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# 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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### How to correct error during the syndrome measurement or recovery process?

Here's the figure 10.21 from Nielsen's Quantum Computation and Quantum Information to explain the fault-tolerant quantum computing, where the circuit in the figure corrects the error that happens in ...
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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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### Proof of upper and lower bound (Gilbert-Varshamov bound) for linear code

I am trying to prove the following bounds for a $[n, k]$ code that can correct $t$ errors \begin{align} 1-H\left(\frac{t}{n}\right)\geq \frac{k}{n}\geq 1-H\left(\frac{2t}{n}\right) \end{align} where \...
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In Nielsen and Chuang's QCQI, I learned that the quantum tomography for n qubit can be described easily in math as we need to measure $Tr(\rho W_k),\forall k$ where $W_k\in\{I,\sigma_x,\sigma_y,\... • 3,019 3 votes 0 answers 75 views ### On the use of$\log(P\otimes Q)= \log P\otimes I+I\otimes\log Q$for relations between entropic quantities. What if$P,Q$are only semidefinite? Many properties of entropic quantities are shown by resorting to related properties of the relative entropy of suitable quantities. For instance, subadditivity of entropy may follow from non ... 3 votes 2 answers 921 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 ... • 51 3 votes 0 answers 88 views ### 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 ... • 1,631 2 votes 0 answers 48 views ### What is the origin of the term POVM? In Nielsen Chuang section 2.2.6 as they are introducing POVMs they write “The acronym POVM stands for ‘Positive Operator-Valued Measure’, a technical term whose historical origins we won’t worry about.... 2 votes 0 answers 43 views ### Reasoning behind unitary freedom in the ensemble for density matrices theorem Although my question has the same title of a different question, it is not a duplicate. I am asking a different question. I don't care why it made it into the book. Here is a theorem from Nielsen &... 2 votes 2 answers 164 views ### Exercise 11.7 in Nielsen & Chuang and basic properties of Shannon entropy I apologize in advance if this question is trivial, I'm aware I'm a total beginner in this field. This is the exercise I would like to solve: As to the first point, what I get is that one should ... 2 votes 0 answers 68 views ### Question about fault-tolerant measurement in Nielsen&Chuang page 490 The fault-tolerant procedures for state preparation and logical operation consider the procedure successful if only a one-qubit error occurs in each encoded block of the output. So, I thought that the ... • 21 2 votes 0 answers 144 views ### Solution Nielsen and Chuang exercize 10.71 Exercise 10.71: Verify that when$M = e^{−iπ/4}SX$the procedure we have described gives a fault-tolerant method for measuring$M$. The book describes a procedure to perform the measurement. Instead ... • 21 2 votes 0 answers 122 views ### Quantum Process Tomography for 2 qubits I need clarification on a few aspects related to Box 8.5 and Exercise 8.34 from the book Quantum Computation and Quantum Information by Nielsen & Chuang . While attempting Exercise 8.34, I ... 2 votes 0 answers 103 views ###$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/... 2 votes 0 answers 112 views ### 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>... 2 votes 0 answers 188 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 ... • 831 2 votes 0 answers 83 views ### 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 ... 2 votes 0 answers 120 views ### 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 ... 2 votes 0 answers 182 views ### 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 ... 2 votes 0 answers 109 views ### 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 ... • 1,099 2 votes 0 answers 59 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 ... 2 votes 1 answer 251 views ### 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 ... • 321 1 vote 1 answer 82 views ### Why is the matrix obtained from the coefficients of orthogonal states unitary? 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 141 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 57 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\... • 417 1 vote 0 answers 214 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 161 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 ... • 831 1 vote 0 answers 57 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 ... • 831 1 vote 0 answers 156 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 ... • 131 1 vote 0 answers 143 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 ... • 678 1 vote 0 answers 571 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 ... • 831 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 ... • 831 1 vote 0 answers 51 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 370 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 ... • 316 0 votes 0 answers 29 views ### Which stabilizer generators of a CSS code follow from which parity check matrices? Do I have the right correspondence below between$X$-type ($Z$-type) stabilizer generators and the rows of the parity check matrices of$C_2^\perp$($C_1$)? I ask because it seems Nielsen and Chuang ... • 203 0 votes 0 answers 148 views ### Show that for pure states the description of the Bloch vector we have given coincides with that in section 1.2$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} $I am having a little bit of difficulty with part (4) of Exercises 2.72 from Nielsen and Chuang's "Quantum ... • 641 0 votes 0 answers 62 views ### Composition of rotations sign I'm solving exercise 4.15 from Nielsen and Chuang: Prove that if a rotation through an angle$\beta_1 $about the axis$\hat{n}_1$is followed by a rotation through an angle$\beta_2$about an axis$\...
$\newcommand{\ket}[1]{\left|#1\right\rangle} \newcommand{\bra}[1]{\left\langle#1\right|}$The Heisenberg Uncertainty principle as formulated in Nielsen and Chuang is  \Delta (C) \Delta (D) \geq \...