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.

Filter by
Sorted by
Tagged with
3 votes
1 answer
108 views

Show that $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)$ implies $U=\begin{bmatrix}[E_1]&\cdots\\ [E_2]&\cdots\\\vdots&\ddots\end{bmatrix}$

In Page 365, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that We have a principal system $Q$ and an environment $E$ and $U$ ...
  • 383
0 votes
1 answer
62 views

Question regarding the measurement of Pauli matrices on a Bell state

Michael A. Nielsen & Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition p.113, Box 2.7 states that "if a measurement of $\vec v\cdot\vec\sigma$ is ...
  • 157
1 vote
1 answer
81 views

How can I represent the completely mixed state as $\frac I2=\frac14(\rho+X\rho X+Y\rho Y+Z\rho Z)$?

Consider the completely mixed state $I/2$. The equation comes from Eq.(8.101) of Nielsen's book: $\frac{I}{2}=\frac{\rho+X\rho X+Y\rho Y+Z\rho Z}{4}$, How comes this equation?
  • 359
2 votes
1 answer
133 views

Heisenberg Uncertainty Principle (Nielsen and Chuang Box 2.4)

I'm trying to follow Nielsen and Chuang Book on Quantum Computation and Quantum Information. There is Box 2.4 on the Heisenberg Uncertainty Principle. I got stuck pretty fast. In that box they define: ...
  • 167
2 votes
2 answers
92 views

Why are orthogonal quantum states represented as collinear in the Bloch sphere?

We know that the angle between two orthogonal qubit states is 90 degrees. Why then, when we use the Bloch sphere, the angle becomes 180 degrees?
  • 75
1 vote
2 answers
66 views

Errors while encoding (3 qubit code)

I am relatively new to quantum error correction, so apologies if this question appears is naive. In the three qubit code, there seems to always be the assumption that errors occur after encoding the ...
1 vote
1 answer
109 views

Show the linearity of $(\langle a_m|\otimes I_B\otimes I_C\otimes \langle d_q|) U(I_{A}\otimes I_B\otimes |0_{C}\rangle\otimes |0_{D}\rangle)$

Suppose a composite system $AB$ initially in an unknown quantum state $\rho$ is brought into contact with a composite system $CD$ initially in some standard state $|0\rangle$, and the two systems ...
  • 383
0 votes
1 answer
58 views

Bit flip error correction syndrome measurements

I'm coming across some confusion in chapter 10.1.1 of Nielsen and Chaung. In terms of the 'recovery' procedure, how can the result of the syndrome measurement be 0, 2 or 3? I am assuming that, for ...
2 votes
1 answer
68 views

How to calculate the action of a channel on part of a quantum state?

As the title shows, but I think we can restrict ourselves into a more specific example. Let's consider depolarizing channel $\varepsilon$: $$\varepsilon(\rho)\equiv p\frac{I}{d}+(1-p)\rho\tag{1}$$ ...
  • 589
2 votes
1 answer
54 views

Writing a Density matrix in terms of the magnitude of the Bloch Vector

Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 ...
  • 313
2 votes
2 answers
53 views

Find an operator-sum representation for a depolarizing channel acting on 2qubit

In Nielsen and Chuang (page:379), it shows how to represent a 1 qubit depolarizing channel in operator-sum representation. $$ \mathcal{E}_1(\rho)=pI/2+(1-p)\rho =(1-3p/4)\rho+p/4(X\rho X+Y\...
  • 29
2 votes
2 answers
55 views

Understanding different forms of an arbitrary Unitary transformation in $\mathcal{H}_2$

I'm working to have a greater understanding of the arbitrary unitary transformation matrix when working in the context of the Bloch sphere. At this time I have found several equivalent ...
  • 313
2 votes
2 answers
90 views

What does the phase $\phi_1$ in a state $|\psi\rangle=a_1|0\rangle+a_2|1\rangle$ with $a_j=r_j e^{i\phi_j}$ say about state $|1\rangle$?

I'm a beginner in quantum computing and this question has been bugging me for quite some time. I have seen in various articles that a qubit is a device whose state can be represented by a unit vector ...
1 vote
1 answer
134 views

Kaye Exercise 7.1.3, Quantum Phase Estimation

Prove that $O(\log_2(r))$ phase estimations with $n = m$ and taking the outcome that occurs most often provides an estimate $\tilde \omega$ of the phase $\omega$ which will with probability at least $...
0 votes
1 answer
44 views

On the Conjugate Transpose Problem of Composite Systems

I've tried two quantum computing textbooks "QUANTUM COMPUTING From Linear Algebra to Physical Realizations" and "quantum information and quanutum computing" , and most only have a ...
  • 388
1 vote
1 answer
73 views

Do the linear operators $M\otimes I$ and $I\otimes N$ commute?

If not, does that mean that when doing partial measurements on two different shares of an entangled state, the results (expressed as a proability mass function) can depend on the order (i.e who ...
0 votes
2 answers
92 views

Some confusion about the quantum expectation formula

The issue is 1.which one is right? \begin{align} & (|{{w}_{1}}...{{w}_{n}}\rangle {{)}^{\dagger }}=\langle {{w}_{1}}...{{w}_{n}}|? \\ & (|{{w}_{1}}...{{w}_{n}}\rangle {{)}^{\dagger }}=\...
  • 388
0 votes
1 answer
65 views

Accuracy of Quantum Phase estimation; Finding the max difference integer, e

Working through Lab 5 in the Qiskit text, I have been attempting to complete Part 1, Step B. I implemented the following code as it seemed, at the time, to be what the question was asking for: ...
  • 313
3 votes
1 answer
91 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 ...
1 vote
1 answer
110 views

Factoring some qubits with Kronecker product

I have this quantum state: $$|\phi \rangle =\frac{1}{4\sqrt{2}}|000\rangle +\frac{3-\sqrt{5}i}{8}|001\rangle +\frac{1}{4\sqrt{2}}|010\rangle +\frac{3-\sqrt{5}i}{8}|011\rangle \\+\frac{1}{4\sqrt{2}}|...
  • 27
2 votes
1 answer
198 views

construction of Y gate from X,Z and H gates

As a part of textbook exercise, Y gate is to be constructed using H,Z and X-gates, just like we have $X = HZH$. is there some way/process/intuition to find such combinations or it is just like we need ...
3 votes
0 answers
32 views

Is there a tutorial for error mitigation in qiskit that does not use the deprecated ignis?

Working through the qiskit text I have come accross measurement error mitigation a few times (Measurement Error Mitigation, Lab 3: Quantum Measurment,...]. In the online textbook the coding indicates ...
  • 313
2 votes
1 answer
52 views

Exercise 1.1 in Quantum Processes Systems, and Information

I was going through Quantum Processes Systems, and Information by Benjamin Schumacher and Michael Westmoreland and could not understand the very first exercise which goes like Exercise 1.1 Identify ...
  • 149
0 votes
1 answer
35 views

How to initialize my circuit with two random complex numbers?

Working through Lab 3 in the Qiskit text, I have been attempting to initialize my one qubit circuit with two random numbers. My first attempt, as directed, is using ...
  • 313
0 votes
1 answer
43 views

measurement probability from density operator?

I've been through this before but I can't fully get my head round this upon review. So the density operator $\hat{\rho}=\sum_j p_j|\psi_j\rangle\!\langle \psi_{j}|$ for pure states $|\psi_{j}>$ at ...
0 votes
0 answers
42 views

Composition of rotations sign

I'm solving exercise 4.15 from Nielsen and Chuang: Prove that if a rotation through an angle $\beta_1 $ about the axis $\hat{n}_1$ is followed by a rotation through an angle $\beta_2$ about an axis $\...
  • 1
1 vote
0 answers
98 views

Jupyter Notebook not rendering latex code

Working through the Qiskit text and came across this bit of coding on the density matrix page: psi_AB.draw('latex', prefix='|\\psi_{AB}\\rangle = ') which is ...
  • 313
0 votes
1 answer
45 views

What is size of the state space?

I'm taking my first Quantum Computing MOOC and the image below is from one of the very first slides: My question is, what is the state space? I've a guess that it is the set of all possible results ...
  • 101
0 votes
0 answers
21 views

Quantum process tomography, for one and two qubit

I'm reading Nielsen and Chuang and I read quantum tomography process given by N&C (box 8.5), which provides an algorithm for determining $\chi$ in terms of block matrices and density matrices. And ...
  • 45
0 votes
1 answer
43 views

How to get the Dirac representation of a general quantum gate?

writing a matrix from bra-ket notations is easier. Going back is like finding prime factors. How to get the bra-ket form of all basic quantum gates in their matrix form in general?
  • 1
1 vote
0 answers
25 views

How can one visual a transformation which affects a component of density matrix?

How to visualize a transformation that looks like $$\rho = \frac{\textbf{I} + r_1 \sigma_1 + r_2 \sigma_2 + r_3 \sigma_3}{2} \rightarrow \frac{\textbf{I} + r_1 \sigma_1 + \lambda ~r_2 \sigma_2 + r_3 \...
  • 149
1 vote
0 answers
41 views

How does one begin to map a circuit to this problem? (Reversible 2-bit demultiplexer using NOT, CNOT, Toffoli & Fredkin)

I am relative beginner to building qc circuits and programming. Please share with me how you would begin to approach this problem. What steps would you take to draw this circuit? Thank you for your ...
0 votes
1 answer
28 views

Which protocol does the Qiskit textbook use for Quantum Key Distribution?

I've worked through a couple different protocols for quantum key distribution, but when I look at the Qiskit textbook on this page I am unable to determine which one was used in this lesson. If anyone ...
  • 313
0 votes
1 answer
77 views

What is the solution to Nielsen and Chuang Exercise 2.65?

Nielsen&Chuang Exercise2.65: Express the states (|0 + |1)/ √ 2 and (|0−|1)/ √ 2 in a basis in which they are not the same up to a relative phase shift. Consider an orthnormal basis :$\begin{cases}...
-1 votes
1 answer
36 views

Which angle convention is used in general equation of quantum state?

In this post, I am unsure which angles are denoted in general equation of quantum state. I realize that $\theta$ is azimuthal angle, while $\phi$ is the ...
0 votes
2 answers
48 views

It is possible to obtain amplitudes from superposition with certain probability?

In many papers and books on quantum theory it is written that it is not possible to obtain amplitudes exactly when we have a qubit in superposition. In a course we said that actually it is possible, ...
1 vote
2 answers
104 views

What projective measurement discriminates between a set of pairwise orthogonal states?

If we know a set of $K$ states $\{\lvert\psi_j\rangle:j\in[K]\}$ such that they are pairwise orthogonal, and we are given an unknown state $\lvert\psi_i\rangle$, what sort of projective measurement ...
0 votes
1 answer
39 views

What’s the point of the amplitudes being complex?

I understand that Introducing complex numbers to the amplitude allows us an extra degree of freedom. Through the rotation of the complex vector, you can encode the same magnitude (1/√2) with an ...
2 votes
1 answer
49 views

Why are qubits represent as unit vectors in an Hilbert space?

Mathematically, our qubits (pure quantum states) can be represented as a normalized vector (of length 1) in a complex Hilbert space. is it because of the sum of squares of vertical and horizontal ...
2 votes
1 answer
39 views

(Proof verification) Kaye Exercise 3.5.5, partial trace in larger system

This is the exercise as stated in Kaye's book, Introduction An Introduction to Quantum Computing: Show that for any density operator $\rho$ on a system $A$, there exists a pure state $|\psi\rangle$ ...
1 vote
1 answer
71 views

Is measurement of $M\otimes I$ equivalent to measurement of $M$ on first subsystem?

Let $\rho$ be a state in $H_A\otimes H_B$ and $\rho_A$ its reduced state on $H_A$ (obtained by tracing out $H_B$). Is the probability distribution corresponding to the observable $M\otimes I$ on $\rho$...
0 votes
0 answers
46 views

Phase damping and cnot gate

Show that a single controlled-NOT gate can be used as a model for phase damping, if we let the initial state of the environment be a mixed state, where the amount of damping is determined by the ...
  • 45
-1 votes
1 answer
42 views

Why the equality regarding Kronecker delta holds?

Can anyone show me why this equality below holds? I understand the matrix form of Kronecker delta is an identity matrix, but why this "coming from nowhere" delta function $\delta_{i,j}$ can ...
2 votes
2 answers
89 views

Why density matrix representation usually written in the following form?

The pure state $|\psi\rangle = \sum \alpha_i |i\rangle$, $\rho = \sum_{i,j}\alpha_i^*\alpha_j |j\rangle\langle i|$, why is the form not be $\rho = \sum_{i}\alpha_i^*\alpha_i |i\rangle\langle i|$ ???
  • 359
1 vote
1 answer
76 views

How to determine with local measurements which Bell state we have?

We have 2-qubit state which we know is 1 of 4 Bell states. Can we determine, using unitary transformations and single-qubit measurements, which Bell state do we have, and if we can, how?
  • 3,124
0 votes
1 answer
67 views

How can orthonormal vectors satisfy $\langle i|j\rangle=\delta_{ij}$?

In the book "Quantum Computation and Quantum Information" ("Mike and Ike") - chapter 2, page 66 - I have encountered the following paragraph: If the vectors i and j are ...
  • 1,237
1 vote
1 answer
39 views

Equivalence between quantum circuit: CNOT changes control and target qubit

It's know that the following two circuits are equal. In fact, answers for this can be found on wikipedia, and on this website. However, I am looking for a more formal answer. I'd like to see the ...
1 vote
1 answer
95 views

Do unitary matrices acting on entangled states always give a quantum state?

I'm trying to understand what happens when Alice(Bob) apply a unitary to her(his) part of an entangled state. Let us consider the following unitary transformations: $$U_1 = \frac{1}{\sqrt{2}} \...
-1 votes
1 answer
208 views

Solution to problem 5.3 Book Quantum Computation and Quantum Information Nielsen Chuang regarding Kitaev's algorithm

Problem 5.3: (Kitaev’s algorithm) Consider the quantum circuit where |u> is an eigenstate of U with eigenvalue $e ^ {2 \pi i \phi} $. Show that the top qubit is measured to be 0 with probability $...
1 vote
1 answer
38 views

Implementing deutsch_problem(seed=None) and deutsch() from qiskit text

I'm attempting to work through the exercise at the end of https://learn.qiskit.org/course/ch-gates/phase-kickback#phase-8-0 where I am to build and implement Deutsch's algorithm. I understand that <...
  • 313

1
2 3 4 5
11