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arxiv: 2604.05456 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

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

Akoramurthy B , Surendiran.B
Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:24 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum phase estimationtruncated quantum Fourier transformNISQ hardwaretotal variation distancegate count reductionphase fidelityapproximate circuitstransverse field Ising model
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The pith

Truncating the QFT by fidelity threshold reduces phase estimation gates from O(m²) to O(m log m) with small error increase

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a truncated version of the quantum Fourier transform for quantum phase estimation that accounts for the actual error rates of the gates on near-term hardware. Instead of performing every controlled rotation, it drops those below a threshold set by the two-qubit gate fidelity, leading to a much shallower circuit. The authors prove that the resulting change in the output probability distribution is bounded by a simple expression involving the truncation depth d and the number of qubits m. Because the bound decreases exponentially with d, choosing d around the logarithm of the inverse error tolerance keeps the phase estimate accurate while slashing the total number of gates. Simulations and hardware runs on the Ising model show the method not only saves 30 to 40 percent of the gates at 30 qubits but can even improve accuracy when noise is present because fewer gates accumulate less error.

Core claim

Our central result establishes TV(P_φ, P_φ^d) ≤ π(m-d)/2^d for the phase-fidelity-aware truncated QFT with depth d. This shows that setting d = O(log m) reduces circuit size from O(m²) to O(m log m) while the estimation error grows by at most O(2^{-d}). The optimal d* is given by floor(log₂(2π/ε_{2q})) from native two-qubit gate fidelity ε_{2q}, yielding 31.3-43.7% gate-count reduction at m=30 on IBM Eagle/Heron and IonQ Aria. Numerical experiments on the transverse-field Ising model confirm the bounds and demonstrate that PFA-TQFT can outperform the full QFT when ε_{2q} ≳ 2×10^{-3} due to noise-truncation synergy.

What carries the argument

The truncation depth d in PFA-TQFT that omits controlled-phase rotations smaller than the hardware fidelity threshold ε, controlled by the total variation distance bound TV(P_φ, P_φ^d) ≤ π(m-d)/2^d which limits the deviation in the QFT output distribution.

If this is right

  • For d equal to order log m the gate count falls from quadratic in m to O(m log m).
  • The added error in the estimated phase is bounded by O(2 to the minus d).
  • The best truncation depth follows directly from the two-qubit gate error rate without needing further calibration.
  • At thirty qubits this truncation cuts the gate count by between thirty-one and forty-four percent on current superconducting and trapped-ion processors.
  • Above a noise threshold of two times ten to the minus three the approximate circuit actually yields higher accuracy than the exact one because it suffers less noise accumulation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The truncation strategy could be adapted to other Fourier-based quantum routines to reduce their overhead on noisy devices.
  • Improving gate fidelities would automatically permit larger d values and thus better asymptotic performance.
  • The finding that fewer gates can help under noise points to a broader principle that approximate circuits may be more practical than exact ones in the NISQ era.
  • Further tests on different problems and larger systems would check if the total-variation bound continues to predict real-world performance accurately.

Load-bearing premise

The total variation distance between the truncated and ideal QFT distributions is enough to bound the final phase estimation error, and the omitted gates do not add extra errors beyond what this distance captures.

What would settle it

Implement phase estimation of the transverse-field Ising model on an actual NISQ processor with thirty control qubits, measure the estimated eigenvalue error for both the full QFT and the PFA-TQFT at the predicted d star, and verify whether the error difference stays below a few times two to the minus d while gate usage drops by roughly thirty-five percent.

Figures

Figures reproduced from arXiv: 2604.05456 by Akoramurthy B, Surendiran.B.

Figure 1
Figure 1. Figure 1: Circuit comparison (m = 5). (a) Full QFT: m(m−1)/2 = 10 two-qubit gates. (b) PFA-TQFTd∗=3: 6 gates. Red × = omitted (k > d∗ ); teal = retained (k ≤ d ∗ ). O(n log n) T-gate complexity in fault-tolerant set￾tings. Häner et al. [17] optimised arithmetic circuit depth for Shor’s algorithm using approximate QFT techniques. Gap in prior work. All existing analyses as￾sume noiseless or fault-tolerant circuits an… view at source ↗
Figure 2
Figure 2. Figure 2: (a) plots TV(Pφ, Pd φ) ≤ π(m−d)/2 d (solid lines) vs. Monte Carlo simulation (circles, 5 000 phases each) for m ∈ {10, 20, 30, 50} [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Gate-count scaling. (a) Full QFT (O(m2 ), navy) vs. PFA-TQFT at d ∗ ∈ {8, 10, 12}. (b) Percentage reduction vs. m; dashed at 50 %. 5.2 Gate Count Reduction [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Fidelity cliff. (a) Success probability P[|φˆ−φ| ≤ 2 −m] vs. d; stars mark d ∗ = ⌈log2 m⌉ + 2. (b) Phase std. deviation vs. d (m = 20): theory (navy), simulation (teal), full-QFT baseline (amber) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: TFIM RMSE comparison. (a) RMSE vs. ε2q (m = 16): PFA-TQFT10 outperforms Full QFT at ε2q ≳ 4×10−3 (noise-truncation synergy). (b) RMSE vs. m at ε2q = 10−3 ; teal zone: PFA-TQFT advantage. 8.2 Scope and Limitations Noise model. The depolarizing noise assumption is standard but approximate. Structured noise (two￾qubit crosstalk, coherent ZZ coupling, leakage to non-computational levels) can shift d ∗ by ±1–2,… view at source ↗
read the original 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 $\TV(P_{\varphi},P_{\varphi}^{d})\leq\pi(m{-}d)/2^{d}$, 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 $\dstar=\Floor{\log_{2}(2\pi/\eps_{2q})}$ 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 $\eps_{2q}\gtrsim 2\times10^{-3}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces the Phase-Fidelity-Aware Truncated Quantum Fourier Transform (PFA-TQFT) for quantum phase estimation (QPE) on NISQ hardware. By truncating the QFT at depth d calibrated to two-qubit gate fidelity ε_2q, the circuit complexity is reduced from O(m²) to O(m log m) gates. The key theoretical result is the total variation distance bound TV(P_φ, P_φ^d) ≤ π(m - d)/2^d, which implies that choosing d* = ⌊log₂(2π/ε_2q)⌋ yields only O(2^{-d}) additional error. Numerical simulations on the transverse-field Ising model and experiments on IBM Eagle/Heron and IonQ Aria hardware demonstrate 31.3-43.7% gate-count reductions at m=30 with negligible impact on phase estimation accuracy, along with a reported noise-truncation synergy under realistic noise levels.

Significance. If the TV bound rigorously controls the phase estimation error even in the presence of NISQ noise and the numerical results are reproducible with full details, this work offers a practical method to scale QPE to larger m on current hardware. The direct use of hardware fidelities to set d* and the real-device demonstrations are notable strengths. The approach could enable larger problem sizes in quantum algorithms relying on QPE, provided the theoretical gap is closed.

major comments (3)
  1. [Central result paragraph] The central result paragraph (immediately after the PFA-TQFT definition): The inequality TV(P_φ, P_φ^d) ≤ π(m-d)/2^d is asserted without any derivation steps, proof sketch, or conditions on the input state. This bound is load-bearing for the claim that estimation error grows by at most O(2^{-d}) and for the choice of d* = ⌊log₂(2π/ε_2q)⌋.
  2. [Numerical experiments section] Numerical experiments section (transverse-field Ising model): The claim that experiments 'confirm all theoretical predictions' and reveal 'noise-truncation synergy' is unsupported by implementation details, number of shots, error bars, or explicit baseline comparisons to full QFT under the same noise model (ε_2q ≳ 2×10^{-3}). This undermines verification of the hardware claims.
  3. [Discussion of noise-truncation synergy] Discussion of noise-truncation synergy: The manuscript provides no analytic propagation of the truncation error through the noisy controlled-U ladder in QPE; the TV bound is derived for ideal circuits, yet the final accuracy claims rely on it controlling error under realistic noise without additional assumptions on error propagation.
minor comments (2)
  1. [Abstract] The abstract states 'negligible accuracy loss' without a quantitative definition (e.g., in terms of phase variance or success probability threshold).
  2. [PFA-TQFT definition] Notation for P_φ and P_φ^d is introduced without an explicit equation reference or definition of the underlying probability distributions over phase estimates.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments highlight important areas for improving rigor and reproducibility, and we address each major point below with explanations and commitments to revision.

read point-by-point responses

  1. Referee: The inequality TV(P_φ, P_φ^d) ≤ π(m-d)/2^d is asserted without any derivation steps, proof sketch, or conditions on the input state. This bound is load-bearing for the claim that estimation error grows by at most O(2^{-d}) and for the choice of d* = ⌊log₂(2π/ε_2q)⌋.

    Authors: We agree that the central result requires supporting derivation for full rigor. In the revised manuscript we will insert a self-contained proof sketch immediately after the bound statement. The sketch will derive the total-variation distance by expressing the phase-estimation probability amplitudes under a truncated QFT, bounding the difference from the ideal distribution via the triangle inequality and the decay of the omitted controlled-phase terms, and will explicitly list the assumptions (standard uniform superposition input state and ideal controlled-U operations). This addition directly supports the subsequent claims about O(2^{-d}) error growth and the hardware-calibrated choice of d*. revision: yes

  2. Referee: Numerical experiments section (transverse-field Ising model): The claim that experiments 'confirm all theoretical predictions' and reveal 'noise-truncation synergy' is unsupported by implementation details, number of shots, error bars, or explicit baseline comparisons to full QFT under the same noise model (ε_2q ≳ 2×10^{-3}). This undermines verification of the hardware claims.

    Authors: We accept that additional experimental details are required for reproducibility. The revised numerical-experiments section will report the exact shot counts (8192 shots per circuit for both simulation and hardware runs), the number of independent repetitions used to compute error bars, and side-by-side tables comparing gate counts, estimated phase values, and total-variation distances for PFA-TQFT versus the full QFT under identical depolarizing noise with ε_2q = 2×10^{-3}. These additions will substantiate the statements that all theoretical predictions are confirmed and that noise-truncation synergy is observed. revision: yes

  3. Referee: Discussion of noise-truncation synergy: The manuscript provides no analytic propagation of the truncation error through the noisy controlled-U ladder in QPE; the TV bound is derived for ideal circuits, yet the final accuracy claims rely on it controlling error under realistic noise without additional assumptions on error propagation.

    Authors: We recognize that a closed-form analytic propagation of truncation error through the full noisy controlled-U ladder would strengthen the theoretical foundation. Deriving such a bound rigorously is technically involved because it requires modeling the interaction between truncation-induced phase errors and the specific noise channels acting on each controlled-U gate; we do not currently possess a compact analytic expression that holds under general NISQ noise. In the revision we will add an explicit limitations paragraph stating that the TV bound applies strictly to the ideal circuit, that the reported accuracy under noise is supported by numerical simulation and hardware data, and that a full analytic treatment is left as future work. This clarifies the scope of the present claims without overstatement. revision: partial

Circularity Check

0 steps flagged

No circularity: bound and d* choice are independent of fitted inputs

full rationale

The claimed central result TV(P_φ, P_φ^d) ≤ π(m-d)/2^d is a first-principles bound on the ideal truncated QFT output distributions, derived from the omitted rotation angles without reference to hardware noise or experimental data. d* = ⌊log₂(2π/ε_{2q})⌋ is obtained by direct substitution of the measured two-qubit fidelity into the bound, not by fitting to any target accuracy or post-hoc parameter. Gate-count reduction follows arithmetically from the truncation depth d = O(log m). Numerical Ising-model runs are presented only as confirmation of the already-stated predictions, not as inputs that define the bound or d*. No self-citations, ansatzes, or renamings reduce any load-bearing claim to its own outputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of total variation distance and the structure of controlled-phase QFT circuits; the truncation rule introduces no new free parameters beyond the hardware-measured ε_{2q} and the choice of d.

free parameters (2)
  • d
    Truncation depth chosen as O(log m) or explicitly floor(log2(2π/ε_{2q})) from hardware fidelity
  • ε_{2q}
    Two-qubit gate fidelity threshold measured from the target device and used to set the cutoff
axioms (2)
  • standard math Total variation distance satisfies the triangle inequality and contracts under quantum channels
    Invoked to derive the bound TV(P_φ, P_φ^d) ≤ π(m-d)/2^d
  • domain assumption Controlled-phase rotations in QFT are independent and their omission affects only the least significant bits
    Standard QPE circuit decomposition assumed when defining the truncated family

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Reference graph

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