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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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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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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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 ...
CommunityBot
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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.
...
glS♦
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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 $...
glS♦
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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 ...
CommunityBot
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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 ...
CommunityBot
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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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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 ...
CommunityBot
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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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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
$$
is introduced which is stated to ...
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Question regarding Quantum Phase Estimation (Nielsen and Chuang exercise 5.8)
I was working through Nielsen and Chuang's book on quantum computing and they state the following result regarding the performance of the Quantum Phase Estimation algorithm,
"... given the input $...
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Show that any measurement where the measurement operators and the POVM elements coincide is a projective measurement
The following question is exercise 2.62 from Nielsen and Chuang's "Quantum Computation and Quantum Information"
Show that any measurement where the measurement operators and the POVM ...
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Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?
I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following:
$$
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)$$
I came across the above, but I'm not sure how it's true. I figured they first partial traced out the B subsystem, and then trace A, but ...
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Verification for calculation on Shor's code
Here I have tried to determine the end result for the qubit states, when we apply an arbitrary gate on the first qubit in the 9 qubit code.
I have followed this diagram:
U's operation on a qubit can ...
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In quantum error correction, what does an "arbitrary error that yields an un-normalized state" mean?
This is from page 434 of Nielsen and Chuang:
. Supposing the state of the encoded qubit is |ψ⟩
before the noise acts, then after the noise has acted the state is
E(|ψ⟩⟨ψ|). To analyze the effects of
...
glS♦
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Non trace-preserving map in axiomatic approach to quantum operations
In Nielsen and Chuang's Quantum Computation and Quantum information there is an axiomatic definition of the quantum operation (as one of the 3 approaches to quantum operations).
A quantum operation is ...
glS♦
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Clarification regarding application of distributive property in "quantum teleportation" example
For context, this is from Page 27 of Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press:
She then sends the ...
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How to show that the three-qubit repetition code only corrects up to 1-bit flip errors?
From Nielsen and Chuang, the error correction criteria is $$P E_i^{\dagger} E_j P=\alpha_{i j} P$$ $P$ is the projector onto the correct codespace, $E_{j}$ are error operations and $\alpha_{i j} $ is ...
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How are quantum error-correction conditions in Nielsen and Chuang implemented in practice?
Quantum error-correction conditions in Nielsen and Chuang, 10th-anniversary edition (Theorem 10.1) state that the error operation $\mathcal{E}$ with operation elements $\{E_i\}$ is correctable if and ...
glS♦
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Unambiguous discrimination using POVM with highest discriminate probability
I was studying Nielsen&Chuang's textbook (about page 92), and come up with a question that I cannot solve it.
Given one of the two state $|\psi_1\rangle=|0\rangle$ and $|\psi_2\rangle=\frac{1}{\...
glS♦
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How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?
The proof of the Fannes' inequality replies on the formula $T(ρ, σ)≥\sum_i|r_i − s_i|$, where $r_i,s_i$ are the eigenvalues of $\rho,\sigma$, in the descending order.
In the proof given in Box 11.2, ...
glS♦
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Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?
In Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says:
A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i,
The ...
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Do the parity matrices of $\text{CSS}(C_1, C_2)$ and $\text{CSS}_{u,v}(C_1, C_2)$ need to be same and the Number of Distinct Equivalent Descriptions?
This question relates to Exercise 10.27 of Nielsen & Chuang, which says:
Show that the codes defined by
$$| 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 ...
CommunityBot
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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 ...
glS♦
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Nielsen Chang exercise 4.10
I' m trying to figure out how to decompose unitary operator U using only Rx and Ry rotations. I understand that I must use result of exercise 4.8 ($ U = e^{i\alpha}R_n(\theta)$). But I don't ...
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Derivation of efficiency of Phase Estimation Algorithm
In the section Performance and requirements of the phase estimation algorithm of Page 224, Quantum Computation and Quantum Information by Nielsen and Chuang
Let $b$ be the integer in the range $0$ to ...
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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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Derivations of Equations (12.205 - 12.207) in Nielsen Chuang
While proving the security of the BB84 protocol, Nielsen and Chuang demonstrate that it is possible to reduce the CSS protocol to the secure BB84 protocol without requiring Alice to reveal the value ...
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How to derive the number of independent parameters in the $\chi$ matrix from the Choi matrix?
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 $d^4−d^2$ ...
glS♦
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Derivation of Equation 10.69 in Nielsen Chuang
In the proof of correctness of CSS codes in Nielsen and Chuang, we see that equation $(10.68)$:
$$
\frac{1}{\sqrt{|C_2|}} \sum\limits_{y \in C_2} (-1)^{(x+y)\cdot e_2} |x+y \rangle
$$
can be ...
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Calculation of $\frac{d}{dt} I_t(A,X)$ in proving the convexity of the relative entropy via Lieb's theorem
In Page 520, Entropy and information, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that
The relative entropy $S(ρ||σ)$ is jointly convex in its arguments, where $S(ρ|...
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What do we mean by family of CSS codes?
In proving the security of BB84 in Nielsen and Chuang (10th anniversary edition - Section 12.6.5), they argue that a codeword in $\text{CSS}(C_1, C_2)$ is represented by
$$\frac{1}{\sqrt{|C_2|}} \sum\...
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How to extract probabilities from Kraus representation?
Consider a quantum operation described by Kraus operators $K_1, ..., K_n$. As I understand the effect of this operation on a density matrix $\rho$ can be described as $ \mathcal{E}(\rho)= \sum_{i}p(i)\...
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In quantum process tomography for one and two qubit, why can we express the $\chi$ matrix in this form?
I'm reading Nielsen and Chuang and I read quantum tomography process given by N&C (box 8.5), which provides an algorithm for determining $\chi$ in terms of block matrices and density matrices. And ...
glS♦
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Proving the inequality $|\mathrm{tr}(AU)|\le \mathrm{tr}|A|$ in Uhlmann's theorem
In Nielsen and Chuang, in the Fidelity section, (Lemma 9.5, page 410 in the 2002 edition), they prove the following.
$$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$
...
glS♦
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The matrix norm $||A||=\max_{\langle u|u\rangle=1}|\langle u|A|u\rangle|$ in the proof of Lieb's theorem
In Exercise A6.4, Appendix 6: Proof of Lieb’s theorem, Page 645, Quantum Computation and Quantum Information by Nielsen and Chuang, A matrix norm of $A$ is defined as $$||A||=\max_{\langle u|u\rangle=...
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