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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3
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2answers
64 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 ...
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2answers
118 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\...
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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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1answer
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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 ...
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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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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
38 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
68 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 ...
5
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1answer
51 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
48 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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58 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
79 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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45 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 \...
5
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2answers
56 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
57 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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96 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
62 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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105 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
47 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
65 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. $$ \mathrm{tr}(AU) = |\mathrm{tr}(|A|VU)| = |\mathrm{tr}(|A|^{1/2}|A|^{1/2}VU)| $$ ...
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1answer
135 views

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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1answer
57 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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54 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
97 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 ...
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1answer
63 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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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 ...
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2answers
107 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: $...
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1answer
185 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 ...
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54 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 ...
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1answer
124 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. ...
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1answer
145 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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220 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 ...
2
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1answer
106 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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363 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
100 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 ...
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Nielsen & Chuang Exercise 2.2 - Matrix representations in different input and output basis [duplicate]

This is a question in the Nielsen and Chuang textbook (Exercise 2.2). Suppose $V$ is a vector space with basis $|0\rangle$ and $|1\rangle$ and $A$ is a linear operator from $V \to V$ such that $...
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1answer
162 views

Measurement of a qubit and storage of the information on a bit

Suppose we have the quantum circuit below with a quantum register of 2 qubits and a classical register of 2 bits. The Hadamard gates and CNOT gate are not important for the question. When we measure a ...
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What do you specify when you physically apply a unitary?

In the Environment and Quantum Operations in Nielsen and Chuang, section 8.2.2, they say that when you apply a unitary on a state, you expect the output as the just the state transformed by the ...
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2answers
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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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1answer
168 views

FANOUT with Toffoli Gate

Figure 1.16: FANOUT with the Toffoli gate, with the second bit being the input to the FANOUT (and the other two bits standard ancilla states), and the output from the FANOUT appearing on the second ...
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1answer
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Default input states of qubits to quantum circuits [closed]

It is conventional to assume that the state input to the circuit is a computational basis state, usually the state consisting of all $|0\rangle$s. This rule is broken frequently in the literature ...
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2answers
535 views

Partial trace over a product of matrices - one factor is in tensor product form

$$Tr(\rho^{AB} (\sigma^A \otimes I/d)) = Tr(\rho^A \sigma^A)$$ 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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1answer
170 views

Partial Trace over a complicated looking state

In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation: $$\varepsilon(\rho) = tr_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$ They ...
4
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1answer
199 views

Construction of ${R_n(\theta)}$ using only the Hadamard and ${\pi/8}$ gates

In the "Quantum Computation and Quantum Information 10th Anniversary textbook by Nielsen & Chuang", they claim that Eqn(4.75) is a rotation about the axis along the direction ( ${cos(\pi/8)}$, ${...
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1answer
182 views

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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3answers
342 views

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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324 views

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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136 views

Nielsen & Chuang Exercise 2.5 - Inner products of complex vectors [closed]

Reproduced from Exercise 2.5 of Nielsen & Chuang’s Quantum Computation and Quantum Information (10th Anniversary Edition): A function $(\cdot, \cdot)$ from $V × V$ to $C$ is an inner product if ...