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Quantum gates seem to be like black boxes. Although we know what kind of operation they will perform, we don't know if it's actually possible to implement in reality (or, do we?). In classical computers, we use AND, NOT, OR, XOR, NAND, NOR, etc which are mostly implemented using semiconductor devices like diodes and transistors. Are there similar experimental implementations of quantum gates? Is there any "universal gate" in quantum computing (like the NAND gate is universal in classical computing)?

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One can replicate any quantum gate or at least get arbitrarily close using sufficient number of CNOT, H, X, Z and $\pi/8$ rotation gates. That is because they form a universal set of quantum gates (refer to: M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2016, page 189). Be careful here. Clearly, we cannot implement any arbitrary quantum gate $U$ with infinite precision. Instead, given $\epsilon>0$, we implement $U_{\epsilon}$, which is $\epsilon$-close to $U$ (refer to: Quantum Mechanics and Quantum Computation MOOC offered by UC Berkely on EdX). This imperfection of quantum gates is one of the main reasons we need error correction codes.

There have been attempts to implement those basic gates. I'm adding some of the recent research works related to these attempts:

As Wikipedia mentions, another set of universal quantum gates consists of the Ising gate and the phase-shift gate. These are the set of gates natively available in some trapped-ion quantum computers (Demonstration of a small programmable quantum computer with atomic qubits).

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