A Unified Benchmark for Exact Dense Quantum State Preparation

Exact, deterministic preparation of an arbitrary dense n-qubit state is the data-loading step for quantum signal and image processing, and, without ancillae, a generic target needs at least ⌈(2n − n − 1)/2⌉ controlled-NOT (CNOT) gates. We evaluate the two new Quantum Signal-induced Heap Transform (QsiHT) syntheses, Real and Complex, built on the discrete signal-induced heap transform (DsiHT), inside a nine-configuration benchmark drawn from the uniformly controlled rotation, multiplexor, isometry, Schmidt, and Quantum Shannon Decomposition (QSD) families. Qiskit’s StatePreparation is the only method benchmarked as shipped rather than re-implemented. Every configuration reproduces its Haar-random targets to machine precision, with the Real Synthesis confined to sign-bearing targets. Under noise, one coarse tier is stable, set by the executed two-qubit count; the O(2n) frontier stays well below QSD’s O(4n), and no finer order reproduces across two IBM Heron r2 processors at n = 3. Both syntheses reach the 2n − n − 1 multiplexor-family floor. That count matches StatePreparation and, as built, undercuts every other from-scratch construction from n = 3 up. Under a uniform optimization-level-3 pass Plesch–Brukner falls below it at n = 4 and on real targets at n = 6, and as built, it keeps the lowest constant on complex data among the from-scratch methods above that floor. At large registers, the Real Synthesis has the cheapest build-time cost and the Complex Synthesis handles arbitrary targets.