# 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 a circuit with controlled $iR_x(\pi \alpha)$ is universal for quantum computation whenever $\alpha$ is irrational

Show that the three qubit gate $G$ defined by the circuit is universal for quantum computation whenever $\alpha$ is irrational. My Observations The unitary gate on the third qubit is activated only ...
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### 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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### Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy

The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is $$S(A,B)\ge|S(A)-S(B)|$$ this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
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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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### 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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### 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 $\left|{k, y \bigoplus X(k)}\right>$...
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### 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 ...
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### Why is the first register of $|x,y\oplus f(x)\rangle$ called "data" register?

When talking about quantum parallelism, in Nielsen and Chuang, it's said that: it is possible to transform this state into $|x, y \oplus f(x)\rangle$, where $\oplus$ indicates addition modulo 2; the ...
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### How to start reading quantum computing papers?

What's the best way to get to a state where you can read quantum computing papers? I find them too dense and full of notation to approach. I am currently working my way through Nielsen and Chuang's ...
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### Construction of arbitrary Normalizer Gates using H, S and CNOT Gates

This question is in reference to Exercise 10.40 of Nielsen and Chuang's textbook, which is an attempt to prove the theorem that any $n$ qubit Normalizer gate can be built out of $H$, $S$, and $CNOT$ ...
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### Question regarding bloch vector solutions for master equation on page 388 of N&C

On page 388 of N&C, you are asked to find the solution to a differential equation for a two-level atom coupled to a vacuum. However, I have no experience with differential equations, so I am ...
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### 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 ...
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### clarifying a step in the proof of Solovay-Kitaev theorem

There is a step in the proof of the proof of Solovay-Kitaev theorem about the existence of a set containing words of at most length length $l_0$ that cover $SU(2)$ . The proof I'm reading in given in ...
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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 $|\psi\rangle = \frac{|... 1 vote 0 answers 39 views ###$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 $$\begin{bmatrix} a & b\\ b^* & 1-a \end{bmatrix}\rightarrow\begin{bmatrix} (a-a_0)e^{-t/... 1 vote 0 answers 70 views ### Why can Shor code fix arbitrary errors? This is taken from Page 434 of Nielsen and Chuang: To simplify the analysis, suppose noise of an arbitrary type is occurring on the first qubit only; we’ll come back to what happens when noise is ... 1 vote 0 answers 38 views ### What do we mean by family of CSS codes? In proving the security of BB84 in Nielsen and Chuang (10th anniversary edition - Section 12.6.5), they argue that a codeword in \text{CSS}(C_1, C_2) is represented by$$\frac{1}{\sqrt{|C_2|}} \sum\... 1 vote 0 answers 164 views ### Prove$\beta=\Lambda\otimes\Lambda$, where$\Lambda=\dfrac{1}{2}\begin{bmatrix}I&X\\X&-I\end{bmatrix}$for single qubit tomography In the Section on single qubit quantum process tomography, Box 8.5, Page 393, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, and in Prescription for experimental ... 1 vote 0 answers 125 views ### In quantum process tomography for one and two qubit, why can we express the$\chi$matrix in this form? I'm reading Nielsen and Chuang and I read quantum tomography process given by N&C (box 8.5), which provides an algorithm for determining$\chi$in terms of block matrices and density matrices. And ... 1 vote 1 answer 122 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 ... 1 vote 0 answers 51 views ### Probability of the case when$r'\neq r$and$r'$is a factor$r$in the order finding algorithm In the Order-Finding algorithm it is stated that it might be that$s$and$r$have a common factor, in which case the number$r'$returned by the continued fractions algorithm be a factor of$r$, and ... 1 vote 0 answers 86 views ### Is there a way to use stabilizer formalism for non-computational basis input states? In Nielsen and Chuang, exercise 10.42 is to use stabilizers to prove the teleportation circuit works as claimed. It has a footnote that it only works given a restricted class of inputs (it doesn't ... 1 vote 0 answers 127 views ### Definition(s) of$\delta$in quantum phase estimation I read the chapter on QPE (quantum phase estimation) in Nielsen and noticed that$\delta$is defined there as follows:$0 \leq \delta \leq 2^{-t}$, see: 5.2.1 Performance and requirements The above ... 1 vote 0 answers 385 views ### Circuit to show that Hadamard, phase, controlled- and Toffoli gates are universal Part 1 The final output state is,$|\psi_{out}\rangle=\frac{1}{4}[|00\rangle(3S+XSX)|\psi\rangle+|01\rangle(S-XSX)|\psi\rangle+|10\rangle(S-XSX)|\psi\rangle+|11\rangle(-S+XSX)|\psi\rangle]$When the ... 1 vote 0 answers 29 views ###$E(U_j,V_j)\leq\Delta/(2m)$if probabilities of outcomes obtained from the approximate circuit is within a tolerance$Δ>0$Suppose we wish to perform a quantum circuit containing$m$gates,$U_1$through$U_m$. Unfortunately, we are only able to approximate the gate$U_j$by the gate$V_j$. In order that the ... 1 vote 0 answers 45 views ### Generalizing quantum parallelism to bits or qubits On pg. 31 in Nielsen and Chuang, it's said that: This procedure can easily be generalized to functions on an arbitrary number of bits, by using a general operation known as the Hadamard transform, or ... 1 vote 0 answers 305 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 ... 0 votes 0 answers 37 views ### Are unital channels always mixed-unitary? How to prove mixed unitary of a channel for a multi-qubit system is not Unital. I am trying to prove Problem 8.3 of Nelson and Chuang's book. Here's a snippet of the text: Shall I need to take two ... 0 votes 0 answers 45 views ### Show that for pure states the description of the Bloch vector we have given coincides with that in section 1.2$\newcommand\bra{\left\langle#1\right|}\newcommand\ket{\left|#1\right\rangle} $I am having a little bit of difficulty with part (4) of Exercises 2.72 from Nielsen and Chuang's "Quantum ... 0 votes 0 answers 51 views ### Proof of the Lieb's theorem Lemma A6.2: Let$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$0 ≤ t ≤ 1$, $$R_1^t R_2^{1−t}... 0 votes 0 answers 50 views ### The matrix norm ||A||=\max_{\langle u|u\rangle=1}|\langle u|A|u\rangle| in the proof of Lieb's theorem In Exercise A6.4, Appendix 6: Proof of Lieb’s theorem, Page 645, Quantum Computation and Quantum Information by Nielsen and Chuang, A matrix norm of A is defined as$$||A||=\max_{\langle u|u\rangle=... 0 votes 2 answers 64 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 ... 0 votes 0 answers 67 views ### Question regarding the trace-preserving quantum operator trace distance In Michael A. Nielsen & Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition, the proof of Theorem 9.2 (Trace-preserving quantum operations are contractive) on ... 0 votes 0 answers 138 views ### Affine map of single qubit quantum operations In my reference, Page 375, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that Lemma: The Pauli matrices, along with the identity matrix$I$, form an ... 0 votes 0 answers 50 views ### Composition of rotations sign I'm solving exercise 4.15 from Nielsen and Chuang: Prove that if a rotation through an angle$\beta_1 $about the axis$\hat{n}_1$is followed by a rotation through an angle$\beta_2$about an axis$\...
$\newcommand{\ket}{\left|#1\right\rangle} \newcommand{\bra}{\left\langle#1\right|}$The Heisenberg Uncertainty principle as formulated in Nielsen and Chuang is  \Delta (C) \Delta (D) \geq \...