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arxiv: 2606.28236 · v1 · pith:OMSKGRV3new · submitted 2026-06-26 · 🪐 quant-ph · hep-ex

Qudit extension of parameterized IQP circuits: A generative quantum machine learning approach to integer data

Robert J. Banks , Arianna Crippa , Matthias Traube , Josua Unger , Christian Ertler , Wolfgang Lechner This is my paper

Pith reviewed 2026-06-29 03:22 UTC · model grok-4.3

classification 🪐 quant-ph hep-ex
keywords quditIQP circuitsgenerative quantum machine learninginteger dataenergy depositsparticle showersCLIC detector
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The pith

Parameterized IQP circuits extended to qudits allow generative learning on integer data while keeping the original distances intact.

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

The paper shows how to adapt instantaneous quantum polynomial circuits from qubits to qudits so they can work directly with integer-valued inputs instead of forcing them into binary strings. Binary mappings erase the natural ordering and spacing between integer values such as energy measurements, which limits their use in generative tasks. The extension encodes each integer into a fixed-length representation, rewrites the circuit gates in qudit language, and supplies a loss function plus covariance calculation for training. The approach is checked against energy deposit patterns from simulated electron showers in a detector calorimeter. If successful, the same construction applies to any other integer dataset that quantum generative models might encounter.

Core claim

By mapping each integer pixel to a fixed-length bit-string and converting the IQP gates to the corresponding qudit operations, the circuit can be trained with a tailored loss function and feature covariance matrix to generate distributions over integer data that respect the original metric structure, as verified on single-particle electron shower energy deposits recorded in the CLIC electromagnetic calorimeter.

What carries the argument

The qudit-adapted parameterized IQP circuit that encodes integers directly via fixed-length bit-strings and applies transformed gates to operate in the qudit formalism.

If this is right

  • The trained circuit produces generative samples of integer energy deposits that preserve the metric structure of the input data.
  • A covariance matrix among features can be computed directly from the circuit output to guide training.
  • The same loss function and circuit adaptation apply to any other non-binary integer dataset.
  • The construction extends existing parameterized IQP generative models beyond binary distributions.

Where Pith is reading between the lines

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

  • Detector simulation tasks that rely on integer energy values could adopt this encoding to reduce preprocessing artifacts.
  • Hardware implementations on qudit-capable devices would need to verify whether the gate transformations retain their theoretical advantage.
  • The same integer-to-qudit mapping could be tested on other discrete data types such as counts or quantized sensor readings.

Load-bearing premise

Converting the gates to qudit form while using fixed-length bit-string encodings for integers keeps the metric distances of the original data better than ordinary binary qubit mappings.

What would settle it

Measure the average distance between generated and true samples under the original integer metric on the calorimeter dataset and compare the qudit model against an otherwise identical binary IQP model; a statistically significant improvement for the qudit version would support the claim.

Figures

Figures reproduced from arXiv: 2606.28236 by Arianna Crippa, Christian Ertler, Josua Unger, Matthias Traube, Robert J. Banks, Wolfgang Lechner.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
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Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
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Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
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Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: illustrates the evolution of the MMD loss func￾tion with a bandwidth of σ = 5 and an initial gate pa￾rameter scale of 0.03. These hyperparameter values were selected due to their observed performance. To monitor convergence, the MMD was evaluated every 100 itera￾tions on a validation dataset. As can be seen in the figure, the training is relatively smooth. The validation of the model is postponed to Sec. I… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
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Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
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Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
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Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
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Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
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Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
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Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
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Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
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Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
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Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
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Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p019_17.png] view at source ↗
read the original abstract

Parameterized Instantaneous Quantum Polynomial (IQP) circuits have proven useful in quantum generative learning models, particularly for binary distributions. However, when applied to non-binary datasets, they exhibit notable limitations: mapping integer values into qubit-compatible binary representations often destroys the original metric structure of the data. In this paper we aim to extend them to a qudits formulation operating on an integer mapping of the data. The IQP quantum circuit is adapted to encode each integer valued pixel into a bit-string of fixed length and quantum gates are transformed to follow the qudit formalism. As a generative machine learning approach, a suitable loss function for the circuit training and the calculation of the covariance matrix among features are developed and validated on the energy deposits from single-particle electron showers in the electromagnetic calorimeter of the CLIC detector. The method proposed in this work can be also extended to other applications that utilize quantum generative machine learning for non-binary data.

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

1 major / 1 minor

Summary. The manuscript proposes extending parameterized Instantaneous Quantum Polynomial (IQP) circuits from qubits to qudits for generative quantum machine learning on integer-valued data. The approach encodes each integer pixel into a fixed-length bit-string, transforms the quantum gates to qudit formalism, introduces a loss function and covariance matrix calculation for training, and validates the method on energy deposits from single-particle electron showers in the CLIC detector electromagnetic calorimeter. The work claims this enables better handling of non-binary data compared to standard binary qubit mappings.

Significance. If the qudit extension demonstrably improves metric preservation and generative performance on integer data without introducing new inconsistencies, it would provide a concrete route for applying IQP-based generative models to non-binary datasets in high-energy physics and similar domains. The use of real detector data for validation is a strength, but the absence of any reported quantitative results, baselines, error bars, or explicit metric comparisons in the abstract prevents a full assessment of impact.

major comments (1)
  1. [Abstract] Abstract: The central motivation is that binary mappings of integers to qubits destroy the original metric structure of the data. However, the described procedure still encodes each integer valued pixel into a bit-string of fixed length before transforming gates to the qudit formalism. This encoding step appears identical to the binary case being criticized, so any claimed advantage must derive solely from the subsequent gate transformation; no analysis, distance metric comparison, or ablation is indicated to show that metric structure is restored or better preserved.
minor comments (1)
  1. The abstract states that validation occurred but supplies no quantitative results, performance metrics, baseline comparisons, or derivation details for the loss function or covariance calculation, making it impossible to evaluate the method's effectiveness from the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying an important point of clarification in the abstract. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central motivation is that binary mappings of integers to qubits destroy the original metric structure of the data. However, the described procedure still encodes each integer valued pixel into a bit-string of fixed length before transforming gates to the qudit formalism. This encoding step appears identical to the binary case being criticized, so any claimed advantage must derive solely from the subsequent gate transformation; no analysis, distance metric comparison, or ablation is indicated to show that metric structure is restored or better preserved.

    Authors: We agree that the initial encoding of each integer pixel into a fixed-length bit-string is a shared preliminary step with binary qubit mappings, and that the abstract does not explicitly demonstrate via distance metrics or ablations how metric structure is restored. The claimed advantage therefore rests on the subsequent transformation of the IQP gates into qudit formalism together with the integer-specific loss function and covariance matrix. These elements are developed in the body of the manuscript and validated on the CLIC electromagnetic calorimeter data. To address the referee's concern directly, we will revise the abstract to state explicitly that metric preservation is intended to arise from the qudit gate transformations and the associated training procedure rather than from the encoding step itself, and we will add a cross-reference to the relevant sections. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method extension is self-contained with external validation.

full rationale

The paper describes a methodological extension of IQP circuits to qudits for integer-valued data, including an encoding step and loss function, validated on external CLIC detector data. No equations, fitted parameters called predictions, or self-citations are visible in the abstract or description that would reduce any claimed result to its own inputs by construction. The derivation chain (if present in full text) appears independent of the enumerated circularity patterns, with the external benchmark providing falsifiability outside any internal fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides insufficient detail to enumerate free parameters, axioms, or invented entities; the central adaptation assumes standard qudit gate transformations without further specification.

axioms (1)
  • domain assumption Qudit formalism can be applied to IQP circuits by transforming gates to follow the qudit version while encoding integers into fixed-length bit-strings
    Invoked as the core adaptation step described in the abstract.

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discussion (0)

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Reference graph

Works this paper leans on

88 extracted references · 5 canonical work pages

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    Adapting the circuit The current binary IQP formulation consists of a train- able circuit diagonal in the computational basis, with a layer of Hadamard gates on each side as shown in Fig. 1. In this paper the circuit is adapted to the case where each integer has been encoded into a bit-string of length b. Therefore, one can think of the circuit as consist...

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    Note that for qudits, these Pauli operators are defined such that ap- plying themdtimes returns the identity operator,X d d = Z d d =I d

    Estimating expectation values Having looked at the generalization of the Hadamard to qudits, we now define the Pauli-Zand Pauli-Xoper- ators to be [28]: Zd = d−1X j=0 ωj d |j⟩ ⟨j|,(4) Xd = d−1X j=0 |j+ 1 modd⟩ ⟨j|,(5) with the latter acting as a cyclic shift. Note that for qudits, these Pauli operators are defined such that ap- plying themdtimes returns t...

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    Selecting a loss-function The loss-function used in Ref. [9] is the Maximum Mean Discrepancy: MMD2(p, qθ) =E a∼Pσ(a) h (⟨Za⟩p − ⟨Za⟩qθ)2 i (9) for qubits in the binary IQP case, wherea∼P σ(a) de- notes sampling each term inafrom an independent and identical Bernoulli trial. The success probability of the Bernoulli trial is parameterized by the bandwidthσ....

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    The form of this diagonal ansatz is left as a design choice

    Ansatz design The IQP circuit contains a parameterized unitary which is diagonal in the computational basis. The form of this diagonal ansatz is left as a design choice. In Sec. III a hardware efficient ansatz [30] is considered, on the as- sumption that the hardware is ultimately qubit based, all one- and two- body qubitZrotations available on the hardwa...

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    Train on classical To evaluate the loss-function:

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    SampleN ops operators from Eq. (12). This value affects the precision of the MMD estimate

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    Each string is of lengthnand each element takes integer values between 0 andd−1

    SampleN s strings from a uniform distribution. Each string is of lengthnand each element takes integer values between 0 andd−1

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    (8) approximate the expectation values of theN ops operators using theN s random strings

    Using Eq. (8) approximate the expectation values of theN ops operators using theN s random strings. Note that the statistical error scales as∼1/ √Ns

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    The loss-function and the gradient can then be used to update the classical optimizer that tunes the parameter- ized quantum gates in the IQP circuit

    While fixing the samples, the gradient of the loss- function can be found with automatic differentia- tion [31]. The loss-function and the gradient can then be used to update the classical optimizer that tunes the parameter- ized quantum gates in the IQP circuit

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    In the next section, the IQP qudit model is applied to calorimeter data, which is mapped onto integers with finite precision

    Deploy on quantum Once the model has been classically trained, each im- age can be generated from sampling the IQP circuit in the computational basis. In the next section, the IQP qudit model is applied to calorimeter data, which is mapped onto integers with finite precision. In App. C, a second application is an- alyzed, namely the Potts model [32], at f...

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    Case study: 6 pixels 0 200 400 600 Iteration 0.0 0.1 0.2 0.3 MMD loss FIG. 5:Loss function for 6 pixels images and qudit dimension d=16:The MMD loss function in the IQP circuits training with kernel bandwidthσ= 5 and scale for the initial gates parameters of 0.03. The first analysis performed was a training on a 6- pixels dataset with qudit dimensiond= 16...

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    A case, where quantum hardware is required for sampling the final probability distribution

    Case study: 12 pixels As a second example of training with the IQP circuit, a larger dataset of 12 pixels withd= 32 is explored, thus requiring a total of 60 qubits. A case, where quantum hardware is required for sampling the final probability distribution. A total number of 1830 variational param- eters are used in the training. As in the previous case, ...

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    error (%) FIG

    Case study: 6 pixels 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5Energy deposition Validation decoded Trained circuit p1 p2 p3 p4 p5 p6 0 1Rel. error (%) FIG. 7:Mean energy deposition for 6 pixels images with d=16:(top panel) Energy deposition values of rescaled integer validation distribution (blue) and generated samples from the trained IQP circuit (orange) with...

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    The trained model is first compared to the withheld validation dataset

    Case study: 12 pixels The same analysis, as in the previous section, is now repeated for the 12 pixels case. The trained model is first compared to the withheld validation dataset. A MMD 2 value of (2.9±0.1)×10 −3 is found. Since, single images using a simulator cannot be generated in this case , we estimate the expectation value of the number operator fo...

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    The Potts model To assess the performance of the method, we consider the Potts model [32] at finite temperature. The Hamil- tonian for the Potts model is given by HPotts =− X ⟨i,j⟩ Ji,j1(S i =S j),(C10) and is defined on a square lattice of sizeL×Lwith peri- odic boundary conditions. EachJ i,j is randomly sampled from a uniform distribution between 0 and ...

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