Hadamard gate as a product of $R_x$, $R_z$ and a phase

Question

I am having problems with this task.

Since the Hadamard gate rotates a state $180°$ about the $\hat{n} = \frac{\hat{x} + \hat{z}}{\sqrt{2}}$ axis, I imagine the solution can be found the following way:

First rotate $\hat{n}$ so it lies in the $z$-$y$ plane: $$R_z(\pi/2)$$ Then rotate $\hat{n}$ so it is parallell with the $z$-axis: $$R_x(\pi/4)$$ Now do the desired rotation about the $z$-axis: $$R_z(\pi)$$ Then rotate $\hat{n}$ back to its original position. The Hadamard gate can then be written: $$H = e^{i\lambda}R_z(-\pi/2)R_x(-\pi/4) R_z(\pi) R_x(\pi/4) R_z(\pi/2) $$ However, this doesn't seem to work. Can someone explain where my logic is flawed? Thank you very much

Check exercise 4.4 of this PDF. It should explain your concern. — cnada, Jan 24 '19 at 12:55
I think you are right but maybe when doing your calculations, you are not dividing by 2 the angles in the rotation matrix definition. — cnada, Jan 25 '19 at 10:44

score 5 · Accepted Answer · answered Jan 25 '19 at 08:14

Why do you say that it doesn't work? If I enter the following code into Mathematica, it works fine:

X = {{0, 1}, {1, 0}};
Z = {{1, 0}, {0, -1}};
FullSimplify[-I MatrixExp[-I Pi Z/4].MatrixExp[-I Pi X/8].MatrixExp[I Pi Z/2].MatrixExp[I Pi X/8].MatrixExp[I Pi Z/4]]

The only thing that I had to remember was to halve the angles compared to how you wrote them (because $R_Z(\pi)$ should give a $Z$ rotation, which is $e^{i\pi Z/2}$). That's the place that I always make the mistake with these calculations...

Thank you very much! Not sure where my mistake was, but I wrote down the matrices in wolframalpha. With python it worked fine! — QCQCQC, Jan 27 '19 at 22:02

score 3 · Answer 2 · answered Jan 25 '19 at 07:55

3

As far as I can tell, it works fine. If I use the channel-state duality to view this sequence of operation's effect in Quirk's output display, it's the Hadamard gate's matrix:

Maybe you're multiplying or dividing by 2 somewhere you shouldn't be, when performing the operations?

answered Jan 25 '19 at 07:55

Craig Gidney

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I am sure thats what i did wrong! Thank you very much – QCQCQC Jan 27 '19 at 22:03

cnada · Answer 3 · 2019-01-25T10:43:19.320

2

Check exercise 4.4 of this PDF. It should explain your concern.

I think you are right but maybe when doing your calculations, you are not dividing by 2 the angles in the rotation matrix definition.

edited Jan 25 '19 at 10:43

answered Jan 24 '19 at 12:55

cnada

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Thank you for the answer. I have looked at that solution. Even though it solves the problem, it doesnt explain why my logic is flawed – QCQCQC Jan 24 '19 at 13:09

Hadamard gate as a product of $R_x$, $R_z$ and a phase

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