Quantum Compilation of Conservative Matrix Field Products

We systematically compare seven quantum gate constructions for implementing products of rank-2 conservative matrix field (CMF) transition matrices on a single system qubit. The constructions span exact block encodings (Halmos dilation, sequential local encodings, Pauli LCU), approximate unitary surrogates (polar factor, Hermitian-dilation evolution), and CMF- specific compilation strategies (flatness-aware path selection, telescoping gauge optimization). Using a deterministic benchmark with three stiffness regimes and a Monte Carlo stress test over 200 random flat rank-2 LDU fields, we evaluate conditional state fidelity, postselection success probability, transpiled circuit depth, and noisy performance under depolarizing error. All exact constructions achieve ideal conditional fidelity 1 to numerical precision. The key trade-off is success probability: for the stiff regime (5, 4), the normalization product drops from 13.55 on the lexicographic path to 3.15 after path and gauge optimization—a 18.5× improvement in postselection success. We then validate on three authentic CMF families (e, ζ(3), π) and a non-flat 2F1 control across 130 trajectories, formulate the gauge problem as constrained gauge optimization, derive the exact postselected Hermitian-dilation operator, and present HD as a true Pareto family in the success–fidelity trade-off.