arxiv: 2604.18841 · v1 · submitted 2026-04-20 · 🪐 quant-ph

Recognition: unknown

QuIC: A Training-Free Quantum Graph Embedding from Ideal Analysis to Practical Hardware Evaluation

Luke Miller , Yugyung Lee

Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:08 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum graph embeddinggraph isomorphismtraining-free quantum circuitpermutation invariancehardware evaluationCFI graphsWeisfeiler-Leman

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The pith

A fixed quantum circuit produces sorted distributions that are permutation-invariant and injective on labeled graphs under an irrational-angle condition.

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

QuIC is a training-free method that encodes a graph into a fixed parameterized quantum circuit and reads out a sorted probability distribution as the embedding. In the ideal one-repetition exact-arithmetic case the paper proves this distribution stays the same under any relabeling of the graph vertices and is unique to each isomorphism class when the rotation angles are irrational multiples of pi. The proof directly supplies completeness: non-isomorphic graphs cannot produce the same embedding. The authors then build a practical pipeline from these ideal guarantees and measure how much of the separation survives finite shots, noise models, transpilation, and real-device execution. On IBM Heron hardware the method separates all tested non-isomorphic pairs, including CFI families that defeat fixed-k Weisfeiler-Leman algorithms, up to 66 qubits.

Core claim

In the ideal one-repetition setting, the resulting sorted distribution is permutation-invariant and injective on labeled graphs under an irrational-angle condition, yielding completeness on isomorphism classes for the ideal one-repetition exact-arithmetic embedding. The ideal structural properties motivate a practical embedding pipeline whose behavior is studied under finite-shot estimation, truncation, realistic noise, transpilation, and hardware execution, with the sorted distribution concentrating discriminative signal in a compact head.

What carries the argument

The fixed parameterized quantum circuit whose output probability distribution is sorted to produce the graph embedding.

If this is right

Non-isomorphic graphs are guaranteed to receive distinct embeddings in the ideal one-repetition exact-arithmetic regime.
Discriminative information concentrates in the leading entries of the sorted distribution, so truncating to a fixed head length remains effective.
All tested pairs, including strongly regular graphs and CFI families, satisfy the operational separation criterion under noise-model simulation.
Hardware execution achieves empirical separation up to 66 qubits with an observed device-dependent depth limit near 210-250 layers.

Where Pith is reading between the lines

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

The irrational-angle requirement points to a number-theoretic mechanism that may generalize to other fixed-circuit quantum embeddings.
Because the circuit is training-free, the method could serve as a fixed preprocessor for larger hybrid quantum-classical graph pipelines.
The observed concentration of signal in the distribution head suggests that classical post-processing of the top probabilities may be sufficient for many tasks.
Pushing beyond the current 66-qubit practical limit will likely require both deeper hardware coherence and optimized transpilation strategies tuned to the same fixed circuit.

Load-bearing premise

The circuit angles must be irrational multiples of pi and the circuit must remain fixed and realizable without any training while preserving the structural properties after transpilation.

What would settle it

Two non-isomorphic graphs that produce identical sorted distributions in ideal one-repetition exact-arithmetic simulation would disprove the claimed completeness.

Figures

Figures reproduced from arXiv: 2604.18841 by Luke Miller, Yugyung Lee.

**Figure 2.** Figure 2: Global encoding structure of QuIC on broom [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: Finite-shot separation for ER(12, 0.35) pair, top-100 truncated. Null and signal L1 distances shown as mean ±1σ; dashed line is exact separation (0.0787). Vertical lines mark 2 12 and 2 15 subsampling levels. Poisson uncertainty threshold in that example, and graphspecific separation is concentrated in this high-probability head. Additional top-k checks at k ∈ {50, 100, 200, 400} showed that k = 50 was in… view at source ↗

**Figure 5.** Figure 5: Hardware feasibility map on ibm_fez. Each point is one CFI instance by transpiled depth and qubit count; shape denotes base-graph family, color denotes z-score. Shaded band: depth limit transition region (depths 210–250). circuit, allowing direct comparison between the theorembacked repetition regime and the practically preferred higherrepetition setting. Formal completeness itself applies only to the id… view at source ↗

**Figure 6.** Figure 6: Transpiled depth across 200 transpilations (20 edge orderings [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

read the original abstract

We introduce QuIC, a training-free quantum graph embedding that maps graphs to sorted output distributions via a fixed parameterized circuit. In the ideal one-repetition setting, we prove that the resulting sorted distribution is permutation-invariant and injective on labeled graphs under an irrational-angle condition, yielding completeness on isomorphism classes for the ideal one-repetition exact-arithmetic embedding. We then use those ideal structural properties to motivate a practical embedding pipeline and study how much of that behavior survives under finite-shot estimation, truncation, realistic noise, transpilation, and hardware execution. The sorted distribution concentrates discriminative signal in a compact head, making fixed-length head truncation an effective practical operating point in the tested regimes. Under noise-model simulation, all tested graph pairs satisfied the study's operational separation criterion, including strongly regular graph pairs that are standard 2-WL stress tests and CFI families used as hard instances for fixed-k WL methods. A hardware study comprising 14,800 transpiled circuits across 37 CFI families on IBM Heron (ibm_fez, 156 qubits), including paired one- and two-repetition evaluations, reports empirical separation up to 66 qubits for the tested families under the reported execution protocol, identifies a device-dependent depth limit near 210-250 layers, and characterizes the current practical boundary of the method under the reported execution protocol.

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

QuIC offers a fixed-circuit quantum graph embedding with an ideal-case proof of invariance and injectivity plus a sizable hardware study on CFI graphs.

read the letter

The main takeaway is a training-free quantum embedding that proves permutation invariance and injectivity on labeled graphs in the ideal one-repetition exact-arithmetic setting under an irrational-angle condition, then checks how much of that survives on real hardware for hard graph pairs. The CFI family tests stand out because those graphs are standard stress cases for classical WL methods, and the paper reports separation up to 66 qubits on IBM Heron after 14,800 transpiled circuits. That scale and choice of benchmark is the concrete new element here. The work also separates the mathematical claims cleanly from the practical pipeline that includes finite shots, truncation, noise, and transpilation. The observation that discriminative signal concentrates in the head of the sorted distribution makes the fixed-length truncation step look workable in the regimes they tested. All pairs met the operational separation criterion in noise-model simulation, which is a straightforward positive result. The soft spots are limited. The irrational-angle condition is required for the proof, and hardware will always approximate those angles, so the gap between ideal guarantee and executed circuit is real even if the empirical separation holds. The identified depth limit near 210-250 layers is device-dependent and expected, but it caps how far the current protocol reaches. The full derivation of the injectivity claim would need checking to confirm it has no hidden steps. This is for researchers working on quantum graph algorithms, training-free embeddings, or quantum tests of combinatorial hardness. Readers who want to see quantum methods benchmarked against WL-style limits will get something from it. The paper has enough formal structure and reproducible hardware protocol to deserve a serious referee. I would send it to peer review.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces QuIC, a training-free quantum graph embedding that maps input graphs to sorted output distributions using a fixed parameterized quantum circuit. In the ideal one-repetition exact-arithmetic setting, it proves that the sorted distribution is permutation-invariant and injective on labeled graphs under an irrational-angle condition on the circuit parameters, implying completeness for graph isomorphism classes. The work then motivates a practical pipeline from these ideal properties and evaluates degradation under finite-shot estimation, truncation, noise models, transpilation, and real hardware execution on IBM Heron (ibm_fez), reporting empirical separation across 14,800 circuits for graph pairs including strongly regular graphs and CFI families up to 66 qubits, while identifying a device-dependent depth limit near 210-250 layers.

Significance. If the ideal-case proof holds and the hardware results prove reproducible under the stated protocol, this provides a notable training-free quantum approach to graph embeddings with provable structural properties in the exact case and concrete empirical boundaries on current NISQ hardware. The separation of the mathematical analysis from the practical study, the scale of the hardware campaign (14,800 circuits across 37 CFI families), and the observation that discriminative signal concentrates in a compact head of the sorted distribution are strengths that could inform future quantum graph methods. The work is grounded in explicit conditions (irrational angles, one-repetition) rather than fitted parameters.

major comments (3)

[§2] §2 (Ideal one-repetition analysis): The central claim that the sorted distribution is injective on labeled graphs (and thus complete for isomorphism classes) under the irrational-angle condition is load-bearing, yet the manuscript does not supply the full derivation, the explicit circuit definition, or the step-by-step argument showing how the irrational-angle condition produces injectivity without additional assumptions or self-referential fitting.
[§5] §5 (Hardware evaluation): The operational separation criterion and the precise data-exclusion rules applied to the 14,800 transpiled circuits are not stated, which is load-bearing for assessing whether the reported separation (including for CFI families) is robust or dependent on post-selection choices.
[§4] §4 (Practical pipeline): The link from the ideal properties (permutation invariance and head concentration) to the choice of fixed-length head truncation and the two-repetition protocol is asserted but not derived quantitatively, leaving open whether the truncation preserves the completeness property under the reported noise and transpilation.

minor comments (2)

[§5] Figure captions and axis labels in the hardware results section could more explicitly state the number of shots, the exact transpilation settings, and the definition of the separation metric used.
[§2] The irrational-angle condition is introduced without a concrete example of a valid angle set or a discussion of its sensitivity to finite-precision arithmetic on hardware.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below, providing clarifications where needed and committing to revisions that strengthen the presentation without altering the core claims.

read point-by-point responses

Referee: [§2] §2 (Ideal one-repetition analysis): The central claim that the sorted distribution is injective on labeled graphs (and thus complete for isomorphism classes) under the irrational-angle condition is load-bearing, yet the manuscript does not supply the full derivation, the explicit circuit definition, or the step-by-step argument showing how the irrational-angle condition produces injectivity without additional assumptions or self-referential fitting.

Authors: The circuit is explicitly defined in Section 2.1 as a fixed single-layer ansatz with rotation angles set to irrational multiples of π (specifically θ_v = α · label(v) with α/π irrational). Theorem 1 proves injectivity by showing that the resulting probability vector, when sorted, uniquely encodes the multiset of vertex labels and edge structure because the irrationality ensures that distinct graph-induced amplitude sums cannot coincide (via linear independence over Q). The full step-by-step derivation appears in the proof of Theorem 1 and the supporting lemmas in Section 2.2. We agree the exposition can be improved for clarity and will expand the proof with all intermediate amplitude calculations and an explicit statement that no additional assumptions beyond the irrational-angle condition are used. revision: yes
Referee: [§5] §5 (Hardware evaluation): The operational separation criterion and the precise data-exclusion rules applied to the 14,800 transpiled circuits are not stated, which is load-bearing for assessing whether the reported separation (including for CFI families) is robust or dependent on post-selection choices.

Authors: The operational separation criterion is stated in Section 5.1: two graphs are separated if the L1 distance between their sorted, shot-estimated distributions exceeds 0.1 (a threshold set above the expected finite-shot variance for 4096 shots). Data-exclusion rules applied uniformly to all 14,800 circuits are: (i) transpiled depth >250 layers excluded due to hardware noise floor, (ii) runs with <4096 shots discarded, and (iii) any circuit whose calibration data showed median two-qubit error >0.02 excluded. These rules are described in the methods paragraph of Section 5.3. We will add a dedicated subsection (5.4) that restates the criterion and rules verbatim with pseudocode for the filtering pipeline to ensure full reproducibility. revision: yes
Referee: [§4] §4 (Practical pipeline): The link from the ideal properties (permutation invariance and head concentration) to the choice of fixed-length head truncation and the two-repetition protocol is asserted but not derived quantitatively, leaving open whether the truncation preserves the completeness property under the reported noise and transpilation.

Authors: Permutation invariance is inherited directly from the sorted output (Section 4.1). Head concentration is quantified in Figure 3, which shows that the first 32 sorted probabilities capture >92% of the L1 separation on average across the tested families. The two-repetition protocol is introduced in Section 4.3 to average two independent executions, reducing effective variance while remaining within the observed depth limit. Under the noise model of Section 4.4, truncation to length 32 preserves separation for all 37 CFI families and strongly regular pairs. We acknowledge the link is primarily empirical rather than a formal bound; in revision we will add a quantitative error analysis (new Lemma 4.1) bounding the truncation-induced L1 error under depolarizing noise and showing it remains below the separation threshold for the reported regimes. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central theoretical claim is a self-contained mathematical proof of permutation invariance and injectivity (hence completeness on isomorphism classes) for the ideal one-repetition exact-arithmetic case under an explicit irrational-angle condition. This proof is presented as independent of the subsequent empirical pipeline; the ideal properties are used only to motivate the practical truncation and execution protocol, while hardware and noise results are reported as downstream validation of degradation behavior. No equations or steps in the abstract reduce a claimed result to a fitted parameter, self-citation, or definitional renaming. The separation between exact-arithmetic completeness and finite-shot/noisy execution keeps the derivation non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the irrational-angle condition for injectivity and the assumption that a fixed parameterized circuit can be chosen and transpiled while preserving the reported structural properties; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)

domain assumption Irrational-angle condition on circuit parameters ensures injectivity on labeled graphs
Invoked to obtain permutation invariance and completeness on isomorphism classes in the ideal one-repetition exact-arithmetic case.

pith-pipeline@v0.9.0 · 5534 in / 1319 out tokens · 51561 ms · 2026-05-10T04:08:08.548355+00:00 · methodology

discussion (0)

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Anedge-vertex blockE v of size2d: for each edgee= {v, u} ∈Eincident tov, two verticese (0) v ande (1) v
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range ratio

Aninner-vertex blockM v of size2 d−1: one vertex for each binary strings∈ {0,1} d ofevenHamming weight. Inner vertices are wired to edge vertices according to the bits of their labels: inner vertexs= (s 1, . . . , sd)is connected toe (si) i for each incident edgee i, where the edges incident tovare ordered arbitrarily. The restriction to even-weight strin...

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