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Suppose I create the very simple circuit

OPENQASM 2.0;
include "qelib1.inc";
qreg q[1];
x q[0];
barrier q[0];
x q[0];

and submit it to an IBM backend. How do I find the buffer time between the end of the first X and the beginning of the second X for a given backend? I've seen the number ~10 ns thrown around in the literature (see this paper above Fig. 1), but this value is not displayed in the calibration data for a device.

Just for clarify, I am NOT referring to the duration of X but the duration of time between the X pulses even "when it appears to be 0."

If there's a way to obtain it via Qiskit on a per-calibration basis, that's even better, but just a reference which verifies that 10 ns is a reasonable value would suffice.

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Short answer: The minimal delay between gates on IBMQ chips is $0$ nano-seconds.

Longer answer:

The barrier operation is not a hardware operation, it just give instructions to the compiler to schedule the circuit correctly. As such, its duration is $0$ nano-seconds as it is never executed on the hardware. You can check this on luciano answer: the barrier operation is not present in the generated schedule.

If your question was rather "what it is the minimum non-zero delay between two gates on IBM chips", the answer is dt = $\frac{2}{9}$ nano-seconds on the most recent backends. You can recover this value for each backend with

from qiskit import schedule, QuantumCircuit, transpile, IBMQ
IBMQ.load_account()
provider = IBMQ.get_provider()
backend = provider.get_backend("ibmq_armonk")

dt = backend.configuration().dt
print(f"dt = {dt*1e9:.2f} ns")

Pulses duration should be a multiple of 16dt but delay duration can be any multiple of dt.

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Not an expert here, but my impression is that the rotations are scheduled one after the other one, with no delay.

Taking your example:

from qiskit import schedule, QuantumCircuit, transpile, IBMQ
IBMQ.load_account()
provider = IBMQ.get_provider()

circuit = QuantumCircuit(1)
circuit.x(0)
circuit.barrier(0)
circuit.x(0)
circuit_t = transpile(circuit, backend)

If we schedule that one on ibmq_lima:

backend = provider.get_backend('ibmq_lima')
print('lima openpulse support:', backend.configuration().open_pulse)

schedule(circuit_t, backend).draw(backend=backend, time_unit='ns')

enter image description here

It seems to me that the distance between pulses is not 10ns.

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