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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 article describing how to make one.

Only some articles speaking about optical implementation of Hadamard gates. (An approach to realize a quantum Hadamard gate through optical implementation)

Maybe the following is the only one I've found with some information, though quite theoretical too (Realization of efficient quantum gates with a superconducting qubit-qutrit circuit)

How are the IBM and Google Hadamard gates fabricated (or created in the laboratory) and operated?

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  • $\begingroup$ See this related question. $\endgroup$
    – Mark Spinelli
    Commented Nov 12, 2020 at 0:58

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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 this picture of the microwaves being sent down a line:

enter image description here

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    $\begingroup$ Then an exact description of that microwaves? $\endgroup$
    – skan
    Commented Nov 11, 2020 at 20:34
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    $\begingroup$ @skan The microwave pulse is calibrated individually to each qubit, and different shaping techniques are used to balance various sources of errors, so there isn't one specific waveform that's used. It just looks like one or two oscillations inside a gaussian envelope, like in the picture. $\endgroup$
    – Craig Gidney
    Commented Nov 11, 2020 at 22:14
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Fundamentally, a device such as an IBM quantum computer interacts according to a Hamiltonian, which might have some time-varying parameters. For example, for a single qubit, it might look like: $$ H=BZ+\Omega(t)X, $$ where $X$ and $Z$ are the standard Pauli matrices, and $B$ is a constant.

The goal is "simply" to specify the function $\Omega(t)$ to generate whatever unitary evolution you want according to the differential equation $$ i\frac{d|\psi\rangle}{dt}=H|\psi\rangle. $$

For the specific case of Hadamard, you can just set $\Omega(t)=B$ (assuming this is possible given the system constraints) for a fixed length of time $t_0$. Your evolution is then $$ U=e^{-iB(X+Z)t_0}=I\cos(B\sqrt{2}t_0)-i\sin(B\sqrt{2}t_0)\frac{X+Z}{\sqrt{2}} $$ So, if I pick $B\sqrt{2}t_0=\pi/2$, we get the evolution $$ U=-i\frac{X+Z}{\sqrt{2}}, $$ which is Hadamard up to an (irrelevant) global phase.

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  • $\begingroup$ I know the theory, but my question is... How can you build one of such gates? How is the geometry and what are the materials used in this gates? Do you have any diagram or photo, not just an equation? $\endgroup$
    – skan
    Commented Nov 11, 2020 at 18:31
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    $\begingroup$ Although the qubits are physical, the gates are not physical in the devices you ask of; they are more akin to "opcodes". An automated waveform generator generates microwave pulse to act on the qubits. $\endgroup$
    – Mark Spinelli
    Commented Nov 12, 2020 at 1:01
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If you wish to see how the microwave pulse in IBM's device, you can use their pulse schedule function to generate the pulse correspond to the circuit you are running. For example, if you run the circuitz;

enter image description here

then on the device, the qubit is being act on by the following microwave pulse:

enter image description here

More details can be found here on the qiskit documentation page

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  • $\begingroup$ Can I also use the simulator "Quirk" to visualize the pulse? $\endgroup$
    – skan
    Commented Nov 12, 2020 at 19:05