When More Becomes Less: Topology-Reversed Three-Qubit Gate Performance on IBM Quantum Processors
The exact Toffoli gate admits a six-CX decomposition, denoted by $CCX_6$, that is optimal under unrestricted two-qubit connectivity. On a linear three-qubit topology, however, $CCX_6$ contains 4 nearest-neighbor CX gates and 2 non-nearest-neighbor CX gates. By contrast, an alternative exact decomposition, denoted by $CCX_8$, uses only 8 nearest-neighbor CX gates. Because CCX is locally equivalent to CCZ, we perform the experiments using the corresponding $\cczs$ and $\cczl$ circuits. We compare these circuits on sampled linear triples of the 156-qubit IBM Quantum Heron processors \texttt{ibm\_fez} and \texttt{ibm\_kingston}. Under the compilation protocol, the nominal $\cczs$ circuit becomes a twelve-CZ implementation, whereas the linear-nearest-neighbor circuit retains eight native CZ gates. Experimentally measured ensemble-feature-selection estimates favor the eight-CZ realization on nearly all retained triples. We test the same ordering by preparing a three-qubit hypergraph state, which probes the coherent conditional phase rather than only computational-basis populations. The measured hypergraph-state infidelity is lower for the $\cczl$ circuit for most triples on both processors. Phase-altered interleaved randomized benchmarking provides a complementary comparison of Clifford surrogates preserving the two compiled entangling structures. Within the scope of the tested circuits and phase-sensitive input state, the results demonstrate that hardware connectivity can reverse the operational ranking of exact decompositions: a circuit with more abstract two-qubit gates can yield the better physical implementation.