All Questions
Tagged with textbook-and-exercises matrix-representation
21
questions
2
votes
1
answer
70
views
In general, what is feasible quantum computation?
I don't really understand what is feasible quantum computation, in my book (Lipton and Regan's Quantum Algorithms via Linear Algebra) they said that:
A quantum computation $C$ on s qubits is feasible ...
- 103
2
votes
1
answer
211
views
Writing a Density matrix in terms of the magnitude of the Bloch Vector
Working with the density matrix and the Bloch sphere, I have been attempting to complete an exercise in Entangled Systems; New Directions in Quantum Physics. If anyone has the book it is Question 4.3 ...
- 472
0
votes
1
answer
76
views
How to get the Dirac representation of a general quantum gate?
writing a matrix from bra-ket notations is easier. Going back is like finding prime factors. How to get the bra-ket form of all basic quantum gates in their matrix form in general?
- 1
4
votes
3
answers
4k
views
How to represent the Hadamard gate as a rotations on the Bloch sphere?
I am new to Quantum Computing, and I have decided to try and learn the quantum gates. I am trying to understand how to represent some basic gates as rotations on the Bloch Sphere. I was able to ...
- 165
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
$$
X = \sum_{M \in \{I, X, Y, Z\}^{n}} x_{M}M_{1}\otimes ... \otimes M_{n}
$...
- 163
-4
votes
1
answer
64
views
Quantum Linear Algebra [closed]
Find a 4 x 4 unitary matrix U such that U = eiA. (Possibly up to multiplying by a unit scalar, U is a matrix seen in the course.) Verify your calculation by showing how if U were given, one can obtain ...
- 1
3
votes
1
answer
88
views
Why can the Hamiltonian $H=P_x(t)X+P_y(t)Y$ make an arbitrary unitary $U=R_x(b)R_y(c)R_x(d)$?
p.281 of Nielsen and Chuang's book says that
A single spin might evolve under the Hamiltonian $H = P_x(t)X + P_y(t)Y$, where $P_{\{xy\}}$ are classically controllable parameters. From Exercise 4.10, ...
1
vote
1
answer
127
views
Vector math of applying an X-gate on an $|i\rangle$ basis state
It is well known that the X-gate will apply a rotation about the x-axis on the bloch sphere.
Knowing this, the $|i\rangle$ state should be converted to the $|-i\rangle$ state on the application of ...
- 13
0
votes
2
answers
554
views
How do I apply a matrix to a ket state?
If we have the following matrix:
$$\frac{1}{\sqrt{2}}\begin{pmatrix}1&1&0&0\\ 1&-1&0&0\\ 0&0&1&-1\\ 0&0&1&1\end{pmatrix}$$
How do we find the output for ...
- 173
3
votes
1
answer
514
views
How does a general rotation $R_\hat{n}(\theta)$ related to $U_3$ gate?
From eqn. $(4.8)$ in Nielsen and Chuang, a general rotation by $\theta$ about the $\hat n$ axis is given by
$$
R_\hat{n}(\theta)\equiv \exp(-i\theta\hat n\cdot\vec\sigma/2) = \cos(\theta/2)I-i\sin(\...
- 2,398
3
votes
1
answer
2k
views
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:
$$
X=|0\rangle\langle1|+|1\rangle\langle0|\\
Y=-i|0\rangle\langle1|+i|1\rangle\langle0|\\
Z=...
- 2,398
3
votes
1
answer
131
views
Mistake in using dirac notation when applying $X$ gate to vector
The X gate is given by $\big(\begin{smallmatrix}
0 & 1 \\
1 & 0
\end{smallmatrix}\big)$ in the computational basis. In the Hadamard basis, the gate is $X_H = \big(\begin{smallmatrix}
1 &...
- 139
4
votes
1
answer
429
views
Difference between change of basis in bra-ket notation and matrix notation
In matrix notation, say I have the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. It is currently represented in the computational basis $\{\begin{bmatrix} 1 \\ 0\end{bmatrix}, \begin{bmatrix} 0 \\ 1\...
- 1,631
4
votes
1
answer
5k
views
How are the Pauli $X$ and $Z$ matrices expressed in bra-ket notation? [duplicate]
For example:
$$\rm{X=\sigma_x=NOT=|0\rangle\langle 1|+|1\rangle\langle 0|=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}}$$
$$\rm{Z=\sigma_Z=signflip=|0\rangle\langle 0|-|1\rangle\langle 1|=\...
- 183
1
vote
1
answer
155
views
How does the graphical notation used to denote doubly-controlled gates work?
$\qquad$
$\qquad$
What is the difference between solid and hollow? How to express the corresponding matrix of these figures? In addition, if they are not adjacent, what should be done in the middle of ...
- 35