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.
43
questions with no upvoted or accepted answers
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views
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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5
votes
0
answers
103
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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4
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0
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208
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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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4
votes
1
answer
190
views
Question regarding Quantum Phase Estimation (Nielsen and Chuang exercise 5.8)
I was working through Nielsen and Chuang's book on quantum computing and they state the following result regarding the performance of the Quantum Phase Estimation algorithm,
"... given the input $...
4
votes
0
answers
127
views
Implementing "computable phase shifts" using T Toffoli ( problem 4.1 from Nielsen and Chuang's)
I am reading/studying the famous Nielsen and Chuang's book and ran into this interesting question and I don't quite understand the $f(x)$. It says it simply maps from $m$ to $n$ bits. But don't we ...
- 1,011
4
votes
0
answers
247
views
What is the thought process for circuit making after seeing input and output of a matrix?
Here is an exercise (4.27) from Nielsen and Chuang and I found the answer (given in the figure below) online without any explanation. The question was to construct a circuit by seeing a matrix (given ...
- 419
3
votes
1
answer
76
views
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 ...
- 31
3
votes
0
answers
61
views
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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3
votes
0
answers
55
views
What does the $I$ mean when measuring ${\rm Tr}(\rho (I\otimes\sigma\otimes\cdots))$ in quantum tomography?
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,\...
- 2,852
3
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0
answers
431
views
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 ...
- 455
3
votes
0
answers
86
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 ...
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2
votes
2
answers
346
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 ...
- 41
2
votes
0
answers
48
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
1
answer
53
views
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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2
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0
answers
81
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
103
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
123
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 $CNOT$ ...
- 66
2
votes
0
answers
103
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 ...
- 1,107
2
votes
0
answers
49
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
132
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 ...
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1
vote
1
answer
36
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Unitarity of a matrix in the EPR experiment
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{|...
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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
$$\begin{bmatrix}
a & b\\
b^* & 1-a
\end{bmatrix}\rightarrow\begin{bmatrix}
(a-a_0)e^{-t/...
- 43
1
vote
0
answers
70
views
Why can Shor code fix arbitrary errors?
This is taken from Page 434 of Nielsen and Chuang:
To simplify the analysis, suppose noise of an arbitrary type is occurring on the first
qubit only; we’ll come back to what happens when noise is ...
- 147
1
vote
0
answers
38
views
What do we mean by family of CSS codes?
In proving the security of BB84 in Nielsen and Chuang (10th anniversary edition - Section 12.6.5), they argue that a codeword in $\text{CSS}(C_1, C_2)$ is represented by
$$\frac{1}{\sqrt{|C_2|}} \sum\...
- 355
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 ...
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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 ...
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1
vote
1
answer
122
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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 ...
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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 ...
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1
vote
0
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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 ...
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1
vote
0
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127
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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 ...
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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 ...
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1
vote
0
answers
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$E(U_j,V_j)\leq\Delta/(2m)$ 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 ...
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1
vote
0
answers
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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 ...
- 316
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0
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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
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0
answers
45
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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 ...
- 127
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0
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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 $0 ≤ t ≤ 1$,
$$
R_1^t R_2^{1−t}...
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0
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The matrix norm $||A||=\max_{\langle u|u\rangle=1}|\langle u|A|u\rangle|$ in the proof of Lieb's theorem
In Exercise A6.4, Appendix 6: Proof of Lieb’s theorem, Page 645, Quantum Computation and Quantum Information by Nielsen and Chuang, A matrix norm of $A$ is defined as $$||A||=\max_{\langle u|u\rangle=...
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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 ...
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votes
0
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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 ...
- 217
0
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0
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138
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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 ...
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0
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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 $\...
- 1
0
votes
0
answers
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Usefulness of Heisenberg Uncertainty Principle
$
\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 \...
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