What are the gates used to implement Shor's algorithm?

I understand theoretically how Shor's algorithm works, but I don't know what specific gates are used (or would be used) to implement it. What would the quantum circuit look like?

There are several correct answers to this question because there are several candidate sets available for use as a universal set of quantum operators and because there isn't an agreed upon best implementation for the primitives needed by Shor's algorithm. Different circuits will trade-off circuit-depth for ancillary qubits and vice versa. Ultimately, the exact implementation is likely to depend on the architecture of the quantum computer.

What that in mind, you can probably find a version of the details you're looking for in Fast Modular Exponentiation Architecture for Shor's Factoring Algorithm by Pavlidis and Gizopoulos. The exact circuit layout is lengthy and complex and probably too much for an answer here. The paper has lots of diagrams though and a quick scan should should provide you with a rough feel for the circuit construction. Unsurprisingly, the circuit makes heavy use of Hadamard gates and controlled phase-shift operators both to implement the inverse QFT and the modular exponentiation circuit.

StackExchange Computer Science Q#64272, answer score: 2