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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Question regarding bloch vector solutions for master equation on page 388 of N&C
On page 388 of N&C, you are asked to find the solution to a differential equation for a two-level atom coupled to a vacuum. However, I have no experience with differential equations, so I am ...
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What are Sideband Pulses
I see sideband pulses used all over the place when looking up implementation techniques especially in Trapped Ion QC. Is there a layman's explanation of what they actually are? I understand what they ...
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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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clarifying a step in the proof of Solovay-Kitaev theorem
There is a step in the proof of the proof of Solovay-Kitaev theorem about the existence of a set containing words of at most length length $l_0$ that cover $SU(2)$ . The proof I'm reading in given in ...
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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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How can I represent the completely mixed state as $\frac I2=\frac14(\rho+X\rho X+Y\rho Y+Z\rho Z)$?
Consider the completely mixed state $I/2$. The equation comes from Eq.(8.101) of Nielsen's book:
$\frac{I}{2}=\frac{\rho+X\rho X+Y\rho Y+Z\rho Z}{4}$, How comes this equation?
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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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Exercise 4.16 in the Nielsen & Chuang book
In the 4.16 exercice in the Quantum Computation and Quantum Information (Michael A. Nielsen & Isaac L. Chuang), I don't understand why the correct answer is not this matrix :
$$
\left[ {\begin{...
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Measurement interpretation - do individual operators get applied?
The description of measurement operators in Nielsen and Chuang is as follows:
Quantum measurements are described by a collection $\{M_m\}$ of
measurement operators. These are operators acting on the ...
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In Deutsch's algorithm, how to we get $(-1)^{f(x)}|x\rangle(|0\rangle-|1\rangle)$ from $|x\rangle(|0\rangle-|1\rangle)$?
While reading "Quantum Computation and Quantum Information" (by Nielsen and Chuang) I came across this line
A little thought shows that if we apply $U_f$ to the state $\vert x \rangle (\dfrac{\...
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Clarification needed for the N&C proof that BQP ⊆ PSPACE
In QCQI by Chuang and Nielsen (page 201), they prove that $\mathsf{BQP} \subseteq \mathsf{PSPACE}$. I can't understand what they say.
They write "Supposing the quantum circuit starts in the state $...
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Can a density operator be written equivalently as $\rho=\sum_i p_i|\psi_i〉\!\langle\psi_i|$ and $\rho=\sum_i\lambda_i|\psi_i\rangle\!\langle\psi_i|$?
My doubt arises from page 99, 101 of the book Quantum Computation and Quantum Information by Michael A.Nielson and Issac L.Chung.
Let {${p_{i}, | \psi_{i} \rangle }$} be an ensemble of pure states.
...
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How to show that the three-qubit repetition code only corrects up to 1-bit flip errors?
From Nielsen and Chuang, the error correction criteria is $$P E_i^{\dagger} E_j P=\alpha_{i j} P$$ $P$ is the projector onto the correct codespace, $E_{j}$ are error operations and $\alpha_{i j} $ is ...
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Derive the Concavity of Quantum Conditional Entropy from Strong subadditivity
In Exercise 11.25, Page 522, Entropy and information, Quantum Computation and Quantum Information by Nielsen and Chuang, it is required to show that the concavity of the conditional entropy may be ...
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Kraus operators required for operations on $d$ dimensional Hilbert space
All quantum operations $\mathcal{E}$ on a system of Hilbert space dimension $d$ can
be generated by an operator-sum representation containing at most $d^2$ elements,
$$
\mathcal{E}(\rho)=\sum_{k=1}^M ...
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Does applying a random Pauli matrix to a density matrix result in the identity?
Nielsen and Chuang's textbook, Equation 8.101 (section 8.3.4 'Depolarizing Channel') shows that applying a random Pauli to a density matrix representing one qubit equals the identity (times one half):
...
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Hamiltonian Simulation Circuit for Grover's Search
The Hamiltonian in the simulation of Grover Search is given as, $H=|x\rangle\langle x|+|\psi\rangle\langle\psi|$. It is said that in order to simulate $H$ we can simulate the Hamiltonians $H_1=|x\...
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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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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
$$\frac1{\sqrt2}(|0,f(0)\rangle+|1,f(1)\...
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Nielsen & Chuang Exercise 2.32: Show that the tensor product of two projectors is a projector
$\newcommand{\bra}[1]{\left<#1\right|}
\newcommand{\ket}[1]{\left|#1\right>}$Here is what I tried:
Given that we have two projectors:
$$
A = \sum_i \ket{i} \bra{i}, \hspace{2em}
B = \sum_j \ket{...
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Proof of quantum data processing inequality in N&C on pg 566
On page 566, it states that using $S(\rho^{'})-S(\rho,\varepsilon) \ge S(\rho)$ and combining this with $S(\rho) \ge S(\rho^{'})-S(\rho,\varepsilon))$, we get $S(\rho^{'})=S(\rho)-S(\rho,\varepsilon)$....
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What kind of transformation does the Y-gate do on the bloch sphere?
I'm going through "Quantum Computation & Quantum Information" by Michael A. Nielsen and Isaac L. Chuang, and as a high school student with no previous knowledge, I cannot understand some things ...
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Kraus decomposition for non trace preserving operation: shouldn't we have $0 \leq \sum_k E_k^{\dagger} E_k \leq I$
In N&Chuang, on page 368 is written the following theorem:
The map $\mathcal{E}$ satisfies axioms A1,A2,A3 if and only if
$$\mathcal{E}(\rho)=\sum_k E_k \rho E_k^{\dagger}$$
Where $\sum_k ...
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Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
I have trouble understanding this proof in Nielsen & Chuang, specifically the identity in $(10.20)$, which reads $$ U_k^\dagger P_k F_l \sqrt{\rho} = U_k^\dagger P_k^\dagger P_k^\dagger F_l P \...
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Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?
In Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says:
A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i,
The ...
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Nielsen Chang exercise 4.10
I' m trying to figure out how to decompose unitary operator U using only Rx and Ry rotations. I understand that I must use result of exercise 4.8 ($ U = e^{i\alpha}R_n(\theta)$). But I don't ...
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Derivation of Equation 10.69 in Nielsen Chuang
In the proof of correctness of CSS codes in Nielsen and Chuang, we see that equation $(10.68)$:
$$
\frac{1}{\sqrt{|C_2|}} \sum\limits_{y \in C_2} (-1)^{(x+y)\cdot e_2} |x+y \rangle
$$
can be ...
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In what sense is $\langle\psi|\rho|\psi\rangle$ the overlap between $|\psi\rangle$ and $\rho$?
The fidelity between a pure state $|\psi\rangle$ and an arbitrary mixed state $\rho$ is given by, $F(|\psi\rangle,\rho)=\sqrt{\langle\psi|\rho|\psi\rangle}$, which is stated to be equal to the square ...
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Why is dual-rail encoding called an error-detecting code for amplitude damping?
Exercise 8.23 : Suppose that a single qubit state is represented by using two qubits, as $|\psi\rangle=a|01\rangle+b|10\rangle$. Show that $\mathcal{E}_{AD}\otimes\mathcal{E}_{AD}$ applied to this ...
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How does a "measurement of $\vec v\cdot\vec \sigma$" on two-qubit states work?
Michael A. Nielsen & Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition p.113, Box 2.7 states that "if a measurement of $\vec v\cdot\vec\sigma$ is ...
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Grover search in Quantum Computation and Quantum Information (Nielsen and Chuang)
in Nielsen and Chuang's book in Box 6.1 (page 256 for 10th edition) there is a example of Grover's search algorithm for 3 qubits (2 as search space + 1 for oracle).
I am currently trying to implement ...
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For bipartite mixed state, if one part is pure, then the global mixed state is a product state?
In Nielsen and Chuang, the chapter about Schmidt decomposition, there is an interesting result states that for a bipartite pure state $|\psi\rangle_{AB}$, if part A is a pure state, then $|\psi\...
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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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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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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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Confused regarding explanation of Schumachers compression in N&C
On page 547 of N&C, for $|\psi_{0}\rangle=|0\rangle$ and $|\psi_{1}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and for $|\tilde{0}\rangle=\cos(\pi/8)|0\rangle+\sin(\pi/8)|1\rangle$ and $|\tilde{1}\...
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Confusion over HSW theorem depicted in Nielsen and Chuang
On page 560, it states that
$$C^{(1)} \geq S(\frac{\varepsilon(|{\psi}\rangle\langle{\psi}|) +\varepsilon(|{\varphi}\rangle\langle{\varphi}|)}{2} - \frac{1}{2}\varepsilon(|{\psi}\rangle\langle{\psi}|)-...
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Nielsen & Chuang Exercise 4.34 "Measuring an operator"
I need help with the exercise 4.34 from Nielsen & Chuang Book.
I am supposed to get a matrix corresponding to the circuit. Thanks
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Measuring Probability of Mixed States
I am a little stuck on understanding the measurement probabilities of a 3 qubit system (QCQI q 4.41).
1)H gates are applied to both $q_1$ and $q_2$
2) $C^{(1,2)}_3(X)$, a Toffoli, controlled by $q_1$...
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Why don't I get what I expect when measuring with respect to a different basis?
$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$
If ...
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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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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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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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Verify that if $A$ and $B$ are diagonal in the same orthonormal basis, then $[A,B]=0$
This is from Nielson and Chuang's textbook "Quantum Computation and Quantum Information".
They state the Simultaneous Diagonalisation Theorem: Suppose $A$ and $B$ are Hermitian operators. ...
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Verification for calculation on Shor's code
Here I have tried to determine the end result for the qubit states, when we apply an arbitrary gate on the first qubit in the 9 qubit code.
I have followed this diagram:
U's operation on a qubit can ...
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Non trace-preserving map in axiomatic approach to quantum operations
In Nielsen and Chuang's Quantum Computation and Quantum information there is an axiomatic definition of the quantum operation (as one of the 3 approaches to quantum operations).
A quantum operation is ...
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Derivations of Equations (12.205 - 12.207) in Nielsen Chuang
While proving the security of the BB84 protocol, Nielsen and Chuang demonstrate that it is possible to reduce the CSS protocol to the secure BB84 protocol without requiring Alice to reveal the value ...
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Calculation of $\frac{d}{dt} I_t(A,X)$ in proving the convexity of the relative entropy via Lieb's theorem
In Page 520, Entropy and information, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that
The relative entropy $S(ρ||σ)$ is jointly convex in its arguments, where $S(ρ|...
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Quick conditions to see that $I - E_1 -E_2$ is positive for $E_1, E_2$ positives (example from Nielsen and Chuang)
In the Nielsen and Chuang ("Quantum Computation and Quantum Information"), section 2.2.6 POVM measurements, they define these three operators:
$E_1 = \frac{\sqrt{2}}{1+\sqrt{2}} |1\rangle \...
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Conditional entropy as relative entropy between probability distributions
Find the expression for the conditional entropy $H(Y|X)$ as a relative entropy between two probability distributions. Use this expression to deduce that $H(Y |X)≥0$, and to find the equality ...
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