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Can anyone explain quantum gates

  1. Mar 3, 2013 #1
    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?
     
  2. jcsd
  3. Mar 3, 2013 #2

    jedishrfu

    Staff: Mentor

  4. Mar 3, 2013 #3
    I don't understand how a 4x4 gate could effect a 2x2 qubit though
     
  5. Mar 3, 2013 #4

    kith

    User Avatar
    Science Advisor

    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.
     
  6. Mar 4, 2013 #5
    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.
     
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