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We know that the symbol of a quantum gate like "x gate", "z gate" is an abstraction notation. For example, the Not-gate in classical computer is composed of 2 transistors. How can we know quantum gate's exact circuit implementation?

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    $\begingroup$ Hi peachnuts, welcome to QCSE! This may depend on how the quantum bits (qubits) are implemented, but remember a quantum circuit does not generally abstractly describe physical gates, like transistors; rather they are implemented more often as well-timed laser pulses/microwave pulses acting on the qubits. For example, Google's paper on their Sycamore processor - which uses superconducting transmon qubits - goes to some length describing how their gates are implemented with microwave pulses on their qubits. $\endgroup$ – Mark S Nov 12 '19 at 18:21
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    $\begingroup$ Mark S is completely right, but if you're looking to build physical intuition you may want to start with something more elementary than transmons, which make use of cQED. If you're looking to ease into things, I went through an implementation of a controlled-not gate based on non-linear optics in this answer, which is about as simple as quantum gates get. $\endgroup$ – Jonathan Trousdale Nov 12 '19 at 21:21
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    $\begingroup$ I'm afraid this is currently too broad to write a definitive complete answer to, as quite simply, it depends on both the gate and the implementation details, of which there are many - if you have a specific physical type of quantum computer in mind (e.g. superconducting transmons), if you edit the question to include this info, it can get reopened $\endgroup$ – Mithrandir24601 Nov 12 '19 at 23:08
  • $\begingroup$ As mentioned above, there are many approaches how to realize quantum computer. I think this paper arxiv.org/abs/quant-ph/0002077 can help you. It describes general requirements on some system to be a quantum computer. $\endgroup$ – Martin Vesely Nov 13 '19 at 13:54