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.
348
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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 ...
- 1,161
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votes
1
answer
543
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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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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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votes
0
answers
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Prove the equality conditions in the triangle inequality $S(A,B)\ge |S(A)-S(B)|$ for the von Neumann entropy
The triangle inequality or Araki-Lieb inequality of the von Neumann entropy is
$$
S(A,B)\ge|S(A)-S(B)|
$$
this is proven by introducing a system $R$ which purifies systems $A$ and $B$. Applying ...
- 831
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votes
0
answers
116
views
Proof of upper and lower bound (Gilbert-Varshamov bound) for linear code
I am trying to prove the following bounds for a $[n, k]$ code that can correct $t$ errors
\begin{align}
1-H\left(\frac{t}{n}\right)\geq \frac{k}{n}\geq 1-H\left(\frac{2t}{n}\right)
\end{align}
where
\...
- 975
4
votes
0
answers
67
views
What does the $I$ mean when measuring ${\rm Tr}(\rho (I\otimes\sigma\otimes\cdots))$ in quantum tomography?
In Nielsen and Chuang's QCQI, I learned that the quantum tomography for n qubit can be described easily in math as we need to measure $Tr(\rho W_k),\forall k$ where $W_k\in\{I,\sigma_x,\sigma_y,\...
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Why is Hilbert space considered especially large?
In Nielsen & Chuang section 1.2 introduces multiple qubits and Hilbert spaces.
More generally, we may consider a system of n qubits. The computational basis states
of this system are of the ...
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votes
4
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116
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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 ...
- 539
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591
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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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3
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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 ...
- 85
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3
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Single-qubit quantum channel from the CNOT gate
I am studying quantum noise, chapter $8$ in Nielsen and Chuang. Section $8.2.2$ introduces an example for the definition of quantum operations, in particular the CX gate is introduced as an example. I ...
- 31
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1
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views
How can I prove inequality from 4.66 to 4.67 in Nielsen and Chuang's book?
I am reading chapter 4 of Nielsen 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|+|...
3
votes
1
answer
161
views
Question about the phase kickback in the phase estimation algorithm [duplicate]
I have an issue with the quantum phase estimation algorithm as explained Nielsen and Chuang.
There was a question very similar to mine asked about this 2 years ago, but my question is different... ...
3
votes
2
answers
335
views
Output of Quantum Phase Estimation Algorithm
In section 5.2.1 of Nielsen Chuang, Performance and Requirements, there is an idea, that what happens if we can't prepare eigen state $|u\rangle$ and instead have a state $|\psi\rangle$ which is ...
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votes
2
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views
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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2
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568
views
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
...
- 437
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votes
2
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376
views
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 ...
- 485
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2
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219
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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 ...
- 485
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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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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 ...
- 197
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1
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329
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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?
- 75
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votes
2
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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 ...
- 75
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votes
2
answers
183
views
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 ...
- 1,646
3
votes
1
answer
141
views
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)\...
- 197
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1
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Heisenberg Uncertainty Principle (Nielsen and Chuang Box 2.4)
I'm trying to follow Nielsen and Chuang Book on Quantum Computation and Quantum Information. There is Box 2.4 on the Heisenberg Uncertainty Principle. I got stuck pretty fast. In that box they define:
...
- 177
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1
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Probability of success proof for Shor's algorithm
In the book "Quantum Computation and Information" by Nielsen and Chuang, Shor's algorithm is presented with a related probability of success theorem and proof found on page 634, Theorem A4....
3
votes
2
answers
159
views
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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1
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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, ...
3
votes
1
answer
490
views
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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1
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views
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(\...
- 2,408
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votes
1
answer
398
views
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 ...
3
votes
2
answers
229
views
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 ...
- 197
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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 ...
- 33
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votes
1
answer
496
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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=\...
- 831
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1
answer
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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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1
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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 ...
3
votes
1
answer
434
views
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|...
- 3,019
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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 ...
- 361
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1
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237
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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\...
- 160
3
votes
2
answers
194
views
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 ...
- 485
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votes
1
answer
195
views
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 ...
- 1,876
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votes
1
answer
123
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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 ...
- 101
3
votes
1
answer
315
views
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}...
- 471
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votes
1
answer
228
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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
3
votes
1
answer
390
views
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