# 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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### Show that $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)$ implies $U=\begin{bmatrix}[E_1]&\cdots\\ [E_2]&\cdots\\\vdots&\ddots\end{bmatrix}$

In Page 365, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that We have a principal system $Q$ and an environment $E$ and $U$ ...
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### Necessity of decoding in fault-tolerant quantum computation

On page 476 of Nielsen/Chuang's book it is stated: The basic idea of fault-tolerant quantum computation is to compute directly on encoded quantum states in such a manner that decoding is never ...
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### Question regarding the measurement of Pauli matrices on a Bell state

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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1 vote
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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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### Heisenberg Uncertainty Principle (Nielsen and Chuang Box 2.4)

I'm trying to follow Nielsen and Chuang Book on Quantum Computation and Quantum Information. There is Box 2.4 on the Heisenberg Uncertainty Principle. I got stuck pretty fast. In that box they define: ...
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### What does the centered dot notation mean in the expression $|+\rangle|+\rangle\otimes(\cdot)$?

In the expression $|+\rangle|+\rangle \otimes(\cdot)$, what does the centered dot represent? I have come across a definition of the centered dot previously, similar to the one given here, but I am ...
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1 vote
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### Show the linearity of $(\langle a_m|\otimes I_B\otimes I_C\otimes \langle d_q|) U(I_{A}\otimes I_B\otimes |0_{C}\rangle\otimes |0_{D}\rangle)$

Suppose a composite system $AB$ initially in an unknown quantum state $\rho$ is brought into contact with a composite system $CD$ initially in some standard state $|0\rangle$, and the two systems ...
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### Bit flip error correction syndrome measurements

I'm coming across some confusion in chapter 10.1.1 of Nielsen and Chaung. In terms of the 'recovery' procedure, how can the result of the syndrome measurement be 0, 2 or 3? I am assuming that, for ...
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1 vote
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### How is the grouping of terms done in the calculations for the teleportation circuit?

I have a CS background and am studying quantum computing by myself. Struggling at the moment with the Dirac notation for the teleportation circuit. Here we go: The circuit starts with the EPR pair ...
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### CNOT gate an elementary example of a single qubit quantum operation

A natural way to describe the dynamics of an open quantum system is to regard it as arising from an interaction between the system of interest and an environment, which together form a closed quantum ...
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### Proof of upper and lower bound (Gilbert-Varshamov bound) for linear code

I am trying to prove the following bounds for a $[n, k]$ code that can correct $t$ errors \begin{align} 1-H\left(\frac{t}{n}\right)\geq \frac{k}{n}\geq 1-H\left(\frac{2t}{n}\right) \end{align} where \...
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