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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Connection between stabilizer generators and parity check matrices in the Steane code
I'm working through Mike and Ike (Nielsen and Chuang) for self-study, and I'm reading about stabilizer codes in Chapter 10. I'm an electrical engineer with somewhat of a background in classical ...
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Simulate hamiltonian evolution
I'm trying to figure out how to simulate the evolution of qubits under the interaction of Hamiltonians with terms written as a tensor product of Pauli matrices in a quantum computer. I have found the ...
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What is the difference between "code space", "code word" and "stabilizer code"?
I keep reading (e.g. Nielsen and Chuang, 2010; pg. 456 and 465) the following three phases; "code space", "code word" and "stabilizer code" - but am having a difficult time finding definitions of them ...
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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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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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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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How can we be sure that for every $A$, $A^\dagger A$ has a positive square root?
In the Polar Decomposition section in Nielsen and Chuang (page 78 in the 2002 edition),
there is a claim that any matrix $A$ will have a decomposition $UJ$ where $J$ is positive and is equal to $\sqrt{...
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What's the 'physical consistency' in the partial trace scenario?
I'm reading 'Why the partial trace' section on page 107 in Nielsen and Chuang textbook. Here's part of their explanations that I don't quite understand:
Physical consistency requires that any ...
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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 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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Nielsen and Chuang's proof for 'approximating arbitrary unitary gates is generically hard'
The following statement is found on the page 199 of Nielsen and Chuang's book (10th Anniversary Edition) in the proof for the fact that 'approximating arbitrary unitary gates is generically hard':
...
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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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Nielsen and Chuang, Exercise 6.5: How to simulate oracle for n+1 qubits using one oracle gate for n qubits and one extra qubit?
In Chapter 6 of "Quantum Computation and Quantum Information" textbook by Nielsen and Chuang, Exercise 6.5 p.255:
We have an oracle gate $O$ for $n$ qubit ($2^n=N$ searching items),
and we would ...
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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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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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Is there any 'official' list of errata for Nielsen & Chuang?
The book Quantum Computing and Quantum Information by Nielsen and Chuang is a well-known and celebrated text book that can act as a resource in a wide variety of topics. Of course, in such a vast ...
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Quantum channel representation of projective measurement
Let $P$ be a projector and $Q = I-P$ be its complement. How to find probability $p$ and unitaries $U_1, U_2$ such that for any $\rho$, $P\rho P + Q\rho Q = p U_1\rho U_1^\dagger + (1-p)U_2\rho U_2^\...
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Constructing a circuit for $C^1(U)$ for rotation operators with TWO single qubit gates and CNOT gate
This is the exercise 4.23 from Nielsen and Chuang, asking that if it is possible to construct $C^1(U)$ for $U=R_{x,y}(\theta)$ with TWO single qubit gates and CNOT gate. My answer is no, and I would ...
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How do we find the stabilizer generators for the three-qubit bit-flip code spanned by $|000\rangle$ and $|111\rangle$?
In Nielsen & Chuang's book "Quantum Computation and Quantum Information" section 10.5.6, page 467 there is the following statement
Consider the familiar three-qubit bit-flip code ...
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Uncomputation in quantum implementation of a classical algorithm
In Nielsens and Chuangs book, they present a way to implement a reversible version of any classical algorithm (section 3.2.5).
In short, they use Fredkin and other simple reversible gates to ...
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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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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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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 ...
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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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Derive phase damping quantum operation
I am reading about the phase damping quantum operation on page 384 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition).
Nielsen & Chuang derived the ...
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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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Why does a quantum operation being trace-preserving imply that $\sum_k E_k^\dagger E_k=I$?
I am reading Nielsen Chuang Chapter 8. They say that if a quantum operation is trace-preserving, then
\begin{equation}
Tr\left(\sum_k E_k^{\dagger}E_k \rho\right) = 1
\end{equation}
which I understand....
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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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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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Confusion on the definition of the phase-damping channel
I am reading about the phase damping channel, and I have seen that some of the different references talking about such channel give different definitions of the Kraus operators that define the action ...
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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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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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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:
$$
F(\rho, \sigma) := \...
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Understanding Steps in Deutsch's Algorithm
I am currently working my way through the book Quantum Computation and Quantum Information by Chuang and Nielsen. So far it has been a joy to read, however I am hung up on a couple aspects of quantum ...
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Error syndromes and recovery procedure in bit flip code
This question relates to exercise 10.4 in Nielsen and Chuang.
For syndrome diagnosis, the textbook provides an example where one has four projectors, by which, you can identify where a one qubit ...
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Problem 2.2 in Nielsen & Chuang - Properties of the Schmidt number
This question has been asked here: "Problem 2.2 in Nielsen & Chuang - Properties of the Schmidt number",
but no answer has been provided there yet, thus I move it here.
The problem is stated ...
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In Nielsen and Chuang, how can $\frac{1}{2(e-1)}$ result from $\frac12\int_{e-1}^{2^{t-1}-1}dl\frac{1}{l^2}$?
From Nielsen and Chuang's book: $\textit{Quantum computation and quantum information}$, how can (5.34) equal (5.33)? I.e.
$$\dfrac{1}{2} \int_{e-1}^{2^{t-1}-1} dl \dfrac{1}{l^2} = \dfrac{1}{2(e-1)}.$$...
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Phase estimation algorithm: probability bound of obtaining $m$
Note: Cross-posted on Physics SE.
Hi, I'm studying the quantum phase estimation algorithm from this book: M.A. Nielsen, I.L. Chuang, "Quantum Computation and Quantum Information", Cambridge Univ. ...
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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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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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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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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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Why does quantum distinguishability ensure no faster-than-light communication?
On page 56-57 in Nielsen and Chuang, for a proposed scenario, it's said that:
if Bob had access to a device that could distinguish the four states $|0\rangle$, $|1\rangle$, $|+\rangle$, $|−\rangle$ ...
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Find the quantum operation corresponding to a given unitary evolution and projective measurement
I'm trying to (understand and) solve this problem from Nielsen and Chuang's Quantum Computation and Quantum Information.
I know the definition of Operation Elements: $\sum_{k} E_k \rho E_k^†$ with $...
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Can quantum circuits/operations have truth tables?
In the caption for the following figure, the word "truth table" is put inside a quotation. I am wondering if this means that the truth table the caption refers to isn't exactly a real truth ...
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How is the partial trace related to the operator sum representation? [duplicate]
In Quantum Computation and Quantum Information by Nielsen and Chuang, the authors introduce operator sum representation in Section 8.2.3. They denote the evolution of a density matrix, when given an ...
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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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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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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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