# Quantum Computation

Is there any rule or technique so that one can design quantum gate operator from matrix operator?

Suppose, what will be the quantum gate operator for this matrix operator :

$$\left( \begin{array}{c c c} -1+2/8& 2/8& 2/8\\ 2/8 & -1+(2/8) & 2/8\\ 2/8 & 2/8 & -1+(2/8) \end{array} \right)$$

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As noted by hwlau, quantum circuits always correspond to unitary operators. For the question to be meaningful, you have to start with some set of gates that you consider building blocks. The rules of combination in linear algebra terms are essentially matrix multiplication and Kronecker product, the first corresponding to sequentially applying two circuits, the other to applying them in parallel.

With a finite set you can not decompose any given unitary operator, but you can approximate any operator as closely as you like, if you choose your basic components carefully.

In this article an algorithm is described.

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More precisely, following the paper, it is said that : It has previously been shown that a general quantum gate (unitary transformations operating on a set of $n$ bits) can be simulated exactly or approximately using a quantum circuit built of elementary gates which operate only on one and two qubits –  Trimok Dec 17 '13 at 18:06
@Trimok: thanks, that is a very good remark. This paper shows how to reduce to a composition of one qubit gates (which are not finitely generated) and CNOT's. If you want to reduce to approximations by a fixed finite set of such components, you will have to decompose further. –  doetoe Dec 18 '13 at 8:14
$$\det\left( \begin{array}{c c c} -1+2/8& 2/8& 2/8\\ 2/8 & -1+(2/8) & 2/8\\ 2/8 & 2/8 & -1+(2/8) \end{array} \right) = -1/4$$