Connectivity and Topology
A coupling map is the graph of qubit pairs that can host a two-qubit gate; any pair off the graph must be routed with SWAPs, and each SWAP costs three CX gates. A single long-range CX on a five-qubit line becomes ten two-qubit gates, so topology directly sets circuit depth and error — it is an executable constraint, not a diagram.
What is a coupling map?
A QPU's topology, written down as a coupling map, is a graph: each node is a physical qubit, and each edge joins a pair of qubits that can take part in a two-qubit gate together. If there is no edge between two qubits, no gate can act directly on that pair — full stop.
Think of it as a road map: two-qubit gates can only travel along existing roads. The analogy is honest as far as it goes. Where it breaks: a coupling map can be directed — an edge may support the gate in one orientation only, like a one-way street — and it is version-specific, because a hardware revision or recalibration can add, remove, or re-rate edges. Always check which device version a map describes.
When your circuit asks for a gate between two unconnected qubits, the compiler must do routing: rearranging where the logical qubits live, using SWAP gates, until the required pair sits on an edge. Routing is not free, and the next example prices it exactly.
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