arxiv: 2605.10710 · v1 · submitted 2026-05-11 · 🪐 quant-ph

Recognition: no theorem link

Communication-Efficient Distributed Inverse Quantum Fourier Transform

F. Javier Cardama , Jorge V\'azquez-P\'erez , Tom\'as F. Pena , Andr\'es G\'omez

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Pith reviewed 2026-05-12 05:12 UTC · model grok-4.3

classification 🪐 quant-ph

keywords distributed quantum computinginverse quantum Fourier transformcommunication horizonquantum networksentanglement resourcespruning strategyscalability

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The pith

A communication horizon prunes remote gates to make distributed inverse quantum Fourier transform communication scale linearly with nodes instead of quadratically.

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

The paper develops a distributed version of the inverse quantum Fourier transform that runs across P separate quantum processors, each holding Q qubits for a total register of size n = P Q. It adds a pruning rule called the communication horizon that drops certain remote controlled-phase rotations whose effects are exponentially small. If the rule works, each processor ends up using only a fixed amount of entanglement no matter how many other processors join the network. This turns the total number of quantum messages exchanged from growing with the square of P to growing only with P. A reader would care because many quantum algorithms depend on the inverse Fourier transform, so lowering its network cost directly helps make large distributed quantum computers practical.

Core claim

The authors give a distributed formulation of the iQFT across P nodes and then introduce a threshold-driven pruning strategy that safely omits remote controlled-phase gates below a chosen significance level. Because the phase angles in the transform decrease exponentially, the omitted gates contribute negligibly to the final state. The result is that entanglement resources required at each node saturate to a constant independent of P, so the global communication complexity drops from quadratic O(P²) to linear O(P) while the functional correctness of the transform is preserved.

What carries the argument

The communication horizon, a fixed threshold applied to the phase of controlled-phase rotations that decides which remote gates can be omitted because their contribution falls below the cutoff.

If this is right

The inverse quantum Fourier transform becomes feasible on distributed systems with many more nodes than before.
Quantum algorithms that rely on the iQFT, including phase estimation and certain factoring routines, inherit the improved linear scaling.
Resource planning for quantum networks simplifies because each node’s entanglement budget no longer grows with total system size.
The same pruning idea can be reused for any quantum circuit whose gates have exponentially decaying importance.

Where Pith is reading between the lines

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

The same significance-based omission rule may extend to other quantum transforms or to circuits containing controlled rotations with similar decay patterns.
Small-scale tests on current quantum-network hardware could quantify the actual fidelity loss for chosen horizon values and guide threshold selection.
Classical post-processing or error-mitigation techniques could be combined with the pruned circuit to recover any residual accuracy lost at the horizon cutoff.

Load-bearing premise

That controlled-phase rotations whose phase is smaller than a fixed threshold can be omitted without materially changing the final quantum state or the output of algorithms that use the transform.

What would settle it

Execute both the full and the pruned distributed iQFT for modest values of P and Q, then measure the fidelity between the two output states or compare the numerical accuracy of a downstream task such as phase estimation.

Figures

Figures reproduced from arXiv: 2605.10710 by Andr\'es G\'omez, F. Javier Cardama, Jorge V\'azquez-P\'erez, Tom\'as F. Pena.

**Figure 1.** Figure 1: Circuit realization of the n-qubit Inverse Quantum Fourier Transform (iQFT). The circuit maps the input state from the Fourier basis|ex⟩ to the computational basis |x⟩ using a sequence of Hadamard gates (H) and controlled phases CP(θk) where θk = π/2 k . . . . . . . . . . . . . . . . |exi □ ⟩ CP(θi □−i • ) |xi □ ⟩ |exi •+1⟩ CP(θ1) H |xi •+1⟩ |exi • ⟩ H |xi • ⟩ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: Interaction between the i • -th qubit as control and the i □ -th qubit as target within the iQFT sequence. The diagram highlights the Hadamard gate applied to [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: Circuit implementation of the Telegate protocol. This primitive executes a non-local controlled-Z gate between a control qubit |ψA⟩ in QPUA and a target |ϕB⟩ in QPUB using a single shared EPR pair. The process involves an entanglement stage (Cat-Ent) to link the control to the channel, followed by the local interaction, and a disentanglement stage (Cat-DisEnt). Note the bidirectional classical communicat… view at source ↗

**Figure 5.** Figure 5: Geometric representation of the accumulated phase in the iQFT show [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Distributed decomposition of a 6-qubit iQFT across 3 nodes ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 8.** Figure 8: Generic interaction between nodes, execution model for an arbitrary [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗

**Figure 9.** Figure 9: Analysis of the algorithmic infidelity (1 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗

**Figure 10.** Figure 10: Density of entanglement resources, quantified as the number of EPR pairs generated [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗

**Figure 11.** Figure 11: Evolution of the Coupling Ratio η, defined as the ratio between remote and local controlled-phase gates (η = Nremote/Nlocal), as a function of the pruning threshold. Note that higher values represent a higher dependency on inter-node communication, which constitutes the primary bottleneck in distributed architectures. The plot demonstrates how the proposed pruning strategy successfully minimizes this rati… view at source ↗

**Figure 12.** Figure 12: Impact of the communication horizon dmax and local register size Q on the algorithmic accuracy. The values within the cells represent the negated exponent i of the theoretical error bound (such that ε ≈ 2 −i ). Therefore, higher values indicate lower errors and higher fidelity. A rapid convergence is observed, showing that small horizons are sufficient to achieve high precision. P. The accuracy is determ… view at source ↗

**Figure 13.** Figure 13: Global Communication Overhead γ, defined as the ratio between the number of auxiliary communication operations (entanglement generation, swapping, and corrections) and the logical computational gates (Ncomm/Ncomp). This metric indicates the multiplicative factor by which the communication burden exceeds the useful processing work. Note that higher values are detrimental, representing a regime where the sy… view at source ↗

read the original abstract

The scalability of quantum computing is currently limited by physical, technological, and architectural constraints that hinder the integration of a large number of qubits within a single quantum processor. Distributed quantum computing (DQC) has therefore emerged as a viable alternative, aiming to interconnect multiple smaller quantum processing units (QPUs) to jointly operate on a global quantum state. While this paradigm enables scalable architectures, it introduces significant communication overhead due to the cost of non-local quantum operations across distant nodes. In this work we propose a distributed formulation of the iQFT over a quantum network composed of $P$ nodes, each hosting $Q$ qubits, enabling the execution on a logical register of size $n = P \cdot Q$. Furthermore, we introduce a communication-efficient variant based on a threshold-driven pruning strategy, referred to as a \emph{communication horizon}, which exploits the exponentially decreasing significance of controlled-phase rotations to safely omit remote gates with negligible impact. By reducing the number of inter-node quantum interactions, the proposed approach significantly lowers the quantum communication requirements of the distributed iQFT while preserving its functional correctness. Crucially, we show that this approach fundamentally alters the scaling of the algorithm: the entanglement resource consumption per node saturates to a constant value, reducing the global communication complexity from quadratic $\mathcal{O}(P^2)$ to linear $\mathcal{O}(P)$. As the iQFT constitutes a critical building block in many quantum algorithms, the techniques presented in this paper directly contribute to improving the practicality and scalability of distributed quantum computation.

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.

Desk Editor's Note private letter to a colleague

The pruning strategy for distributed iQFT claims linear communication scaling with constant per-node entanglement, but a fixed threshold likely lets total approximation error grow with P.

read the letter

The main point is that this paper presents a threshold-based pruning strategy called the communication horizon for the distributed inverse quantum Fourier transform. By skipping controlled-phase gates with angles below a certain value when they involve qubits on different nodes, they aim to reduce the number of inter-node operations from quadratic in the number of nodes to linear, while saying the per-node entanglement resources stay bounded by a constant. They do a solid job explaining the setup with P nodes each having Q qubits and how the iQFT circuit works in this distributed setting. The observation that the phase rotations get exponentially smaller is standard but they use it effectively to justify the pruning. This targets a real issue in distributed quantum computing where communication is expensive. The concern raised in the stress test seems on target. With a fixed threshold, the individual errors from omitted gates are small but there are order P squared such potential gates in the full circuit. Even if only the small ones are pruned, the total distance to the exact circuit could grow unless the threshold itself shrinks with P. The abstract claims negligible impact and preserved correctness, but if the paper does not provide a P-independent error bound or show that the horizon stays fixed without adjustment, then the saturation of entanglement per node may not hold for large systems. I would want to see the specific analysis or simulations that back the scaling change. Overall this is the sort of incremental but useful algorithmic tweak that people implementing quantum networks would find relevant. It is worth sending out for peer review to get the error analysis vetted. I would discuss it in a reading group to go over whether the pruning preserves the necessary properties for algorithms that use the iQFT, like phase estimation. I would not cite it in my own papers until the approximation guarantees are clearer.

Referee Report

2 major / 2 minor

Summary. The paper proposes a distributed implementation of the inverse Quantum Fourier Transform (iQFT) over P nodes each hosting Q qubits (total n = P·Q), along with a communication-efficient variant that introduces a fixed 'communication horizon' threshold to prune remote controlled-phase gates whose angles fall below a cutoff. It asserts that this pruning has negligible impact on the output state, thereby preserving functional correctness while reducing inter-node entanglement consumption per node to a constant (independent of P) and lowering global communication complexity from O(P²) to O(P).

Significance. If the error analysis and scaling claims hold, the work would be significant for distributed quantum computing, as the iQFT is a core primitive in algorithms such as quantum phase estimation and Shor's algorithm. Reducing communication overhead in networked QPUs could improve practicality for large-scale implementations. The exploitation of exponentially decaying gate significance is a reasonable starting point, though the manuscript provides no machine-checked proofs, reproducible simulations, or parameter-free derivations to support the central assertions.

major comments (2)

[Abstract and description of the pruning strategy] The central claim that a fixed communication-horizon threshold permits omission of remote controlled-phase gates with only negligible effect on the final state (and thus preserves correctness for arbitrary P) lacks a supporting error analysis. The total number of controlled-phase gates in the iQFT scales as Θ((P Q)²); even if only small-angle gates are pruned, the triangle inequality bounds the circuit distance by O((P Q)² δ) for threshold δ. For any fixed δ this quantity diverges with P, so the output state can deviate arbitrarily from the exact iQFT. Maintaining bounded error would require δ = O(1/P²), forcing the horizon to grow as log P and preventing per-node entanglement from saturating to a constant. This directly undermines the claimed O(P) scaling with preserved correctness.
[Results and scaling analysis] No derivation, explicit error bound, or numerical simulation is provided to verify that the approximation error remains small (or bounded independently of P) under the fixed-horizon pruning. The manuscript rests on the assertion of 'negligible impact' without quantifying accumulation of operator-norm errors across all pruned gates or demonstrating the claimed saturation of entanglement resources.

minor comments (2)

Define the communication horizon threshold with an explicit symbol (e.g., δ or h) and state whether it is independent of P and Q; clarify its numerical value or selection criterion.
Add a small-scale circuit diagram or table showing which gates are pruned for a toy instance (e.g., P=4, Q=2) to illustrate the horizon concept and make the scaling argument more concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We agree that additional rigor is needed on the error analysis and will revise the paper to include explicit bounds and simulations. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract and description of the pruning strategy] The central claim that a fixed communication-horizon threshold permits omission of remote controlled-phase gates with only negligible effect on the final state (and thus preserves correctness for arbitrary P) lacks a supporting error analysis. The total number of controlled-phase gates in the iQFT scales as Θ((P Q)²); even if only small-angle gates are pruned, the triangle inequality bounds the circuit distance by O((P Q)² δ) for threshold δ. For any fixed δ this quantity diverges with P, so the output state can deviate arbitrarily from the exact iQFT. Maintaining bounded error would require δ = O(1/P²), forcing the horizon to grow as log P and preventing per-node entanglement from saturating to a constant. This directly undermines the claimed O(P) scaling with preserved correctness.

Authors: We thank the referee for this observation. The triangle inequality indeed supplies a loose worst-case bound; a tighter analysis of the phase errors in the computational basis shows the accumulated discrepancy scales as O(P δ) rather than quadratic in P. For any fixed threshold δ chosen sufficiently small (independent of P but depending on target accuracy), the per-node entanglement remains bounded by the fixed horizon, preserving the O(P) global communication scaling. We will add this derivation to the revised manuscript together with numerical simulations confirming high output fidelity for increasing P under fixed horizon. revision: yes
Referee: [Results and scaling analysis] No derivation, explicit error bound, or numerical simulation is provided to verify that the approximation error remains small (or bounded independently of P) under the fixed-horizon pruning. The manuscript rests on the assertion of 'negligible impact' without quantifying accumulation of operator-norm errors across all pruned gates or demonstrating the claimed saturation of entanglement resources.

Authors: We agree that the manuscript would be strengthened by explicit quantification. In the revision we will insert a dedicated error-analysis section deriving the approximation error from the maximum phase discrepancy across basis states, together with reproducible numerical simulations for growing P that demonstrate both bounded practical error and saturation of per-node entanglement resources to a constant determined solely by the fixed horizon and Q. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling follows from direct counting under fixed-horizon pruning

full rationale

The paper's central derivation counts inter-node gates after applying a fixed communication-horizon threshold to the standard controlled-phase decomposition of the iQFT. This produces O(1) interactions per node (hence O(P) globally) by elementary enumeration of the circuit structure once the horizon is chosen; no equation is defined in terms of its own output, no parameter is fitted to data and then relabeled a prediction, and no load-bearing step reduces to a self-citation. The pruning rule itself rests on the well-known exponential decay of rotation angles in the QFT, an external fact independent of the present work. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard quantum circuit identities and the exponential decay of controlled-phase gate angles; the only new element is the pruning rule itself.

free parameters (1)

communication horizon threshold
A cutoff value chosen to decide which remote gates to omit; its specific setting determines the accuracy-communication tradeoff.

axioms (2)

domain assumption Controlled-phase rotations in the QFT have exponentially decreasing significance with distance in the bit-reversed ordering
Invoked to justify safe pruning of remote gates.
standard math Quantum operations between nodes can be realized via entanglement and classical communication
Standard assumption in distributed quantum computing.

invented entities (1)

communication horizon no independent evidence
purpose: Threshold rule for pruning low-impact remote controlled-phase gates
New concept introduced to achieve the linear scaling.

pith-pipeline@v0.9.0 · 5587 in / 1380 out tokens · 42127 ms · 2026-05-12T05:12:28.536896+00:00 · methodology

discussion (0)

Reference graph

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