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.
323
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Nielsen & Chuang Exercise Question on CSS code
I was reading the CSS ( Steane Code) from the Nielsen & Chuang book. It asked in Ex. 10.27 to prove that: suppose $C_1$ and $C_2$ are $[n,k_1]$ and $[n,k_2]$classical linear codes such that $C_2\...
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Nielsen & Chuang Exercise 2.1 - "Linear dependence: example" [closed]
Reproduced from Exercise 2.1 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):
Show that $(1, −1)$, $(1, 2)$ and $(2, 1)$ are linearly dependent.
...
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2
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Why are rotations represented by exponentials of Pauli matrices?
I'm self-studying Quantum Computation from Nielsen and Chuang's book. In section 4.2 they discuss that for any unit vector $\hat n$, the rotation operator $R_{\hat n}(\theta) = \exp(-i\theta\hat n \...
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Why is the operation in Nielsen and Chuang's Section 8.5 not a quantum operation?
In Section 8.5 of the 10th anniversary edition, Nielsen and Chuang discuss the limitations of the quantum operations framework. They give an example of a qubit prepared in an unknown state $\rho$, ...
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The construction of every element of the Clifford group using H,S and CNOT circuits
I am trying to understand the following theorem: Every element $U\in C_n$ of the Clifford group can be constructed using $H, S, CNOT$ gates.
In Nielsen and Chuang's book this is left as an exercise (...
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519
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Construction of ${R_n(\theta)}$ using only the Hadamard and ${\pi/8}$ gates
In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen & Chuang", they claim that Eqn(4.75) is a rotation about the axis along the direction
( ${cos(\pi/8)}$, ${...
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301
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Why does the $\chi$ matrix have $d^4-d^2$ independent parameters?
In the section on Quantum process tomography, Page 391, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang. it is given that
In general, $\chi$ will contain $d^4−d^2$ ...
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292
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Why does the bit flip channel produce a uniform contraction of $1-2p$?
Studying the bit flip channel using the Nielsen & Chuang's.
And ran into the picture with the caption stating $yz$ plane is uniformly contracted by a factor of $1-2p$. I don't quite understand how ...
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Prove that rank one projectors have the same partial trace iff they differ by a local unitary operation
In nielsen and chuang's QCQI book, there is a theorem called Unitary freedom in the ensemble for density matrices, which states that the sets $|\psi_i\rangle$ and $|\phi_i\rangle$ generate the same ...
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Exercise 4.41 in N&C book QCQI: how can i implement $R_z(\theta)$ using the circuit shown and $Z$?
I'm studying Nielsen and Chuang's book.
I cannot solve one of the questions in the exercise 4.41.
The question is the last one that is
Explain how repeated use of this circuit and Z gates may be used ...
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Quantum indistinguishability using density operators
There is something that bugs me concerning the use of density matrices. For instance, to argue that quantum teleportation does not spread an information faster than light, Nielsen and Chuang state the ...
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How does $U_f$ in Deutsch's Algorithm affect the state $|x\rangle$?
I am trying to read Nielsen's and Chuang's book on quantum computing and I am having problem understanding Deutsch's algorithm.
According to my understanding of the algorithm, the state $|x\rangle$ ...
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Question about Nielson & Chuang Problem 9.2
I am working on the following problem from the book "Quantum Computation and Quantum Information" by Nielsen and Chuang.
Problem 9.2: Let $\mathcal{E}$ be a trace-preserving quantum
...
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When proving the Solovay-Kitaev theorem, why do we consider a small neighborhood $S_\epsilon$ of the identity?
There are number of points I haven't understood or am confused in the proof of Solovay-Kitaev theorem. The proof I'm reading in given in the Appendix 3 of Neilson and Chuang's book, Quantum ...
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Cost of Modular Exponentiation in Shor's algorithm
In the Shor's algorithm, we need to compute the sequence of controlled $U^{2^j}$ operations used by the phase estimation procedure, where $U$ is defined as
$$
U|y\rangle=|xy\;(\mod N)\rangle\text{ ...
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Equivalent statement of the unitary freedom of Kraus operator?
There is a well-known form of the unitary freedom of Kraus operators, which can be found in Nielsen's book, stating that two sets of Kraus operators describe the same physical process of the system(...
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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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0
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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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Premise of the proof of the No-Cloning Theorem
I have seen two similar proofs of the no-cloning theorem. They assume (to the contrary) that there exists a unitary operator $U$ such that $U |\psi\rangle |0 \rangle = | \psi \rangle | \psi \rangle$, ...
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Necessity of decoding in fault-tolerant quantum computation
On page 476 of Nielsen/Chuang's book it is stated:
The basic idea of fault-tolerant quantum computation is to compute directly on encoded quantum states in such a manner that decoding is never ...
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Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem
In Nielsen and Chuang, in the Fidelity section, (Lemma 9.5, page 410 in the 2002 edition), they prove the following.
$$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$
...
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In the proof of the joint entropy theorem, why are $p_i\lambda_i^j$ the eigenvalues?
From section 11.3.2 of Nielsen & Chuang:
(4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|...
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Why isn't the circuit performing a measurement in the Bell basis?
Nielsen and Chuang (on page 188 exercise 4.33) says that the circuit including CNOT and Hadamard is performing a measurement in the Bell basis. But I can't see how.
The matrix representing the ...
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What kind of errors does the master equation in the Lindblad form describe, continuous errors or discrete errors?
I noticed that in page 427 in Nielsen & Chuang's book Quantum Computation and Quantum Information, quantum error correction is possible because errors can be discretized.
In other hand, the ...
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Exercise 10.34 of Nielsen and Chuang : $-I$ not element of stabilizer group iff $g_j^2=I$ and $g_j \neq I$ where $g_j$ generators
I am stuck with this exercice of Nielsen and Chuang:
Let $S = \langle g1,... ,gl \rangle $.Show that $−I$ is not an element of S if and only if $g^2_j = I$ for all $j$,and $g_j \neq − I$ for all $j$...
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In what sense do repeated applications of Grover's operator rotate the state closer to the target?
I'm studying the quantum search algorithm on this book:
M.A. Nielsen, I.L. Chuang, "Quantum Computation and Quantum Information", Cambridge Univ. Press (2000) [~p. 252].
To sum up we have a state:
$...
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Regrouping the terms in expression 1.31 in Quantum Computing and Quantum Information, Nielsen and Chuang
I'm trying to reproduce the passage from expression 1.31 to 1.32 in the book Quantum Computing and Quantum Information, by Michael Nielsen and Isaac Chuang.
Expression 1.31 is:
$$|\psi_2\rangle = \...
4
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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 ...
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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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Identity for linear codes and their duals: why do we have $\sum_y (-1)^{x\cdot y}=|C|\delta_{x\in C^\perp}$?
I've come across this exercise plenty of times and I still don't understand how to do it. (Here it is from N.C. Ex.10.25)
Let $C$ be a linear code (Lets suppose its a binary code, i.e. a $k$-...
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Why do we need to reverse the order of qubits in Quantum Fourier Transform? [duplicate]
Looking at Qiskit's QFT tutorial, their implementation of QFT requires you to swap the qubits at the end (Nielsen and Chuang do this too). I'm wondering why this is the case. Can we flip the gates ...
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What is the technique for calculating $\text{Tr}_b[{U(\rho\otimes\rho_b)U^{\dagger}}]$?
I am stuck on calculating $\mathcal{E}(\rho)=\text{Tr}_b[{U(\rho\otimes\rho_b)U^{\dagger}}]$. For example, in the case when $U$ is the CNOT matrix $$U=\begin{pmatrix} 1 & 0 & 0 & 0\\\ 0 &...
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What's the difference between $p(i|m)$ and $p(m|i)$ in measurement?
Suppose we perform a measurement described by measurement operators $M_m$. If the initial state is $|{\psi_i}\rangle$, then the probability of getting result $m$ is
$$
\begin{align}
p(m|i)=\| M_m|\...
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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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Nielsen and Chuang ex 2.73
I've been trying to solve exercise 2.73 (p.g 105), and I'm not sure if i'v been overthinking it and the answer is as simple as i've described below or if I am missing something, or i'm just wrong!
Ex ...
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What does it look like to split an EPR pair?
I am reading Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang, and I am confused about a concept presented in Section 1.3.7: Quantum Teleportation. The book ...
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Show that $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)$ implies $U=\begin{bmatrix}[E_1]&\cdots\\ [E_2]&\cdots\\\vdots&\ddots\end{bmatrix}$
In Page 365, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that
Given a trace-preserving quantum operation expressed in the ...
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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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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:
$$...
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Understanding the 3rd step of Nielsen and Chuang's description of the quantum order-finding algorithm
In Nielsen and Chuang's description of Quantum order-finding algorithm, the 3rd step of the procedure says
$$\frac1{\sqrt{2^t}}\sum_{j=0}^{2^t-1}|j\rangle|x^j\mod N\rangle \approx \frac1{\sqrt{r2^t}}\...
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How to derive the expression for the probability in quantum phase estimation? ((5.27) Nielsen & Chuang)
I'm trying to understand the QPE algorithm that is presented in the
Nielsen and Chuang textbook. More precisely, I do not understand Equation $(5.27)$.
Context: In the following, let $b$ be a natural ...
- 41
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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 ...
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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 ...
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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 ...
- 41
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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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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
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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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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,\...
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Why is Hilbert space considered especially large?
In Nielsen & Chuang section 1.2 introduces multiple qubits and Hilbert spaces.
More generally, we may consider a system of n qubits. The computational basis states
of this system are of the ...
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