Cosine Response Spectrum for Number-Theoretic Quantum State Preparation and Verification

I propose the Cosine Response Spectrum (CRS) framework—a resource-efficient approach for preparing quantum states whose amplitudes exhibit number- theoretic periodic structures, specifically tailored for noisy intermediate-scale quantum (NISQ) devices. CRS employs a single ancilla qubit and controlled- phase gates to directly encode a continuous cosine function into the amplitudes of a target quantum state. The encoding parameters are uniquely determined by modulo-30 orbit regularization theory. Under a 210-period encoding scheme, the experimentally measured interference pattern displays nine equally spaced prob- ability peaks, each corresponding to composite numbers containing the prime factor 7 on orbit r = 1. I conducted 80 independent experimental runs across eight distinct configu- rations on a trapped-ion quantum processor. Using phase-folded cosine fitting (PCF), I reconstructed the global interference pattern; all nine peaks are statis- tically significant at the individual level. In ten independent runs (test T7), the PCF fit achieves significance levels ranging from 60.1σ to 75.8σ, with highly consistent amplitude and phase estimates across runs. The measured output exhibits a directly observable cosine waveform, characterized by complementary peak–valley structure across two symmetric branches. PCF serves as a practical and robust tool for verifying NISQ-era algorithms, while CRS provides a hardware-efficient testbed for function-encoding quantum cir- cuits, particularly on trapped-ion platforms. All experimental data and analysis scripts are publicly available under CC BY 4.0.