Two-edge-colour width-2 schedule for the XY-Trotter chain — 2026-07-21

Status: design + classical evidence complete. Addresses audit AUD-7 (B-5): hand-scheduled, genuinely width-2 chain dynamics instead of a deep serial synthesis. The hardware submission remains an owner-gated QPU run (AUD-5/7). Code: src/scpn_quantum_control/analysis/two_colour_schedule.py, tests tests/test_two_colour_schedule.py (24).

Problem

The max-width Kuramoto-XY campaign emits the 1-D chain edge-by-edge (a single PauliEvolutionGate that the transpiler synthesises serially), so each Trotter step is laid out as n−1 sequential two-qubit interactions and the two-qubit depth grows like O(reps · n). A referee can object that the reported "width" is a property of a deep serial circuit, not of parallel dynamics.

The 2-edge-colouring

The path graph is 2-edge-colourable: even-indexed edges {(0,1),(2,3),…} form colour A and odd-indexed edges {(1,2),(3,4),…} form colour B. Every edge within a colour acts on disjoint qubits, so a colour is a single parallel layer. Each Trotter step becomes: one single-qubit rz layer, then colour A, then colour B — a constant number of two-qubit layers independent of n.

This is the same first-order Trotter approximation (a reordering of the same rz / rxx·ryy terms), and every gate conserves total excitation number, so the scheduled circuit realises genuine parity-preserving width-2 dynamics.

Classical evidence (measured, this module)

• Excitation-number conservation preserved: the exact opposite-parity probability of the 2-colour circuit is < 1e-9 for every tested width, initial state, and depth — identical to the serial baseline. The reschedule does not change the physics.
• Constant two-qubit depth per step (genuine width-2): for a fixed Trotter-step count the 2-colour two-qubit depth is invariant across n = 4, 8, 16, 32 (measured: 20 layers at 5 steps = 4 layers/step), and scales linearly with the step count. The serial baseline grows with n (30 → 78 two-qubit layers for n = 8 → 32 at 5 steps), so the reduction factor increases with chain length (≈1.5× at n=8, ≈3.9× at n=32, and larger still at the campaign widths). Measured conservatively against a clean edge-by-edge rxx·ryy baseline; against the campaign's transpiled PauliEvolutionGate synthesis the reduction is larger.

What this unblocks

A hardware run of the 2-colour-scheduled circuits would test the same XY dynamics at a fraction of the depth, separating genuine width from scheduling-induced noise. That submission is owner-gated (QPU); this module and its classical evidence are the design deliverable.

Authored by Anulum Fortis & Arcane Sapience (protoscience@anulum.li) Seat: 7f6b