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A common task to perform during quantum computation on the surface code is moving qubits from one place to another. There are standard ways to do this within the surface code, but I was wondering what the actual fundamental limits are. If we forget about the fact that we're using the surface code, and just focus on the fact that we have a planar grid of noisy qubits with nearest-neighbor connections, and a fast classical computer noting measurements and generally helping out, how fast can we move quantum information across that patch?

Given an operation failure rate $\epsilon$, a patch of length L and height H, and the ability operations in parallel with some duration T, how long does it take to move N qubits from the left side of the patch to the right side of the patch?

1In the setting you describe in the last paragraph, why not prepare entangled states beforehand and teleport? – Norbert Schuch – 2018-11-18T11:04:04.780

@NorbertSchuch Sure, that's a valid strategy, except you don't start with pre-prepared entanglement between the left and right halve sides. You have to set it up by communicating quantum information over the patch. Which comes back to the original question of what the quantum bandwidth is. – Craig Gidney – 2018-11-18T19:43:21.927

Well, I'm trying to understand the rules of the game more precisely. Is it a fair setting to say that input data (qubits) are provided on the qubits in the leftmost column on demand, and read out on demand from the rightmost column, and you want to know the rate at which you can transfer a large number N>>L,H of qubits? Are all operations (=two-qubit Hamiltonians) allowed? – Norbert Schuch – 2018-11-18T22:19:13.143

@NorbertSchuch Within the LxH patch, all single-qubit operations including measurement are allowed. All two-qubit operations are allowed, but only between adjacent qubits. Every operation takes time T. Operations can be performed in parallel if they affect disjoint qubits. The sender and receiver are on opposite sides of the patch. They are arbitrarily powerful quantum computers. They can communicate classically, and they can interact with the outermost layer of qubits on their side of the patch. There is an arbitrarily powerful classical computer to process measurement results and choreograph – Craig Gidney – 2018-11-19T04:46:53.520

Ah. So any operation, but constant time? Why not interactions which I can switch on/off? – Norbert Schuch – 2018-11-19T11:18:12.620

An initial question would be to see how fast one can transport information error free. One could e.g. think of some neat trick like swapping and while swapping transporting half of an entangled pair backwards. This would give twice the "naive" capacity (i.e., one could transport 2

H qubits per time unit T , rather than 1H). I strongly suspect this is optimal. @DaftWullie should know more about these kind of things. – Norbert Schuch – 2018-11-19T11:21:05.990@NorbertSchuch I agree that "two way communication" is probably relevant. I don't think just swapping will be optimal, because it's not taking advantage of the fact that you can do measurement in the middle of the patch (a somewhat unusual property for a quantum channel). As for turning local interactions on/off instead of applying local unitaries, that would be acceptable. – Craig Gidney – 2018-11-19T17:00:13.473