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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Exercise 4.6 in Quantum Computing and Quantum Information Nielsen and Chuang
Question 4.6: One reason why the
$R_\hat{n}(θ)$ operators are referred to as rotation operators is the following fact, which
you are to prove. Suppose a single qubit has a state represented by the ...
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In Bell's inequalities, what is the meaning of assuming that the physical properties $P_Q,P_R,P_S,P_T$ have definite values?
Two assumptions behind Bell inequalities (Page 117 Nielsen Chuang)
(1) The assumption that the physical properties $P_{Q}$, $P_{R}$, $P_{S}$, $P_{T}$ have definite values
$Q$,$R$, $S$, $T$ which exist ...
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Nielsen & Chuang Exercise 2.55: Prove that $\exp \left[ -\frac{iH(t_2 - t_1)}{\hbar} \right]$ is unitary
$\newcommand{\expterm}[0]{\frac{-iH(t_2 - t_1)}{\hbar}}
\newcommand{\exptermp}[0]{\frac{iH(t_2 - t_1)}{\hbar}}$Nielsen & Chuang (10th edition, page 82) states that $H$ is a fixed Hermitian ...
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Do any two distinct pure states form a basis?
$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$
In ...
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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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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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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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Viewing two-qubit measurement as a projective measurement
I am following Nielsen and Chuang, section 2.2.5:
A projective measurement is described by an observable, $M$, a Hermitian
operator on the state space of the system being observed. The
...
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Time Evolution Operator of Rabi Oscillations
I am referring to Exercise 7.18 of "Quantum Computing and Information 10th Anniversary Edition" by Nielsen and Chuang. The exercise wants me to show that the time evolution operator related to Rabi ...
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Angular Error associated with Quantum Search Algorithm
Chapter 6.3 of "Quantum Computation and Quantum Information 10th Anniversary Edition" textbook by Nielsen and Chuang talks about using the Quantum Counting Algorithm to find the number of solutions to ...
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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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How can the entropy of quantum states increase after projective measurements?
I'm reading Nielsen and chuang 11.3.3 Measurements and Entropy.
It says after measurement, one's entropy increases.
How is this possible? Shouldn't measurement decrease one's uncertainty?
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How to prove generalized Uhlmann's theorem?
I think the Uhlmann theorem should be in general of this form:
Let $\rho$ and $\sigma$ be density operators acting on $A$, with Schmidt degrees at most $r$, and let $B$ be another Hilbert space with ...
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In quantum process tomography, how does $\chi$ characterize a quantum process?
I'm working through Nielsen and Chuang and I'm pretty confused by the discussion of quantum process tomography. I'm trying to work through an example of 1-qubit state tomography given by N&C (box ...
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Phase shifter acting on double rail states
In Nielsen and Chuang, it is stated that the photonic phase shift gate acts on the single photon states as $P|0\rangle \ = \ |0\rangle$ and $P|1\rangle \ = \ e^{i\Delta}|1\rangle$, where $\Delta \ = \ ...
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Nielsen & Chuang Exercise 2.3 - “Matrix representation for operator products” [closed]
Reproduced from Exercise 2.3 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition):
Suppose $A$ is a linear operator from vector space $V$ to vector space ...
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Are the states in the convex decomposition of a density matrix necessarily orthogonal?
In Nielsen and Chuang's QC&QI, I do not see a statement one way or another. In Steeb and Hardy's Problems and Solutions, orthogonality is asserted. If the $p_i$ in $\sum_i p_i |\psi_i\rangle\...
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When discussing error correction, what are the objects in the expression $PE_i^\dagger E_j P=\alpha_{ij} P$?
I've started reading the book "Quantum Computation and Quantum Information" by Michael A. Nielsen and Issac L. Chuang, specifically chapter 10 (about quantum error correction), and I'm ...
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Why can the Hamiltonian $H=P_x(t)X+P_y(t)Y$ make an arbitrary unitary $U=R_x(b)R_y(c)R_x(d)$?
p.281 of Nielsen and Chuang's book says that
A single spin might evolve under the Hamiltonian $H = P_x(t)X + P_y(t)Y$, where $P_{\{xy\}}$ are classically controllable parameters. From Exercise 4.10, ...
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How to derive the rotations caused by the H gate?
In Nielsen and Chuang, there's the following paragraph:
The Hadamard operation is just a rotation of the sphere about the ˆy axis by 90◦, followed by a rotation about the ˆx axis by 180◦.
I am ...
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How does a general rotation $R_\hat{n}(\theta)$ related to $U_3$ gate?
From eqn. $(4.8)$ in Nielsen and Chuang, a general rotation by $\theta$ about the $\hat n$ axis is given by
$$
R_\hat{n}(\theta)\equiv \exp(-i\theta\hat n\cdot\vec\sigma/2) = \cos(\theta/2)I-i\sin(\...
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Understanding the outer products in density matrices
I don't understand a simple property of the outer product when doing density matrices. I am studying nielsen and chuang's book.
At equation 2.197 they do show the density matrix of the state of ...
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Proof of the 4.11 exercise in the Nielsen & Chuang book
In question 4.11 in Nielsen and Chuang's book, it states that there is a formula to describe any unitary matrix $U$ with two vectors $\vec{n}$ and $\vec{m}$ in the following way:
$$U=\exp(i \alpha) ...
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Is the phase-estimation a specific case of the Hidden Subgroup Problem?
I read Nielsen & Chuang and I have difficulties understanding the links between the Hidden Subgroup Problem and the Phase Estimation.
In Exercise 5.14 (Section 5.3.1 "Application: order-...
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Proof of Nielsen's theorem (Theorem 12.15) given in Nielsen-Chuang (assumption of invertibility)
Theorem 12.15 of Nielsen and Chuang's 10th anniversary edition is Nielsen's Theorem (1999). In particular, it says,
Theorem 12.15: A bipartite pure state $\mid \psi \rangle$ may be transformed to ...
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Expansion of multi-qubit density matrix in the Pauli matrix basis
The single qubit density matrix can be expanded as
$$
\rho=\frac{tr(\rho)I+tr(X\rho)X+tr(Y\rho)Y+tr(Z\rho)Z}{2}
$$
which can be shown as,
$\rho$ is a positive operator with $tr(\rho)=1$, ie.,
$\rho=\...
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Phase estimation algorithm: Modulo part in Nielsen and Chuang
In Nielsen and Chuang the explanation of phase estimation states:
We have the following state:
$$\frac{1}{2^{t/2}} \sum\limits_{k=0}^{2^t-1} e^{2 \pi i \varphi k}|k\rangle$$
Now we apply the inverse ...
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In the Deutsch-Jozsa algorithm, why is the resulting amplitude for the constant and balanced cases $\pm 1$ and $0$, respectively?
I am currently learning from Nielsen and Chuang and I am currently learning about Deutsch-Jozsa algorithm. However, I am stumped with the mathematics of the algorithm at the following section:
I ...
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Show that for any measurement operator $M_m$ there exists unitary $U_m$ such that $M_m=U_m\sqrt{E_m}$ with $E_m$ POVM
Exercise 2.63 of Nielsen & Chuang asks one to show that if a measurement is described by measurement operators $M_m$, there exists unitary $U_m$ such that $M_m = U_m \sqrt{E_m}$ where $E_m$ are ...
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Showing measurement of a Hermitian Unitary operator gives final states as eigenvectors
This is related to exercise 4.34,
The operation described can be written as $(H \otimes I)C^1(U)(H \otimes I)(|0\rangle \otimes |\psi\rangle)$
I can get to the point where the state of the system is ...
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Nielsen and Chuang, Exercise 6.12: How to simulate the specific Hamiltonian in the search algorithm by the Oracle gates?
In Chapter 6 of "Quantum Computation and Quantum Information" Textbook by Nielsen and Chuang, Exercise 6.12:
Exercise 6.12: (Alternative Hamiltonian for quantum search) Suppose:
$$H=|x\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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What does measurement mean in quantum error correction(syndrome diagnosis)?
In the case of the simple three-qubit repetition code, the encoding consists of the mappings $|0\rangle \rightarrow\left|0_{\mathrm{L}}\right\rangle \equiv|000\rangle$ and $|1\rangle \rightarrow\left|...
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Knill-Laflamme condition derivation in Nielsen&Chuang: issue to understand a part of the proof
I have some trouble to understand the proof in Nielsen&Chuang about Knill-Laflamme conditions.
The conditions:
Let $C$ be a quantum code and $P$ the projector onto $C$. Suppose
$\mathcal{E}$ is a ...
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How would I theorise a quantum query algorithm in O(1)?
I am currently attempting to solve a problem from Nielsen-Chuang, and I can't seem to figure out how I would do this;
I'm trying to implement Grover's algorithm to solve the problem of differentiating ...
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Quantum Phase Estimation Circuit and Modular Exponentiaton
In Nielsen and Chuang, it is stated that the effect of phase estimation circuit is mapping state $|j\rangle |u\rangle$ to $|j\rangle U^j |u\rangle$.
Here is my solution:
Consider the first $CU^{2^0}...
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Quantum addressing scheme
Nielsen explains how a search algorithm can access a classic database. I have a few questions. I hope you can help me a bit :) I work with a few quotes from the book.
The principle of operation is a ...
user4961
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Understanding the quantum circuit for the oracle in Grover's algorithm given in N&C
In the chapter about the Grover algorithm, it is suggested that the gate which executes the phase shift is given in the following form:
Now I have looked at this gate in detail and come to the ...
user4961
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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 ...
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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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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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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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Nielsen & Chuang Exercise 2.5 - Inner products of complex vectors [closed]
Reproduced from Exercise 2.5 of Nielsen & Chuang’s Quantum Computation and Quantum Information (10th Anniversary Edition):
A function $(\cdot, \cdot)$ from $V × V$ to $C$ is an inner product if ...
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How to compute the unitary from the $\chi$ matrix obtained from QPT
I am trying to do quantum process tomography for one qubit and obtain the unitary for the gates that are applied on the qubit. I have studied the theory on process tomography from mike and ike and the ...
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How to understand intuitively the concavity of the binary entropy?
In Nielsen and Chuang's Quantum Computation and Quantum Information book, introducing the binary entropy, they gave an intuitive example about why binary entropy is concave:
Alice has in her ...
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What does the centered dot notation mean in the expression $|+\rangle|+\rangle\otimes(\cdot)$?
In the expression $|+\rangle|+\rangle \otimes(\cdot)$, what does the centered dot represent? I have come across a definition of the centered dot previously, similar to the one given here, but I am ...
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Understanding the definition of entropy in the joint entropy theorem derivation
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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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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What we get when measure $|0\rangle$ under computational basis?
It is said if we have been given the state $|0\rangle$, the measurement will yield $0$ with probability $1$ in Nielsen's book.
So here, the measurement will yield $0$ refers to we will get state $|0\...
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What is the meaning of $\langle e_k|U|e_0\rangle$ when $U$ acts on a larger Hilbert space than that in which $|e_0\rangle$ and $|e_k\rangle$ live?
In Nielsen and Chuang, 10th Anniversary Edition, there is a definition of the operator sum representation of a quantum operation: $\mathcal{E}(\rho)=\sum_{k}\langle e_k|U[\rho\otimes|e_0\rangle\langle ...
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