How to synthesize quantum circuits?

[nashed]

I wonder if anyone can explain how to actually synthesis a quantum circuit, what I mean by this is that given some requirements, I am able to write the matrix form of the circuit needed, but I have no idea how to turn that into a list of gates.

For example:
Build a circuit that performs the exclusive or operation on three qubits $$ |abc\rangle \rightarrow |ab\rangle\otimes|a\oplus b\oplus c\rangle $$ looking at a truth table for the circuit I can see that when a and b are the same then c is conserved, otherwise c is flipped so I can deduce that if I connect c to a controlled not where a is the control and another controlled not where b is the control I get the desired operation.
That said if I try and write the matrix form of the circuit I have no idea how to manipulate it so it looks like the repeated application of controlled not gates.

BTW this is not a homework question but I am not sure this is the correct forum either so feel free to tell me and I"ll delete the post.

 

[Strilanc]

This kind of information is covered in quantum computing textbooks, and those might be your best bet.

Other than that, you could try reading this blog post about cutting down a NOT with many controls into single-control NOTs and single-qubit operations.

You will probably also find my drag-and-drop quantum circuit simulator Quirk very useful. For example, I bet the truth table you have looks quite a lot like the grid on the right: