# Quantum gates

1. Jan 16, 2009

### Amith2006

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:
http://arxiv.org/PS_cache/quant-ph/pdf/9503/9503016v1.pdf
Lemma 5.1
For a unitary 2x2 matrix W, a $$\wedge$$$$_{1}$$(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 $$\wedge$$$$_{1}$$(W) gate. How do u read it?$$\wedge$$ 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.