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I know that there is connectivity restriction when implementing physical CNOT gate. But, I wonder if there is a limit to the number of physical CNOT gates that can be performed at the same time. For example, suppose six qubits are positioned in a row and only nearest qubit physical CNOT is allowed. q1 q2 q3 q4 q5 q6 I think that CNOT(q1,q2), CNOT(q3,q4), CNOT(q5,q6) gates can be implemented at the same time. However, is there such a that because at most two CNOT gate is implementable, CNOT(5,6) cannot be implemented with CNOT(q1,q2) and CNOT(q3,q4)? If so, why this happens?

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    $\begingroup$ If two-qubit gates act on different pairs of qubits, they can be executed simultaneously (in parallel). There is nothing wrong with that in principle. Counting all two-qubit gates that can be parallelized as a single gate gives the depth of the circuit. Depth can be as important as the gate count itself. $\endgroup$ May 11 at 12:02
  • $\begingroup$ Thanks for answering. I agree that it's obvious in quantum information theory but is it still obvious in experiment? Also, I wonder if it can be implemented experimentally in infinity regardless of the qubit system. $\endgroup$ May 11 at 12:34
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    $\begingroup$ Well, why not? If you have some particular concerns in mind, do share them, and people familiar with how real hardware operates may help. Also I'm not sure what do you mean by 'in infinity'. $\endgroup$ May 11 at 12:58
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    $\begingroup$ I concern that if the number of cnot gate is restricted, when performing syndrome extraction in quantum error correcting codes, the time for syndrome measurement can be large. (e.g, SC-17 has eight stabilizers, and we need to 32(8X4) CNOT for syndrome extraction. In many papers, these 8 stabilizers can work simultaneously. But when there is CNOT restriction, we might need more time and should consider this as error probability because some data qubits are idle). $\endgroup$ May 12 at 5:50

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In terms of physical realizations, your ability to execute parallel CNOTs (or other 2 qubit gates) on distinct pairs of qubits is platform-dependent.

Trapped neutral atoms use optical driving to implement 2 qubit gates, and the drive laser can be split to realize many gates simultaneously. Bluvstein et al have a really good publication with a video (see ancillary files at the Arxiv link) that shows how pairs of atoms are rearranged to implement the necessary 2 qubit gates to implement a toric error correction code.

On the other hand, if your qubit platform is carbon-13 nuclear spins in diamond where a nitrogen-vacancy (NV) center mediates your two qubit gates, then you can only implement them in series. For example, see this paper which describes how one NV is used to entangle pairs and larger sets of nuclear spins sequentially.

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  • $\begingroup$ Thanks for answering. Could you explain what if the system is superconducting qubit? CNOT gates are simultaneously realized? And another question is that if the qubit platform is carbon-13 nuclear spins, only one CNOT gate can be implemented at once? $\endgroup$ May 12 at 5:43
  • $\begingroup$ For C-13 nuclear spins, yes, you can only implement one CNOT at a time. I'm not as familiar with SC qubit systems, but I believe you can drive CNOTs simultaneously on multiple pairs by driving the cavities coupling each pair with separate microwave pulses. $\endgroup$
    – Chris E
    May 13 at 5:09

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