Fermionic gates

[kevinsung]

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The reason we have XXYYGate is that the interaction a_p^\dagger a_q + h.c. (h.c. stands for Hermitian conjugate) translates to XXYYGate under the Jordan-Wigner transform when p and q are adjacent. But there are many other kinds of interactions that arise in fermionic simulations, and these are usually first expressed as sums of fermionic operators. For example, to implement UCC we need the interaction a_p^\dagger a_q^\dagger a_r a_s + h.c. We need a gate for this interaction too.

I think it's most natural to express these gates in the fermionic picture, so perhaps we need a notion of a fermionic gate that can be transformed into a qubit gate in various ways. That way we could study different transformations too. Furthermore, a hypothetical simulator specialized for fermionic circuits would want to take as input circuits expressed using fermionic gates.

[kevinsung]

For starters, here are circuits for the term a^\dagger_p a^\dagger_{p+1} a_{p+2} a_{p+3} + h.c. =(JWT) - |0011><1100| - |1100><0011|, compiled by @Strilanc .