Phase-Fidelity-Aware Truncated Quantum Fourier Transform for Scalable Phase Estimation on NISQ Hardware

Quantum phase estimation~(QPE) is central to numerous quantum algorithms, yet its standard implementation demands an $\text{calO}(m^2)$-gate quantum Fourier transform~(QFT) on $m$ control qubits-a prohibitive overhead on near-term noisy intermediate-scale quantum (NISQ) devices. We introduce the \emph{Phase-Fidelity-Aware Truncated QFT} (PFA-TQFT), a family of approximate QFT circuits parameterised by a truncation depth~$d$ that omits controlled-phase rotations below a hardware-calibrated fidelity threshold~$\text{eps}$. Our central result establishes $\text{TV}(P_{\phi },P_{\phi }^d)\le \pi (m-d)\mathrm{/}{2}^d$, showing that for $d=\text{calO}(\log m)$ circuit size collapses from $\text{calO}(m^2)$ to $\text{calO}(m\log m)$ while estimation error grows by at most $\text{calO}(2^{-d})$. We characterise $\text{dstar}=\text{Floor}{\log }_2(2\pi \mathrm{/}{\text{eps}}_{2q})$ directly from native gate fidelities, demonstrating 31.3 -43.7% at m = 30, gate-count reduction on IBM Eagle/Heron and IonQ~Aria with negligible accuracy loss. Numerical experiments on the transverse-field Ising model confirm all theoretical predictions and reveal a \emph{noise-truncation synergy}: PFA-TQFT outperforms full QFT under NISQ noise ${\text{eps}}_{2q}\gtrsim 2\times {10}^{-3}$.