All Questions
Tagged with textbook-and-exercises pauli-gates
22
questions
-2
votes
1
answer
59
views
help understanding gate to hamiltonian and representation
So I have this question: Given an operator, find some Hamiltonian implementing this operator/gate. I have realized that this is a swap gate and I know the matrix for it. I also know that $U = \text{...
- 1
0
votes
1
answer
41
views
Understanding the error operator representation $E = i^{\lambda}X(a)Z(b)$
Question regarding exercise $27.3.2$ in "Concise Encyclopedia of Coding Theory".
The exercise states:
We write $E = X((0,1))Z((0,0))$ and $E' = iX((0,1))Z((1,1))$. We
choose the ordering $(...
- 631
0
votes
2
answers
569
views
How to prove the matrix identities $HXH = Z$ and $HZH = X$?
As we know Hadamard gates are used to bring quantum bits into superposition states.
I’m trying to understand how identities $HXH = Z$ & $HZH = X$ w.r.t rotation.
- 145
0
votes
1
answer
125
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What state do you get applying the pauli Y gate to $|\pm\rangle$? [duplicate]
I know it's a basic question but what state gives when you apply pauli $Y$ gate over states $+$ and $-$?
If I apply $Y|+i⟩ = |+i⟩$ or $Y|0⟩ = i|1⟩$, but I don't understand what do you get when you do $...
3
votes
2
answers
80
views
Do the operators $\Phi(\sigma_k)$ form a basis if $\sigma_k$ do?
Assume we have a quantum channel $\Phi$. The single qubit Pauli basis is $\sigma_0, \sigma
_1, \sigma_2, \sigma_3$. Now we apply $\Phi$ to Pauli basis and get $\gamma_0=\Phi(\sigma_0), \gamma_1 = \Phi(...
- 579
5
votes
1
answer
361
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Regarding the inductive proof that any Clifford gate can be made of Hadamard, phase and c-not
In Exercise 10.40 of Nielsen and Chunang's textbook, the reader is supposed to construct an inductive proof of Theorem 10.6 that any Clifford gate can be made of Hadamard, phase and c-not. There it is ...
- 51
2
votes
1
answer
2k
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construction of Y gate from X,Z and H gates
As a part of textbook exercise, Y gate is to be constructed using H,Z and X-gates, just like we have $X = HZH$. is there some way/process/intuition to find such combinations or it is just like we need ...
- 577
2
votes
1
answer
466
views
How does the Pauli Y gate act in the $|+\rangle, |-\rangle$ basis?
The X gate in the $|+\rangle$, $|-\rangle$ basis becomes the Z gate and vice versa.
What is the Pauli Y gate as a matrix transformation in the $|+\rangle$, $|-\rangle$ basis?
2
votes
2
answers
180
views
How to prove that the trace of n-qubit matrices satisfies ${\rm Tr}(XY)=2^n\sum_{M\in\{I,X,Y,Z\}^n} x_M y_M$?
It is known that for n-qubit matrices X, Y $\in \mathbb{C}^{2^{n}\times 2^{n}}$ (and Pauli matrices $I, X, Y, Z$) such that
$$
X = \sum_{M \in \{I, X, Y, Z\}^{n}} x_{M}M_{1}\otimes ... \otimes M_{n}
$...
- 163
3
votes
1
answer
88
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Why can the Hamiltonian $H=P_x(t)X+P_y(t)Y$ make an arbitrary unitary $U=R_x(b)R_y(c)R_x(d)$?
p.281 of Nielsen and Chuang's book says that
A single spin might evolve under the Hamiltonian $H = P_x(t)X + P_y(t)Y$, where $P_{\{xy\}}$ are classically controllable parameters. From Exercise 4.10, ...
0
votes
1
answer
697
views
In what sense are Pauli matrices measurement operators?
Neilson and Chuang's textbook shows a nice example of measuring in the $Z$ basis on page 89 in section 2.2.5. The Hermitians for measuring in the $Z$ basis, $|0\rangle\langle 0|$ and $|1\rangle\langle ...
- 1,391
4
votes
3
answers
963
views
Can we write Pauli-Y gate without even complex part?
I was just curious, why is the quantum gate Y-gate (Pauli-Y gate) written in terms of complex numbers? We can actually write Pauli-Y gate as
$$
Y = i * \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{...
- 159
2
votes
1
answer
215
views
Why can any density operator be written this way? (quantum tomography)
From page 24 of the thesis "Random Quantum States and Operators", where $(A,B)$ is the Hilbert-Schmidt inner product:
\begin{aligned}
\rho &=\left(\frac{1}{\sqrt{2}} I, \rho\right) \frac{...
- 1,391
6
votes
2
answers
977
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How do I decompose the given $4\times 4$ matrix in terms of Pauli matrices? [duplicate]
I have been working on a question where I have to decompose this matrix in terms of Pauli Matrices:
\begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\...
- 63
3
votes
1
answer
2k
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Is there a different way to represent Pauli gates in X basis?
It's easy to see that in computational basis, Pauli matrices could be represented in the outer product form:
$$
X=|0\rangle\langle1|+|1\rangle\langle0|\\
Y=-i|0\rangle\langle1|+i|1\rangle\langle0|\\
Z=...
- 2,398