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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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2 Answers

The matrix you quote has the following determinant

$$ \det\left( \begin{array}{c c c} -1+2/8& 2/8& 2/8\\ 2/8 & -1+(2/8) & 2/8\\ 2/8 & 2/8 & -1+(2/8) \end{array} \right) = -1/4 $$

which is not unitary required by quantum mechanics. It implies that your matrix is not possible to be constructed using any standard quantum gates which require unitary. Note that a 3x3 matrix needs at least a qtrit or two qubits.

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ok. if the matrix is unitary how can one construct operator using standard quantum gates? –  safat siddiqui Dec 17 '13 at 18:39
    
@user35949 Have you taken a look of the paper cited by doetoe? Most of papers are talking about how to minimize the number of quantum gates needed. There might be a simple but costly method that I don't remember. –  hwlau Dec 17 '13 at 19:12
    
These lecture notes www-bcf.usc.edu/~tbrun/Course/lecture12.pdf describe such a straightforward but costly method to reduce to CNOT's and single qubit gates –  doetoe Dec 18 '13 at 10:36
    
Yup, the construction does cost exponential number of gates as doetoe mentioned in the notes. @user35949 –  hwlau Dec 18 '13 at 10:46
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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
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