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:

• CNOT: Construction of controlled-NOT gate based on microwave-activated phase (MAP) gate in two transmon system and Rydberg atoms based mesoscopic $\text{CNOT}^{\text{N}}$ gate using STIRAP

• Hadamard (H): An approach to realize a quantum Hadamard gate through optical implementation

• Phase flip (Z): One-step implementation of a multiqubit controlled-phase-flip gate & Realizing quantum controlled phase-flip gate through quantum dot in silicon slow-light photonic crystal waveguide

• Bit Flip (X): Chiral Spin Flipping Gate Implemented in IBM Quantum Experience

• $\pi/8$ rotation: Qudit versions of the qubit "pi-over-eight" gate

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).

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:

• CNOT: Construction of controlled-NOT gate based on microwave-activated phase (MAP) gate in two transmon system and Rydberg atoms based mesoscopic $\text{CNOT}^{\text{N}}$ gate using STIRAP

• Hadamard (H): An approach to realize a quantum Hadamard gate through optical implementation

• Phase flip (Z): One-step implementation of a multiqubit controlled-phase-flip gate & Realizing quantum controlled phase-flip gate through quantum dot in silicon slow-light photonic crystal waveguide

• Bit Flip (X): Chiral Spin Flipping Gate Implemented in IBM Quantum Experience

• $\pi/8$ rotation: Qudit versions of the qubit "pi-over-eight" gate

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).