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](https://en.wikipedia.org/wiki/Quantum_logic_gate#Universal_quantum_gates) 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](https://www.edx.org/course/quantum-mechanics-quantum-computation-uc-berkeleyx-cs-191x)). This imperfection of quantum gates is one of the main reasons we need [error correction codes](https://en.wikipedia.org/wiki/Quantum_error_correction).


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

- CNOT: [Construction of controlled-NOT gate based on microwave-activated phase (MAP) gate in two transmon system](https://arxiv.org/abs/1803.04105) and [Rydberg atoms based mesoscopic $\text{CNOT}^{\text{N}}$ gate using STIRAP](https://arxiv.org/abs/1803.02844)

- Hadamard (H): [An approach to realize a quantum Hadamard gate through optical implementation](http://ieeexplore.ieee.org/document/5612120/)

- Phase flip (Z): [One-step implementation of a multiqubit controlled-phase-flip gate](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.73.012323) & [Realizing quantum controlled phase-flip gate through quantum dot in silicon slow-light photonic crystal waveguide](https://arxiv.org/abs/0803.1017)

- Bit Flip (X): [Chiral Spin Flipping Gate Implemented in IBM Quantum Experience](http://naturalspublishing.com/Article.asp?ArtcID=13584)

- $\pi/8$ rotation: [Qudit versions of the qubit "pi-over-eight" gate](https://arxiv.org/abs/1206.1598)

As Wiki 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](https://www.nature.com/articles/nature18648)).