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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1answer
58 views

How does a general rotation $R_\hat{n}(\theta)$ related to $U_3$ gate?

From eqn. $(4.8)$ in Nielsen and Chuang, a general rotation by $\theta$ about the $\hat n$ axis is given by $$ R_\hat{n}(\theta)\equiv \exp(-i\theta\hat n\cdot\vec\sigma/2) = \cos(\theta/2)I-i\sin(\...
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1answer
45 views

Why does $\sum_n \langle n|M_m\rho M_m^\dagger|n\rangle$ simplify to $\langle \psi|M_m^\dagger M_m|\psi\rangle$?

I was trying to derive the formula for $p(m)$ in exercise 8.2 on page 357 in Nielsen & Chuang. But I am wondering what rule I can apply to simplify this $$\mathrm{tr}(\mathcal{E}_m(\rho) )= \...
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1answer
35 views

How would I compute a density matrix of a complex qubit mixed state?

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed state, $$ \frac{1}{9}\begin{bmatrix} 5 & 1 & −i \\ 1 & 2 &...
2
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1answer
40 views

How would I compute a density matrix of a 2 qubit mixed state?

I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed states, how would I do this? $$ |00> \;with \;probability \; 2/4 \\ ...
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1answer
57 views

Derivation of Equation $8.7$ in Nielsen Chuang [duplicate]

Equation \eqref{eq:sp1} represents the reduced state of the system after tracing over environment.(Page number 358) $$\mathcal{E}(\rho) = \mathrm{tr}_{env}(\lbrack U(\rho \otimes \rho_{env} )U^{\...
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4answers
56 views

In Bell's inequalities, what is the meaning of assuming that the physical properties $P_Q,P_R,P_S,P_T$ have definite values?

Two assumptions behind Bell inequalities (Page 117 Nielsen Chuang) (1) The assumption that the physical properties $P_{Q}$, $P_{R}$, $P_{S}$, $P_{T}$ have definite values $Q$,$R$, $S$, $T$ which exist ...
3
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1answer
43 views

How do we find the stabilizer generators for the three-qubit bit-flip code spanned by $|000\rangle$ and $|111\rangle$?

In Nielsen & Chuang's book "Quantum Computation and Quantum Information" section 10.5.6, page 467 there is the following statement Consider the familiar three-qubit bit-flip code ...
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2answers
369 views

What's the 'physical consistency' in the partial trace scenario?

I'm reading 'Why the partial trace' section on page 107 in Nielsen and Chuang textbook. Here's part of their explanations that I don't quite understand: Physical consistency requires that any ...
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0answers
29 views

What are Sideband Pulses

I see sideband pulses used all over the place when looking up implementation techniques especially in Trapped Ion QC. Is there a layman's explanation of what they actually are? I understand what they ...
3
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1answer
75 views

What's the difference between $p(i|m)$ and $p(m|i)$ in measurement?

Suppose we perform a measurement described by measurement operators $M_m$. If the initial state is $|{\psi_i}\rangle$, then the probability of getting result $m$ is $$ \begin{align} p(m|i)=\| M_m|\...
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1answer
177 views

What does $M_m |\psi_i\rangle$ mean in the equation $p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$?

I have trouble understanding two equations in the Nielsen & Chuang textbook. Suppose we perform a measurement described by the operator $M_m$. If the initial state is $|\psi_i\rangle$, then the ...
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2answers
94 views

Output of Quantum Phase Estimation Algorithm

In section 5.2.1 of Nielsen Chuang, Performance and Requirements, there is an idea, that what happens if we can't prepare eigen state $|u\rangle$ and instead have a state $|\psi\rangle$ which is ...
3
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1answer
145 views

Application of QFT to Order-finding

In the Nielsen & Chuang book, section 5.3.1 page 226, there is a statement which goes like this:- (statement-1) The quantum algorithm for order-finding is just the phase estimation algorithm ...
2
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1answer
74 views

EPR Experiment: What does it mean for Alice to measure $\vec{v} \cdot \vec{\sigma}$ on her qubit?

I am trying to understand Box 2.7 on page 113 of Quantum Computation and Quantum Information book by Nielsen and Chuang. They start out with following wave function: \begin{equation} \psi = \frac{|01\...
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44 views

Complexity of controlled operations in a two-level unitary operation

In Neilsen and Chuang, chapter 4.5.2 (~p.193), why did the authors come to the conclusion that complexity of operations $C^n(X)$ and $C^n(\tilde{U})$ is $O(n)$? Did they assume using work qubits? If ...
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49 views

Standard form Shor's code

I'm trying to solve exercise 10.57 in Nielsen-Chuang, where you have to obtain the standard form check matrix of Shor's code. I followed the procedure laid out in the earlier chapter but then realised ...
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1answer
49 views

How do get from $|0\rangle=\alpha|a\rangle+\beta|b\rangle$, $|1\rangle=\gamma|a\rangle+\delta|b\rangle$ to an expression for $|01\rangle-|10\rangle$?

My question is linked to the Nielsen Chuang book. Particularly equation 2.216 on basis change from $|0\rangle$, $|1\rangle$ to orthonormal $|a\rangle$ and $|b\rangle$. How do we get the equation from ...
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1answer
124 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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1answer
54 views

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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1answer
75 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
65 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 ...
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1answer
72 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 ...
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1answer
73 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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3answers
328 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
69 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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1answer
52 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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1answer
35 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
39 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 ...
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1answer
153 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 ...
5
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1answer
196 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
81 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
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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\...
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1answer
82 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
57 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
51 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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2answers
111 views

Nielsen and Chuang: Demonstration of equation 2.12 [duplicate]

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
39 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
165 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
43 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
270 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 ...
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1answer
75 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
58 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
131 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
147 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
54 views

Prove that the partial trace is a quantum operation, finding its Kraus representation

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
114 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
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1answer
107 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\...
5
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1answer
138 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
318 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
93 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}...