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 does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?
In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation:
$$\mathcal E(\rho) = \mathrm{Tr}_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$...
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How to perform quantum state tomography on two qubits?
I would like to do a quantum tomography on two qubit states.
Recently, I successfully did so for one qubit based on Nielsen-Chuang. They advise to use this formula for one qubit density operator ...
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How to perform Quantum Process Tomography for three qubit gates?
I am trying to perform Quantum process tomography (QPT) on three qubit quantum gate. But I cannot find any relevant resource to follow and peform the experiment. I have checked Nielsen and Chuang's ...
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Nielsen & Chuang Exercise 2.2 - “Matrix representations: example” [closed]
Reproduced from Exercise 2.2 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):
Suppose $V$ is a vector space with basis vectors $|0\rangle$ and $|1\...
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General parametrisation of an arbitrary $2 \times 2$ unitary matrix
From Nielsen & Chuang's Quantum Computation and Quantum Information (QCQI):
Since $U$ is unitary, the rows and columns of $U$ are orthonormal, form which it follows that there exist real numbers $...
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Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$
$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$
I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
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How to find the operator sum representation of the depolarizing channel?
In Nielsen and Chuang (page:379), it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity ...
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Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?
I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following:
$$
F(\rho, \sigma) := \...
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Intuitive role of the polar decomposition in proof of Uhlmann's theorem for fidelity
I have read the Wikipedia article which relates the polar decomposition to a complex number being split into its modulus and phase but this analogy isn't very intuitive to me.
In Nielsen and Chuang, ...
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Are three POVM measurements on a single qubit physically realizable?
In Nielsen and Chuang Quantum Computation and Quantum Information book section 2.2.6, a POVM of three elements are used to measure a single qubit in order to know for sure whether the state is $|0\...
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Understanding Steps in Deutsch's Algorithm
I am currently working my way through the book Quantum Computation and Quantum Information by Chuang and Nielsen. So far it has been a joy to read, however I am hung up on a couple aspects of quantum ...
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Why is Deutsch's gate universal?
(This is related to Exercise 4.44 in Nielsen and Chuang)
Deutsch quantum gate is basically a $iR_x(\alpha \pi)$ gate with two control qubits. The constant $\alpha$ is an irrational number that allows ...
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Procedures and intuition for designing simple quantum circuits?
I'm working my way through one of the quantum circuits sections in Nielsen and Chuang and I'm struggling to get a feel for the basics of circuit construction. For example, one of the exercises is as ...
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How does the spectral decomposition of the Choi operator relate to Kraus operators?
In Nielsen and Chuang's QCQI, there is a proof states that
Theorem 8.1: The map $\mathcal{E}$ satisfies axioms A1, A2 and A3 if and only if
$$
\mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger}
$$...
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How to visualize Hadamard gate as $X$-$Z$-$X$ decomposition?
In the book Quantum Computation and Quantum Information by Nielsen and Chuang, chapter 4, exercise 4.4 (pg. 175), the author has asked to express Hadamard gate as product of $R_x$, $R_z$ rotations and ...
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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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How to find the matrix representation of an operator from its action on a basis?
First, I apologize if something is poorly written but English is not my first language.
I know that these exercises have been solved in this question. But I do not agree. Inner product and concrete ...
user12503
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Question Regarding Quantum Period-Finding Fourier Transform Approximation
I am following the 5.4.1 Period-Finding Algorithm in Nielsen and Chuang as shown below:
My confusion lies with the second expression of point 3 in the procedure. Why is the second expression an ...
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Partial Cyclic Permutation with only Toffoli and CNOT gates?
I have been trying to solve this puzzle of constructing this transformation from CNOT and Toffoli gates as mentioned in NC page 193 (Ex 4.27)
Here is what I have done:
First observation is it has 8 ...
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Correct Formulation of N&C Exercise 4.11 and other textbooks misquoting
Inspired by the comments in this question How to approximate $Rx$, $Ry$ and $Rz$ gates?, there is the errata for question 4.11 pg 176 in N&C. The original form states that for any non parallel $m$ ...
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Given $n-k$ stabiliser generators, how can we find an additional $k$ commuting generators?
I am trying to understand "Stabilizer codes construction" in Nielsen & Chuang (page 465). Below, we're working in a Hilbert space of dimension $2^n$, and $G_n$ is the $n$-qubit Pauli group.
A ...
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Nielsen&Chuang 5-qubit quantum error-correction encoding gate
$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$
In ...
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What does $M_m |\psi_i\rangle$ mean in the equation $p(m|i)=\langle\psi_i|M_m^\dagger M_m|\psi_i\rangle$?
I have trouble understanding two equations in the Nielsen & Chuang textbook. Suppose we perform a measurement described by the operator $M_m$. If the initial state is $|\psi_i\rangle$, then the ...
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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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How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?
How can one prove that $tr(A\mid\psi\rangle\langle\psi\mid)=\langle\psi\mid A\mid\psi\rangle$? In Nielsen/Chuang they mention this is due to Gram-Schmidt decomposition but I can’t understand how.
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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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No-cloning theorem and distinguishing between two non-orthogonal quantum states revisited
There are a couple of posts on this question, but I think they are not satisfactory. The question is Nielsen and Chuang's QCQI, Exercise 1.2, page 57, which asks "Explain how a device which, upon ...
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Prove that there exist tripartite $|\psi\rangle$ which cannot be written as $|\psi\rangle=\sum_i\lambda_i|i_A\rangle|i_B\rangle|i_C\rangle$
Exercise 2.77 in Nielsen and Chuang asks to show by example that there exist tripartite states $| \psi \rangle_{ABC} $ which cannot be written as
$$| \psi \rangle = \sum_i \lambda_i | i_A \rangle | ...
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Problem with building quantum circuit for Hamiltonian operation
In the book, Nielsen and Chuang, there is a section on quantum simulation of the quantum search algorithm. Hamiltonion operator is defined as follows-
$$
H = |x\rangle\langle x| + |\psi\rangle\langle\...
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Show that $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)$ implies $U=\begin{bmatrix}[E_1]&\cdots\\ [E_2]&\cdots\\\vdots&\ddots\end{bmatrix}$
In Page 365, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that
Given a trace-preserving quantum operation expressed in the ...
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Application of QFT to Order-finding
In the Nielsen & Chuang book, section 5.3.1 page 226, there is a statement which goes like this:- (statement-1)
The quantum algorithm for order-finding is just the phase estimation
algorithm ...
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Why isn't the circuit performing a measurement in the Bell basis?
Nielsen and Chuang (on page 188 exercise 4.33) says that the circuit including CNOT and Hadamard is performing a measurement in the Bell basis. But I can't see how.
The matrix representing the ...
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Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem
In Nielsen and Chuang, in the Fidelity section, (Lemma 9.5, page 410 in the 2002 edition), they prove the following.
$$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$
...
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In the proof of the joint entropy theorem, why are $p_i\lambda_i^j$ the eigenvalues?
From section 11.3.2 of Nielsen & Chuang:
(4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|...
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Show that quantum channels act as affine transformations in the Bloch sphere
I am referring to Equation (8.89) to (8.92) in Chapter 8 of "Quantum Computing and Information 10th Anniversary Edition" by Nielsen and Chuang. This section deals with the geometric picture ...
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In the Deutsch-Jozsa algorithm, why is the resulting amplitude for the constant and balanced cases $\pm 1$ and $0$, respectively?
I am currently learning from Nielsen and Chuang and I am currently learning about Deutsch-Jozsa algorithm. However, I am stumped with the mathematics of the algorithm at the following section:
I ...
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How can I prove inequality from 4.66 to 4.67 in Nielson and Chuang's book?
I am reading chapter 4 of Nielson and Chuang's QCQI book.
I cannot prove the inequality from (4.66) to (4.67) in page 195.
That inequality is the following:
$$ |\langle\psi|U^\dagger M|\Delta\rangle|+|...
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Method to find $r$ in the case when $r'$ returned by the continued fractions procedure is a factor of $r$
In the Quantum Computation and Quantum Information (10th ed.) textbook by Nielsen and Chuang, section 5.3.1 (titled "Application: order-finding") describes how phase estimation can be used to find the ...
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Prove $|η(r) − η(s)| ≤ η(|r − s|)$ when $|r − s| ≤ 1/2$ [closed]
Background
If $\rho$ and $\sigma$ are density matrices such that the trace distance between them satisfies $T(\rho,\sigma)\leq1/e$. Then the Fannes' inequality states that $$|S(\rho)-S(\sigma)|\leq T(...
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How to implement the exponential of an outer product?
In exercise 6.7 page 258 in Nielsen and Chuang book, they have a Hamiltonian $H = \left| x \right\rangle \!\!\left\langle x \right| + \left| \psi \right\rangle \!\!\left\langle \psi \right|$ and the ...
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Showing measurement of a Hermitian Unitary operator gives final states as eigenvectors
This is related to exercise 4.34,
The operation described can be written as $(H \otimes I)C^1(U)(H \otimes I)(|0\rangle \otimes |\psi\rangle)$
I can get to the point where the state of the system is ...
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Question Regarding Simulating Hamiltonian With Quantum Circuit
There have been a few other questions about this section of Nielsen and Chuang, but when working through the output of the circuit, there are some inconsistencies that are probably due to some mistep/...
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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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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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Amplitude Damping of a Harmonic Oscillator
Exercise 8.21 of Nielsen and Chuang asks us to show that the operation elements for a harmonic oscillator (system) coupled to another harmonic oscillator (environment) is
$E_k = \sum_n \sqrt{(^n_k)}\...
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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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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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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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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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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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