# Qutrit Teleportation

## Main Question or Discussion Point

Ive been reading nielson and chuangs book "Quantum computation and quantum information" and find the the teleportation of a qubit very intuitiuve.
I would like to know how mathematically the same process would apply to a qutrit = a|0> + b|1> + c|2>.
can one use the exact same circuit ( ie a qutrit version of a hadamard and a qutrit version of a c-not ) to teleport this qutrit.

You can build the circuit using my qudit simulator. the abitrary qutrit inputs can be made by enabling the option " allow Abitrary inputs" on the simulator.

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

Actually the reason i asked this is beacause i would like a killer application for my simulator. If you can think of any i would greatly appreciate it.
In other words a useful circuit for dimensions other than qubits.

Regards
Damien Fitzgerald

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The teleportation circuit can be generalized to arbitrary dimensional systems (i.e. qudits). Denote the general state to be teleported by |\psi> and define the maximally entangled two-qudit state

|\phi> = sum_{j=0}^{d-1} |jj>

Now do a joint measurement of the qudit |\psi> and one particle of |\phi> in the maximally entangled basis

|\phi_{km}> = sum_{j=0}^{d-1} e^{2*pi*i*k/d} |j j+m>

where k,m = 0,1,...,d-1 and + inside the Ket denotes addition modulo d. Now, the remaining qudit from the maximally entangled pair will be in one of d^2 states given by |\psi> rotated by a unitary that depends on the measurement outcome.

You can work out the details of this using the gates I described to you in response to your previous posting, i.e. the gates to recover the |\psi> are just the generalization of the Pauli gates I gave there.

To work out a circuit version of this, you will need to generalize the Hadamard gates to Fourier transform gates i.e.

|j> -> \sum_{k=0}^{d-1} e^{2*pi*i*j*k/d} |k>

and the CNOT becomes a more general entangling gate, which you should be able to work out.

As regards 'killer applications' for your simulator, I am not really sure if there are any. Certainly, the details of this teleportation circuit can be easily worked out on a piece of paper. You are looking for something that is too complicated to work out by hand, but is not so large that it would require a real quantum computer to do it in a reasonable amount of time. Maybe including some realistic noise in your simulator and then simulating error correction and fault tolerant computation might be useful. This might be nice because robustness to noise is supposed to be an advantage of using qudits over qubits.

I would like to know how can I generalize the CNot gate for higher dimensions?
I'm trying to entangle the qudits, but dunno how to generalize the CNot.
I would really appreciate it if anyone can help me!