Questions tagged [hadamard]

Single qubit Hadamard gate transforms standard basis states (zero and one states) to their superpositions (plus and minus states)

Single qubit Hadamard gate transforms $\mid 0\rangle$ state to $|+\rangle=\frac{\mid 0\rangle + \mid 1\rangle}{\sqrt{2}}$ state and $\mid 1\rangle$ state to $|-\rangle=\frac{\mid 0\rangle - \mid 1\rangle} {\sqrt{2}}$ state and so, in matrix form, can be written as $$H = |+\rangle\langle 0|+|-\rangle\langle 1|= \frac{1}{\sqrt{2}}\begin{pmatrix}1&1 \\\ 1&-1\end{pmatrix}$$.curcuit symbol for Hadamard gate taken from https://www.kisspng.com/png-quantum-logic-gate-hadamard-matrix-hadamard-transf-4947412/

The $n$-qubit Hadamard gate can then be written as $$H^{\otimes n} = H\otimes H^{\otimes \left(n-1\right)}$$ and is $n$ individual Hadamard gates acting on $n$ different qubits.

For more info, one can visit Hadamard matrix and Hadamard transform.

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

I am trying to solve this exercise from Qiskit's textbook, problem set 2 "Basic Synthesis of Single-Qubit Gates": Show that the Hadamard gate can be written in the following two forms \begin{equation} H = \frac{X + Z}{\sqrt{2}} \equiv \exp\left(i…
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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 $H^1$. $$H = \begin{bmatrix}1&1\\1&-1\end{bmatrix}$$ $$H^1 = \begin{bmatrix}-1&1\\1&1\end{bmatrix}$$ From my maths it's a unitary matrix, so it's a…
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How are the IBM's and Google's Hadamard gates fabricated and operated?

There are thousands of articles, books and web sites describing the Hadamard Gate from a theoretical point of view. But I haven't been able to find any photo about any real implementeation of a Hadamard Gate on superconducting circuits nor any…
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Hadamard Overlap Test

I am trying to understand a test called Hadamard Overlap Test, which consists of a destructive swap test (section IV of swap test and Hong-Ou-Mandel effect are equivalent) right after a Hadamard test. The circuit is from the Variational Quantum…
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Is there a fast sparse Hadamard transform?

Suppose I give you an $n$-qubit state vector as a classical list of numbers (or as an oracle that can query the amplitudes). I tell you this state vector will contain exactly $k$ non-zero amplitudes, after you apply a Hadamard transform to it. You…
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Why isn't $Ry(\pi/2)$ gate equivalent to Hadamard gate?

I've been experimenting with quantum circuits and can't quite fathom how the difference between states comes together. Speaking in terms of simulations using qiskit, the following code yelds the same results: circuit =…
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Why can the QFT be replaced by Hadamard gates?

I'm studying Shor's Algorithm. In the book, author explains QFT can be replaced by Hadamard gates? Why this process is possible?? Thank you everybody. This is QPE. I attach part of book!!
유도경
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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 commonly used subcircuit?
neutrino
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Can we use Hadamard test to estimate phases?

There have been some questions discussing the Hadamard test and quantum phase estimation (QPE), but I did not find the answer to the following question. Suppose we are given $|\psi\rangle$, which is an eigenstate of $U$ such that $U|\psi\rangle =…
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How to visualize Hadamard gate as $X$-$Z$-$X$ decomposition?

In the book Quantum Computation and Quantum Information by Nielsen and Chuang, chapter 4, exercise 4.4 (pg. 175), the author has asked to express Hadamard gate as product of $R_x$, $R_z$ rotations and $e^{i\phi}$ for some angle $\phi$. I have found…
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Hadamard Test to calculate imaginary part

I am trying to understand the Hadamard Test by finding the average value of $U_1$, which is a diagonal matrix with $1$ everywhere except on the first element. I performed the regular Hadamard Test as presented in the wiki page: and so far so good,…
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Finding a global phase that transform the Hadamard gate to an element of $SU(2)$ and propose an evolution operator which implements 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 $U(2)$. Find the global phase with which one needs to…
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Can we learn anything interesting about a claw by taking the square-root-of-NOT of each qubit?

Consider being given a circuit for a two-to-one Boolean function $f$ from $n$ (qu)bits to $m\ge n-1$ (qu)bits, and prepare the following state: $$\frac{1}{\sqrt {2^n}}\sum_0^{2^n-1}|x\rangle|f(x)\rangle.$$ Upon measuring the second register in the…
Mark Spinelli
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How to generalize the relationship HXH = Z for higher dimensions

Concerning the Hadamard gate and the Pauli $X$ and $Z$ gates for qubits, it is straightforward to show the following relationship via direct substitution: $$ HXH = Z.\tag{1}$$ And I would like to demonstrate this relationship for higher dimensions…
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Why is implementation of controlled Hadamard on IBM Q so complex?

With reference to question how to implement CCH gate I easily realized that CH gate can be implemented with $\mathrm{Ry}$ gates and $\mathrm{CNOT}$ followingly: Note $\theta = \frac{\pi}{4}$ for first $Ry$ gate and $\theta = -\frac{\pi}{4}$ for…
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