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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2 votes
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Solution to Nielsen & Chuang Exercise 5.3 (FFT)

Can somebody help me with the solution of Nielsen Chuang, where we are supposed to derive the FFT from the equation (5.4): $$|j_1,\ldots,j_n\rangle\rightarrow\frac{\big(|0\rangle+e^{2\pi i 0.j_n}|1\...
2 votes
3 answers
109 views

Quantum computing coding setup for a beginner

I want to dive into Quantum computing but I don’t have a personal computer for coding. I have an iPad and I know some quantum computing basics. I had earlier used IBM quantum experience (IBM’s cloud ...
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4 votes
0 answers
52 views

Question regarding Quantum Phase Estimation (Nielsen and Chuang exercise 5.8)

I was working through Nielsen and Chuang's book on quantum computing and they state the following result regarding the performance of the Quantum Phase Estimation algorithm, "... given the input $...
2 votes
1 answer
130 views

How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?

The proof of the Fannes' inequality replies on the formula $T(ρ, σ)≥\sum_i|r_i − s_i|$, where $r_i,s_i$ are the eigenvalues of $\rho,\sigma$, in the descending order. In the proof given in Box 11.2, ...
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1 vote
1 answer
91 views

Conditional entropy as relative entropy between probability distributions

Find the expression for the conditional entropy $H(Y|X)$ as a relative entropy between two probability distributions. Use this expression to deduce that $H(Y |X)≥0$, and to find the equality ...
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-3 votes
2 answers
102 views

What is the definition of expectation value in Nielsen and Chuang?

As someone who learned Quantum Mechanics prior to studying Quantum Computing, I am having problems with this book. For example, equation 2.92 is incorrect. That is the definition of expected value, ...
3 votes
1 answer
77 views

Proof of Nielsen's theorem (Theorem 12.15) given in Nielsen-Chuang (assumption of invertibility)

Theorem 12.15 of Nielsen and Chuang's 10th anniversary edition is Nielsen's Theorem (1999). In particular, it says, Theorem 12.15: A bipartite pure state $\mid \psi \rangle$ may be transformed to ...
1 vote
0 answers
54 views

Why is dual-rail encoding called an error-detecting code for amplitude damping?

Exercise 8.23 : Suppose that a single qubit state is represented by using two qubits, as $|\psi\rangle=a|01\rangle+b|10\rangle$. Show that $\mathcal{E}_{AD}\otimes\mathcal{E}_{AD}$ applied to this ...
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0 votes
3 answers
79 views

What is the result of applying a CNOT to $|01\rangle$?

I can't get my head around this. I'm trying to implement a simple CNOT-Gate. I have to qubits: c = $|0\rangle$ = [1, 0] t = $|1\rangle$ = [0, 1] For all combinations of states $|00\rangle$, $|10\...
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-1 votes
1 answer
84 views

Why is $|P_0- P_1|=1$?

I have a question we have $ |0 \rangle, |1 \rangle, |+ \rangle$ and $|- \rangle, $ defined as usual. Let $P_0$= probability that a state be in 0, $P_+$= probability that a state be in +, and same ...
1 vote
1 answer
73 views

Nielsen & Chuang, 10th anniversary edition, page 85, equation 2.92

I don't understand why the stated equation is not equal to the expectation value of $m^2$
2 votes
2 answers
86 views

What is the Bloch sphere representation of $\rho\to\mathcal{E}(\rho) = |+\rangle\langle+|ρ|+\rangle\langle+| + |−\rangle\langle−|ρ|−\rangle\langle−|$?

Suppose a projective measurement is performed on a single qubit in the basis $|+\rangle, |−\rangle$, where $|±\rangle \equiv (|0\rangle\pm |1\rangle)/\sqrt{2}$. In the event that we are ignorant of ...
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0 votes
0 answers
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Question regarding the trace-preserving quantum operator trace distance

In Michael A. Nielsen & Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition, the proof of Theorem 9.2 (Trace-preserving quantum operations are contractive) on ...
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1 vote
2 answers
69 views

How to perform a basis change on a 2x2 density operator?

I have an ensemble described by following density operator: $$ P=3/8 |+\rangle\langle+| + 5/8 |-\rangle\langle-| $$ I am trying to write this operator in $\{|0\rangle, |1\rangle\}$ basis. I know that ...
2 votes
1 answer
93 views

How to compute the maximum possible coherence of a two-particle Bell state?

I am reading through some notes and am stuck on a bit of math that shows the max possible coherence. Our wave function is $|\psi\rangle =\frac{|01\rangle+|10\rangle}{2}$ and doing $|\psi\rangle \...
2 votes
0 answers
52 views

How to use stabilizers to simulate the circuit?

Sorry for this easy questions. That is a simple textbook question but I don't know how to solve it. I have the following circuit: The question was "use stabilizers to simulate the circuit" ...
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0 votes
1 answer
59 views

What does the notation $\sum_\lambda \lambda \left | \lambda \right > \left < \lambda \right |$ mean?

In many places, I see a form of representing a matrix with: $A = \sum_\lambda \lambda \left | \lambda \right> \left < \lambda \right |$ Where $\lambda$ is an eigenvalue, and $\left | \lambda \...
0 votes
0 answers
59 views

How to represent the following regular quantum circuits with tensor and concatenation symbols?

I am fascinated by such a quantum structure as above, how should the regular distribution for Toffoli and SWAP gates be described by the formula? Is the following correct? $\prod\limits_{n}{{{I}_{{{2}^...
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0 votes
1 answer
50 views

Is the mathematical expression between the CNOT gates arranged in this way a tensor product or a concatenated product?

Is the mathematical expression between the CNOT gates arranged in this way a tensor product or a concatenated product? That is \begin{matrix} \text{CNO}{{\text{T}}^{\otimes n}} & or & \prod\...
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0 votes
1 answer
164 views

About swap gate [closed]

As I was reading Qiskit textbook and there is this problem ... How can I swap a $|+\rangle$ to a $|-\rangle$ Qiskit textbook --> Basic Circuit Identities
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4 votes
1 answer
200 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 $U$ ...
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1 vote
1 answer
101 views

How does a "measurement of $\vec v\cdot\vec \sigma$" on two-qubit states work?

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 ...
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1 vote
1 answer
102 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?
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1 answer
148 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: ...
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4 votes
1 answer
165 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?
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1 vote
0 answers
33 views

Construct the unitaries U by decomposition in TSP Qiskit? [duplicate]

I'm stuck in here and really don't understand this stuff. In order to construct the unitary matrices for Quantum Phase Estimation, Qiskit decomposes the diagonal unitary matrices U(j) by tensor ...
1 vote
2 answers
88 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
125 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 ...
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0 votes
1 answer
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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
99 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 $\varepsilon$: $$\varepsilon(\rho)\equiv p\frac{I}{d}+(1-p)\rho\tag{1}$$ ...
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2 votes
1 answer
74 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 ...
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2 votes
2 answers
87 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\...
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2 votes
2 answers
76 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 ...
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2 votes
2 answers
115 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
158 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
46 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 ...
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1 vote
1 answer
84 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 }}=\...
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0 votes
1 answer
102 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: ...
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4 votes
1 answer
131 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
126 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}}|...
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2 votes
1 answer
394 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
72 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 ...
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2 votes
1 answer
130 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 ...
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1 answer
64 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 ...
  • 428
0 votes
1 answer
48 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
46 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 $\...
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3 votes
1 answer
191 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 ...
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0 votes
3 answers
117 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 ...
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1 vote
0 answers
54 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 ...

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