Questions tagged [hadamard]

The single qubit Hadamard gate is equivalent to 1/sqrt(2) (sigmax + sigmaz)

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Is quantum superposition state a truth or an assumption?

Please, be patient with my question I already read that there is a heuristic that makes superposition a fact of reality. In addition, this superposition, when observed, it has a state of 0 or 1. ...
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2answers
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What is the matrix representation for $n$-qubit gates?

Let's say I have more than one qbits $|0\rangle|1\rangle$ and I want to perform a $H$ on both of them. I know the matrix representation for the Hadamard on a single qbit is $$\frac{1}{\sqrt{2}}\begin{...
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3answers
138 views

How do I prove that the Hadamard satisfies $H\equiv e^{i\pi H/2}$?

How can I demonstrate on the exponential part equality of the Hadamard matrix: $$H=\frac{X+Z}{\sqrt2}\equiv\exp\left(i\frac{\pi}{2}\frac{X+Z}{\sqrt2}\right).$$ In general, how can I demonstrate on: $\...
7
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2answers
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Could the Hadamard gate have been constructed differently with similar characteristics?

Say we had a Hadamard-like gate with the -1 in the first entry instead of the last. Let's call it $H^1$. $$H = \begin{bmatrix}1&1\\1&-1\end{bmatrix}$$ $$H^1 = \begin{bmatrix}-1&1\\1&...
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3answers
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Can everything in QM be described with degrees instead of matrices and vectors?

I found this explanation. "The Hadamard gate can also be expressed as a 90º rotation around the Y-axis, followed by a 180º rotation around the X-axis. So $H=XY^{1/2}H = X Y^{1/2}H=XY^{1/2}$." Can ...
2
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1answer
40 views

Analysis of the second Hadamard in the Detusch-Josza Algorithm

Consider the Deutsch-Josza, algorithm, which first initializes the state $|0 \rangle^{\otimes n} | 1 \rangle$, creates a superposition using the the Hadamard gate and the $U_f$ to get into the state: $...
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2answers
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Prove by induction $H^{\otimes n} \left| 0 \right>^{\otimes n} = \frac{1}{\sqrt{2^n}} \sum_{i=0}^{2^n -1} \left| i \right>$

Let H be the Hadamard operator. $$ H = (\left| 0 \right> \left< 0 \right| + \left| 0 \right> \left< 1 \right| + \left| 1 \right> \left< 0 \right| -\left| 1 \right> \left< 1 \...
2
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3answers
175 views

How do 2 Hadamard gates act on a single qubit?

When I perform $2$ Hadamard $H$ gates on a single qubit, why is the probability of getting $0$ as the outcome 100%? Why is it not 50% 0 and 50% 1 instead? Why is the second $H$ gate not putting the ...
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1answer
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How to understand a phase operation between 2 Hadamard gates?

I would like to understand this image, of a "payload preparation" gate. A single H gate will create a superposition, while the phase will rotate 45 degrees. What does the second H gate do in this ...
5
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1answer
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Finding a global phase that transform the Hadamard gate to an element of $SU(2)$ and propose an evoultion operator which implents the operation

I was looking back over an old assignment and I came across a question I wasn't quite sure how to do the problem statement is as follows: The Hadamard rotation is an element of the group $U(2)$. (i)...
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2answers
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Projecting $\lvert ++ \rangle$ on Bell Basis

I understand that, projecting $\lvert 00\rangle$ on the Bell states would produce $\lvert\Phi^+\rangle$. Because, $$ CNOT(H\lvert0\rangle \otimes \lvert0\rangle) = \frac{1}{\sqrt{2}}(\lvert00\rangle +...