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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Definition(s) of $\delta$ in quantum phase estimation

I read the chapter on QPE (quantum phase estimation) in Nielsen and noticed that $\delta$ is defined there as follows: $0 \leq \delta \leq 2^{-t}$, see: 5.2.1 Performance and requirements The above ...
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49 views

Factoring Decision Problem - why not in P? [closed]

Nielsen and Chuang, 10th Anniversary Edition, page 142, refers to the following (classical) computation problem: Given a composite integer m and L <m, does m have a non-trivial factor less than L? ...
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72 views

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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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 In order to obtain Eq. 5.27 we have ...
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76 views

Equating the state of the Phase Estimation algorithm to $\frac{1}{2^{t/2}}\sum_{k=0}^{2^t-1} e^{2\pi i\phi k}|k\rangle$

It is stated in the Phase Estimation algorithm in Page 222, Quantum Computation and Quantum Information by Nielsen and Chuang that It seems to say that taking the inverse Quantum Fourier transform of ...
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58 views

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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35 views

Making sense of the terms Polynomial and Exponential Precision in a Quantum Circuit

The quantum circuit construction of the quantum Fourier transform apparently requires gates of exponential precision in the number of qubits used. However, such precision is never required in any ...
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39 views

What is meant by a "projection operator" in the book "Quantum Computation and Quantum Information"?

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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45 views

What is the physical process causing a Bell state to be shared? [duplicate]

Nielsen and Chuang, QCQI, page 57, last paragraph, says "suppose Alice and Bob share between them a Bell state." I know how to prepare a Bell state, but what would be the physical process ...
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34 views

Can every unitary on $\mathcal{H}\otimes \mathcal{K}$ be modelled by quantum operations on $\mathcal{H}$?

In section 8.2.3 of Nielsen and Chuang, they discuss how unitary dynamics of a system and environment arise from quantum operations (i.e. Kraus operators $E_k$ such that $\sum_k E_k^*E_k=I$). ...
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120 views

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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Phase estimation algorithm: Bounding of probability in Nielsen and Chuang

I am currently studying the Quantum Phase Estimation (QPE) algorithm as described in Nielsen and Chuang, pages 223-224. We have the following situation there, we have the state: $$\frac{1}{2^t} \sum\...
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27 views

prove that $E(U^m_{\Delta t},e^{-2miH\Delta t})\leq m\alpha \Delta t^3$ where $U_{\Delta t}=e^{-2iH\Delta t}+O(\Delta t^3)$

Let $H=\sum_{k=1}^LH_k$ and define $U_{\Delta t}=[e^{-iH_1\Delta t}e^{-iH_2\Delta t}\cdots e^{-iH_L\Delta t}][e^{-iH_L\Delta t}e^{-iH_{L-1}\Delta t}\cdots e^{-iH_1\Delta t}]=e^{-2iH\Delta t}+O(\Delta ...
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58 views

Formulation of quantum phase estimation in Nielsen and Chuang is different then from other sources?

In a chapter of Quantum Computation and Quantum Information by Nielsen and Chuang (10th edition) about quantum phase estimation I get a little confused. Namely: Before applying inverse QFT our quantum ...
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96 views

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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Prove a circuit with controlled $iR_x(\pi \alpha)$ is universal for quantum computation whenever $\alpha$ is irrational

Show that the three qubit gate $G$ defined by the circuit is universal for quantum computation whenever $\alpha$ is irrational. My Observations The unitary gate on the third qubit is activated only ...
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Circuit to show that Hadamard, phase, controlled- and Toffoli gates are universal

Part 1 The final output state is, $|\psi_{out}\rangle=\frac{1}{4}[|00\rangle(3S+XSX)|\psi\rangle+|01\rangle(S-XSX)|\psi\rangle+|10\rangle(S-XSX)|\psi\rangle+|11\rangle(-S+XSX)|\psi\rangle]$ When the ...
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Show that there are unitaries $U_m$ such that $M_m=U_m \sqrt{E_m}$, for any measurement $M_m$ and associated POVM $E_m$

Nielsen and Chuang's QCQI, section 2.2.6, page 92, asks Suppose a measurement is described by measurement operators $M_m$. Show that there exist unitary operators $U_m$ such that $M_m=U_m\sqrt{E_m}$, ...
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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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Prove $E(R_n(\alpha),R_n(\theta)^n)<\epsilon/3$ from $E\big(R_n(\alpha),R_n(\alpha+\beta)\big)=|1-\exp(i\beta/2)|$

where $E(U,V)=\max_{|\psi\rangle}||(U-V)|\psi\rangle ||=||U-V||$ is the error when $V$ is implemented instead of $U$. See page 196, Quantum Computation and Quantum Information by Nielsen and Chuang. I ...
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$E(U_j,V_j)\leq\Delta/(2m)$ if probabilities of outcomes obtained from the approximate circuit is within a tolerance $Δ>0$

Suppose we wish to perform a quantum circuit containing $m$ gates, $U_1$ through $U_m$. Unfortunately, we are only able to approximate the gate $U_j$ by the gate $V_j$ . In order that the ...
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318 views

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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44 views

In Schumacher’s noiseless channel coding theorem, how do we get the exponents in $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\otimes n(1−p)/2}$?

On pg. 55 in Nielsen and Chuang, it's said that: the $|0\rangle + |1\rangle$ product can be well approximated by a superposition of states of the form $|0\rangle ^{\otimes n(1−p)/2}|1\rangle ^{\...
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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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65 views

Measurement of single qubit operator $U$ which is both Hermitian and unitary with eigenvalues $±1$

Suppose we have a single qubit operator $U$ with eigenvalues $±1$, so that $U$ is both Hermitian and unitary, so it can be regarded both as an observable and a quantum gate. Suppose we wish to measure ...
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41 views

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, ...
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Why can the state $\sum_x|x, f(x)\rangle$ be written without normalization factor?

It is stated on page 32 of Nielsen and Chuang that: In our single qubit example: , measurement of the state gives only either |0, f(0)> or |1, f(1)>! Similarly, in the general case, measurement ...
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27 views

Derivation of the state after applying $U_f$ in the Deutsch–Jozsa algorithm

On page 35 in Nielsen and Chuang, it's said that for the following quantum circuit implementing the general Deutsch–Jozsa algorithm: Next, the function $f$ is evaluated (by Bob) using $U_f$, giving $$...
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Why is the first register of $|x,y\oplus f(x)\rangle$ called "data" register?

When talking about quantum parallelism, in Nielsen and Chuang, it's said that: it is possible to transform this state into $|x, y \oplus f(x)\rangle$, where $\oplus$ indicates addition modulo 2; the ...
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32 views

Generalizing quantum parallelism to bits or qubits

On pg. 31 in Nielsen and Chuang, it's said that: This procedure can easily be generalized to functions on an arbitrary number of bits, by using a general operation known as the Hadamard transform, or ...
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39 views

Derivation of the effect of the Hadamard transform on a state |x⟩ in the Deutsch–Jozsa algorithm

On pg. 35 of Nielsen and Chuang, there's the following paragraph: By checking the cases $x=0$ and $x=1$ separately we see that for a single qubit $H|x\rangle=\sum_x (-1)^{xz}|z\rangle/\sqrt{2}$. I'm ...
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265 views

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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105 views

Prove that rank one projectors have the same partial trace iff they differ by a local unitary operation

In nielsen and chuang's QCQI book, there is a theorem called Unitary freedom in the ensemble for density matrices, which states that the sets $|\psi_i\rangle$ and $|\phi_i\rangle$ generate the same ...
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In Nielsen & Chuang, shouldn't $P=\sum_{i=1}^k|i\rangle\!\langle i|$ in (2.35) equal the identity?

Nielsen and Chuang define Projectors as: An operator $A$ whose adjoint is $A$ is known as a Hermitian or self-adjoint operator. An important class of Hermitian operators is the projectors. Suppose $...
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40 views

Why do we want the no error limit to be 1?

In a textbook by Nielsen and Chuang, there's the following paragraph: The idea of quantum data compression is that the compressed data should be recovered with very good fidelity. Think of the ...
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1answer
40 views

Reason for sending numbers from 0 to $2^n − 1$ in Deutsch–Jozsa algorithm

In Nielsen and Chuang, when talking about the Deutsch–Jozsa algorithm. The Deutsch’s problem is described as the following game. Alice, in Amsterdam, selects a number x from 0 to $2^n − 1$, and mails ...
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94 views

How to use the output state $\frac1{\sqrt2}(|0,f(0)\rangle+|1,f(1)\rangle)$ given by this quantum circuit?

In Nielsen and Chuang, for the following circuit that demonstrates quantum parallelism, I have the following question: since the output state of the circuit is $$\frac1{\sqrt2}(|0,f(0)\rangle+|1,f(1)\...
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207 views

How to understand intuitively the concavity of the binary entropy?

In Nielsen and Chuang's Quantum Computation and Quantum Information book, introducing the binary entropy, they gave an intuitive example about why binary entropy is concave: Alice has in her ...
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85 views

Copying quantum state [closed]

I'm confused about the last complete sentence in the following paragraphs. If ab=0, that means either a or b equals to 0. As a result, doesn't $|\psi\rangle|\psi\rangle$ equal to either $b^2|11\rangle$...
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102 views

Why does $a ⊕ (a ⊕ b) = b$ and $(a ⊕ b) ⊕ b = a$? [closed]

For the following circuit that swap two qubits The sequence of gates is said to have the following sequence of effects on a computational basis state |a, b> where all additions are done modulo 2. ...
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Is where measurement is done the requirement for what gets to be called the computational basis?

In Nielsen and Chuang, chapter 1.3.3 is named as "Measurements in bases other than the computational basis". This name confuses me - after the measurement is done on a new base, doesn't this ...
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50 views

The use of modulo 2 in state representation after CNOT

The following circuit with a CNOT gate has the following effect on a computational basis state $|a, b\rangle$, where all additions are done modulo 2. Why is the state of the second qubit changed to $...
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104 views

How to describe a known quantum state using classical information?

In Nielsen and Chuang, it's said that to describe a known quantum state precisely takes an infinite amount of classical information since $|\psi\rangle$ takes values in a continuous space (from the ...
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280 views

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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70 views

Why can every Bell state be written as $|\beta_{xy}\rangle=\frac1{\sqrt2}(|0,y\rangle + (-1)^x|1,\bar y\rangle)$?

In Nielsen and Chuang, there's the following paragraph: The mnemonic notation $|\beta_{00}\rangle, |\beta_{01}\rangle, |\beta_{10}\rangle, |\beta_{11}\rangle$ may be understood via the equations $$ |\...
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1answer
112 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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1answer
237 views

Why is H gate called a ‘square-root of NOT’ gate?

In Nielsen and Chuang, there's the following paragraph: I understand that \begin{align*} \sqrt{NOT} = \frac{1}{2}\left( {\begin{array}{*{20}{c}} \sqrt 2 e^{i\pi / 4}&\sqrt 2 e^{-i\pi / 4}\\ \sqrt ...
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31 views

Write the difference of 2 density operators in terms of a spectral decomposition

An exercise question (9.7) from Quantum computation and Quantum Information by Michael E. Nielson and Isaac L. Chuang says that I can write the difference of any 2 arbitrary density operators $\rho,\...
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75 views

Equivalent statement of the unitary freedom of Kraus operator?

There is a well-known form of the unitary freedom of Kraus operators, which can be found in Nielsen's book, stating that two sets of Kraus operators describe the same physical process of the system(...
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69 views

How to start reading quantum computing papers?

What's the best way to get to a state where you can read quantum computing papers? I find them too dense and full of notation to approach. I am currently working my way through Nielsen and Chuang's ...

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