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Quantum gates

by Amith2006
Tags: gates, quantum
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Jan16-09, 06:36 AM
P: 421
1. The problem statement, all variables and given/known data
With reference to a research paper on "Elementary gates for Quantum computation", I'm unable to understand certain concepts given in it. I am providing a link to this paper which is:
Lemma 5.1
For a unitary 2x2 matrix W, a [tex]\wedge[/tex][tex]_{1}[/tex](W) gate can be simulated by a network of the form,
where A,B and C belong to SU(2)(Lie group) , if and only if W belongs to SU(2).

Before getting into the problem, I would like to get familiar with the notations in it. They speak of simulation of general [tex]\wedge[/tex][tex]_{1}[/tex](W) gate. How do u read it?[tex]\wedge[/tex] is the boolean AND. In this Lemma are they trying to prove that any 2 bit gate can be simulated using 3 one bit gates and 2 CNOT gates provided det(W)=1? Also, the whole problem is divided into 2 parts namely the if part and the only if part. What is the logic behind this?

2. Relevant equations

A.B.C=I & A.X.B.X.C=W where X is a Pauli X matrix.

3. The attempt at a solution
I understand that it is a controlled unitary operation. We consider here 2 cases of applying first a 0 to the top bit and then a 1 to the top bit.There is no change in the output from the lower bit if the top bit is 0 as is the case for a controlled operation and hence A.B.C=I is applied otherwise A.X.B.X.C=W applied.
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