Can anyone explain quantum gates

[pleasehelpmeno]

Hi I am studying Quantum computing and basically have no understanding of quantum gates and my lecturer is not very helpful.

I don't understand why a 4x4 quantum gate would ever effect a two qubit system because surely that is at best only a 2x2 matrix, assuming they effect each other through multipication.

How can there be a gate which flips the first qubit and leave the second one unchanged, surely they would both be affected?

 

[jedishrfu]

this may help

http://en.wikipedia.org/wiki/Quantum_gate

 

[pleasehelpmeno]

I don't understand how a 4x4 gate could effect a 2x2 qubit though

 

[kith]

The state space of a single qubit is 2-dimensional. Composite states are given by the direct product. The direct product of two 2-dimensional states is a 4-dimensional state.

|1> can be written as column vector (1,0). |11> = |1>⊗|1> can be written as (1,0,1,0).

Simple quantum gates act on single qubits, so they are represented by 2x2 matrices. If a quantum gate operates on two qubits, it has to be represented by a 4x4 matrix, because it has to act on a column vector with four components. Try for example to construct the matrix representation of a "double not" gate to see how this works.

 

[Zarqon]

For 2 qubits, you have 4 possible states: |00>, |01>, |10> and 11>. Thus, any matrix operator that tries to relate a vector of size 4 onto another vector of size 4 (2 qubits still remain after the operation) have to be a 4x4 matrix.