Can Quantum Gates Be Generalized for Higher Dimensions?

[damo642]

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=
3)...etc
Obviously in this scenario 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
gate.

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


http://www.compsoc.nuigalway.ie/~damo642/QuantumSimulator/QuantumSimulator/QuantumQuditSimulator.htm

Thanks in Advance

Damien

 

[slyboy]

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:
quant-ph/0108062
quant-ph/0210049

 

[R0man]

Hi Damien, it's great to hear that you are working on a quantum computer simulator! Generalised quantum gates are a very important aspect of quantum computing and it's great that you are incorporating them into your simulator.

To answer your question, the matrix representation for a certain gate in a given dimension can be found by using the tensor product of the base matrices. For example, the matrix representation for the Pauli Y gate in a 3-dimensional system (qutrit) can be found by taking the tensor product of the Pauli Y gate in a 2-dimensional system (qubit) with the identity matrix in the third dimension.

As for the other gates you mentioned, they can also be represented using tensor products or a combination of tensor products and unitary matrices. For example, the Toffoli gate can be represented as a tensor product of two CNOT gates and the Phase gate can be represented as a tensor product of the identity matrix and the phase gate in a qubit system.

I would suggest looking into papers and books on quantum computing for more information on the matrix representations of these gates in different dimensions. Also, there are many online resources and forums where you can ask for help and collaborate with others working on quantum computing.

Good luck with your simulator and keep up the great work!