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I have a matrix $M= \begin{pmatrix} a - ib & 0 \\ 0 & a + ib \end{pmatrix}$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$. I need to represent this matrix in terms of the quantum logic gates. Is the following representation valid \begin{equation} M = a \mathbb{I} - ib \sigma_Z, \end{equation}

where $ \mathbb{I} $ and $\sigma_Z$ are the identity matrix and Pauli-z gate, respectively. Can this be implemented in a lab?

1Does $a^2 + b^2 = 1$? What 'fundamental logic gates' are you using? – Niel de Beaudrap – 2019-08-26T13:03:42.913

@NieldeBeaudrap, thanks. But $a^2+b^2 \ne 1$. I am talking about the quantum logic gates. Maybe the word fundamental is confusing in my question. – Rob – 2019-08-27T03:37:41.230

3if $a^2+b^2\neq1$ then the matrix is not unitary, and as such, non-reversible. Depending on your definition of "quantum logic gate", this might not qualify as one. Are you sure that's what you want? This aside, sure, you can write $M$ using that formula with Pauli operators – glS – 2019-08-27T07:13:57.563