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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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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1answer
81 views

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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How can I check if the following are possible states of a qubit?

I would like to check if state $$ |\psi\rangle = \cos{\theta}|0\rangle + i \sin{\theta}|1\rangle $$ is properly defined. But when I calculated $\langle \psi | \psi \rangle$ is get $$ \cos^2{\theta}...
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200 views

What kind of errors does the master equation in the Lindblad form describe, continuous errors or discrete errors?

I noticed that in page 427 in Nielsen & Chuang's book Quantum Computation and Quantum Information, quantum error correction is possible because errors can be discretized. In other hand, the ...
5
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1answer
137 views

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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1answer
66 views

Qubit measurement of the state $\frac{1}{\sqrt2}|00\rangle+\frac{i}{2}|01\rangle-\frac{1}{2}|11\rangle$

If we measure the first qubit and obtain $|0\rangle$, what does the second qubit collapses to? $$ \left| \varphi \right>=\frac{1}{\sqrt{2}} \left| 00 \right> + {\frac{i}{2}} \left| 01\right> ...
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95 views

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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2answers
141 views

How can I find the $\theta$ and $\phi$ values of a qubit on the Bloch sphere?

If I have the following state: $$ \left| \varphi \right>=\frac{1}{\sqrt{2}}\left(\left(\frac{1+i}{\sqrt{2}} \right)\left| 0 \right> + \left| 1\right>\right) $$ How can I find the $\theta$ ...
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1answer
79 views

Understanding how unitary operation changes state of system

While reading "Quantum Computation and Quantum Information" (by Nielsen and Chuang) I came across this line A little thought shows that if we apply $U_f$ to the state $\vert x \rangle (\dfrac{\...
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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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1answer
50 views

Quantum adder mod 4 [closed]

Does anyone has (or is aware of) a solution to Nielsen's QCQI 4.36 exercise: construct a circuit to add two-qubit numbers x and y modulo 4 vs $|x,y\rangle \rightarrow |x, x+y \mod 4\rangle$
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115 views

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\...
3
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107 views

How is partial trace related to operator sum representation?

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 ...
2
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2answers
127 views

POVM number of measurement

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\...
3
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2answers
106 views

Premise of the proof of the No-Cloning Theorem

I have seen two similar proofs of the no-cloning theorem. They assume (to the contrary) that there exists a unitary operator $U$ such that $U |\psi\rangle |0 \rangle = | \psi \rangle | \psi \rangle$, ...
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3answers
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+100

Atom magnetic moment caused by orbiting electron

In Nielsen and Chuang Quantum Computation and Quantum Information book section 1.5.1 describing the Stern-Garlach experiment, it says: "Hydrogen atoms contain a proton and an orbiting electron. You ...
4
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2answers
116 views

Nielsen & Chuang Exercise Question on CSS code

I was reading the CSS ( Steane Code) from the Nielsen & Chuang book. It asked in Ex. 10.27 to prove that: suppose $C_1$ and $C_2$ are $[n,k_1]$ and $[n,k_2]$classical linear codes such that $C_2\...
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0answers
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Clarification of bra-ket notation [duplicate]

How do I get from equation 1.31 to equation 1.32? It seems like some terms are changing.
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2answers
42 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 ...
2
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1answer
74 views

Can someone show the linear algebra calculations for X, H, and CNOT gates?

I am on Ch.1 of the Mike & Ike book. On page 18, the text shows an X gate that essentially flips the $\alpha$ and $\beta$ amplitudes. The text shows the $X$ matrix but it doesn't show those for ...
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1answer
53 views

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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1answer
53 views

Exercise 10.34 of Nielsen and Chuang : $-I$ not element of stabilizer group iff $g_j^2=I$ and $g_j \neq I$ where $g_j$ generators

I am stuck with this exercice of Nielsen and Chuang: Let $S = \langle g1,... ,gl \rangle $.Show that $−I$ is not an element of S if and only if $g^2_j = I$ for all $j$,and $g_j \neq − I$ for all $j$...
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Help with understanding Unitary Operator of Quantum Optical Fredkin Gate

I'm referring to the textbook "Quantum Computation and Quantum Information" 10th Anniversary Edition by Nielsen and Chuang. Chapter 7.4 has a Box 7.4 which introduces the Quantum Optical Fredkin Gate....
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62 views

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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1answer
94 views

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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1answer
93 views

Amplitude Damping of a Harmonic Oscillator

Exercise 8.21 of Nielsen and Chuang asks us to show that the operation elements for a harmonic oscillator (system) coupled to another harmonic oscillator (environment) is $E_k = \sum_n \sqrt{(^n_k)}\...
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47 views

Operation Elements for Amplitude Damping Channel

To find operation elements for the Amplitude Damping channel, Nielsen and Chuang (in Section 8.3.5 of my copy) use the action of a beamsplitter on an initial state $ \alpha |0\rangle + \beta |1\rangle$...
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1answer
24 views

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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2answers
66 views

Operation Elements in Operator-sum Representation

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 $...
2
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1answer
86 views

Hamiltonian for Single-photon, Single-atom QED Cavity

Equation 7.71 of Nielsen and Chuang's Quantum Computation and Quantum Information gives the Hamiltonian for a two level atom and single mode photons in a cavity as: $H = \hbarωN + δZ + g(a^†σ_− + aσ_+...
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101 views

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 ...
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1answer
74 views

Construction of Phase Shift Operation used in Quantum Search

In Chapter 6 of "Quantum Computation and Quantum Information" Textbook by Nielsen and Chuang, Box 6.1 gives a circuit example of Quantum Search Algorithm done on a two-bit sized search space. The ...
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2answers
109 views

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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1answer
55 views

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 ...
2
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1answer
70 views

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. $$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$ ...
5
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1answer
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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': ...
2
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1answer
59 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 ...
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2answers
57 views

Proof of joint entropy theorem

From section 11.3.2 of Nielsen & Chuang: (4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\...
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1answer
104 views

Grover's algorithm and RSA from Nielsen

Nilsen states that one can define a function for the oracle in the Grover algorithm, which is constructed as follows. So there is a number $m$ that consists of $p$ and $q$ (both primes) $m = pq$. Now ...
2
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1answer
64 views

Eigenstate of unitary operator used for Order-Finding

In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen and Chuang", chapter 5.3.1 introduces the concept of solving the Order-Finding Problem. (Eqn 5.36) states ...
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74 views

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 ...
3
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2answers
115 views

In what sense do repeated applications of Grover's operator rotate the state closer to the target?

I'm studying the quantum search algorithm on this book: M.A. Nielsen, I.L. Chuang, "Quantum Computation and Quantum Information", Cambridge Univ. Press (2000) [~p. 252]. To sum up we have a state: $...
5
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1answer
198 views

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 ...
3
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1answer
95 views

N&C quantum circuit for Grover's algorithm

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 ...
6
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1answer
133 views

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. ...
2
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1answer
189 views

Clarification needed for the N&C proof that BQP ⊆ PSPACE

In QCQI by Chuang and Nielsen (page 201), they prove that $\mathsf{BQP} \subseteq \mathsf{PSPACE}$. I can't understand what they say. They write "Supposing the quantum circuit starts in the state $...
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2answers
271 views

Why isn't the circuit performing a measurement in the Bell basis?

Nielsen and Chuang (on page 188 exercise 4.33) says that the circuit including CNOT and Hadamard is performing a measurement in the Bell basis. But I can't see how. The matrix representing the ...
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1answer
113 views

If CNOTs and single qubit gates are universal then why do we need to prove that controlled U operations can be composed by them as well?

In the book by Chuang and Nielsen they prove that controlled U operations can be made out of CNOTs and single qubit gates. But then they go on to prove that they are universal by showing that every n ...
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467 views

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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1answer
104 views

Why does $|P_U − P_V |$ equal $\langle \psi |U^{\dagger} M U|\psi\rangle −\langle \psi |V^{\dagger} M V |\psi\rangle$?

In QC and QI by Chuang and Nielsen, they state that the $P_U$ of operation $U$ acting on $\psi$ can be reached by $\langle \psi |U^{\dagger} M U |\psi\rangle$. Where $P_U$ (or $P_V$) is the ...