Questions tagged [nielsen-and-chuang]

For questions about exercises or passages from the popular quantum computing textbook *Quantum Computation and Quantum Information* by Michael Nielsen and Isaac Chuang.

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Prove that $\text{Tr}(M|ψ\rangle\langleϕ|)=\langleϕ|M|ψ\rangle$

Question: I am studying alone, and I found p.76 of the book quantum computation and quantum information of nielsen &c huang that: $$\text{Tr}(M |\psi\rangle \langle\psi)=\langle\psi| M |\psi\...
1 vote
2 answers
209 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 ...
1 vote
1 answer
55 views

Why is the matrix obtained from the coefficients of orthogonal states unitary?

I'm having troubles in understanding a statement in Box 2.7 at page 113 in the Nielsen & Chuang. Firstly, it assumed to be working with a two-qubits quantum system in state $|\psi\rangle = \frac{|...
1 vote
1 answer
375 views

How to construct a quantum circuit for quantum Fourier transform in a prime dimensional Hilbert space?

This problem is given as a problem in Nielsen and Chuang. Consider a Hilbert space of dimension $p$ where $p$ is a prime number. Quantum Fourier transform (QFT) in this space is defined as $$ |j\...
2 votes
1 answer
166 views

Trace Distance in Bloch sphere, what is the vector of Pauli matrices?

While reading Chapter 9.2.1 Trace distance in "Quantum Computation and Quantum Information," I encountered a question. What is the vector of Pauli matrices referring to? $$ \vec{\sigma} = (\...
1 vote
1 answer
49 views

How to find projection operators for spectral decomposition

I am a little bit confused about the spectral decomposition for the observable $Z_{1}Z_{2}$ in Section $10.1$ of Nielsen and Chunag's "Quantum Computation and Quantum Information". The idea ...
2 votes
0 answers
100 views

Solution Nielsen and Chuang exercize 10.71

Exercise 10.71: Verify that when $M = e^{−iπ/4}SX$ the procedure we have described gives a fault-tolerant method for measuring $M$. The book describes a procedure to perform the measurement. Instead ...
0 votes
0 answers
15 views

Which stabilizer generators of a CSS code follow from which parity check matrices?

Do I have the right correspondence below between $X$-type ($Z$-type) stabilizer generators and the rows of the parity check matrices of $C_2^\perp$ ($C_1$)? I ask because it seems Nielsen and Chuang ...
4 votes
2 answers
279 views

Proof of the 4.11 exercise in the Nielsen & Chuang book

In question 4.11 in Nielsen and Chuang's book, it states that there is a formula to describe any unitary matrix $U$ with two vectors $\vec{n}$ and $\vec{m}$ in the following way: $$U=\exp(i \alpha) ...
3 votes
2 answers
646 views

How to choose a suitable number of iterations for Grover's algorithm?

In Nielsen and Chuang (2010), section 6.1.1. it is written: "For an N item search problem with M solutions, it turns out that we need only apply the search oracle ...
0 votes
2 answers
90 views

Is there a criteria to ensure a one-qubit operator is exactly of the form $R_n(\theta)$ (i.e without a global phase $e^{i\alpha}$)?

Reading the Nielsen and Chuang, I saw that every unitary operator $U$ can be written as $e^{i\alpha} R_n(\theta)$ for some well chosen $n \in \mathbb{R}^3$ and $0 \leq \theta < 2\pi$. I would like ...
4 votes
1 answer
120 views

How to correct error during the syndrome measurement or recovery process?

Here's the figure 10.21 from Nielsen's Quantum Computation and Quantum Information to explain the fault-tolerant quantum computing, where the circuit in the figure corrects the error that happens in ...
1 vote
2 answers
77 views

Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?

I have trouble understanding this proof in Nielsen & Chuang, specifically the identity in $(10.20)$, which reads $$ U_k^\dagger P_k F_l \sqrt{\rho} = U_k^\dagger P_k^\dagger P_k^\dagger F_l P \...
2 votes
2 answers
82 views

How is the $\beta$-matrix interpreted in single qubit QPT?

In Chapter 8 of Quantum Computation & Quantum Information by Nielsen & Chuang, more precisely Box 8.5, there is an example of quantum process tomography for a single qubit. (The same ...
5 votes
2 answers
351 views

What is the thought process for circuit making after seeing input and output of a matrix?

Here is an exercise (4.27) from Nielsen and Chuang and I found the answer (given in the figure below) online without any explanation. The question was to construct a circuit by seeing a matrix (given ...
5 votes
1 answer
117 views

Question about Nielson & Chuang Problem 9.2

I am working on the following problem from the book "Quantum Computation and Quantum Information" by Nielsen and Chuang. Problem 9.2: Let $\mathcal{E}$ be a trace-preserving quantum ...
2 votes
0 answers
69 views

Quantum Process Tomography for 2 qubits

I need clarification on a few aspects related to Box 8.5 and Exercise 8.34 from the book Quantum Computation and Quantum Information by Nielsen & Chuang . While attempting Exercise 8.34, I ...
2 votes
0 answers
81 views

Grover's algorithm for multiple solutions complexity

I'm reading Nielsen&Chuang book (for myself) and I'm completely stuck with one of the problems, 6.3(Database retrieval): Given a quantum oracle which returns $\left|{k, y \bigoplus X(k)}\right>$...
1 vote
1 answer
137 views

DFT like operation in the third step of Period finding and Discrete Logarithm algorithm

In the third step of the algorithm for discrete logarithm, the state $$ |\hat{f}(l_1,l_2)\rangle=\frac{1}{\sqrt{r}}\sum_{j=0}^{r-1}e^{-2\pi il_2j/r}|{f}(0,j)\rangle $$ is introduced which is stated to ...
4 votes
1 answer
282 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
2 answers
74 views

What we get when measure $|0\rangle$ under computational basis?

It is said if we have been given the state $|0\rangle$, the measurement will yield $0$ with probability $1$ in Nielsen's book. So here, the measurement will yield $0$ refers to we will get state $|0\...
0 votes
1 answer
192 views

Unital channel which is not mixed unitary

How to prove that for a multi-qubit system a unital channel is not necessarily mixed unitary? This is Problem 8.3 in Nielsen and Chuang. Here's a snippet of the text: Shall I need to take two ...
0 votes
2 answers
84 views

Explanation of the 2.60 equation page 76 in the Nielsen and Chuang [duplicate]

In the Nielsen and Chuang book page 76, equation 2.60 says that we can rewrite the trace $$Tr(A \left|\psi\right>\left<\psi\right|)$$ as follow : $$Tr(A \left|\psi\right>\left<\psi\right|) ...
1 vote
3 answers
131 views

Exercise 4.16 in the Nielsen & Chuang book

In the 4.16 exercice in the Quantum Computation and Quantum Information (Michael A. Nielsen & Isaac L. Chuang), I don't understand why the correct answer is not this matrix : $$ \left[ {\begin{...
4 votes
1 answer
208 views

Implementing "computable phase shifts" using T Toffoli ( problem 4.1 from Nielsen and Chuang's)

I am reading/studying the famous Nielsen and Chuang's book and ran into this interesting question and I don't quite understand the $f(x)$. It says it simply maps from $m$ to $n$ bits. But don't we ...
7 votes
1 answer
626 views

Correct Formulation of N&C Exercise 4.11 and other textbooks misquoting

Inspired by the comments in this question How to approximate $Rx$, $Ry$ and $Rz$ gates?, there is the errata for question 4.11 pg 176 in N&C. The original form states that for any non parallel $m$ ...
1 vote
1 answer
249 views

In what sense is $\langle\psi|\rho|\psi\rangle$ the overlap between $|\psi\rangle$ and $\rho$?

The fidelity between a pure state $|\psi\rangle$ and an arbitrary mixed state $\rho$ is given by, $F(|\psi\rangle,\rho)=\sqrt{\langle\psi|\rho|\psi\rangle}$, which is stated to be equal to the square ...
1 vote
1 answer
119 views

Prove that classical counting requires $k=\Omega(N)$ oracle calls

Consider a classical algorithm for the counting problem which samples uniformly and independently $k$ times from the search space, and let $X_1, ... ,X_k$ be the results of the oracle calls, that is, $...
2 votes
1 answer
110 views

How is the size of the circuit derived, in proving the threshold theorem?

In chapter 10.6.1 in Nielsen and Chuang, the section on concatenated codes and the threshold theorem (pages 480-481) states: The size of the simulating circuit goes as $d^k$ times the size of the ...
1 vote
1 answer
96 views

Can a density operator be written equivalently as $\rho=\sum_i p_i|\psi_i〉\!\langle\psi_i|$ and $\rho=\sum_i\lambda_i|\psi_i\rangle\!\langle\psi_i|$?

My doubt arises from page 99, 101 of the book Quantum Computation and Quantum Information by Michael A.Nielson and Issac L.Chung. Let {${p_{i}, | \psi_{i} \rangle }$} be an ensemble of pure states. ...
3 votes
0 answers
475 views

Show that quantum channels act as affine transformations in the Bloch sphere

I am referring to Equation (8.89) to (8.92) in Chapter 8 of "Quantum Computing and Information 10th Anniversary Edition" by Nielsen and Chuang. This section deals with the geometric picture ...
4 votes
2 answers
153 views

How to derive the expression for the probability in quantum phase estimation? ((5.27) Nielsen & Chuang)

I'm trying to understand the QPE algorithm that is presented in the Nielsen and Chuang textbook. More precisely, I do not understand Equation $(5.27)$. Context: In the following, let $b$ be a natural ...
2 votes
0 answers
73 views

$T_1$ and $T_2$ time with amplitude damping

Exercise 8.30 of Nielson & Chuang's QCQI says Equation 7.144, which is mentioned in the text, is $$\begin{bmatrix} a & b\\ b^* & 1-a \end{bmatrix}\rightarrow\begin{bmatrix} (a-a_0)e^{-t/...
14 votes
1 answer
5k views

General parametrisation of an arbitrary $2 \times 2$ unitary matrix

From Nielsen & Chuang's Quantum Computation and Quantum Information (QCQI): Since $U$ is unitary, the rows and columns of $U$ are orthonormal, form which it follows that there exist real numbers $...
0 votes
0 answers
119 views

Show that for pure states the description of the Bloch vector we have given coincides with that in section 1.2

$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} $ I am having a little bit of difficulty with part (4) of Exercises 2.72 from Nielsen and Chuang's "Quantum ...
5 votes
2 answers
435 views

Why are rotations represented by exponentials of Pauli matrices?

I'm self-studying Quantum Computation from Nielsen and Chuang's book. In section 4.2 they discuss that for any unit vector $\hat n$, the rotation operator $R_{\hat n}(\theta) = \exp(-i\theta\hat n \...
0 votes
1 answer
126 views

Show that any measurement where the measurement operators and the POVM elements coincide is a projective measurement

The following question is exercise 2.62 from Nielsen and Chuang's "Quantum Computation and Quantum Information" Show that any measurement where the measurement operators and the POVM ...
6 votes
1 answer
355 views

Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?

I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following: $$ F(\rho, \sigma) := \...
1 vote
1 answer
59 views

Verify that if $A$ and $B$ are diagonal in the same orthonormal basis, then $[A,B]=0$

This is from Nielson and Chuang's textbook "Quantum Computation and Quantum Information". They state the Simultaneous Diagonalisation Theorem: Suppose $A$ and $B$ are Hermitian operators. ...
8 votes
3 answers
931 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 $$ \mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger} $$...
0 votes
0 answers
59 views

Proof of the Lieb's theorem

Lemma A6.2: Let $R1 , R2 , S1 , S2 , T1, T2$ be positive operators such that $0 = [R1, R2 ] = [S1, S 2 ] = [T1, T2 ]$, and $$ R1 ≥ S1 + T1\\ R2 ≥ S2 + T2 $$ Then for all $0 ≤ t ≤ 1$, $$ R_1^t R_2^{1−t}...
1 vote
0 answers
112 views

Why can Shor code fix arbitrary errors?

This is taken from Page 434 of Nielsen and Chuang: To simplify the analysis, suppose noise of an arbitrary type is occurring on the first qubit only; we’ll come back to what happens when noise is ...
7 votes
3 answers
3k views

Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$

$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$ I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
1 vote
1 answer
49 views

Verification for calculation on Shor's code

Here I have tried to determine the end result for the qubit states, when we apply an arbitrary gate on the first qubit in the 9 qubit code. I have followed this diagram: U's operation on a qubit can ...
0 votes
1 answer
52 views

In quantum error correction, what does an "arbitrary error that yields an un-normalized state" mean?

This is from page 434 of Nielsen and Chuang: . Supposing the state of the encoded qubit is |ψ⟩ before the noise acts, then after the noise has acted the state is E(|ψ⟩⟨ψ|). To analyze the effects of ...
1 vote
1 answer
55 views

Non trace-preserving map in axiomatic approach to quantum operations

In Nielsen and Chuang's Quantum Computation and Quantum information there is an axiomatic definition of the quantum operation (as one of the 3 approaches to quantum operations). A quantum operation is ...
0 votes
1 answer
80 views

Clarification regarding application of distributive property in "quantum teleportation" example

For context, this is from Page 27 of Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press: She then sends the ...
1 vote
2 answers
166 views

How to show that the three-qubit repetition code only corrects up to 1-bit flip errors?

From Nielsen and Chuang, the error correction criteria is $$P E_i^{\dagger} E_j P=\alpha_{i j} P$$ $P$ is the projector onto the correct codespace, $E_{j}$ are error operations and $\alpha_{i j} $ is ...
2 votes
2 answers
149 views

How are quantum error-correction conditions in Nielsen and Chuang implemented in practice?

Quantum error-correction conditions in Nielsen and Chuang, 10th-anniversary edition (Theorem 10.1) state that the error operation $\mathcal{E}$ with operation elements $\{E_i\}$ is correctable if and ...
2 votes
1 answer
281 views

Unambiguous discrimination using POVM with highest discriminate probability

I was studying Nielsen&Chuang's textbook (about page 92), and come up with a question that I cannot solve it. Given one of the two state $|\psi_1\rangle=|0\rangle$ and $|\psi_2\rangle=\frac{1}{\...

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