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

Abstract: Quantum phase estimation~(QPE) is central to numerous quantum algorithms, yet its standard implementation demands an $\calO(m^{2})$-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~$\eps$. Our central result establishes $\TV(P_{\varphi},P_{\varphi}^{d})\leqπ(m{-}d)/2^{d}$, showing that for $d=\calO(\log m)$ circuit size collapses from $\calO(m^{2})$ to $\calO(m\log m)$ while estimation error grows by at most $\calO(2^{-d})$. We characterise $\dstar=\Floor{\log_{2}(2π/\eps_{2q})}$ 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 $\eps_{2q}\gtrsim 2\times10^{-3}$.

• The paper introduces the Phase-Fidelity-Aware Truncated Quantum Fourier Transform (PFA-TQFT), optimizing QFT for NISQ devices by aligning truncation depth with hardware limits.
• PFA-TQFT reduces gate count from O(m²) to O(m log m) by omitting gates with negligible phase contribution, effectively leveraging NISQ noise constraints.
• This framework identifies a 'fidelity cliff,' highlighting conditions where the truncated method outperforms traditional QFT under realistic NISQ noise settings.

Phase-Fidelity-Aware Truncated QFT: A Hardware-Calibrated Paradigm for NISQ-Optimal Quantum Phase Estimation

Problem Motivation and Prior Constraints

Quantum Phase Estimation (QPE) is foundational to several quantum algorithms, including quantum simulation, Shor's algorithm, and the Harrow-Hassidim-Lloyd (HHL) algorithm. The canonical QFT used in QPE requires $O(m^2)$ two-qubit controlled-phase gates for $m$ control qubits, directly conflicting with the circuit depth and error rates tolerated by contemporary NISQ devices. While prior work, beginning with Coppersmith's truncation of small-angle rotations [quant-ph/0201067], Barenco et al. [Phys. Rev. A 54, 139], and more recent analytical advances [quant-ph/0403071, (Häner et al., 2018)], established that controlled-phase rotations below a threshold can be omitted, these works lack:

1. Closed-form bounds for truncation-induced distributional phase error under realistic (noisy) hardware assumptions,
2. A device-calibrated, analytically optimal truncation depth,
3. An analysis accounting for the interplay (and possible synergy) between truncation and noise—specifically, conditions under which aggressive truncation can outperform the full QFT under NISQ noise.

The present work addresses all three and provides a framework for optimal QFT truncation targeted to hardware characteristics.

PFA-TQFT Framework: Construct and Theoretical Guarantees

The authors introduce the Phase-Fidelity-Aware Truncated QFT (PFA-TQFT), explicitly parameterized by a truncation depth $d$ derived from the two-qubit gate fidelity $\varepsilon_{2q}$ of a given hardware platform. Controlled-phase gates $R_k$ with rotation angles $\theta_k < 2\pi/2^{d}$ are omitted, and $d$ is chosen such that their omission incurs phase errors no greater than that of a noisy gate implementation.

The main structural and analytical results can be summarized as follows:

• Definition: For an $m$-qubit QFT, the PFA-TQFT applies only those controlled-phase gates with $k \leq d$, where $d = \lfloor \log_2(2 \pi/\varepsilon_{2q}) \rfloor$.
• Error Bound: The total variation distance (TVD) between the output distributions of the full and truncated QFT satisfies $m$0 for all eigenphases $m$1 and all $m$2 (Theorem 1).
• Resource Scaling: The gate count reduces from $m$3 to $m$4, i.e., for $m$5, the scaling is $m$6.
• Hardware Calibration: The optimal truncation depth $m$7 is platform-specific. For instance, with $m$8 for IBM Eagle, $m$9 yields a $d$0 gate-count reduction at $d$1.
• Fidelity Cliff and Success Probability: There exists a sharp "fidelity cliff" at $d$2, below which performance collapses and above which the success probability saturates to the full QFT.

This hardware-calibrated truncation criterion is both analytically motivated and operationally simple, involving only a one-line formula derived from device-specific error rates.

Figure 1: Gate-level comparison between a full QFT and PFA-TQFT ($d$3) for $d$4; PFA-TQFT omits four small-angle gates, yielding significant gate reduction even for modest system sizes.

Numerical and Analytical Validation

Distributional and Fidelity Analysis

The TVD bound is benchmarked against Monte Carlo simulations across a range of $d$5 and $d$6. The theoretical bound matches the empirical maxima within a constant factor. The relation between truncation depth, system size, TVD, and phase estimation failure probability is quantitatively validated.

Figure 2: (a) TVD error bound versus truncation depth $d$7 across several $d$8; theoretical curves and empirical simulations agree. (b) Hardware-calibrated $d$9 as a function of two-qubit gate error, with demarcations for specific platforms.

A "fidelity cliff" arises as the truncation depth is reduced below $\varepsilon_{2q}$0: phase estimation accuracy drops precipitously, while beyond this threshold accuracy plateaus.

Figure 3: (a) Success probability for accurate phase estimation as a function of truncation depth $\varepsilon_{2q}$1, revealing a sharp transition ("cliff") near $\varepsilon_{2q}$2.

Resource Efficiency

PFA-TQFT delivers substantial two-qubit gate reduction while keeping the approximation-induced error within the hardware's inherent noise floor:

Figure 4: (a) Gate count scaling with $\varepsilon_{2q}$3 for full QFT and PFA-TQFT at different truncation depths; (b) Percent reduction in gate count increases with system size.

For IBM Eagle r3, Heron r2, IonQ Aria, and IQM Garnet, a single calibration line produces the device-optimal $\varepsilon_{2q}$4, as shown empirically and in tabulated results.

Figure 5: (a) Percentage gate count reduction and phase error overhead per platform at $\varepsilon_{2q}$5. (b) Comparison of absolute gate counts for full QFT versus PFA-TQFT across platforms.

Noise-Truncation Synergy

A central and perhaps counterintuitive result is that—in the presence of realistic NISQ noise—PFA-TQFT can outperform even the full QFT in estimation accuracy. For sufficiently large $\varepsilon_{2q}$6 (e.g., $\varepsilon_{2q}$7 as seen on certain NISQ devices), the noise accumulated by the additional omitted gates in the full QFT exceeds the approximation error from PFA-TQFT truncation. This regime is quantitatively characterized both by RMSE on the 1D transverse-field Ising model and by analytic crossover thresholds.

Figure 6: (a) RMSE in QPE for the TFIM as a function of $\varepsilon_{2q}$8. PFA-TQFT ($\varepsilon_{2q}$9) outperforms full QFT at high error rates, marking the practical advantage of noise-truncation synergy. (b) Scaling with $R_k$0 at fixed noise rate.

Comparison with Alternative QPE Implementations

PFA-TQFT is compared to semiclassical, Bayesian, and iterative QPE methods. Unlike these, it is a single-shot, feedback-free approach with explicit hardware-aware error bounds. While semiclassical or iterative QPE can further reduce quantum resources, they require extensive classical feedback and are not directly compatible with certain applications (e.g., those demanding low-latency quantum-to-quantum processing).

Implications and Future Research Directions

Practical Impacts

• QPE-based applications: By transforming the QFT subroutine from an $R_k$1 to an $R_k$2 component, PFA-TQFT directly extends the viable use of QPE (and thus, quantum simulation, ground state energy estimation, and more) to larger problem sizes on NISQ hardware.
• Compiler Design: The PFA-TQFT criterion is straightforward to incorporate into quantum circuit compilers (Qiskit, Cirq, tket), enabling automatic truncation optimally tuned to device characterizations.
• Quantum Algorithm Engineering: In HHL and Shor's algorithm, where QPE has traditionally been a depth bottleneck, PFA-TQFT reduces required gates by more than an order of magnitude, matching hardware noise limits.

Theoretical Implications

The closed-form TVD bound rigorously links circuit approximation to distributional error in phase estimation, not merely global fidelity or operator norm, and does so under a practically relevant noise model.

The observed noise-truncation synergy demonstrates that, under decoherence-dominated regimes, intentional circuit approximation can become an error-mitigation strategy—a paradigm shift from the traditional fault-tolerance view.

Future Directions

• Adaptive per-qubit truncation in response to spatially varying error rates across devices.
• Extensions to structured noise models (Pauli, amplitude damping, crosstalk).
• Integration with error mitigation frameworks (e.g., zero-noise extrapolation).
• Experimental demonstrations on cloud-accessible IBM and IonQ systems.
• Full integration into quantum software stacks for turn-key, hardware-aware QFT truncation.

Conclusion

PFA-TQFT provides a robust, hardware-calibrated approximate QFT variant that is theoretically well-founded and operationally simple. By ensuring that circuit complexity aligns with the error budget of practical NISQ hardware, this approach makes QPE-based quantum algorithms tractable at scales otherwise inaccessible. The synergy between approximation and hardware-induced noise further refines the role of compilation in the NISQ era, establishing truncation not only as a compromise but as a principled optimization.

References: (2604.05456), [quant-ph/0201067], [quant-ph/0403071], (Häner et al., 2018), [PhysRevA.54.139].