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Generalised Quantum Gates

  1. Mar 30, 2004 #1
    Hi i am coding a quantum computer simulator.
    the simulator will be able to work in dimensions other than qubits.
    in other words the user can select either qubits(d=2), qutrits(d=
    Obviously in this senario one must have the generalised versions of
    all the gates

    So far i have have found generalised versions of-

    Hadamard gate
    Not Gate
    C-Not Gate
    Swap Gate
    Pauli X Gate

    Ive been researching this for some months now and am finding it
    impossible to
    find generalised versions of any of the other common quantum gates. eg
    pauli y gate toffoli gate, controlled swap gate ,controlled unitary
    gate Phase gate and pi/8 gate or any other useful gates.

    Essentially all i need to know is for any given dimension ( be it
    qubits or qutrits etc) what is the matrix representation for a certain

    ps. The current Version of my software can be found on :


    Thanks in Advance

  2. jcsd
  3. Mar 30, 2004 #2
    Did you see my reply on sci.physics.research? If not, I have pasted it below.

    There is no unique generalization of these gates and the one that you
    choose usually depends on the application you have in mind. For
    example, the Pauli operators are Hermitian, unitary and form a basis
    for the space of single qubit operators, but there is generally no set
    of operators with all these three properties in higher dimensions. A
    unitary generalization that is often used is:

    X|j> = |j+1 (mod d)> Z|j> = w |j>

    where w is a primitive dth root of unity. Then the operators
    (X^n)(Z^m) form a unitary basis, analogous to the Pauli operators.

    One possibility for a generalized controlled unitary gate which is
    often used is

    |i>|j> -> |i> U^i |j>

    but there are many other possible generalizations.

    I imagine you are looking to implement a universal set for qudits, in
    which case you should take a look at:
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