Help: Applying Quantum Gates in Order

In Quantum Computing --> Information --> Quiz 3: Gates Galore --> Problem 10, three quantum gates are applied, in the following order: "first X, then Z, and then H".

I tried to solve this by turning HZX into a single operation using matrix multiplication. However, I seem to have ordered my multiplication incorrectly. I thought that the order would be (X times Z) times H, which works out to the correct answer of 12[1111]\frac{1}{\sqrt2}\begin{bmatrix} -1 & 1\\ 1 & 1 \end{bmatrix}

However, the order shown in the explanation (which gives the same answer) is H times (Z times X).

Which is correct, and why?

P.S. Quantum Computing does not show up as a Topic when making a post, so I labelled this "Computer Science".

Note by Adrian Self
8 months, 2 weeks ago

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As I understand, we usually understand 0\ket{0} to be the vector [01]\begin{bmatrix}0\\ 1\end{bmatrix}. So, in order to apply the function (or linear transformation) XX on 0\ket{0}, one says X0X\ket{0}. Seen this way, we need to figure out HZX0HZX\ket{0}

Agnishom Chattopadhyay Staff - 7 months, 2 weeks ago

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Interestingly, the next problem is, "Would the resulting state always be the same if we had applied the gates in a different order?"

Adrian Self - 8 months, 2 weeks ago

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In general, matrix multiplication is non-commutative (meaning that the order is important). Try out the order shown in the problem and see if it works out to your answer. If it doesn't, there is probably a typo in the quiz.

Steven Chase - 8 months, 2 weeks ago

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