# 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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### Exercise 4.6 in Quantum Computing and Quantum Information Nielsen and Chuang

Question 4.6: One reason why the $R_\hat{n}(θ)$ operators are referred to as rotation operators is the following fact, which you are to prove. Suppose a single qubit has a state represented by the ...
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### 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 ...
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### Nielsen & Chuang Exercise 2.55: Prove that $\exp \left[ -\frac{iH(t_2 - t_1)}{\hbar} \right]$ is unitary

$\newcommand{\expterm}{\frac{-iH(t_2 - t_1)}{\hbar}} \newcommand{\exptermp}{\frac{iH(t_2 - t_1)}{\hbar}}$Nielsen & Chuang (10th edition, page 82) states that $H$ is a fixed Hermitian ...
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### Do any two distinct pure states form a basis?

$\newcommand{\bra}{\left<#1\right|}\newcommand{\ket}{\left|#1\right>}\newcommand{\bk}{\left<#1\middle|#2\right>}\newcommand{\bke}{\left<#1\middle|#2\middle|#3\right>}$ In ...
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### 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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### Method to find $r$ in the case when $r'$ returned by the continued fractions procedure is a factor of $r$

In the Quantum Computation and Quantum Information (10th ed.) textbook by Nielsen and Chuang, section 5.3.1 (titled "Application: order-finding") describes how phase estimation can be used to find the ...
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In the case of the simple three-qubit repetition code, the encoding consists of the mappings $|0\rangle \rightarrow\left|0_{\mathrm{L}}\right\rangle \equiv|000\rangle$ and $|1\rangle \rightarrow\left|... 3 votes 1 answer 148 views ### Knill-Laflamme condition derivation in Nielsen&Chuang: issue to understand a part of the proof I have some trouble to understand the proof in Nielsen&Chuang about Knill-Laflamme conditions. The conditions: Let$C$be a quantum code and$P$the projector onto$C$. Suppose$\mathcal{E}$is a ... 3 votes 1 answer 121 views ### How would I theorise a quantum query algorithm in O(1)? I am currently attempting to solve a problem from Nielsen-Chuang, and I can't seem to figure out how I would do this; I'm trying to implement Grover's algorithm to solve the problem of differentiating ... 3 votes 1 answer 281 views ### Quantum Phase Estimation Circuit and Modular Exponentiaton In Nielsen and Chuang, it is stated that the effect of phase estimation circuit is mapping state$|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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Nielsen explains how a search algorithm can access a classic database. I have a few questions. I hope you can help me a bit :) I work with a few quotes from the book. The principle of operation is a ... 331 views

### Understanding the quantum circuit for the oracle in Grover's algorithm given in N&C

In the chapter about the Grover algorithm, it is suggested that the gate which executes the phase shift is given in the following form: Now I have looked at this gate in detail and come to the ... 532 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 ...
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### Prove $|η(r) − η(s)| ≤ η(|r − s|)$ when $|r − s| ≤ 1/2$ [closed]

Background If $\rho$ and $\sigma$ are density matrices such that the trace distance between them satisfies $T(\rho,\sigma)\leq1/e$. Then the Fannes' inequality states that |S(\rho)-S(\sigma)|\leq T(...
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### 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 ...
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### What do you specify when you physically apply a unitary?

In the Environment and Quantum Operations in Nielsen and Chuang, section 8.2.2, they say that when you apply a unitary on a state, you expect the output as the just the state transformed by the ...
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### Nielsen & Chuang Exercise 2.5 - Inner products of complex vectors [closed]

Reproduced from Exercise 2.5 of Nielsen & Chuang’s Quantum Computation and Quantum Information (10th Anniversary Edition): A function $(\cdot, \cdot)$ from $V × V$ to $C$ is an inner product if ...
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### How to compute the unitary from the $\chi$ matrix obtained from QPT

I am trying to do quantum process tomography for one qubit and obtain the unitary for the gates that are applied on the qubit. I have studied the theory on process tomography from mike and ike and the ...
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### How to understand intuitively the concavity of the binary entropy?

In Nielsen and Chuang's Quantum Computation and Quantum Information book, introducing the binary entropy, they gave an intuitive example about why binary entropy is concave: Alice has in her ...
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### What does the centered dot notation mean in the expression $|+\rangle|+\rangle\otimes(\cdot)$?

In the expression $|+\rangle|+\rangle \otimes(\cdot)$, what does the centered dot represent? I have come across a definition of the centered dot previously, similar to the one given here, but I am ...