# 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.

375 questions
Filter by
Sorted by
Tagged with
25 views

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?
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? ...
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 ...
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 ...
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 ...
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 ...
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 ...
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?
36 views

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 \...
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 ...
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
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 ...
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 ...
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 $M_m=U_m\sqrt{E_m}$, ...
61 views

61 views

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://...
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 ...
26 views

38 views

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 $E[M]=\... 1answer 89 views ### What happens measuring the first qubit of a$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\...
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 ...
### 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=|\...