Questions tagged [hadamard]

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

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Are composite gates within superconducting hardware implemented as a single pulse or as a series of pulses?

If we have for example a gate $U^{\otimes2}$, then within superconducting hardware, is the $U$ applied onto the first qubit and then the second or is a pulse corresponding to a composite gate (tensor ...
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Is the CNOT in the standard three-qubit circuit for the GHZ state necessary?

This is a very basic question about the GHZ state. I know the standard construction: A Hadamard on one qubit, and then CNOT gates with targets on all the other ones. However, why can't I just have $n$...
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why is $H^{⊗2}$ used to denote the parallel action of two Hadamard gates?

Why is the tensor product used here, what's its meaning? I learned tensor products as an operation between 2 matrices, and have an effect such as the follows: How does the tensor product above relate ...
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Derivation of the effect of the Hadamard transform on a state |x⟩ in the Deutsch–Jozsa algorithm

On pg. 35 of Nielsen and Chuang, there's the following paragraph: By checking the cases $x=0$ and $x=1$ separately we see that for a single qubit $H|x\rangle=\sum_x (-1)^{xz}|z\rangle/\sqrt{2}$. I'm ...
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Applying Hadamard gate to $\sqrt{3/4}|0\rangle + \sqrt{1/4}|1\rangle$

[I am just transferring this from Stack Overflow. It might need editing.] ———— [The reader can skip to “It all sounds fine…”, before the spreadsheet representation.] I am trying to figure out quantum ...
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Derivation for the result of performing the Hadamard transform on $|0\rangle^{\otimes n}$ being $2^{-n/2}\sum_x|x\rangle$

It's said that the result of performing the Hadamard transform on n qubits initially in the all |0> state is $$ \frac{1}{\sqrt{2^n}}\sum_x|x\rangle $$ where the sum is over all possible values of x....
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Represent the $n$-qubit $2^n\times2^n$ size Hadamard/quantum Fourier transform unitary square matrix as product of $k$ two-level unitary matrices

I wish to know if it is possible to express the n-qubit Hadamard unitary square matrix of size $2^n * 2^n$ as a product of 'k' two-level unitary square matrices where 'k' is of the order of polynomial ...
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Why is H gate called a ‘square-root of NOT’ gate?

In Nielsen and Chuang, there's the following paragraph: I understand that \begin{align*} \sqrt{NOT} = \frac{1}{2}\left( {\begin{array}{*{20}{c}} \sqrt 2 e^{i\pi / 4}&\sqrt 2 e^{-i\pi / 4}\\ \sqrt ...
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How to derive the rotations caused by the H gate?

In Nielsen and Chuang, there's the following paragraph: The Hadamard operation is just a rotation of the sphere about the ˆy axis by 90◦, followed by a rotation about the ˆx axis by 180◦. I am ...
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Initial state preparation for Hadamard test

I thought I understood Hadamard test but it seems to be shaky. I understand that to get the expectation value $\langle\psi\ | V^\dagger|{\bf Q}|V|\psi\rangle$ we need to have gate $V$ (in blue) below ...
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Doing $|0\rangle$ then Hadamard gate then measurement

I am starting to use quantum experience and following exactly first example from lecture. After initializing the qubit to $|0\rangle$, then applying a Hadamard gate, the probability for measuring $|1\...
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How do you represent a Hadamard gate as a product of $R_x$ and $R_y$ gates?

I'm looking for a representation of Hadamard gate that uses only $R_x(x)$ and $R_y(y)$ gates. The values $x$ and $y$ may be the same, but they don't necessarily need to be.
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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 ...
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How does a Hadamard discrete-time quantum walk result in a skewed distribution?

I was reading this tutorial about discrete random walk and got confused by the following paragraph. After the succession of Hadamard applications ($H$), I wonder how do we get skewed distribution. I ...
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What is the relation between Hadamard transformation and QFT?

I am new to the field and I can't help having a feeling that Hadamard and Fourier Transform are somehow related, but it is not clear to me how. Any explanation on how these two are related would be ...
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Analysis of the second Hadamard in the Detusch-Jozsa Algorithm

Consider the Deutsch-Jozsa, 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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Is the tensor product of 2 Hadamard gates entangled?

Assume that you have a system of two qubits in the state $|11 \rangle$. Apply $H \otimes H$, where $H$ is the Hadamard matrix. Is the state $(H \otimes H)|11\rangle$ entangled? I know if we take the ...
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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. ...
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What is the outcome when you apply 2 hadamard gates on CNOT

So when I run through risk, it displayed it had an equal 25% chance to get 00 01 10 11 respectively. I know how the CNOT output looks like when you apply hadamard gate on control part before CNOT, but ...
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How does $U_f$ act on a qudit state in the Deutsch-Jozsa Algorithm

The problem starts with the given the input state $|\psi_{in} \rangle = |0 \rangle |1 \rangle$, I'm asked to calculate $|\psi'\rangle = H_d \otimes H_d |\psi_{in} \rangle$ where $H_d$ is the Hadamard ...
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Understanding Deutsch Algorithm

From the image below, if we focus on the first qubit, we know after Hadamard (state 1) $|0\rangle$ will become $|+\rangle$ and the second qubit $|1\rangle$ will become $|-\rangle$. What exactly would ...
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What is the intuition of using Hadamard gate in quantum fourier transform?

According to this answer by rrtucci, I still cannot catch the spirit of QFT algorithm. So I would like to ask why are we using the Hadamard gate when computing the Fourier Transform? Moreover, what ...
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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 ...
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How do two H gates act on two entangled qubits?

In this circuit, if the two qubits are initial in state 0, then after the oracle they are entangled and in state: $0.5 * (|00\rangle+|01\rangle+|10\rangle-|11\rangle)$ My question is how do the two H ...
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Is it possible to tune the amplitude of superposition generated by Hadamard gates?

I had a question earlier about generating the superposition of all the possible states: Here. In that case, we could apply $H^{\otimes n}$ to the state $|0\rangle^{\otimes n}$, and each state has the ...
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What happens if $|\psi\rangle$ = $|0\rangle$ or $|\psi\rangle$ = $|1\rangle$ is passed as an input to two Hadamard gates in sequence?

I'm a computer science student and soon I will have a math exam. I'm really struggling with this preparation question. Also, includes the following: How does this demonstrate that we need the “ket” ...
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How to describe the state of a qubit passing through two Hadamard gates? [duplicate]

Describe the state of the qubit at points A, B and C. How does this demonstrate that we need the “ket” (or the vector) representation of qubits, rather than just describing them in terms of ...
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What is a complexity of producing arbitrary equally distributed superposition?

In the article An Optimized Quantum Maximum or Minimum Searching Algorithm and its Circuits I found statement (pg. 4): Preparing an initial state takes $\log_2(N)$ steps. In this case the initial ...
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Is there an error on Qiskit.org textbook with the superdense coding section?

The textbook on Qiskit.org has When the H-gate is applied to first qubit, it creates superposition and we get the state $|0+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |01\rangle)$ Shouldn't it be: $$|...
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What Hamiltonians generate Hadamard and CNOT? [closed]

Find a $2 \times 2$ Hamiltonian $H_H$ such that $e^{iH_H}$ equals the Hadamard matrix and a $4 \times 4$ Hamiltonian $H_{CNOT}$ such that $e^{-iH_{CNOT}}$ equals the matrix of the CNOT gate. I have ...
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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 ...
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Show that the two circuits are equivalent mathematically

This exercise wants me to prove the equivalence of the two circuits using their mathematical representations. Circuit 1: Circuit 2: Circuit 1 (q1 CNOT ...
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How does the outcome of measurement of a qubit change when we use different basis despite the system hasn't changed? [closed]

Let's assume that the quantum state of the system is written in a standard basis {$|0\rangle, |1\rangle$} and when we performed a measurement we got $|0\rangle$ as an outcome of measurement so we ...
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Show that the Hadamard gate is equivalent to a 180 degree rotation of a certain axis

Show that the Hadamard gate is equivalent to a 180 degree rotation about the axis defined by $(\vec{e_x} - \vec{e_z}) / \sqrt{2}$ where $\vec{e_x}$ and $\vec{e_z}$ are unit vectors pointing along the ...
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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 &...
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Measuring in the computational basis in the single qubit gate QFT implementation

I've come across this paper about a single-qubit-gate-only QFT implementation. In the paper it is claimed that measuring a qubit after applying the Hadamard gate (it isn't called Hadamard gate in the ...
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A CNOT between two Hadamard gates: why does the CNOT changed the output of the second Hadamard gate?

Applying the Hadamard gate twice in a row, it restores the original input: https://algassert.com/quirk#circuit={%22cols%22:[[%22H%22],[%22H%22]]} However, if a CNOT control is added between the two ...
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Transforming $|100\rangle$ state into $|000\rangle + |111\rangle$ state using only Hadamard and CNOT gates

Hi, How to convert $|100\rangle$ 3-qubit quantum state into $\frac{1}{\sqrt{2}}(|000\rangle + |111\rangle)$ state using only Hadamard and CNOT gates? Also, is output state an entangled one?
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Definitions of $D_y$ gate in Hamiltonian simulation: are they the same?

I'm reading a Hamiltonian simulation example proposed in this paper. From their notation, the operator $D_y$ (sometimes it's called $H_y$) serves the function to diagonalize the Pauli matrix $\sigma_y(...
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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 ...
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Transforming $|01 \rangle + |10 \rangle - |11 \rangle \to |01 \rangle - |10 \rangle + |11 \rangle$

How to convert from current state: $$|\psi \rangle =\dfrac{ |01 \rangle + |10 \rangle - |11 \rangle}{\sqrt{3}}$$ into a target state $$|\phi \rangle = \dfrac{|01 \rangle - |10 \rangle + |11 \rangle}{\...
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Why 2 $H$ gates in series create a probability of 100% for one value of the qubit and 0% of the second value of the qubit?

Why 2 $H$ gates in series create a probability of 100% for one value of the qubit and 0% of the second value of the qubit since an $H$ gate acts like a superposition generator?
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Confused about the application of Hadamard gate to uncorrelated qubits [duplicate]

Why does applying the following circuit on a $00$ state produce $|0\rangle \otimes |+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |01\rangle)$. Shouldn't it produce $ |+\rangle \otimes |0\rangle = \...
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Is there a clear boundary between quantum coupling and quantum entanglement?

I have a few questions in understanding the difference between coupling and entanglement in quantum systems: Is there a clear boundary between quantum coupling and quantum entanglement? If two quantum ...
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How do I apply a Hadamard gate on a given qubit, in matrix formalism?

Hadamard gate matrix is: $$\frac{1}{\sqrt2}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}$$ The matrix for $|0\rangle$ is: $$\begin{bmatrix}1 \\ 0\end{bmatrix}$$ I am unable to understand, how ...
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Is there a gate that puts a qubit into superposition with a not so purely probabalistic (50 50) outcome?

I know that a Hadamard states is a purely probabalistic one; e.g. $$H\vert 0\rangle=a\vert 0\rangle+b\vert 1\rangle$$ where $a^2=0.5$ and $b^2=0.5$. Are there any states in which the ...
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How do I apply the Hadamard gate to one qubit in a two-qubit pure state?

So in lectures I see lots of these: And somehow I intuitively understand it (at least for the 1 qubit case), but I don't understand the math – especially for 2 qubits.
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How to translate the Hadamard gate matrix into Dirac notation?

Hadamard gate matrix is: $$\frac{1}{\sqrt 2}\begin{bmatrix}1 && 1 \\ 1 && -1\end{bmatrix}$$ The Dirac notation for it is: $$\frac{|0\rangle+|1\rangle}{\sqrt 2}\langle0|+\frac{|0\...
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What are the $|+\rangle$ and $|-\rangle$ states?

In the Gates Glossary of IBM Quantum Experience it states H gate The H or Hadamard gate rotates the states |0⟩ and |1⟩ to |+⟩ and |−⟩, respectively. It is useful for making superpositions. ...
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Calculation of the system states and the individual wire states in a quantum circuit

I am bit confused with calculating the overall state of a quantum gate and the individual wire states. For example, lets say there are two Qubits, where Q1 is in $\frac{1}{\sqrt{2}}(\vert 0\rangle + \...