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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Solution to problem 5.3 Book Quantum Computation and Quantum Information Nielsen Chuang regarding Kitaev's algorithm
Problem 5.3: (Kitaev’s algorithm) Consider the quantum circuit
where |u> is an eigenstate of U with eigenvalue $e ^ {2 \pi i \phi} $. Show that the top qubit is measured to be 0 with probability $...
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Equating the state of the Phase Estimation algorithm to $\frac{1}{2^{t/2}}\sum_{k=0}^{2^t-1} e^{2\pi i\phi k}|k\rangle$
It is stated in the Phase Estimation algorithm in Page 222, Quantum Computation and Quantum Information by Nielsen and Chuang that
It seems to say that taking the inverse Quantum Fourier transform of ...
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Pauli Matrix tolerant and encoding
I'm stumped by these questions in Chuang's book.
If I have a state $|ψ\rangle=1/2\sqrt2(1+M_0)(1+M_1)(1+M_2)|0\rangle_7$, where
$$M_0=X_0X_4X_5X_6; M_1=X_1X_3X_5X_6; M_2=X_2X_3X_4X_6$$
I rewrite its ...
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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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Minimum Multi-Degree Polynomials representing Boolean Functions
In the 10th Anniversary Edition of Nielsen and Chuang Quantum Computation and Quantum Information textbook, Chapter 6.7 talks about Black Box algorithm limits.
It is given:
$f:\{0,1\}^n \...
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Explanation of the 2.60 equation page 76 in the Nielsen and Chuang [duplicate]
In the Nielsen and Chuang book page 76, equation 2.60 says that we can rewrite the trace $$Tr(A \left|\psi\right>\left<\psi\right|)$$ as follow :
$$Tr(A \left|\psi\right>\left<\psi\right|) ...
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Show that for pure states the description of the Bloch vector we have given coincides with that in section 1.2
$\newcommand\bra[1]{\left\langle#1\right|}\newcommand\ket[1]{\left|#1\right\rangle} $
I am having a little bit of difficulty with part (4) of Exercises 2.72 from Nielsen and Chuang's "Quantum ...
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Proof of the Lieb's theorem
Lemma A6.2: Let $R1 , R2 , S1 , S2 , T1, T2$ be positive operators such that
$0 = [R1, R2 ] = [S1, S 2 ] = [T1, T2 ]$, and
$$
R1 ≥ S1 + T1\\
R2 ≥ S2 + T2
$$
Then for all $0 ≤ t ≤ 1$,
$$
R_1^t R_2^{1−t}...
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In quantum error correction, what does an "arbitrary error that yields an un-normalized state" mean?
This is from page 434 of Nielsen and Chuang:
. Supposing the state of the encoded qubit is |ψ⟩
before the noise acts, then after the noise has acted the state is
E(|ψ⟩⟨ψ|). To analyze the effects of
...
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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 $$||A||=\max_{\langle u|u\rangle=...
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Is there a criteria to ensure a one-qubit operator is exactly of the form $R_n(\theta)$ (i.e without a global phase $e^{i\alpha}$)?
Reading the Nielsen and Chuang, I saw that every unitary operator $U$ can be written as $e^{i\alpha} R_n(\theta)$ for some well chosen $n \in \mathbb{R}^3$ and $0 \leq \theta < 2\pi$.
I would like ...
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Composition of rotations sign
I'm solving exercise 4.15 from Nielsen and Chuang:
Prove that if a rotation through an angle $\beta_1
$ about the axis
$\hat{n}_1$ is followed by a rotation through an angle $\beta_2$ about an axis $\...
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What is the solution to Nielsen and Chuang Exercise 2.65?
Nielsen&Chuang Exercise2.65: Express the states (|0 + |1)/
√
2 and (|0−|1)/
√
2 in a basis in
which they are not the same up to a relative phase shift.
Consider an orthnormal basis :$\begin{cases}...
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Usefulness of Heisenberg Uncertainty Principle
$
\newcommand{\ket}[1]{\left|#1\right\rangle}
\newcommand{\bra}[1]{\left\langle#1\right|}
$The Heisenberg Uncertainty principle as formulated in Nielsen and Chuang is
$$
\Delta (C) \Delta (D) \geq \...
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Prove that a C2(U) gate (for any single qubit unitary U) can be constructed using at most eight one-qubit gates, and six controlled- NOTs [duplicate]
Its a Problem from
Michael Nielsen quantum computation and quantum information
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Nielsen and Chuang: Demonstration of equation 2.12 [duplicate]
Reproduced from Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition) in page 64:
We've seen that matrices can be regarded as linear operators. [...] Suppose $...
user12503
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Default input states of qubits to quantum circuits [closed]
It is conventional to assume that the state input to the circuit is a
computational basis state, usually the state consisting of all
$|0\rangle$s. This rule is broken frequently in the literature ...
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The use of modulo 2 in state representation after CNOT
The following circuit with a CNOT gate has the following effect on a computational basis state $|a, b\rangle$, where all additions are done modulo 2.
Why is the state of the second qubit changed to $...
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What is the definition of expectation value in Nielsen and Chuang?
As someone who learned Quantum Mechanics prior to studying Quantum Computing, I am having problems with this book. For example, equation 2.92 is incorrect. That is the definition of expected value, ...
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Copying quantum state [closed]
I'm confused about the last complete sentence in the following paragraphs. If ab=0, that means either a or b equals to 0. As a result, doesn't $|\psi\rangle|\psi\rangle$ equal to either $b^2|11\rangle$...
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Why does $a ⊕ (a ⊕ b) = b$ and $(a ⊕ b) ⊕ b = a$? [closed]
For the following circuit that swap two qubits
The sequence of gates is said to have the following sequence of effects on a computational basis state |a, b>
where all additions are done modulo 2.
...
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what are the angles 𝜃, and ϕ values of the following quantum state? [duplicate]
I need to find the coordinate 𝜃 and ϕ values of the quantum state on the bloch sphere
$$
\left| \varphi \right>=\frac{1+i}{\sqrt{3}} \left| 0 \right> + {\sqrt{\frac{1}{3}}} \left| 1\right>
$$...
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