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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63 views

In Nielsen and Chuang, how can $\frac{1}{2(e-1)}$ result from $\frac12\int_{e-1}^{2^{t-1}-1}dl\frac{1}{l^2}$?

From Nielsen and Chuang's book: $\textit{Quantum computation and quantum information}$, how can (5.34) equal (5.33)? I.e. $$\dfrac{1}{2} \int_{e-1}^{2^{t-1}-1} dl \dfrac{1}{l^2} = \dfrac{1}{2(e-1)}.$$...
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In Uhlmann's theorem, should the polar decomposition be written as $A=|A|V$ or $A=V|A|$?

In the proof of Uhlmann's theorem, the book writes the polar decomposition: $A = |A|V$, with $|A| = \sqrt{A^\dagger A}$. Shouldn't it be $V|A|$ instead? The former case is $A^\dagger A = V^\dagger|A||...
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60 views

CSS Code and Fault-tolerant Problem in Nielsen and Isaac Chuang‘s book

Could someone help me with $Problem-10.52$ in Nielsen and Isaac Chuang‘s book? The screenshot is shown below. I have no idea about that. Hope someone can give me some suggestions. By the way, if I ...
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1answer
60 views

Pauli Matrix tolerant and encoding

I'm stumped by these questions in Chuang's book. If I have a state $|ψ\rangle=1/2\sqrt2(1+M_0)(1+M_1)(1+M_2)|0\rangle_7$, where $$M_0=X_0X_4X_5X_6; M_1=X_1X_3X_5X_6; M_2=X_2X_3X_4X_6$$ I rewrite its ...
2
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1answer
48 views

Nielsen & Chuang Exercise 6.13: Standard deviation of classical counting algorithm

$\newcommand{\expectation}[1]{\mathop{\mathbb{E}} \left[ #1 \right] } \newcommand{\Var}{\mathrm{Var}}$ From Nielsen & Chuang 10th edition page 261: Consider a classical algorithm for the counting ...
2
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1answer
70 views

von Neumann entropy in a limiting case

I am stuck with a question from the book Quantum theory by Asher Peres. Excercise (9.11): Three different preparation procedures of a spin 1/2 particle are represented by the vectors $\begin{pmatrix} ...
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284 views

Nielsen&Chuang 5-qubit quantum error-correction encoding gate

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

Does composition of two single qubit rotations yield a single rotation around a unit vector?

$ \newcommand{\coefcos}[0]{c_1 c_2 - s_1 s_2 \hat{n}_1 \cdot \hat{n}_2} \newcommand{\coefsin}[0]{s_1 c_2 \hat{n}_1 + c_1 s_2 \hat{n}_2 - s_1 s_2 \hat{n}_2 \times \hat{n}_1}$This question relates to ...
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48 views

Show that for any measurement operator $M_m$ there exists unitary $U_m$ such that $M_m=U_m\sqrt{E_m}$ with $E_m$ POVM

Exercise 2.63 of Nielsen & Chuang asks one to show that if a measurement is described by measurement operators $M_m$, there exists unitary $U_m$ such that $M_m = U_m \sqrt{E_m}$ where $E_m$ are ...
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30 views

Where do 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 ...
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0answers
38 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
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149 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
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1answer
120 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 ...
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1answer
73 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{...
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1answer
47 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
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1answer
78 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 ...
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1answer
56 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 $S(\rho^{'})=S(\rho)-S(\rho,\varepsilon)$....
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1answer
48 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}\...
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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 $...
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1answer
34 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 ...
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2answers
149 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 ...
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1answer
42 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}|)-...
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2answers
253 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 ...
4
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1answer
64 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 $m$ ...
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1answer
55 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 ...
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1answer
113 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
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2answers
135 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 ...
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2answers
49 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: $$...
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1answer
107 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 $q_1$...
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1answer
100 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
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1answer
101 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
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1answer
233 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 ...
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1answer
77 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}...
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1answer
65 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
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1answer
40 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 $k$-...
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1answer
104 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/...
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3answers
129 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 ...
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1answer
187 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 ...
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1answer
67 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 ...
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2answers
218 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 ...
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1answer
88 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) := \...
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1answer
37 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 ...
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0answers
58 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 ...
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2answers
498 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 ...
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1answer
55 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^{-...
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2answers
534 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 ...
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2answers
130 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 ...
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3answers
229 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 | ...
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1answer
184 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 ...
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1answer
79 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 $|x\rangle$ ...