# Is there the number of CNOT restriction?

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?

• 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. May 11, 2022 at 12:02
• 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. May 11, 2022 at 12:34
• 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'. May 11, 2022 at 12:58
• 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). May 12, 2022 at 5:50