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Is it possible to approximately compile Toffoli using H and CSWAP?
TL;DR: Nope. The triple tensor product of the $+1$ eigenstate of the Hadamard gate is a $+1$ eigenstate of every product of the controlled-SWAP gate and the Hadamard. This is not the case for Toffoli, ...
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Is it possible to approximately compile Toffoli using H and CSWAP?
You can exactly implement a Toffoli using CSWAPs catalyzed by a singlet state:
The singlet state is the -1 eigenstate of the swap gate, and you can get phase kickback from a swap by using cswap, so ...
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