Quantum Computing Help: Function Gates & Matrix Representation

[pleasehelpmeno]

Hi

This is help from lectures on quantum computing that I missed due to illness and now my professor is away.

1) Can anyone help me on what a function gate is and its matrix representation?
I realize that the function is not something like x^2 but something that takes a value from 0 or 1, but other than that I really don't understand, can anyone give me examples say for f(0)=1 or f(1)=0?

2) Can this matrix representation have any kind of circuit diagram feature?

 

[NegativeDept]

Forgive me if I restate obvious things, but I think it helps to keep language as simple as possible when dealing with confusing subjects like QM.

1. A quantum state is represented by a complex vector.
2. If you choose a basis for a vector space, you can represent a vector by its components with respect to that basis.
3. I find it easier to think of a state vector as a column of complex numbers. Each number is a component of the state vector with respect to whatever basis you chose. (If you use a numerical program to simulate quantum systems, you almost have to think this way.)
4. Things you can do to a quantum state are represented by linear operators. If you choose a basis, then every linear operator can be represented by a matrix. What that operator does to a vector is calculated by multiplying (matrix) * (column of components).
5. Certain special matrices show up a lot in quantum computing. Each represents something that a quantum computer might want to do to a state. Some of those matrices are named after digital circuits because they do a weird quantum-y version of a digital computer operation.

Here are some examples of the special matrices I mean:
http://en.wikipedia.org/wiki/Quantum_gate