ...
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1answer
61 views

Why can we write $\rho=\sum_\mu q_\mu|\varphi_\mu\rangle\!\langle\varphi_\mu|$ iff $q\preceq \mathrm{spec}(\rho)$?

Exercise 2.6 in Preskill's notes (chapter 2, around page 48, pdf available here) asks to prove that an arbitrary state $\rho=\sum_i p_i |\alpha_i\rangle\!\langle\alpha_i|$, where $p_i$ and $|\alpha_i\...
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1answer
82 views

Prove $E(R_n(\alpha),R_n(\theta)^n)<\epsilon/3$ from $E\big(R_n(\alpha),R_n(\alpha+\beta)\big)=|1-\exp(i\beta/2)|$

where $E(U,V)=\max_{|\psi\rangle}||(U-V)|\psi\rangle ||=||U-V||$ is the error when $V$ is implemented instead of $U$. See page 196, Quantum Computation and Quantum Information by Nielsen and Chuang. I ...
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1answer
28 views

Knill Laflamme conditon

In Preskill's notes on quantum error correcting codes in Section 7.2, there seems to be no condition on the environment part of the state, i.e. $|0\rangle_E$ in $|\psi\rangle \otimes |0\rangle_E$. ...
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1answer
44 views

In Schumacher’s noiseless channel coding theorem, how do we get the exponents in $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\otimes n(1−p)/2}$?

On pg. 55 in Nielsen and Chuang, it's said that: the $|0\rangle + |1\rangle$ product can be well approximated by a superposition of states of the form $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\...
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1answer
61 views

Decomposition of U1 gate $U_1(\lambda)$ , Phase Shift gate $\phi(\delta) $, and Swap gate [closed]

Can we express U1 gate $U_1(\lambda)$ , Phase Shift gate $\phi(\delta) $, and Swap gate $$ U_1(\lambda) = \begin{pmatrix}1 & 0 \\ 0 & e^{i\lambda}\end{pmatrix}$$ $$ \phi(\delta) = \begin{...
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1answer
65 views

How to compute the measurement probabilities of $|\phi\rangle=\sum_k c_k |k\rangle$ in a rotated basis $V|k\rangle$?

I came across the following question and have some conceptual questions. Consider a general quantum state $|\phi\rangle$ of dimension $N$ spanned by some standard basis $\{|k\rangle,k=0,1,...N-1\}$. ...
3
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0answers
43 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,\...
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1answer
26 views

How can I compute $P_{|b\rangle}(|0\rangle)$?

I've just started to learn Quantum Computing and, to do it, I'm reading the course "Introduction from Quantum Computing" by IBM. Now, I'm reading the chapter "Entangled states", ...
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0answers
38 views

from hello_qiskit import run_puzzle results in error

I am working my way through the beta version of the Qiskit textbook. Everything was going fine until I got to the Visualizing Entanglement section and had some exercises to complete. All of the ...
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1answer
64 views

Measurement of single qubit operator $U$ which is both Hermitian and unitary with eigenvalues $±1$

Suppose we have a single qubit operator $U$ with eigenvalues $±1$, so that $U$ is both Hermitian and unitary, so it can be regarded both as an observable and a quantum gate. Suppose we wish to measure ...
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0answers
60 views

How can I create a Qiskit function producing a quantum circuit such that $\langle Z^{\otimes n}\rangle=J_{n-1}/2^{n-1}$?

Problem: Create a function circ(n) that returns an $n$-qubits quantum circuit that, when measured in the $Z^{\otimes n}$ basis, should yield the following ...
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3answers
119 views

Why can no pair of single qubits look like $\frac{1}{\sqrt2}(|00\rangle+|11\rangle$)?

I've just started to learn Quantum Computing and, to do it, I'm reading the course "Introduction from Quantum Computing" by IBM. Now, I'm reading the chapter "Entangled states", ...
2
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1answer
40 views

Why can the Hamiltonian $H=P_x(t)X+P_y(t)Y$ make an arbitrary unitary $U=R_x(b)R_y(c)R_x(d)$?

p.281 of Nielsen and Chuang's book says that A single spin might evolve under the Hamiltonian $H = P_x(t)X + P_y(t)Y$, where $P_{\{xy\}}$ are classically controllable parameters. From Exercise 4.10, ...
3
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1answer
237 views

Using distinguishability of non-orthogonal states to create a cloning device

Consider the following problem from Nielsen and Chuang's Quantum Computation and Quantum Information: Explain how a device which, upon input of one of two non-orthogonal quantum states $\left|\psi\...
1
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1answer
35 views

Qiskit noise model question (from textbook)

I'm reading the chapter Introduction to Quantum Error Correction using Repetition Codes and a code example demonstrates how to add depolarizing and pauli error. I have several questions. Is it not ...
1
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2answers
219 views

How does Equation (9) follows from these definitions?

How does Equation (9) follows from these definitions? This is my working out for the first identity. It appears that the identity is not true. For the second identity, I used sympy to check. The ...
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2answers
34 views

Compute ${\rm tr}(a_k a_{k'}\rho)$ with $\rho=e^{-\beta H}/Z(\beta)$ Gibbs state and $a_k$ ladder operators

Consider a harmonic oscillator with hamiltonian $H=\sum_k\omega_k a_k^\dagger a_k$ and a state $\rho=\frac{e^{-\beta H}}{Z(\beta)}$ where $Z(\beta)=\text{tr}[{e^{-\beta H}}]$. The quantity $$A:=\sum_{...
0
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1answer
57 views

Why in quatum coin toss, when starting from state $|1\rangle$, we get $|-\rangle$?

I was reading qiskit's text book, there I found that for a Double quantum coin toss, we have negative probability amplitude for $|1\rangle$ state when we starts from $|1\rangle$ state. Link : https://...
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2answers
32 views

Why can the state $\sum_x|x, f(x)\rangle$ be written without normalization factor?

It is stated on page 32 of Nielsen and Chuang that: In our single qubit example: , measurement of the state gives only either |0, f(0)> or |1, f(1)>! Similarly, in the general case, measurement ...
1
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1answer
26 views

Derivation of the state after applying $U_f$ in the Deutsch–Jozsa algorithm

On page 35 in Nielsen and Chuang, it's said that for the following quantum circuit implementing the general Deutsch–Jozsa algorithm: Next, the function $f$ is evaluated (by Bob) using $U_f$, giving $$...
1
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1answer
75 views

Quantum channels and maps: Confusing terminology

1. On page 73 of John Watrous' famous book, a quantum channel is defined as a linear map $$\Phi: L(\mathcal{X})\rightarrow L(\mathcal{Y})$$ Now $L(\mathcal{X})$ stands for $L(\mathcal{X},\mathcal{X})$...
5
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2answers
264 views

How does the spectral decomposition of the Choi operator relate to Kraus operators?

In Nielsen and Chuang's QCQI, there is a proof states that Theorem 8.1: The map $\mathcal{E}$ satisfies axioms A1, A2 and A3 if and only if $$ \mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger} $$...
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2answers
38 views

Find expectation value of the observable $X_1\otimes Z_2$ for a maximally entangled two-qubit system

In this exercise I need to find the expectation value of the observable $M=X_1 \otimes Z_2$ for two qubit system measured in the state $\dfrac{|00\rangle + |11\rangle}{\sqrt{2}}$. I know that $E[M]=\...
2
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1answer
89 views

What happens measuring the first qubit of a $GHZ$ state in the basis $\{|+\rangle, |-\rangle\}$?

This exercise ask me to explain what happens when the first qubit is measured in the diagonal base ($|+\rangle,|-\rangle$), considering this state: $|GHZ\rangle=\dfrac{1}{\sqrt{2}} (|000\rangle+|111\...
5
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1answer
105 views

Prove that rank one projectors have the same partial trace iff they differ by a local unitary operation

In nielsen and chuang's QCQI book, there is a theorem called Unitary freedom in the ensemble for density matrices, which states that the sets $|\psi_i\rangle$ and $|\phi_i\rangle$ generate the same ...
4
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1answer
130 views

Can all mixed states be written as a convex combination $\rho=\sum_j p_j |\psi_j\rangle\langle \psi_j|$?

States belonging to some space $\mathcal H$ can be described by density operators $\rho\in L(\mathcal H)$ that are positive and have trace one. Pure states are the ones that can be written as $\rho=|\...

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