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 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 ...
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Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
I have trouble understanding this proof in Nielsen & Chuang, specifically the identity in $(10.20)$, which reads $$ U_k^\dagger P_k F_l \sqrt{\rho} = U_k^\dagger P_k^\dagger P_k^\dagger F_l P \...
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How is the $\beta$-matrix interpreted in single qubit QPT?
In Chapter 8 of Quantum Computation & Quantum Information by Nielsen & Chuang, more precisely Box 8.5, there is an example of quantum process tomography for a single qubit. (The same ...
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What is the thought process for circuit making after seeing input and output of a matrix?
Here is an exercise (4.27) from Nielsen and Chuang and I found the answer (given in the figure below) online without any explanation. The question was to construct a circuit by seeing a matrix (given ...
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Question about Nielson & Chuang Problem 9.2
I am working on the following problem from the book "Quantum Computation and Quantum Information" by Nielsen and Chuang.
Problem 9.2: Let $\mathcal{E}$ be a trace-preserving quantum
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Quantum Process Tomography for 2 qubits
I need clarification on a few aspects related to Box 8.5 and Exercise 8.34 from the book Quantum Computation and Quantum Information by Nielsen & Chuang .
While attempting Exercise 8.34, I ...
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Grover's algorithm for multiple solutions complexity
I'm reading Nielsen&Chuang book (for myself) and I'm completely stuck with one of the problems, 6.3(Database retrieval):
Given a quantum oracle which returns $\left|{k, y \bigoplus X(k)}\right>$...
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DFT like operation in the third step of Period finding and Discrete Logarithm algorithm
In the third step of the algorithm for discrete logarithm, the state
$$
|\hat{f}(l_1,l_2)\rangle=\frac{1}{\sqrt{r}}\sum_{j=0}^{r-1}e^{-2\pi il_2j/r}|{f}(0,j)\rangle
$$
is introduced which is stated to ...
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Question regarding Quantum Phase Estimation (Nielsen and Chuang exercise 5.8)
I was working through Nielsen and Chuang's book on quantum computing and they state the following result regarding the performance of the Quantum Phase Estimation algorithm,
"... given the input $...
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What we get when measure $|0\rangle$ under computational basis?
It is said if we have been given the state $|0\rangle$, the measurement will yield $0$ with probability $1$ in Nielsen's book.
So here, the measurement will yield $0$ refers to we will get state $|0\...
glS♦
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Unital channel which is not mixed unitary
How to prove that for a multi-qubit system a unital channel is not necessarily mixed unitary? This is Problem 8.3 in Nielsen and Chuang. Here's a snippet of the text:
Shall I need to take two ...
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Explanation of the 2.60 equation page 76 in the Nielsen and Chuang [duplicate]
In the Nielsen and Chuang book page 76, equation 2.60 says that we can rewrite the trace $$Tr(A \left|\psi\right>\left<\psi\right|)$$ as follow :
$$Tr(A \left|\psi\right>\left<\psi\right|) ...
glS♦
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Exercise 4.16 in the Nielsen & Chuang book
In the 4.16 exercice in the Quantum Computation and Quantum Information (Michael A. Nielsen & Isaac L. Chuang), I don't understand why the correct answer is not this matrix :
$$
\left[ {\begin{...
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Unitarity of a matrix in the EPR experiment
I'm having troubles in understanding a statement in Box 2.7 at page 113 in the Nielsen & Chuang.
Firstly, it assumed to be working with a two-qubits quantum system in state $|\psi\rangle = \frac{|...
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Implementing "computable phase shifts" using T Toffoli ( problem 4.1 from Nielsen and Chuang's)
I am reading/studying the famous Nielsen and Chuang's book and ran into this interesting question and I don't quite understand the $f(x)$. It says it simply maps from $m$ to $n$ bits. But don't we ...
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Proof of the 4.11 exercise in the Nielsen & Chuang book
In question 4.11 in Nielsen and Chuang's book, it states that there is a formula to describe any unitary matrix $U$ with two vectors $\vec{n}$ and $\vec{m}$ in the following way:
$$U=\exp(i \alpha) ...
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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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In what sense is $\langle\psi|\rho|\psi\rangle$ the overlap between $|\psi\rangle$ and $\rho$?
The fidelity between a pure state $|\psi\rangle$ and an arbitrary mixed state $\rho$ is given by, $F(|\psi\rangle,\rho)=\sqrt{\langle\psi|\rho|\psi\rangle}$, which is stated to be equal to the square ...
glS♦
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Prove that classical counting requires $k=\Omega(N)$ oracle calls
Consider a classical algorithm for the counting problem which samples uniformly and
independently $k$ times from the search space, and let $X_1, ... ,X_k$ be the results of
the oracle calls, that is, $...
glS♦
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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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Can a density operator be written equivalently as $\rho=\sum_i p_i|\psi_i〉\!\langle\psi_i|$ and $\rho=\sum_i\lambda_i|\psi_i\rangle\!\langle\psi_i|$?
My doubt arises from page 99, 101 of the book Quantum Computation and Quantum Information by Michael A.Nielson and Issac L.Chung.
Let {${p_{i}, | \psi_{i} \rangle }$} be an ensemble of pure states.
...
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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 ...
glS♦
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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 ...
glS♦
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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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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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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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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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Why is dual-rail encoding called an error-detecting code for amplitude damping?
Exercise 8.23 : Suppose that a single qubit state is represented by using two qubits, as $|\psi\rangle=a|01\rangle+b|10\rangle$. Show that $\mathcal{E}_{AD}\otimes\mathcal{E}_{AD}$ applied to this ...
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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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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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Show that any measurement where the measurement operators and the POVM elements coincide is a projective measurement
The following question is exercise 2.62 from Nielsen and Chuang's "Quantum Computation and Quantum Information"
Show that any measurement where the measurement operators and the POVM ...
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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) := \...
glS♦
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Verify that if $A$ and $B$ are diagonal in the same orthonormal basis, then $[A,B]=0$
This is from Nielson and Chuang's textbook "Quantum Computation and Quantum Information".
They state the Simultaneous Diagonalisation Theorem: Suppose $A$ and $B$ are Hermitian operators. ...
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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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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 $0 ≤ t ≤ 1$,
$$
R_1^t R_2^{1−t}...
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Why can Shor code fix arbitrary errors?
This is taken from Page 434 of Nielsen and Chuang:
To simplify the analysis, suppose noise of an arbitrary type is occurring on the first
qubit only; we’ll come back to what happens when noise is ...
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Partial trace over a product of matrices - prove that ${\rm Tr}(\rho^{AB}(\sigma^A\otimes I))={\rm Tr}(\rho^A\sigma^A)$
$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$
I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
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Verification for calculation on Shor's code
Here I have tried to determine the end result for the qubit states, when we apply an arbitrary gate on the first qubit in the 9 qubit code.
I have followed this diagram:
U's operation on a qubit can ...
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In quantum error correction, what does an "arbitrary error that yields an un-normalized state" mean?
This is from page 434 of Nielsen and Chuang:
. Supposing the state of the encoded qubit is |ψ⟩
before the noise acts, then after the noise has acted the state is
E(|ψ⟩⟨ψ|). To analyze the effects of
...
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Non trace-preserving map in axiomatic approach to quantum operations
In Nielsen and Chuang's Quantum Computation and Quantum information there is an axiomatic definition of the quantum operation (as one of the 3 approaches to quantum operations).
A quantum operation is ...
glS♦
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Clarification regarding application of distributive property in "quantum teleportation" example
For context, this is from Page 27 of Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press:
She then sends the ...
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How to show that the three-qubit repetition code only corrects up to 1-bit flip errors?
From Nielsen and Chuang, the error correction criteria is $$P E_i^{\dagger} E_j P=\alpha_{i j} P$$ $P$ is the projector onto the correct codespace, $E_{j}$ are error operations and $\alpha_{i j} $ is ...
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How are quantum error-correction conditions in Nielsen and Chuang implemented in practice?
Quantum error-correction conditions in Nielsen and Chuang, 10th-anniversary edition (Theorem 10.1) state that the error operation $\mathcal{E}$ with operation elements $\{E_i\}$ is correctable if and ...
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Unambiguous discrimination using POVM with highest discriminate probability
I was studying Nielsen&Chuang's textbook (about page 92), and come up with a question that I cannot solve it.
Given one of the two state $|\psi_1\rangle=|0\rangle$ and $|\psi_2\rangle=\frac{1}{\...
glS♦
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How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?
The proof of the Fannes' inequality replies on the formula $T(ρ, σ)≥\sum_i|r_i − s_i|$, where $r_i,s_i$ are the eigenvalues of $\rho,\sigma$, in the descending order.
In the proof given in Box 11.2, ...
glS♦
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Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?
In Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says:
A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i,
The ...
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Do the parity matrices of $\text{CSS}(C_1, C_2)$ and $\text{CSS}_{u,v}(C_1, C_2)$ need to be same and the Number of Distinct Equivalent Descriptions?
This question relates to Exercise 10.27 of Nielsen & Chuang, which says:
Show that the codes defined by
$$| x + C_2 \rangle = \frac{1}{\sqrt{C_2}} \sum\limits_{y \in C_2} (-1)^{u \cdot y} | x+y+v ...
CommunityBot
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Register size in factoring 15 using Shor's algorithm
In Nielsen and Chuang's book: Quantum computation and quantum information (2016), there is an example in Box 5.4 which shows how to factor $15$ using Shor's algorithm. I am confused about a ...
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Proof of the optimality of Grover's algorithm
I am currently working on the proof of Grover's algorithm, which states that the runtime is optimal.
In Nielsen they say, the idea is to check whether $D_k$ is restricted and does not grow faster than ...
glS♦
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Derive the Concavity of Quantum Conditional Entropy from Strong subadditivity
In Exercise 11.25, Page 522, Entropy and information, Quantum Computation and Quantum Information by Nielsen and Chuang, it is required to show that the concavity of the conditional entropy may be ...
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