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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How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?
The proof of the Fannes' inequality replies on the formula $T(ρ, σ)≥\sum_i|r_i − s_i|$, where $r_i,s_i$ are the eigenvalues of $\rho,\sigma$, in the descending order.
In the proof given in Box 11.2, ...
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In the order finding circuit, why is the equal superposition of the controlled unitary eigenstates the $|1\rangle$ state?
In Nielsen and Chuang's "Quantum Computation and Quantum Information" when the quantum order finding process is being presented (specifically page 227, equation 5.44) we are told that by &...
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Question about the phase kickback in the phase estimation algorithm [duplicate]
I have an issue with the quantum phase estimation algorithm as explained Nielsen and Chuang.
There was a question very similar to mine asked about this 2 years ago, but my question is different... ...
2
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Nielsen & Chuang Theorem 2.6 Proof
I got a problem in understanding the proof of the Theorem 2.6 (Unitary freedom in the ensenble for density matrices), 2.168 and 2.169 in the Nielsen and Chuang book
Equation 2.168
Suppose $|{\tilde\...
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Can someone show the linear algebra calculations for X, H, and CNOT gates?
I am on Ch.1 of the Mike & Ike book. On page 18, the text shows an X gate that essentially flips the $\alpha$ and $\beta$ amplitudes. The text shows the $X$ matrix but it doesn't show those for ...
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How Does the Transformation $|x \rangle | 0 \rangle \rightarrow | x \rangle | Hx \rangle $ Avoid Violating the No-Cloning Theorem?
This question relates to Nielsen & Chuang, Exercise 10.26, which says
Suppose $H$ is a parity check matrix. Explain how to compute the transformation $|x \rangle | 0 \rangle \rightarrow | x \...
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How to extract probabilities from Kraus representation?
Consider a quantum operation described by Kraus operators $K_1, ..., K_n$. As I understand the effect of this operation on a density matrix $\rho$ can be described as $ \mathcal{E}(\rho)= \sum_{i}p(i)\...
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What is the Bloch sphere representation of $\rho\to\mathcal{E}(\rho) = |+\rangle\langle+|ρ|+\rangle\langle+| + |−\rangle\langle−|ρ|−\rangle\langle−|$?
Suppose a projective measurement is performed on a single qubit in the basis $|+\rangle, |−\rangle$, where $|±\rangle \equiv (|0\rangle\pm |1\rangle)/\sqrt{2}$. In the event that we are ignorant of ...
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Heisenberg Uncertainty Principle (Nielsen and Chuang Box 2.4)
I'm trying to follow Nielsen and Chuang Book on Quantum Computation and Quantum Information. There is Box 2.4 on the Heisenberg Uncertainty Principle. I got stuck pretty fast. In that box they define:
...
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Probability of success proof for Shor's algorithm
In the book "Quantum Computation and Information" by Nielsen and Chuang, Shor's algorithm is presented with a related probability of success theorem and proof found on page 634, Theorem A4....
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Can every unitary on $\mathcal{H}\otimes \mathcal{K}$ be modelled by quantum operations on $\mathcal{H}$?
In section 8.2.3 of Nielsen and Chuang, they discuss how unitary dynamics of a system and environment arise from quantum operations (i.e. Kraus operators $E_k$ such that $\sum_k E_k^*E_k=I$). ...
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Show that there are unitaries $U_m$ such that $M_m=U_m \sqrt{E_m}$, for any measurement $M_m$ and associated POVM $E_m$
Nielsen and Chuang's QCQI, section 2.2.6, page 92, asks
Suppose a measurement is described by measurement operators $M_m$. Show that there exist unitary operators $U_m$ such that $M_m=U_m\sqrt{E_m}$, ...
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Reason for sending numbers from 0 to $2^n − 1$ in Deutsch–Jozsa algorithm
In Nielsen and Chuang, when talking about the Deutsch–Jozsa algorithm. The Deutsch’s problem is described as the following game.
Alice, in Amsterdam, selects a number x from 0 to $2^n − 1$, and mails ...
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Why can every Bell state be written as $|\beta_{xy}\rangle=\frac1{\sqrt2}(|0,y\rangle + (-1)^x|1,\bar y\rangle)$?
In Nielsen and Chuang, there's the following paragraph:
The mnemonic notation $|\beta_{00}\rangle, |\beta_{01}\rangle, |\beta_{10}\rangle, |\beta_{11}\rangle$ may be understood via the equations
$$ |\...
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Unambiguous discrimination using POVM with highest discriminate probability
I was studying Nielsen&Chuang's textbook (about page 92), and come up with a question that I cannot solve it.
Given one of the two state $|\psi_1\rangle=|0\rangle$ and $|\psi_2\rangle=\frac{1}{\...
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Why is the subscript like this in the equation $\sum_i |\psi_i\rangle \langle\psi_i| = \sum_{ijk} u_{ij} u_{ik}^{*}|\phi_j\rangle \langle\phi_k|$?
In Nielsen's book when proving "Unitary freedom in the ensemble for density matrices"(Theorem 2.6):
$$\text{Suppose }|\widetilde{\psi_i}\rangle = \sum\limits_{j}u_{ij} |\widetilde{\phi_j}\...
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Output of Quantum Phase Estimation Algorithm
In section 5.2.1 of Nielsen Chuang, Performance and Requirements, there is an idea, that what happens if we can't prepare eigen state $|u\rangle$ and instead have a state $|\psi\rangle$ which is ...
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von Neumann entropy in a limiting case
I am stuck with a question from the book Quantum theory by Asher Peres.
Excercise (9.11):
Three different preparation procedures of a spin 1/2 particle are represented by the vectors $\begin{pmatrix}
...
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How are quantum error-correction conditions in Nielsen and Chuang implemented in practice?
Quantum error-correction conditions in Nielsen and Chuang, 10th-anniversary edition (Theorem 10.1) state that the error operation $\mathcal{E}$ with operation elements $\{E_i\}$ is correctable if and ...
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Solution to Nielsen & Chuang Exercise 5.3 (FFT)
Can somebody help me with the solution of Nielsen Chuang, where we are supposed to derive the FFT from the equation (5.4):
$$|j_1,\ldots,j_n\rangle\rightarrow\frac{\big(|0\rangle+e^{2\pi i 0.j_n}|1\...
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How to prove the strong convexity of the trace distance?
On page $408$ of Nielsen & Chuang in the step going from equation $(9.48)$ to $(9.49)$, I don't see how:
$$\sum\limits_i (p_i - q_i)tr(P \sigma_i) \leq D(p_i, q_i)$$
I proceed as follows:
$$\sum\...
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How is the size of the circuit derived, in proving the threshold theorem?
In chapter 10.6.1 in Nielsen and Chuang, the section on concatenated codes and the threshold theorem (pages 480-481) states:
The size of the simulating circuit goes as $d^k$ times the size of the
...
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Are the eigenvalues of projectors always zero and/or one?
Nielsen and Chuang, page 87, defining projective measurements, refers to projectors with "eigenvalue m." However, exercise 2.16 on page 70 seems to imply that the eigenvalue is always one or ...
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Is there a way to prove that the number of gates in Exercise 4.22 of Nielsen and Chuang's book is the smallest possible number?
I've been going over Nielsen and Chuang's Quantum Computation and Quantum Information and I ran into Exercise 4.22, which says,
Prove that a $C^{2}(U)$ gate (for any single qubit unitary $U$) can be ...
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Why is H gate called a ‘square-root of NOT’ gate?
In Nielsen and Chuang, there's the following paragraph:
I understand that
\begin{align*} \sqrt{NOT} =
\frac{1}{2}\left( {\begin{array}{*{20}{c}}
\sqrt 2 e^{i\pi / 4}&\sqrt 2 e^{-i\pi / 4}\\
\sqrt ...
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Can we write the density operator as a sum of mixed states?
In every resource I find (like Nielsen and Chuang or online courses), the density operator is defined as follows: we consider a sequence of pure states $\left|\psi_i\right\rangle$ with associated ...
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Why does $\sum_n \langle n|M_m\rho M_m^\dagger|n\rangle$ simplify to $\langle \psi|M_m^\dagger M_m|\psi\rangle$?
I was trying to derive the formula for $p(m)$ in exercise 8.2 on page 357 in Nielsen & Chuang. But I am wondering what rule I can apply to simplify this
$$\mathrm{tr}(\mathcal{E}_m(\rho) )= \...
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How would I compute a density matrix of a complex qubit mixed state?
I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed state,
$$
\frac{1}{9}\begin{bmatrix}
5 & 1 & −i \\
1 & 2 &...
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How would I compute a density matrix of a 2 qubit mixed state?
I am currently reading Nielsen & Chuang, and one of the questions asks to calculate a density matrix with the following mixed states, how would I do this?
$$
|00> \;with \;probability \; 2/4 \\
...
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In Uhlmann's theorem, should the polar decomposition be written as $A=|A|V$ or $A=V|A|$?
In the proof of Uhlmann's theorem, the book writes the polar decomposition: $A = |A|V$, with $|A| = \sqrt{A^\dagger A}$.
Shouldn't it be $V|A|$ instead?
The former case is $A^\dagger A = V^\dagger|A||...
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Conditional version of the triangle inequality for Von Neumann entropy
I'm trying to solve problem 11.3 in Nielsen Chuang:
(3) Prove the conditional version of the triangle inequality:
$$
S(A,B|C)\geq S(A|C)-S(B|C)
$$
But the inequality seems incorrect. For example,...
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Find the Kraus operators of the amplitude damping channel, partial tracing after evolution through a beamsplitter
To find operation elements for the Amplitude Damping channel, Nielsen and Chuang (in Section 8.3.5 of my copy) use the action of a beamsplitter on an initial state $ \alpha |0\rangle + \beta |1\rangle$...
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Hamiltonian for Single-photon, Single-atom QED Cavity
Equation 7.71 of Nielsen and Chuang's Quantum Computation and Quantum Information gives the Hamiltonian for a two level atom and single mode photons in a cavity as:
$H = \hbarωN + δZ + g(a^†σ_− + aσ_+...
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Eigenstate of unitary operator used for Order-Finding
In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen and Chuang", chapter 5.3.1 introduces the concept of solving the Order-Finding Problem.
(Eqn 5.36) states ...
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If CNOTs and single qubit gates are universal then why do we need to prove that controlled U operations can be composed by them as well?
In the book by Chuang and Nielsen they prove that controlled U operations can be made out of CNOTs and single qubit gates. But then they go on to prove that they are universal by showing that every n ...
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FANOUT with Toffoli Gate
Figure 1.16: FANOUT with the Toffoli gate, with the second bit being the input to the FANOUT (and the other two bits standard ancilla states), and the output from the FANOUT appearing on the second ...
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Nielsen & Chuang Exercise 2.4 - “Matrix representation for identity” [closed]
Reproduced from Exercise 2.4 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):
Show that the identity operator on a vector space $V$ has a matrix ...
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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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Nielsen and Chuang: Solving equation of motion for amplitude damping
I would like to know how to obtain a solution to the equation of motion given in Section 8.4.1 Master equations of Nielsen and Chuang, 10th edition.
The equation of motion that allows getting the ...
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In Nielsen & Chuang, shouldn't $P=\sum_{i=1}^k|i\rangle\!\langle i|$ in (2.35) equal the identity?
Nielsen and Chuang define Projectors as:
An operator $A$ whose adjoint is $A$ is known as a Hermitian or self-adjoint operator. An important class of Hermitian operators is the projectors. Suppose $...
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Quantum parallelism and Deutsch's algorithm - what is $U_f$ really? [closed]
I'm trying to understand quantum parallelism ideas leading the Deutsch's algorithm. The circuit in question is
I understand that we end up with
$$|\psi_3 \rangle = \pm | f(0) \oplus f(1) \rangle \...
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EPR Experiment: What does it mean for Alice to measure $\vec{v} \cdot \vec{\sigma}$ on her qubit?
I am trying to understand Box 2.7 on page 113 of Quantum Computation and Quantum Information book by Nielsen and Chuang. They start out with following wave function:
\begin{equation}
\psi = \frac{|01\...
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Standard form Shor's code
I'm trying to solve exercise 10.57 in Nielsen-Chuang, where you have to obtain the standard form check matrix of Shor's code. I followed the procedure laid out in the earlier chapter but then realised ...
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CSS Code and Fault-tolerant Problem in Nielsen and Isaac Chuang‘s book
Could someone help me with $Problem-10.52$ in Nielsen and Isaac Chuang‘s book?
The screenshot is shown below. I have no idea about that.
Hope someone can give me some suggestions.
By the way, if I ...
user13341
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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 ...
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$T_1$ 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/...
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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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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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