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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?

glS
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Craig Gidney
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1 Answers1

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1 Cell Movement

The steps to move qubit in only 1 cell are:

1 - Finish full-cycle finish in all array
2 – before next cycle start – instruct software – "don’t mesure Z stab below"
3 – Also 2 X stabs are converted – 4 CNOTs -> 3
4 – Get X6 (measure X of Q6) used to track erros + influence new X_L
5 – turn on the Z stab above + arms of X stabs
6 – perform surface code cycle
7 – wait (d-1) cycles to establish stab value in time

enter image description here

L Cells Movement

Multi_cell movement, done very quick because you turn the stabilizers off very quickly, in total $d+1$ cycles (2 for the turn on and off, + $(d-1)$ to stabilize fault-tolerance), and not dependent on the length of movement $L$. The time per each stabilizer cycle is limited mainly by the measurement time. All you have to make sure that your $\epsilon$ - the rate of physical failure, will be below the threshold

refer to the article "Surface codes: Towards practical large-scale quantum computation" to see more details

enter image description here

Ron Cohen
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