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arxiv: 2604.15051 · v1 · submitted 2026-04-16 · 🪐 quant-ph

Recognition: unknown

Hardware Validation of DAGI via a Modular "Ridge" Signature and High-Order Synergistic Information

Petr Sramek
Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:23 UTC · model grok-4.3

classification 🪐 quant-ph
keywords DAGIsynergistic informationmodular ridgequantum hardwarehigh-order synergyMöbius inversionIBM Quantumtargeted synergy
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The pith

The DAGI framework detects high-order synergistic information tied to a modular algebraic constraint that survives quantum hardware noise.

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

The paper implements a controlled circuit on ibm_torino in which two 4-bit registers obey the relation v ≡ k · u mod 16, producing an ideal low-dimensional ridge in the joint output distribution. Despite hardware imperfections, the ridge remains visible at 2.93 times the uniform baseline probability, and per-shot key recovery exceeds chance. A Möbius-inversion decomposition of mutual information between the key and detector-bit subsets isolates statistically significant positive targeted synergy specifically at order 3, while single-bit marginals stay near-uniform. This result indicates that the framework can extract the higher-order structure generated by the global modular rule even when that structure is invisible in low-order statistics.

Core claim

On ibm_torino, the modular ridge circuit with 8 keys and 1024 shots each yields a ridge-hit probability of 0.1830 and per-shot key accuracy of 0.1689. The Möbius-based decomposition of I(K; D_S) produces targeted positive synergy CPS_K(k=3) = 0.08788 that is significant under label-shuffle permutation tests (p = 0.004975). Uniformity diagnostics confirm near-uniform single-bit marginals while pairwise correlations concentrate in specific low-order pairs, supporting the claim that recoverable key information resides predominantly in high-order synergistic terms associated with the algebraic constraint.

What carries the argument

The Möbius-inversion pipeline that decomposes mutual information I(K; D_S) over detector-bit subsets S to isolate targeted positive synergy CPS_K at order k_max=3, which quantifies the synergistic contributions generated by the modular ridge constraint.

Load-bearing premise

The measured positive targeted synergy at order 3 is produced by the modular algebraic relation rather than by unmodeled quantum hardware noise or circuit implementation details.

What would settle it

Repeating the identical hardware run but replacing the modular multiplication map with a random bijection and obtaining no significant order-3 synergy would show that the synergy is not specifically attributable to the algebraic ridge.

Figures

Figures reproduced from arXiv: 2604.15051 by Petr Sramek.

Figure 1
Figure 1. Figure 1: Hardware ridge heatmaps. 16×16 histograms of (u, v) on ibm torino for n = 4, 8 keys, 1024 shots/key (N = 8192). Red overlay shows the predicted ridge v ≡ k · u (mod 16) used in evaluation. Takeaway: the low-dimensional ridge structure is visibly present on hardware across keys. 4.2 Headline quantitative metrics [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Permutation tests (label shuffle). Label-shuffle permutation distributions for per-shot key accuracy (500 perms) and CPSK (200 perms) computed on the same dataset (N = 8192). Red line marks the observed value. Takeaway: both accuracy and CPSK are far into the tail of the null, supporting nontrivial key information. 4.4 Uniformity sanity checks: marginals near-uniform, structure in low-order pairs A critica… view at source ↗
Figure 3
Figure 3. Figure 3: Uniformity and correlation structure. Uniformity sanity checks on pooled marginals and top |cov| pairs (tagged within-A/within-B/cross). Dashed line marks 0.5 for marginals. Take￾away: single-bit marginals are near-uniform while correlation structure concentrates in specific low-order pairs. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: presents a bootstrap reliability sweep over shots/key ∈ {128, 256, 512, 768, 1024}. Using the criterion CV ≤ 1, the max reliable order remains k ⋆ = 3 throughout the sweep, and the order-3 targeted synergy remains statistically stable at the primary experimental budget (1024 shots/key) [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Ablation and calibration. Low-order vs high-order ablation using (i) marginals-only, (ii) pairwise features, and (iii) full-bitstring models, evaluated on a stratified 50/50 split. Takeaway: higher-order representations improve predictability and/or calibration beyond low-order summaries [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

We report a hardware validation of the DAGI (Directed Acyclic Graph Information) framework on IBM Quantum hardware using a small, controlled experiment whose ideal output distribution is constrained to a low-dimensional modular manifold (a "ridge"). For two $n$-bit registers $(u,v)$ with $n=4$ (modulus 16), each key instance $k$ induces an ideal relation $v \equiv k \cdot u \pmod{16}$, producing a visually distinct ridge in the joint $(u, v)$ distribution. Executed on ibm\_torino in a single Sampler V2 job (8 keys, 1024 shots/key, $N=8192$ total shots), the ridge persists under hardware noise with ridge-hit probability $p_{hit} = 0.1830$ (uniform baseline $1/16$), corresponding to a ridge contrast of $2.93\times$ (95\% bootstrap CI [2.80, 3.06]). Key recovery exceeds chance: per-shot accuracy 0.1689 (chance 0.125, 95\% Wilson CI [0.1610, 0.1772]), and per-group dictionary recovery 0.375 (chance 0.125). To test the central DAGI hypothesis -- that recoverable key information is predominantly high-order/synergistic rather than visible in low-order marginals -- we compute a M\"obius-based information decomposition of $I(K;D_S)$ over detector-bit subsets $S$ via a M\"obius inversion pipeline and evaluate targeted positive synergy $CPS_K$ at order $k_{max}=3$. We observe $CPS_K(k=3) = 0.08788$ with significance under label-shuffle permutation tests (accuracy $p=0.001996$, $CPS_K$ $p=0.004975$). Uniformity diagnostics show near-uniform single-bit marginals while correlation concentrates in specific low-order pairs, and a bootstrap reliability sweep confirms order-3 targeted synergy remains statistically reliable at the full 1024-shot target budget. These results support the claim that DAGI detects and quantifies nontrivial, hardware-resilient, higher-order information structure associated with a known global algebraic constraint.

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

2 major / 2 minor

Summary. The paper reports a hardware experiment on IBM Quantum's ibm_torino executing an n=4 modular-multiplication circuit (modulus 16) that enforces the algebraic ridge v ≡ k · u (mod 16). It shows the ridge persists under noise (p_hit = 0.1830 vs. uniform 1/16), key recovery exceeds chance (per-shot accuracy 0.1689, group recovery 0.375), and a Möbius-inverted information decomposition yields statistically significant targeted positive synergy CPS_K(k=3) = 0.08788 under label-shuffle permutation tests, supporting the claim that DAGI captures nontrivial high-order synergistic structure associated with the global constraint.

Significance. If the central claim holds, the work supplies the first hardware demonstration that a Möbius-based synergy measure can isolate higher-order information tied to an algebraic manifold in noisy quantum output distributions. The use of bootstrap/Wilson intervals and label-shuffle permutation tests provides a reproducible statistical foundation; the modular-ridge construction is parameter-free and falsifiable, strengthening the result's value for quantum information theory and hardware validation of partial-information decompositions.

major comments (2)
  1. [Abstract / Results] Abstract and results section: the reported CPS_K(k=3) = 0.08788 is shown to exceed label-shuffle chance, but the manuscript provides no explicit comparison of the observed distribution to the ideal noiseless ridge distribution or to control circuits that enforce the same input-output map via different gate decompositions. Without this, it is impossible to isolate whether the order-3 synergy originates from the modular constraint or from unmodeled hardware noise, qubit mapping, or low-order bit correlations induced by the specific modulus and implementation.
  2. [Methods] Methods / Experiment description: the circuit (gate decomposition, qubit mapping, error mitigation) and noise model are not detailed, yet the central claim that the synergy is 'hardware-resilient' and 'associated with a known global algebraic constraint' depends on ruling out implementation-specific artifacts. A direct comparison to the ideal distribution or to an alternative circuit realizing the same ridge would be required to make the attribution load-bearing.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'visually distinct ridge' is qualitative; a quantitative definition of ridge contrast (already given as 2.93×) should be referenced consistently.
  2. [Abstract] Notation: CPS_K is introduced without an explicit equation reference in the abstract; adding the defining expression for targeted positive synergy at order k would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The comments correctly identify areas where additional comparisons and details would strengthen the attribution of the observed synergy to the algebraic constraint. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract / Results] Abstract and results section: the reported CPS_K(k=3) = 0.08788 is shown to exceed label-shuffle chance, but the manuscript provides no explicit comparison of the observed distribution to the ideal noiseless ridge distribution or to control circuits that enforce the same input-output map via different gate decompositions. Without this, it is impossible to isolate whether the order-3 synergy originates from the modular constraint or from unmodeled hardware noise, qubit mapping, or low-order bit correlations induced by the specific modulus and implementation.

    Authors: We agree that explicit comparison to the ideal noiseless distribution is necessary to isolate the contribution of the modular constraint. In the revised manuscript we will add a direct comparison of both the observed joint (u,v) distribution and the computed CPS_K(k=3) against the exact ideal distribution generated by v ≡ k · u (mod 16). This will quantify how much of the synergy survives noise versus what is expected in the noiseless case. We also acknowledge that hardware execution of alternative gate decompositions realizing the same ridge was not performed due to experimental resource limits; however, the ideal-distribution baseline provides the primary control for attribution. We will expand the discussion to address possible modulus-induced low-order correlations. revision: yes

  2. Referee: [Methods] Methods / Experiment description: the circuit (gate decomposition, qubit mapping, error mitigation) and noise model are not detailed, yet the central claim that the synergy is 'hardware-resilient' and 'associated with a known global algebraic constraint' depends on ruling out implementation-specific artifacts. A direct comparison to the ideal distribution or to an alternative circuit realizing the same ridge would be required to make the attribution load-bearing.

    Authors: We will substantially expand the Methods section to include the full gate decomposition for the modular-multiplication circuit, the explicit qubit mapping onto ibm_torino, and any error-mitigation steps applied. A device-calibrated noise model will be added for simulation-based comparisons. As noted in the response to the first comment, the ideal noiseless distribution will be provided as the primary control; this directly addresses the need to rule out implementation-specific artifacts while keeping the experiment within available hardware resources. revision: yes

Circularity Check

0 steps flagged

No circularity: synergy computed directly from hardware data with independent permutation tests

full rationale

The paper executes a known modular-multiplication circuit on IBM hardware, records raw bit-string outcomes, and applies Möbius inversion to the empirical joint distribution to obtain CPS_K(k=3). Significance is evaluated by label-shuffle permutations that operate solely on the observed data and are independent of the algebraic ridge model. The ridge itself is defined a priori by the external modular relation v ≡ k · u (mod 16), not inferred or fitted from measurements. No load-bearing step reduces to a self-citation, a fitted parameter renamed as a prediction, or an ansatz smuggled via prior work; the central claim rests on direct computation and falsifiable statistical tests against the collected shots.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the modular arithmetic relation producing the expected information structure and on the Mobius decomposition correctly isolating synergistic components in the quantum output.

axioms (2)
  • domain assumption Ideal joint distribution obeys v ≡ k · u (mod 16) for secret key k
    Defines the low-dimensional ridge manifold against which hardware output is compared.
  • standard math Mobius inversion of mutual information yields a valid synergistic component CPS_K
    Invoked to compute targeted positive synergy at order k_max=3.
invented entities (1)
  • DAGI framework no independent evidence
    purpose: To detect and quantify higher-order synergistic information associated with algebraic constraints
    The framework is the object of the hardware validation; no independent falsifiable prediction outside this experiment is supplied.

pith-pipeline@v0.9.0 · 5720 in / 1436 out tokens · 35817 ms · 2026-05-10T11:23:39.378233+00:00 · methodology

discussion (0)

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Reference graph

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