Cosine Response Spectrum (CRS): Full-Quantum Cosine Wave Generation, Dual-Platform Real-Machine Verification, and Quantum Number-Theoretic Applications

This paper proposes the Cosine Response Spectrum (CRS) method, which encodes a continuous cosine function in real time into quantum state ampli- tudes using only a single ancilla qubit and controlled-phase gates, without requiring a Quantum Fourier Transform. The mathematical rigor of the cir- cuit has been verified across three independent simulation platforms—Qiskit, MindQuantum, and pyQPanda—with a precision of 10^−17. In a total of 17 real-machine tests on a superconducting platform (Tianyan 176) and a trapped-ion platform (Qudoor Quantum), the trapped-ion system achieved a 90% single-peak emergence probability and, for the first time, simultaneous dual-peak passage (4/10). The post-selection pass rate dropped from 60% on the superconducting platform to 49% on the trapped-ion platform, indicating significantly reduced channel contamination. The CRS method maintains a 90% emergence probability even on a legacy device with a two-qubit gate fi- delity of only 95%, demonstrating exceptional robustness to hardware noise. The noise analysis reveals that the extremely short T2 decoherence time of the superconducting platform (approximately 21 μs) is the physical root cause of the inflated pass rate and signal washout, while the trapped-ion platform’s long coherence time of approximately 600 ms compensates for its lower gate fi- delities. Additional post-calibration comparison tests on the Tianyan 176 fur- ther confirm that although improvements in gate fidelity can purify the post- selection channel, they cannot enhance signal emergence. This establishes T2 decoherence as the dominant bottleneck for the superconducting platform. With its streamlined circuit structure and low resource requirements, CRS provides an efficient building block for quantum state preparation, quantum number-theoretic computation, and cross-platform benchmarking in the NISQ era.