After meeting with Mei:
the brute-force search done by both of you can be translated to a long formula. However, I think by writing the Toffoli as |11><11|\otimes X + (\id[4] - |11><11|)\otimes \id[2], i.e. by explicitly using the structure of the Toffoli, I think one can derive a much shorter formula. Similar to the fact that the CNOT formula is much shorter than enumerating its update on all Paulis.
Or maybe one could build the large formula and then give it to a (CNF)-minimizer?
After meeting with Mei:
the brute-force search done by both of you can be translated to a long formula. However, I think by writing the Toffoli as |11><11|\otimes X + (\id[4] - |11><11|)\otimes \id[2], i.e. by explicitly using the structure of the Toffoli, I think one can derive a much shorter formula. Similar to the fact that the CNOT formula is much shorter than enumerating its update on all Paulis.
Or maybe one could build the large formula and then give it to a (CNF)-minimizer?