Could somebody explain in which way are we able to achieve superposition with Pauli $X$, $Y$, $Z$ matrices? In case of Hadamard gate $H$ we change coefficients to $1/\sqrt{2}$ directly, in case of $X$ and $Y$ rotation gates we apply angle of rotation around $X$ and $Y$ and transform qubit to superposition state. But none of Pauli matrices are applying superposition state to qubit; am I missing something?
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$\begingroup$ Are you asking how a Hadamard gate can be decomposed as $X$ and $Z$ rotations? $\endgroup$– R.WCommented Apr 2, 2022 at 16:48
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$\begingroup$ @R.W I'm trying to understand how I should apply these matrices to |0> and get (|0> + |1>)/√2 (for example) $\endgroup$– Michael TorstonCommented Apr 2, 2022 at 17:06
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2$\begingroup$ Yes we do not have a single qubit pauli matrice that puts qubit in the superposition as you mentioned it can be done using a rotation an $X$ gate $\endgroup$– FarhadCommented Apr 2, 2022 at 19:42
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2 Answers
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You can't decompose a Hadamard gate into Pauli gates. Pauli gates, and products thereof, are all phased permutations. But the Hadamard gate is not.
answered Apr 2, 2022 at 19:52
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Pauli matrices are bases for the well-known 1 Qubit gates, and you can build from them using the summing of Pauli matrices any 1 Qubit gate.
But it does not mean that if you will operate each one separately in a sequence, you can get any transformation.
You can think of them as 180 degrees rotations around the Bloch sphere, which is not enough to get to any point on the Bloch sphere.
But if you make superposition and not composition (which is just math, and not a real operation which is composed of 2): $$ H = 1/\sqrt{2} (Z+X)$$
But there is no combination that will give you $H$ this way:
$$H = X*Z*..Y*..I..*X...$$
