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.

Filter by
Sorted by
Tagged with
0
votes
1answer
22 views

Where does errors occur during error syndrome and recovery?

I'm studying quantum error correction on Nielsen's Quantum Computation and Quantum Information, but I don't understand how it identifies the number of places where an error may occur, it uses ...
1
vote
0answers
33 views

Deformation of the Bloch sphere and contraction of its planes under the action of channels

On pg 376-377 of N&C, it gives 3 different diagrams showing how the various axis of the Bloch sphere will be contracted under the action of the channels, limiting the possible states after it's ...
3
votes
1answer
124 views

Nielsen & Chuang Exercise 2.55: Prove that $\exp \left[ -\frac{iH(t_2 - t_1)}{\hbar} \right]$ is unitary

$\newcommand{\expterm}[0]{\frac{-iH(t_2 - t_1)}{\hbar}} \newcommand{\exptermp}[0]{\frac{iH(t_2 - t_1)}{\hbar}}$Nielsen & Chuang (10th edition, page 82) states that $H$ is a fixed Hermitian ...
3
votes
1answer
81 views

Is there any 'official' list of errata for Nielsen & Chuang?

The book Quantum Computing and Quantum Information by Nielsen and Chuang is a well-known and celebrated text book that can act as a resource in a wide variety of topics. Of course, in such a vast ...
1
vote
1answer
64 views

Nielsen & Chuang Exercise 2.32: Show that the tensor product of two projectors is a projector

$\newcommand{\bra}[1]{\left<#1\right|} \newcommand{\ket}[1]{\left|#1\right>}$Here is what I tried: Given that we have two projectors: $$ A = \sum_i \ket{i} \bra{i}, \hspace{2em} B = \sum_j \ket{...
1
vote
1answer
42 views

Nielsen & Chuang Theorem 2.6 Proof

I got a problem in understanding the proof of the Theorem 2.6 (Unitary freedom in the ensenble for density matrices), 2.168 and 2.169 in the Nielsen and Chuang book Equation 2.168 Suppose $|{\tilde\...
3
votes
1answer
71 views

Deriving $\left( A | v \rangle \right)^\dagger = \langle v | A^\dagger$ without using $A^\dagger=\left(A^* \right)^T$

From Nielsen & Chuang (10th edition), page 69: Suppose $A$ is any linear operator on a Hilbert space, $V$. It turns out that there exists a unique linear operator $A^\dagger$ on $V$ such that for ...
1
vote
1answer
51 views

Proof of quantum data processing inequality in N&C on pg 566

On page 566, it states that using $S(\rho^{'})-S(\rho,\varepsilon) \ge S(\rho)$ and combining this with $S(\rho) \ge S(\rho^{'})-S(\rho,\varepsilon))$, we get $S(\rho^{'})=S(\rho)-S(\rho,\varepsilon)$....
1
vote
1answer
45 views

Confused regarding explanation of Schumachers compression in N&C

On page 547 of N&C, for $|\psi_{0}\rangle=|0\rangle$ and $|\psi_{1}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and for $|\tilde{0}\rangle=\cos(\pi/8)|0\rangle+\sin(\pi/8)|1\rangle$ and $|\tilde{1}\...
-1
votes
2answers
95 views

Nielsen and Chuang: Demonstration of equation 2.12

Reproduced from Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition) in page 64: We've seen that matrices can be regarded as linear operators. [...] Suppose $...
0
votes
1answer
32 views

Question regarding part of the proof for the typical subspace theorem

Part three (going by N&C page 544) states that $$tr(S(n)\rho^{\otimes n})=tr(S(n)\rho^{\otimes n}P(n,\epsilon))+tr(S(n)\rho^{\otimes n}(I-P(n,\epsilon))).$$ Now I understand how the term on the ...
1
vote
2answers
143 views

How to find the matrix representation of an operator from its action on a basis?

First, I apologize if something is poorly written but English is not my first language. I know that these exercises have been solved in this question. But I do not agree. Inner product and concrete ...
1
vote
1answer
30 views

Confusion over HSW theorem depicted in Nielsen and Chuang

On page 560, it states that $$C^{(1)} \geq S(\frac{\varepsilon(|{\psi}\rangle\langle{\psi}|) +\varepsilon(|{\varphi}\rangle\langle{\varphi}|)}{2} - \frac{1}{2}\varepsilon(|{\psi}\rangle\langle{\psi}|)-...
5
votes
2answers
193 views

Procedures and intuition for designing simple quantum circuits?

I'm working my way through one of the quantum circuits sections in Nielsen and Chuang and I'm struggling to get a feel for the basics of circuit construction. For example, one of the exercises is as ...
3
votes
1answer
55 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
1answer
51 views

What kind of transformation does the Y-gate do on the bloch sphere?

I'm going through "Quantum Computation & Quantum Information" by Michael A. Nielsen and Isaac L. Chuang, and as a high school student with no previous knowledge, I cannot understand some things ...
0
votes
1answer
84 views

Nielsen & Chuang Exercise 4.34 “Measuring an operator”

I need help with the exercise 4.34 from Nielsen & Chuang Book. I am supposed to get a matrix corresponding to the circuit. Thanks
2
votes
2answers
122 views

How to implement the exponential of an outer product?

In exercise 6.7 page 258 in Nielsen and Chuang book, they have a Hamiltonian $H = \left| x \right\rangle \!\!\left\langle x \right| + \left| \psi \right\rangle \!\!\left\langle \psi \right|$ and the ...
1
vote
2answers
46 views

Proving that Partial Trace is a Quantum Operation

I am referring to Nielsen and Chuang Quantum Computation and Quantum Information 10th Anniversary Edition Textbook, Chapter 8.3. A linear operator $E_i:H_{QR}\longrightarrow H_Q $ is defined by: $$...
1
vote
1answer
99 views

Measuring Probability of Mixed States

I am a little stuck on understanding the measurement probabilities of a 3 qubit system (QCQI q 4.41). 1)H gates are applied to both $q_1$ and $q_2$ 2) $C^{(1,2)}_3(X)$, a Toffoli, controlled by $q_1$...
3
votes
1answer
97 views

Problem with building quantum circuit for Hamiltonian operation

In the book, Nielsen and Chuang, there is a section on quantum simulation of the quantum search algorithm. Hamiltonion operator is defined as follows- $$ H = |x\rangle\langle x| + |\psi\rangle\langle\...
3
votes
0answers
68 views

Question Regarding Quantum Period-Finding Fourier Transform Approximation

I am following the 5.4.1 Period-Finding Algorithm in Nielsen and Chuang as shown below: My confusion lies with the second expression of point 3 in the procedure. Why is the second expression an ...
4
votes
1answer
126 views

Deutsch's algorithm in Qiskit

I am trying to understand and implement the Deutsch algorithm. I follow the logic from Nielsen book and I started to implement it in Qiskit. For implementing the oracle, I use a CNOT gate and now I ...
1
vote
1answer
59 views

Quantum Phase Estimation Circuit and Modular Exponentiaton

In Nielsen and Chuang, it is stated that the effect of phase estimation circuit is mapping state $|j\rangle |u\rangle$ to $|j\rangle U^j |u\rangle$. Here is my solution: Consider the first $CU^{2^0}...
3
votes
1answer
57 views

Showing measurement of a Hermitian Unitary operator gives final states as eigenvectors

This is related to exercise 4.34, The operation described can be written as $(H \otimes I)C^1(U)(H \otimes I)(|0\rangle \otimes |\psi\rangle)$ I can get to the point where the state of the system is ...
2
votes
1answer
34 views

Identity for linear codes and their duals: why do we have $\sum_y (-1)^{x\cdot y}=|C|\delta_{x\in C^\perp}$?

I've come across this exercise plenty of times and I still don't understand how to do it. (Here it is from N.C. Ex.10.25) Let $C$ be a linear code (Lets suppose its a binary code, i.e. a $k$-...
2
votes
1answer
79 views

Question Regarding Simulating Hamiltonian With Quantum Circuit

There have been a few other questions about this section of Nielsen and Chuang, but when working through the output of the circuit, there are some inconsistencies that are probably due to some mistep/...
3
votes
3answers
117 views

Do any two distinct pure states form a basis?

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ In ...
2
votes
1answer
152 views

How to perform Quantum Process Tomography for three qubit gates?

I am trying to perform Quantum process tomography (QPT) on three qubit quantum gate. But I cannot find any relevant resource to follow and peform the experiment. I have checked Nielsen and Chuang's ...
1
vote
1answer
64 views

Why don't I get what I expect when measuring with respect to a different basis?

$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ If ...
3
votes
2answers
172 views

Nielsen and Chuang ex 2.73

I've been trying to solve exercise 2.73 (p.g 105), and I'm not sure if i'v been overthinking it and the answer is as simple as i've described below or if I am missing something, or i'm just wrong! Ex ...
4
votes
1answer
77 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) := \...
4
votes
1answer
32 views

What does it look like to split an EPR pair?

I am reading Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang, and I am confused about a concept presented in Section 1.3.7: Quantum Teleportation. The book ...
0
votes
0answers
53 views

Not sure what do Nielsen and Chuang mean by number of operations

I am reading Nielsen and Chuang's "Quantum Computation and Quantum Information". One important concept about algorithms is how the number of operations scales with the length of the input. I realized ...
2
votes
2answers
410 views

Why is Hilbert space considered especially large?

In Nielsen & Chuang section 1.2 introduces multiple qubits and Hilbert spaces. More generally, we may consider a system of n qubits. The computational basis states of this system are of the ...
1
vote
1answer
48 views

Readout using an NMR spectrum

I'm trying to understand how to read results from an NMR quantum computer. According to Nielsen and Chuang: The principal output is the free induction decay signal $$V(t) = V_0\text{Tr}\left[e^{-...
6
votes
2answers
456 views

Why is Deutsch's gate universal?

(This is related to Exercise 4.44 in Nielsen and Chuang) Deutsch quantum gate is basically a $iR_x(\alpha \pi)$ gate with two control qubits. The constant $\alpha$ is an irrational number that allows ...
3
votes
2answers
124 views

Viewing two-qubit measurement as a projective measurement

I am following Nielsen and Chuang, section 2.2.5: A projective measurement is described by an observable, $M$, a Hermitian operator on the state space of the system being observed. The ...
3
votes
3answers
197 views

No Schmidt decomposition for tripartite states

Exercise 2.77 in Nielsen and Chuang asks to show by example that there exist tripartite states $| \psi \rangle_{ABC} $ which cannot be written as $$| \psi \rangle = \sum_i \lambda_i | i_A \rangle | ...
2
votes
1answer
170 views

Understanding the outer products in density matrices

I don't understand a simple property of the outer product when doing density matrices. I am studying nielsen and chuang's book. At equation 2.197 they do show the density matrix of the state of ...
4
votes
1answer
69 views

How does $U_f$ in Deutsch's Algorithm affect the state $|x\rangle$?

I am trying to read Nielsen's and Chuang's book on quantum computing and I am having problem understanding Deutsch's algorithm. According to my understanding of the algorithm, the state $|x\rangle$ ...
2
votes
1answer
89 views

Understanding Steps in Deutsch's Algorithm

I am currently working my way through the book Quantum Computation and Quantum Information by Chuang and Nielsen. So far it has been a joy to read, however I am hung up on a couple aspects of quantum ...
1
vote
0answers
75 views

Affine Map of 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 of single ...
2
votes
1answer
56 views

Conditional version of the triangle inequality for Von Neumann entropy

I'm trying to solve problem 11.3 in Nielsen Chuang: (3) Prove the conditional version of the triangle inequality: $$ S(A,B|C)\geq S(A|C)-S(B|C) $$ But the inequality seems incorrect. For example,...
4
votes
1answer
112 views

Constructing a circuit for $C^1(U)$ for rotation operators with TWO single qubit gates and CNOT gate

This is the exercise 4.23 from Nielsen and Chuang, asking that if it is possible to construct $C^1(U)$ for $U=R_{x,y}(\theta)$ with TWO single qubit gates and CNOT gate. My answer is no, and I would ...
3
votes
2answers
141 views

Problem 2.2 in Nielsen & Chuang - Properties of the Schmidt number

This question has been asked here: "Problem 2.2 in Nielsen & Chuang - Properties of the Schmidt number", but no answer has been provided there yet, thus I move it here. The problem is stated ...
0
votes
2answers
60 views

How can I check if the following are possible states of a qubit?

I would like to check if state $$ |\psi\rangle = \cos{\theta}|0\rangle + i \sin{\theta}|1\rangle $$ is properly defined. But when I calculated $\langle \psi | \psi \rangle$ is get $$ \cos^2{\theta}...
4
votes
1answer
231 views

What kind of errors does the master equation in the Lindblad form describe, continuous errors or discrete errors?

I noticed that in page 427 in Nielsen & Chuang's book Quantum Computation and Quantum Information, quantum error correction is possible because errors can be discretized. In other hand, the ...
5
votes
1answer
249 views

Uncomputation in quantum implementation of a classical algorithm

In Nielsens and Chuangs book, they present a way to implement a reversible version of any classical algorithm (section 3.2.5). In short, they use Fredkin and other simple reversible gates to ...
0
votes
1answer
88 views

Qubit measurement of the state $\frac{1}{\sqrt2}|00\rangle+\frac{i}{2}|01\rangle-\frac{1}{2}|11\rangle$

If we measure the first qubit and obtain $|0\rangle$, what does the second qubit collapses to? $$ \left| \varphi \right>=\frac{1}{\sqrt{2}} \left| 00 \right> + {\frac{i}{2}} \left| 01\right> ...