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I want to measure the depth of a circuit, but I do not know which quantum gates should be used when the circuit is decomposed. For example, $ZZ$ rotation (i.e. $e^{i\theta \sigma_z^i\sigma_z^{i+1}}$) can be decomposed as $\text{CNOT}\,R_z(\theta)\,\text{CNOT}$. So, I think the depth of $ZZ$ rotation is 3. However, if $ZZ$ rotation gate exists, then the depth should be 1.

My question is as follows: Which gate set should be used when I measure the depth of a circuit?

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There's no universal convention. Normally there will be some context that suggests a gate set to use (e.g. you plan to run on certain hardware that only supports certain gates, or you plan to use a particular error correcting code where certain gates are transversal).

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As the above answer mentioned depending on hardware it varies. It can be verified on Qiskit and cirq as they both support depth view, try comparing there standards.

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