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While the QFT and Hadamard transforms are different, their action on the input state $|00\ldots 0\rangle$ is identical; both produce the uniform superposition of all states. So, if you've got a choice of which to use, you shoulduse the one that is the easiest to implement: the Hadamard transform. Hadamard Transform: $$H|0\rangle=\frac{|0\rangle+|1\rangle}{\... 10 The reason for this is because the inverse of Hadamard gate is itself. That is, giving that$$ H = \dfrac{1}{\sqrt{2}} \begin{pmatrix} 1& 1\\ 1 & -1 \\ \end{pmatrix}$$then$$ H^{-1} = \dfrac{1}{\sqrt{2}} \begin{pmatrix} 1& 1\\ 1 & -1 \\ \end{pmatrix} $$so therefore,$$HH|\psi\rangle = H H^{-1} |\psi \rangle = I |\psi \rangle = |\psi\rangle ...

10

While Hadamard gate is defined as $$H= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix},$$ $y$-rotation by $\pi/2$ leads to gate $$Ry(\pi/2)= \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}.$$ So, there is a difference in position of -1 in the second column. Application of the $X$ gate returns the -1 in $... 9 First of all, note that the statement, as written, is wrong (or rather, it is correct only as long as the "$\equiv$" symbol is taken to mean "equal up to a phase"). An easy way to see it is by computing the determinant of$H=e^{i\pi H/2}$, which gives$-1=1$(using$\det[\exp(A)]=\exp[\operatorname{Tr}(A)] $for all$A$and$\operatorname{Tr}(H)=0). Now, ... 8 If you're not concerned with global phase then the following works using only two rotation gates: \begin{align} R_y\left(-\frac{\pi}{2}\right) R_x\left(\pi\right) &= \exp \left(i\frac{\pi}{4}Y\right) \exp \left(-i\frac{\pi}{2}X\right) \\&= \left(\cos \frac{\pi}{4} I + i\sin \frac{\pi}{4} Y\right) \left(-iX\right) \\&=\begin{pmatrix} \cos \frac{\... 6 For questions like this, the conventional physics notation is easier to work with than the QIT gate notation. Define\vec \sigma = (\sigma_1,\sigma_2,\sigma_3)$to represent the three Pauli matrices $$\sigma_1 = X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \;\;\; \sigma_2 = Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \;\;\; \... 6 The Hadamard gate has close ties to the discrete Fourier transform. Consider the DFT for an N-level system:$$\vert j \rangle = \frac{1}{\sqrt{N}} \sum\limits_{k=0}^{N-1} e^{\frac{i2 \pi j k}{N}} \vert k \rangle.$$For N=2 this is simply$$\vert j \rangle = \frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\ 1 & -1 \end{bmatrix} \, \vert k \rangle = H \... 6 You can use$Ry$gate to prepare a qubit in superposition with arbitrary probabilities. When you apply the gate on qubit in state$|0\rangle$, you get a qubit in superposition $$|\psi\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle.$$ By chaning angle$\theta$you can set any probability you want. For setting$\theta = \pi/2you will get ... 6 First remember that each matrix element can be written as outer products in Dirac notation: |0\rangle\langle 0| = \begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix},|1\rangle\langle 1| = \begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix},|1\rangle\langle 0| = \begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix}, |0\rangle\langle 1| = \begin{bmatrix}0 &... 6 A Hadamard gate isn't usually a physical object that you pass qubits through. In the case of superconducting qubits, the Hadamard gate is performed by bouncing microwaves off of the qubits. It doesn't look like anything. So you're not going to find a picture of a superconducting Hadamard gate on a chip. The closest thing to that would be one of the blips in ... 6 As mentioned in the other answer, the Hadamard gate is a pi rotation (180 degree) around the X + Z axis. That is, it is a 180 degree rotation around the purple axis indicated in the below figure: And the Rx(\pi/2) follows by a Rz(\pi/2) then follow by Rx(\pi/2) does exactly this rotation as well. To have better visualization of this, suppose I ... 6 You could almost do it. In fact, the following circuit would work: You can convince yourself that what you do is in fact applying the unitary \mathbf{VUV^\dagger}, which will allow you to evaluate the real part of \langle\psi|\mathbf{V^\dagger UV}|\psi\rangle. Indeed, the standard Hadamard test allows you to evaluate the real part of \langle\psi|\... 5 The |+⟩ and |-⟩ are states given by the following decomposition in the Z-basis: \begin{equation} \begin{aligned} |+⟩ &= \frac{1}{\sqrt{2}} \Big(|0⟩ + |1⟩\Big)\\ |-⟩ &= \frac{1}{\sqrt{2}} \Big(|0⟩ - |1⟩\Big) \end{aligned} \end{equation} As from quantum mechanics, you can see that there's a 50% chance for both states to be found in the |0⟩ or |1⟩ ... 5 The intuition, roughly speaking, is that the only way that you're going to get some difference between classical and quantum computing is if you are able to prepare qubits in a superposition. If you remain in a basis state the whole time, you just have a classical computation. So, you need superposition. Hadamard is the gate that prepares superpositions at ... 5 The Hadamard gate is:\frac{1}{\sqrt 2} \left(|0\rangle \langle 0 | + |0\rangle\langle 1| + |1\rangle \langle 0| - |1\rangle \langle 1|\right)$$And since |+\rangle is \frac{1}{\sqrt 2}\left(|0\rangle + |1\rangle \right), you can work out that H(|+\rangle) = |0\rangle So,$$CNOT(H|+\rangle \otimes |+\rangle)= CNOT(|0\rangle \otimes |+\... 5 If you look at the formula you want to prove term-by-term, you'll notice that the sum and the(-1)^f(x)$part is the same in both formulas; you just need to show that $$H^{\otimes n} |x\rangle = \frac{1}{\sqrt{2^n}} \left( \sum_{y=0}^{2^n-1} (-1)^{x \cdot y} |y\rangle \right )$$ You can either show this strictly by induction (similar to this question but ... 5 If you apply the same gate twice on a state$|\psi\rangle$you will get out the state$|\psi\rangle$if the hermitian of the gate is the same as the gate. In your case, you applied the Hadamard gate,$H$, twice on (presumably) the$|0\rangle$state, and so you got$|0\rangle$as your result. This happens because it is a requirement for quantum computing ... 5 Hadamard gate can be interpreted as a rotation in 3D Euclidean space (on Bloch sphere) by angle$\pi$around X+Z axis. The qubit rotation by angle$\theta$around axis pointed by unit vector$\textbf{n}=\{n_x,n_y,n_z\}$is described by rotation operator ($X$,$Y$and$Zare Pauli matrices) \begin{align} R_{\textbf{n}}(\theta)=&n_xe^{-i\frac{\theta}{2}X}... 5 Welcome to QCSE. You already know thata^2=b^2=0.5$. For a single qubit gate akin to the Hadamard gate you can achieve any two probabilities you want, as long as they add to$1$. For example one trick that I learned was that you could choose ratios of Pythagorean triples, i.e. numbers$a$,$b$,$c$such that$a^2+b^2=c^2$. Let's have a gate called$\...

5

The issue is that you are using noisy hardware with imperfect operations and measurements. In particular, the most likely problem here is that after you prepare a qubit it immediately begins decaying towards the ground state $|0\rangle$ via interactions with the environment. Each qubit will be slightly more likely to be measured as 0 instead of 1 than you'd ...

5

If you initialize three qubits to $|0\rangle$, apply a Hadamard gate to each, then measure each in the computational basis, the result will be an independent coin flip for each bit: that is, any of 000, 001, 010, ..., 111, each with probability 1/8. If you measure all three bits of the GHZ state in the computational basis, you'll get either 000 or 111, each ...

5

So QPE using $\mathcal{O}(1/\epsilon)$ queries to $U$ outputs an estimate of the eigenphase $\theta$ given a corresponding eigenvector with additive error and $\Omega(1)$ probability. The method using separate Hadamard Tests, that you mentioned, works. However, it requires $\mathcal{O}(1/\epsilon^2)$ samples to just get $\epsilon$-estimates of the real and ...

5

This depends on what superposition you're in. Recall that the not gate can be written in the form $$\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)$$ so it has eigenvectors $(1,\pm 1)$. The whole point of an eigenvector is that when the matrix acts on it, it's unchanged. So, the state $H|0\rangle$ (Hadamard acting on a 0 input state) is $(... 5 It's true that the$|\pm\rangle=\tfrac{1}{\sqrt{2}}(|0\rangle\pm|1\rangle$states are orthonormal eigenvectors of the$X$gate (meaning they are "unneffected" by$X$), but this does not mean that$X$doesn't have any effect on states in other superpositions. For instance, consider the state$|\psi\rangle=\frac{1}{\sqrt{4}}|0\rangle+\frac{\sqrt{3}}{...

4

Most probably because your input state was $|0\rangle$. The Hadamard gate has the property that $H^2=I$, thus acting on an arbitrary input $|\psi\rangle$ with the Hadamard gate twice in a row results in the output being identical to the input (that is, $|\psi\rangle$).

4

You did more than you needed to in part i. You effectively did part ii already. Because $$g = \frac{u}{\sqrt{\det{u}}}$$ is in $SU(2)$, the phase you need to multiply by is $\frac{1}{\sqrt{\det{u}}}$. $$\det H = -1/2-1/2 = -1$$ So to get in $SU(2)$, you need to multiply $\sqrt{-1}=\pm i$. That is seen as the $e^{\pi i /2}$ you see as the prefactor ...

4

Whether or not quantum superposition is a "truth", is a philosophical question. Quantum theory is simply an axiomatic mathematical model of the universe that happens to give correct experimental predictions (at least, for a fairly broad range) for several physical phenomena. It certainly might be possible to come up with a different mathematical model of the ...

4

Hint: Consider the series expansion of the exponential in the case of matrices $M$ that satisfy $M^2=-1$ (in your case $M=i \frac{X+Z}{\sqrt(2)}$. You should find something akin to Euler's formula which renders the proof trivial.

4

Generally, quantum states are determined up to a phase - i.e. up to multiplication by scalar. So $\frac{\left|0\right\rangle -\left|1\right\rangle }{\sqrt{2}},\frac{-\left|0\right\rangle +\left|1\right\rangle }{\sqrt{2}}$ are essentially the same state. It is a good question how the statevector simulator chooses to normalize its statevector representation.

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Summarization based on discussion with user met927: Transpiled circuit form depends on used backend - it is different for simulator and real quantum processor: On simulator, the $\mathrm{CH}$ gate is transpiled to the circuit shown above On real quantum processor, the gate is implemented with two $\mathrm{U2}$ gates and $\mathrm{CNOT}$ (i.e. like in the ...

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