So the usual quantum circuit I know for quantum teleportation allows for the teleportation of the state on a single qubit, in the following way:

Quantum circuit for quantum teleportation of 1-qubit state How easy is to generalize this algorithm to allow for the teleportation of the state of 2 or $n$-qubits?

Note that I am not considering qudits.

  • $\begingroup$ @AdamZalcman those are qudits not qubits $\endgroup$
    – Mauricio
    Oct 20 at 7:06
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    $\begingroup$ A qudit is a quantum system with a finite number of levels $d$. A system of $n$ qubits is a quantum system with $2^n$ levels and thus can be viewed as a qudit with $d=2^n$. In this sense, qudits are in fact not only more general than qubits but also more general than finite collections of qubits. $\endgroup$ Oct 20 at 7:17
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    $\begingroup$ see also quantumcomputing.stackexchange.com/a/4216/55 $\endgroup$
    – glS
    Oct 20 at 9:20
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    $\begingroup$ judging from the accepted answer, you are asking about a gate decomposition to perform teleportation of $n$-qubit states, correct? That would make the question different than the one about qudits, but also the current title a bit misleading $\endgroup$
    – glS
    Oct 20 at 9:21

You can simply teleport a state per partes, i.e. each qubit separately.

To do so, firstly you have to prepare as many Bell states as a number of qubits teleported. Then entangle each qubit of the teleported state with your part of some Bell state and run the protocol to teleport the qubit to other side. This has to be done for each qubit of the teleported state.

Here is an example of teleporting two-qubits entangled state $\cos(\pi/8)|01\rangle + \sin(\pi/8)|10\rangle$:

enter image description here

Here is a commented code in QASM for the circuit above

include "qelib1.inc";

qreg aSrc[2];
qreg aEnt[2];
qreg bEnt[2];
creg c[2];

//teleported state psi prep.
ry(pi/4) aSrc[0];
x aSrc[1];
cx aSrc[0],aSrc[1];
//entangling Alice and Bob
h aEnt[0]; //first qubit
cx aEnt[0],bEnt[0];
h aEnt[1]; //second qubit
cx aEnt[1],bEnt[1];
//connecting psi to quantum channel
//between Alice and Bob
cx aSrc[0],aEnt[0]; //first qubit
h aSrc[0];
cx aSrc[1],aEnt[1]; //secpmd qubit
h aSrc[1];
//sending classical bits - teleport.
//(here used q-gates for 
//the circuit simplification)
cz aSrc[0],bEnt[0];//first qubit
cx aEnt[0],bEnt[0];
cz aSrc[1],bEnt[1];//second qubit
cx aEnt[1],bEnt[1];
//measurement at Bob
measure bEnt[0] -> c[1];
measure bEnt[1] -> c[0];

And finally here are results of the teleportation on IBM Quantum simulator (1024 shots):

enter image description here

Clearly, you can extend this approach to general $n$-qubits state. To be honest, I do not know if this approach is the most efficient in terms of Bell states used.


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