The Hilbert-Pólya conjecture proposes that the non-trivial zeros of the Riemann zeta function correspond to the eigenenergy spectrum of a quantum system. Although experiments based on analog quantum probes have determined some low-order zeros, the classical computational bandwidth bottleneck—caused by synthesizing complex macroscopic driving waveforms—has severely hindered the extension of this approach to high-frequency spectra. Here, we propose a native digital quantum emulation framework that maps a continuous $\phi^4$ topological potential well into a Pauli tensor network based on first principles, physically deploying it on a real superconducting quantum processor (Rigetti Ankaa-3). Using a 3-qubit array, we accurately extracted the first non-trivial zero (14.134725). Addressing the "semiclassical aliasing" spectrum divergence triggered by limited local phase space capacity and depleted coherence time (~100 µs) when climbing to higher-order zeros, we introduce a classical-quantum hybrid Quantum Subspace Expansion (QSE) protocol. By injecting an adiabatic geometric phase into an expanded 8-qubit quantum manifold, the system effectively corrected the truncation divergence, accurately restoring the Weyl logarithmic density law (Mean Absolute Error MAE = 0.569), and physically quantified the decoherence leakage rate (~10%) and shot thermal drift on the actual hardware. This native digital method bypasses the computational paradox of classical pre-computation, predicting that in the future, only 14 fault-tolerant logical qubits will be needed to natively resolve the first 10,000 high-frequency zeros. This study provides a viable path for natively emulating quantum chaos on universal quantum architectures.