Quantum Gate Scheduler: 2D Greedy List-Scheduling

The 3D quantum gate scheduler shows the two textbook extremes of compiling a program of G gate operations across a qubit grid: a sequential schedule (one gate per time step) and an idealized parallel schedule assuming the load splits perfectly evenly across all Q qubits, ceil(G/Q). This 2D companion computes something genuinely different: it randomly assigns each of the G gates to a qubit — the way a real, non-uniform program actually behaves — and then schedules the parallel case with an authentic greedy list-scheduling pass, the same family of algorithm real transpilers use, rather than assuming the ideal bound holds. The result, verified numerically against the idealized formula, is a real circuit depth that is almost always higher than ceil(G/Q) once the gate-to-qubit load is uneven, and a Gantt-style timeline beneath the qubit map — a genuinely 2D-native diagnostic with no 3D equivalent — shows exactly which qubit's gate backlog is setting the floor under how parallel the schedule can actually get. The same fidelity model, F(N) = (1-ε)^N · exp(-N·t_gate/T2), is then applied to this honest, load-imbalanced depth, so the fidelity gain readout reflects what a real transpiler delivers rather than a best-case number.