Distant quantum gates between uncoupled qubits

Question

Is there any formalism to perform quantum gates between two qubits (let's say in a superconducting quantum network) to perform a quantum gate between two qubits which are not directly coupled? I want to ask the same question for any known physical model of quantum computing. Another way of saying is that, if it is possible to indirectly couple two qubits via some unitary transformations on the Hamiltonian of a system? One example could be to ask if in a linear chain of three qubits $A,B$ and $C$, is it possible to do a SWAP gate between $A$ and $C$ not directly involving $B$, meaning without first swapping with $B$. I am not asking in the sense of quantum circuit where the answer could be possibly a 'no'. I am asking more from a Hamiltonian point of view.

Answer 1

This depends on exactly what your criteria are. Let me give the easiest solution to your 3-qubit case: apply the Hamiltonian $$ H=(X_1Y_2-X_1Y_2)+(X_2Y_3-X_2Y_3) $$ for a time $\pi/(2\sqrt{2})$. Apart from an extra $-1$ phase on the $|010\rangle$ and $|101\rangle$ terms (in essence, a controlled-phase between each pair of qubits), this does exactly what you want. It might be possible, in this 3-qubit case, to avoid some of those phases with a slightly different Hamiltonian, I'd have to experiment a bit.

This is essentially the content of a paper by Bose.

Of course, you impose in your question that B would not be involved, while in this, during the time evolution, the state of B does leave the B qubit even though it comes back later. This is always going to be a feature of such solutions, so I assume you're going to allow it.

If you want to do the transfer over greater distances, this is basically the study of "perfect state transfer" (full disclosure: this is very much my area of research). The transfer over greater distances doesn't, as a general rule, leave the intermediate qubits unmoved (but you can easily think of doing two swap pulses: one for all N qubits on a line, and one for all $N-2$ qubits in the middle to move them back to where they started). One advantage of this method compared to doing a whole bunch of swap gates in the gate model is that you can achieve the transformation in about half the time. A second advantage is that there's effectively a controlled-phase gate automatically built in to this sort of interaction. You may want to check out these papers:

• perfect state transfer in single excitation subspace only, but long distance
• consideration of higher excitation subspaces.
• My review paper (shameless plug)

Note that you might get annoyed by all the many controlled-phase gates that go on during the swapping process. If you undo the swap, it undoes all of them, so it's not so bad in that sense. However, you also have the option of encoding. If you encode a qubit as $(\alpha|00\rangle+\beta|11\rangle)$, that defeats all those controlled-phase gates.