Understanding Elementary Gates For Quantum Computation

[Amith2006]

Homework Statement

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?

Homework Equations

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

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.

 

[Valkarie]

So, this circuit gives us the matrix W which should be unitary and hence det(W)=1. That is why we require that W belongs to SU(2).

 

[Mene]

Dear student,

I understand your confusion and I will try my best to clarify the concepts mentioned in the research paper for you. Let's start with the notation \wedge_{1}(W). This is not the boolean AND, but rather a notation used in quantum computation to represent a controlled unitary operation. In this case, the gate W is controlled by the first bit, and the output of the operation depends on the state of the first bit. This gate is also known as a controlled-U gate, where U is the unitary operation controlled by the first bit.

Now, in Lemma 5.1, the authors are proving that any 2x2 unitary matrix W can be simulated by a network of the form shown in the paper, where A, B, and C are all 2x2 unitary matrices. This means that any 2-bit gate can be simulated by using 3 one-bit gates and 2 CNOT gates, as you correctly stated.

The authors also divide the proof into two parts - the "if" part and the "only if" part. The "if" part shows that if W belongs to the special unitary group SU(2), then it can be simulated using the network shown. The "only if" part shows that if W can be simulated using the network, then it must belong to SU(2). This is a common approach in mathematics and is used to prove the equivalence between two statements.

In terms of the logic behind this, the authors are essentially showing that any 2-bit gate can be broken down into simpler operations, namely one-bit gates and CNOT gates. This is important because it allows for the implementation of quantum algorithms using a limited set of basic operations, making it easier to design and build quantum computers.

I hope this helps to clarify some of the concepts mentioned in the paper. It is a complex topic and may take some time to fully understand, but keep working at it and don't hesitate to ask for further clarification if needed. it is important to have a thorough understanding of the concepts and theories in your field of study. Keep up the good work!