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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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: $\...
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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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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!!
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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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1answer
101 views

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 ...
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0answers
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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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2answers
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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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1answer
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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 ...
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1answer
179 views

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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How should I understand the change of qubit's basis as a rotation?

I have a little difficulty with understanding. How do I properly visualize the change of qubit's basis as a rotation? Let's say that we have classical basis vectors, $|0\rangle$ and $|1\rangle$. Now, ...
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Why do multi-bit hadamards expands to what they do?

I'm a Computer Scientist undergrad student studying for an exam in Quantum computing. In all of the algorithms I have been studying (Deutsch–Jozsa, Simons, Shors, Grovers) I constantly see multi-qubit ...
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1answer
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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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231 views

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 ...
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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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3answers
889 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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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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2answers
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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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1answer
520 views

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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1answer
60 views

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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1answer
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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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1answer
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$H = e^{i\pi/4} \sqrt{iNOT}$?

In the paper Valley qubit in Gated MoS$_2$ monolayer quantum dot, a description of how a $NOT$ gate would be performed on a qubit in the described device is given. The authors say that in the ...
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1answer
148 views

Generate the state $\frac{-|0\rangle + |1\rangle}{\sqrt{2}}$ with qiskit: problem with Pauli-Z behavior

I want to construct the following state of a qubit using a quantum circuit: $\frac{-|0\rangle + |1\rangle}{\sqrt{2}}$ When I use the following qiskit code in Python: ...
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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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1answer
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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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1answer
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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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1answer
56 views

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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2answers
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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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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 +...
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2answers
126 views

Why do we divide by $\sqrt2$ in the qubit states $\lvert\pm\rangle=\frac{1}{\sqrt2}(\lvert0\rangle\pm\lvert1\rangle)$?

I have a very basic question. I have found qubits are represented as complex vectors. I get it totally. I understand bracket notation and vector\matrix algebra. However, I cannot move further from ...
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1answer
49 views

Randomness using simple parallel Hadamard circuit

I've recently tried to build a Random generator using 5 hadamard gates (shown as U2 below) measured to 5 classical bits in parallel as shown in the circuit image. I've executed this circuit for 8192 ...
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1answer
101 views

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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1answer
49 views

Prove that QFT and Walsh-Hadamard gates give the same output when acting on $\lvert x\rangle\lvert 0\rangle$ [duplicate]

I know that $QFT_n|0\rangle$ is equivalent to $H_n|0\rangle$ (mathematical proof). And it is also easy to prove that $QFT_1$ is equivalent to $H_1$ (applied to one QuBit). From looking at the circuit ...
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1answer
109 views

N-Qubit Hadamard vs Quantum Fourier Transform

Both Simon's algorithm and the algorithm for period finding begin by placing qubits in the equal superposition state, but Simon's algorithm uses the n-qubit Hadamard $H^{\otimes n}$ while the period ...
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1answer
78 views

Hadamard gate with two inputs in the circuit for the BB84 protocol?

I am reading the book "Quantum Computing verstehen" by Matthias Homeister. At the moment i'm having a look at the BB84 protocol (which is described in kind of an abstract way). In this chapter a ...
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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 ...
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1answer
42 views

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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1answer
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Probabilities does not sum up to 1 in simple circuit

I have an issue, perhaps with normalization with the following state. For $\alpha^2 + \beta^2 =1 $, the probabilities in this state does not sum up to 1. $$|\psi\rangle := \frac{1}{2}\left[\alpha\...
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1answer
50 views

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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1answer
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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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1answer
57 views

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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1answer
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How can I make qiskit output raw data?

I am new to quantum computing, and I want to make a program to output 0 or 1 randomly by Hadamard gate, and use that information to make a GUI interface. For example, a coin flip program that output "...
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3answers
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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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474 views

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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2answers
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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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3answers
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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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2answers
230 views

How to construct a IBM Quantum Experience circuit for the following state transformation?

Please help me in building IBM Quantum Experience circuit for: $$ M|0\rangle = \frac{1}{2}(|0\rangle+|1\rangle+|2\rangle+|3\rangle) $$ Edit: Is it possible to make a circuit for a general ...
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1answer
168 views

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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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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2answers
167 views

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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291 views

Simple algebraic explanation for normalizing states

I'm wondering how a set of three 0-state qubits, each prepared identically, like so: When considered together, may produce the fraction: along with their combined states. This is the entire circuit....