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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Prove a circuit with $iR_x(\pi \alpha)$ 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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39 views

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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Quantum Operation - non trace increasing

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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75 views

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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301 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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41 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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61 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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39 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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1answer
22 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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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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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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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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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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66 views

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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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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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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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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2answers
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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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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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1answer
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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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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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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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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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1answer
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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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104 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
224 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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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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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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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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In what sense are Pauli matrices measurement operators?

Neilson and Chuang's textbook shows a nice example of measuring in the $Z$ basis on page 89 in section 2.2.5. The Hermitians for measuring in the $Z$ basis, $|0\rangle\langle 0|$ and $|1\rangle\langle ...
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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?
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Exercise 4.41 in N&C book QCQI: how can i implement $R_z(\theta)$ using the circuit shown and $Z$?

I'm studying Nielsen and Chuang's book. I cannot solve one of the questions in the exercise 4.41. The question is the last one that is Explain how repeated use of this circuit and Z gates may be used ...
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Understanding the definition of entropy in the joint entropy theorem derivation

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 $\left|...
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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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1answer
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How can I prove inequality from 4.66 to 4.67 in Nielson and Chuang's book?

I am reading chapter 4 of Nielson 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|+|...
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1answer
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Regrouping the terms in expression 1.31 in Quantum Computing and Quantum Information, Nielsen and Chuang

I'm trying to reproduce the passage from expression 1.31 to 1.32 in the book Quantum Computing and Quantum Information, by Michael Nielsen and Isaac Chuang. Expression 1.31 is: $$|\psi_2\rangle = \...
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1answer
75 views

In the amplitude amplification algorithm in Nielsen and Chuang's book, why is the error probability $M/N$?

I was able to follow til the yellow-highlighted sentences, which seems to be important to understand. Why M/N is the probability of error? If so 1 - M/N would be the probability of success?
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Construction of arbitrary Normalizer Gates using H, S and CNOT Gates

This question is in reference to Exercise 10.40 of Nielsen and Chuang's textbook, which is an attempt to prove the theorem that any $n$ qubit Normalizer gate can be built out of $H$, $S$, and $CNOT$ ...
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Measurement interpretation - do individual operators get applied?

The description of measurement operators in Nielsen and Chuang is as follows: Quantum measurements are described by a collection $\{M_m\}$ of measurement operators. These are operators acting on the ...
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How to prove generalized Uhlmann's theorem?

I think the Uhlmann theorem should be in general of this form: Let $\rho$ and $\sigma$ be density operators acting on $A$, with Schmidt degrees at most $r$, and let $B$ be another Hilbert space with ...
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1answer
65 views

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 ...
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1answer
66 views

Quantum parallelism and Deutsch's algorithm - what is $U_f$ really? [closed]

I'm trying to understand quantum parallelism ideas leading the Deutsch's algorithm. The circuit in question is I understand that we end up with $$|\psi_3 \rangle = \pm | f(0) \oplus f(1) \rangle \...
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1answer
56 views

Umambiguous discrimination using POVM with highest discriminate probability

I was studying Nielsen&Chuang's textbook (about page 92), and come up with a question that I cannot solve it. Given one of the two state $|\psi_1\rangle=|0\rangle$ and $|\psi_2\rangle=\frac{1}{\...
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51 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|...

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