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Quantum Fidelity Landscape-Guided Prior Calibration for Single-Circuit QGAN Image Generation

Xue Yang, Rigui Zhou, Dax Enshan Koh, Siong Thye Goh, Yitao Tang, ShiZheng Jia, Young-Wook Cho, Hongyu Chen

arXiv:2609.36702v1quant-phcs.CV

TL;DR

Patch-based QGAN image generation can weaken global consistency and increase quantum resource costs, motivating end-to-end, single-circuit pixel generation. BasicQGAN calibrates the quantum prior’s QFL to the data-induced QFL before adversarial training, and the reported experiments show competitive generation with fewer resources than representative patch-based quantum generators.

  • Problem

    Patch-based QGAN image models generate separate local patches that can weaken global consistency, while their circuit and parameter costs grow rapidly with resolution.

    Source: §1 · pp. 2–3

  • Method

    BasicQGAN aligns prior-induced and data-induced QFL distributions before training a single-circuit generator adversarially with a WGAN-GP critic.

    Source: §1 · pp. 4–5

  • Results

    BasicQGAN is competitive on 16 × 16 MNIST and obtains lower FID than PQWGAN on Uppercase Letters and Geometric Shapes, using fewer qubits and trainable parameters than representative patch-based generators.

    Source: §1 · p. 4, §6 · p. 25

  • Takeaways & Limitations

    The results support QFL-guided prior calibration as a practical basis for compact, single-circuit QGAN image generation within the evaluated scope.

    Source: §6 · p. 25

    Experiments use ideal noise-free simulation and mainly small-scale grayscale images; performance degrades at 28 × 28 resolution, and real NISQ hardware remains unevaluated.

    Source: §6 · p. 25

Abstract

from arXiv · show

Quantum Generative Adversarial Networks (QGANs) have emerged as representative generative models in the Noisy Intermediate-Scale Quantum (NISQ) era and have attracted increasing attention in quantum machine learning. However, most existing QGAN methods rely on patch-based decomposition strategies, which weaken the global consistency of generated images and increase quantum resource overhead. In this work, we investigate a simpler approach: pixel-level, end-to-end image generation using a single-quantum-circuit QGAN. By analyzing the structural matching relationship between the quantum prior and the target data distribution in Hilbert space, we provide a new theoretical perspective for understanding the training behavior of naive end-to-end QGANs. Specifically, we introduce the Quantum Fidelity Landscape (QFL), defined as the pairwise-fidelity structure induced by an ensemble of quantum states and preserved under shared unitary transformations of the quantum generation process. We show that, under a fixed Lipschitz readout, this invariant imposes a one-sided bound on decoded sample separation, motivating calibration of the prior-induced QFL before adversarial training. To validate this theoretical insight, we propose BasicQGAN, a QGAN framework incorporating quantum prior calibration. Before adversarial optimization, BasicQGAN aligns the prior-induced QFL with the data-induced QFL. Experimental results on small-scale grayscale image datasets show that BasicQGAN achieves stable and effective end-to-end pixel-level image generation while requiring fewer qubits and trainable parameters than representative patch-based quantum generators. Furthermore, experiments with different initial quantum-state ensembles show that QFL-calibrated ensembles achieve better generative performance.

1. Introduction

The paper investigates why single-circuit, pixel-level QGANs can be difficult to train and proposes calibrating the quantum prior before adversarial optimization. It introduces QFL to connect prior-state geometry with decoded sample separation, and reports effective generation with compact quantum resources.

1. Introduction

  • Patch-based QGANs generate and stitch local patches, which can weaken global consistency and make circuit and parameter costs grow rapidly with resolution.The paper therefore studies end-to-end, pixel-level generation with a single parameterized circuit.Source: §1 · pp. 2–3
  • BasicQGAN generates competitive 16 × 16 MNIST samples, outperforms patch-based PQWGAN across MNIST digit classes, and obtains lower FID on Uppercase Letters and Geometric Shapes.It also requires fewer qubits and trainable parameters than representative patch-based quantum generators.Source: §1 · p. 4
  • The Quantum Fidelity Landscape (QFL) is the ensemble’s pairwise-fidelity structure, which remains invariant under a shared unitary applied by the generator.A different state-space distribution can induce a markedly different QFL histogram.Source: §1 · pp. 3–4
  • For fixed Lipschitz readouts, QFL imposes a one-sided bound on decoded sample separation, with an explicit Euclidean bound available for amplitude readout.Thus, high-fidelity state pairs cannot map arbitrarily far apart in output space.Source: §1 · pp. 4–5
  • BasicQGAN calibrates the prior-induced QFL to the data-induced QFL offline, then uses a single-circuit generator in WGAN-GP adversarial training.Calibration aligns empirical QFL distributions before adversarial optimization.Source: §1 · pp. 4–5
  • Prior ablations and auxiliary-system extensions are used to further validate QFL calibration.The contribution summary reports these as additional tests of calibration effectiveness and robustness.Source: §1 · p. 5

2. Background and Related Work

The background establishes fidelity properties used to connect shared quantum evolution and measurement to classical output distinguishability. Related work covers adversarial objectives, quantum fidelity in QGANs, image-generation architectures, and latent-prior approaches.

2.1. Quantum-Information Preliminaries

  • Fidelity is invariant under a common unitary, while measurement connects quantum-state distinguishability to classical outcome distributions; these properties support the paper’s analysis.The background identifies unitary invariance and measurement’s relation to distinguishability as serving distinct roles.Source: §2.1 · pp. 5–6

2.2. Related Work

  • WGAN-GP uses a gradient penalty to enforce the critic’s Lipschitz constraint, and BasicQGAN adopts it for online adversarial training.WGAN replaces the original GAN divergence with the Wasserstein-1 distance.Source: §2.2 · p. 6
  • Earlier QGAN work applies fidelity in online losses, whereas QFL describes ensemble pairwise-fidelity geometry rather than an online loss between individual real and generated states.The passage contrasts QFL with QuGAN’s swaptest-based fidelity losses.Source: §2.2 · p. 6
  • Patch-based image QGANs reduce qubit needs per subcircuit but do not explicitly model dependencies across patches, and total circuit and parameter requirements grow with patch count.Other approaches generate lower-dimensional representations that classical transformations or decoders map back to image space.Source: §2.2 · pp. 6–7
  • Prior QGANs commonly sample prescribed latent distributions, while VAE-QWGAN fits a Gaussian mixture to classically encoded data; this paper instead focuses on encoded states’ pairwise-fidelity geometry.The cited approaches also include alternative mechanisms for processing latent noise.Source: §2.2 · p. 7

3. Method: BasicQGAN

BasicQGAN studies how prior-state pairwise fidelity constrains decoded outputs, then calibrates that geometry before adversarial training. Its method combines QFL matching with a fixed amplitude decoder and a shared quantum generator.

3.1. Problem Setting and Overview

  • The framework first calibrates prior geometry, then uses the calibrated state ensemble in a single-circuit generator with a fixed readout for end-to-end image generation.The analysis asks which relations among input samples a shared unitary can change.Source: §3.1 · pp. 7–8

3.2. QFL as a Prior-Determined Invariant

  • A shared unitary preserves every pairwise fidelity, so optimizing generator parameters cannot reshape the prior ensemble’s QFL even though individual states may move.The QFL is the ensemble’s pairwise-fidelity structure.Source: §3.2 · pp. 8–9

3.3. From QFL to Classical Output Separation

  • Under a stable fixed readout, high-fidelity prior pairs cannot decode arbitrarily far apart, but low fidelity permits rather than guarantees larger output separation.QFL alone does not determine the generated distribution or replace an expressive circuit and successful adversarial optimization.Source: §3.3 · p. 10

3.4. Amplitude-Readout Specialization

  • For amplitude readout, square-root probabilities represent amplitude magnitudes and yield a fidelity–Euclidean relation, but decoding constrains rather than reconstructs output geometry.The implemented normalization may distort pairwise Euclidean distances by fixed dimension-dependent factors.Source: §3.4 · pp. 11–12

3.5. QFL-Guided Quantum Prior Calibration

  • Offline calibration aligns empirical prior- and data-induced QFL distributions using KDE discrepancy and mean/variance matching, rather than matching finite QFL matrices entry by entry.The calibrated preparation parameters are then held fixed during online adversarial training.Source: §3.5 · pp. 12–13
  • Empirical QFL histograms converge toward the population histogram as ensemble size increases; the study uses 1,500 states for remaining QFL calculations based on stability and cost.The consistency statement applies when a common bin partition is used across ensemble sizes.Source: §3.5 · p. 14

3.6. Single-Circuit Generation and Online Training

  • BasicQGAN prepares N = log2 d qubits for a d-pixel image, applies shared trainable RY and nearest-neighbor CNOT layers, and decodes the complete pixel vector without patches or a trainable classical decoder.The online stage fixes the calibrated prior and optimizes the generator and discriminator with WGAN-GP.Source: §3.6 · p. 16

4. Experiments

The experiments assess single-circuit, end-to-end pixel-level generation on small grayscale image benchmarks, comparing BasicQGAN with quantum and classical baselines. Image quality and diversity are evaluated using FID, where lower scores indicate better performance.

4. Experiments

  • Four curated datasets cover two MNIST resolutions, Geometric Shapes, and Uppercase Letters, all as small-scale grayscale benchmarks.The datasets are derived from MNIST, EMNIST, and HDS repositories.Source: §4 · p. 16
  • PQWGAN(Global) provides a controlled naive end-to-end baseline, while patch-based PQWGAN and classical WGAN-GP provide additional comparisons.PQWGAN(Global) uses Patch=1, removing patch-wise decomposition; PQWGAN is described as a fully quantum generator, and WGAN-GP as a classical analogue of BasicQGAN.Source: §4 · p. 17
  • FID evaluates generated-image visual quality and diversity, with lower scores indicating better performance.The FID definition uses feature means and covariance matrices for real and generated images.Source: §4 · p. 17
  • All models were trained for 50 epochs with batch size 5 in simulations using PyTorch and PennyLane.The initial-state preparation algorithm and adversarial components were optimized using Adam.Source: §4 · p. 17

5. Results

The results compare BasicQGAN with global and patch-based quantum generators and WGAN-GP across resolutions, datasets, prior choices, and an auxiliary-system variant. BasicQGAN performs competitively at 16 × 16 and on two additional datasets, while 28 × 28 generation shows more noise and higher FID; its generator also uses fewer quantum resources than the compared quantum models.

5.1. Naive End-to-End Baseline

  • On 16 × 16 MNIST, BasicQGAN produces clearer samples and lower FID than PQWGAN(Global), which uses the same generator circuit with a conventional prior.The controlled comparison supports the paper’s diagnosis that prior mismatch, not simply use of one circuit, is a central training difficulty.Source: §5.1 · p. 18, Figure 5 · p. 18, Figure 6 · p. 18

5.2. Impact of Prior QFL on Generation Performance

  • The calibrated prior achieves the lowest FID, 20.38, among evaluated prior constructions; broad prior ranges fail training, while an extremely narrow range causes severe mode collapse.The ablation evaluates initial-parameter ranges U[0, π], U[0, 1], and U[0, 0.01] against the proposed preparation algorithm.Source: §5.2 · p. 19, Figure 7 · p. 19, Table 1 · p. 19

5.3. Multi-Scale Performance

  • At 16 × 16, BasicQGAN and WGAN-GP produce smooth MNIST samples and similar class-wise FID, both outperforming PQWGAN; at 28 × 28, BasicQGAN is noisier and has higher FID than the baselines.The paper attributes the higher-resolution degradation mainly to increased QFL matching error and possible limits in circuit expressive capacity.Source: §5.3 · p. 20, Figure 8 · p. 20, Figure 9 · p. 21

5.4. Cross-Dataset Generalization

  • Across Uppercase Letters and Geometric Shapes, BasicQGAN generates smoother samples and achieves lower FID than PQWGAN.The reported FID comparison is lower for BasicQGAN across classes in both datasets.Source: §5.4 · p. 21, Figure 11 · p. 21, Figure 12 · p. 22

5.5. Resource Efficiency

  • At 16 × 16 and 28 × 28, BasicQGAN requires fewer qubits and trainable parameters than PQWGAN(Global) and patch-based PQWGAN.The resource gap from patch-based PQWGAN becomes more pronounced at higher resolution; BasicQGAN also uses fewer trainable parameters than WGAN-GP.Source: §5.5 · p. 22, Table 2 · p. 23

5.6. Auxiliary-System Ablation

  • With an appended ancilla qubit, BasicQGAN retains recognizable MNIST samples and obtains FID scores comparable to the no-ancilla model.Appending an identically initialized auxiliary state leaves the prior-induced QFL unchanged.Source: §5.6 · p. 23, Figure 14 · p. 24, Figure 15 · p. 24

6. Conclusion and Limitations

The paper connects quantum-prior geometry to decoded sample separation and uses that analysis to motivate QFL calibration before adversarial training. In the evaluated settings, BasicQGAN produces diverse, competitive samples with compact quantum resources, but the validation remains limited in scale and hardware setting.

6. Conclusion and Limitations

  • The Quantum Fidelity Landscape is the pairwise-fidelity structure preserved by shared unitary evolution, and under a fixed Lipschitz readout it constrains decoded sample separation.The constraint is one-sided: high-fidelity pairs cannot be mapped arbitrarily far apart in output space.Source: §6 · p. 24
  • BasicQGAN calibrates the prior-induced QFL against the data-induced QFL before WGAN-GP adversarial training.The conclusion reports that this approach avoids observed training failure or mode collapse and produces diverse samples in the evaluated settings.Source: §6 · pp. 24–25
  • BasicQGAN achieves competitive generation on 16 × 16 MNIST and two additional grayscale datasets while using fewer qubits and trainable parameters than representative patch-based quantum generators.The same calibration principle also applies when the generator is extended with an ancilla qubit.Source: §6 · p. 25
  • Experiments use ideal noise-free simulation and mainly small grayscale images; real-hardware behavior remains unevaluated, and performance degrades at 28 × 28.The conclusion identifies higher-resolution or multi-channel generation as settings needing further development.Source: §6 · p. 25

Appendix A. Proof of QFL Invariance

The appendix establishes QFL invariance by applying unitary invariance of quantum fidelity to every pair in the ensemble. Consequently, generator updates preserve both the complete QFL matrix and its empirical off-diagonal distribution.

Appendix A. Proof of QFL Invariance

  • The appendix uses density operators on a finite-dimensional Hilbert space and the squared Uhlmann fidelity convention.It identifies Tr as matrix trace and A† as conjugate transpose.Source: Appendix A · p. 26
  • Quantum fidelity is invariant under a shared unitary, so each pair of initial states has the same fidelity after generator evolution.The proof applies the property to every pair of indices and every generator parameter setting.Source: Appendix A · p. 26
  • Because every QFL-matrix entry is preserved, generator updates also preserve the empirical distribution of off-diagonal fidelities.Source: Appendix A · p. 26

Appendix B. Proof of the General Output-Separation Bound

The proof chains stable readout, measurement contractivity, a fidelity inequality, and an earlier proposition to obtain the general one-sided bound on decoded output separation. Its uniformity across trainable decoders depends on controlling their Lipschitz constants.

Proof of the General Output-Separation Bound

  • The proof represents measurement outcomes as diagonal density operators, making their trace distance equal to the classical total variation distance.It then applies measurement contractivity and the upper Fuchs–van de Graaf inequality.Source: Appendix B · pp. 27–28
  • The proof derives Eq. (9) by combining stable readout, measurement contractivity, the upper Fuchs–van de Graaf bound, and Proposition 3.2.The steps connect differences between decoded outputs to the fidelity-based bound.Source: Appendix B · pp. 28–29
  • For a trainable decoder family, the bound holds for each fixed decoder, while uniformity over training requires a finite common Lipschitz constant.Without that control, the learned map may amplify arbitrarily small nonzero differences between measurement vectors.Source: Appendix B · p. 29

Appendix C. Proof of the Magnitude-Readout Bound

The proof relates quantum fidelity to Euclidean distance between amplitude-magnitude vectors, then accounts for BasicQGAN’s maximum-normalization decoder. The resulting bound applies directly to the implemented readout under its real-amplitude circuit setting.

Proof of the Magnitude-Readout Bound

  • The proof bounds Euclidean separation between amplitude-magnitude vectors using the squared fidelity of the corresponding pure states.The magnitude vectors are formed from the absolute values of the computational-basis amplitudes.Source: Appendix C · pp. 31–32
  • With only RY and CNOT gates and a real initial state, the ideal amplitudes are real, so BasicQGAN’s Re(cik) readout equals their magnitude up to floating-point error.For complex-valued gates, the passage specifies using |cik| = √pik instead.Source: Appendix C · p. 32
  • Maximum normalization is a fixed bi-Lipschitz transformation on the nonnegative unit sphere, so it distorts Euclidean distances by bounded factors rather than preserving them exactly.The numerical stabilizer changes the constants but preserves sample direction and adds no trainable transformation.Source: Appendix C · pp. 33–34

Appendix D. Consistency of Finite-Ensemble QFL Statistics

The appendices establish consistency of finite-ensemble QFL statistics and provide supplementary stability, alignment, training, dataset, and generation evidence. These materials extend the main analysis across MNIST classes and resolutions and document BasicQGAN’s experimental setup.

Appendix D. Consistency of Finite-Ensemble QFL Statistics

  • Empirical QFL histograms, fixed-grid KDE values, moments, and the composite calibration loss converge almost surely to their population counterparts under the stated fixed settings.The histogram result uses a common finite bin partition; the KDE result uses a fixed bounded kernel, bandwidth, and finite evaluation grid.Source: Appendix D · pp. 35–36

Appendix E. Additional Quantum Fidelity Landscape (QFL) Stability Validation

Appendix F. Achieving QFL Alignment on Diverse Datasets using the Quantum Fidelity Landscape (QFL) optimization algorithm

  • The initial-state ensemble preparation algorithm presents QFL-alignment results for every MNIST digit at both 16 × 16 and 28 × 28 resolutions.Each figure compares the optimized initial-state ensemble with the true-sample ensemble.Source: Appendix F · p. 38, Figure F.18 · p. 39, Figure F.19 · p. 40

Appendix G. The Objective Function and Training Algorithm of BasicQGAN

  • BasicQGAN follows WGAN-GP training with alternating discriminator and generator updates, repeating the two steps until a Nash equilibrium is reached.The discriminator is trained with generator parameters fixed, then the generator is trained with discriminator parameters fixed.Source: Appendix G · p. 41

Appendix H. Dataset Descriptions

  • The experiments construct four datasets from public repositories including MNIST, EMNIST, and HDS, with representative samples and a dataset-summary table provided.The passage describes the datasets as following common QGAN benchmarking practice on small-scale grayscale data.Source: Appendix H · p. 42, Figure H.20 · p. 43, Table H.3 · p. 43

Appendix I. Additional Experimental Results and Training Details

  • Supplementary figures show generated samples across MNIST, geometric shapes, and uppercase letters, while MNIST per-digit FID curves decrease and gradually converge at both reported resolutions.The FID curves cover 16 × 16 and 28 × 28 MNIST.Source: Appendix I · p. 43, Figure I.21 · p. 44, Figure I.22 · p. 44

Appendix J. Analysis of quantum circuit layers

The depth comparison identifies 40 layers as the strongest tested configuration. Greater depth initially improves sample clarity and diversity, but the reported outcomes decline at 60 and 80 layers.

Appendix J. Analysis of quantum circuit layers

  • 40 layers achieve the best reported FID, while increasing depth to 60 or 80 layers reduces sample diversity and raises FID.The qualitative comparison covers 10, 20, 40, 60, and 80 layers; the table reports the corresponding FID scores.Source: Appendix J · p. 47, Figure J.26 · p. 47, Table J.4 · p. 47
  • Circuits with 40 layers generate clearer and more diverse images than circuits with 10 or 20 layers.The passage associates this improvement with increased expressive power from deeper parameterized circuits.Source: Appendix J · p. 47, Figure J.26 · p. 47
  • The 40-layer configuration is the default setting used for BasicQGAN in the qualitative depth comparison.Source: Figure J.26 · p. 47

Appendix K. Analysis of different learning rate

Learning rates of 0.5 and 0.3 yield the smallest QFL Wasserstein distances, while 0.3 gives a smoother average-fidelity curve; the authors select 0.3 for matching quality and stability.

Appendix K. Analysis of different learning rate

  • The analysis uses 1D Wasserstein distance to quantify the discrepancy between the optimized initial-state QFL and the true-sample QFL.The stated computation follows the sum form, omitting 1/n scaling.Source: Appendix K · pp. 47–48
  • 0.0890 and 0.0882 are the smallest Wasserstein distances, at learning rates 0.5 and 0.3, respectively; the smoother curve at 0.3 motivates its selection.The authors consider both QFL matching quality and training stability when choosing the learning rate.Source: Appendix K · p. 48
  • Figure K.27 compares optimized initial-state and true-sample QFL histograms across learning rates, reports Wasserstein distances, and plots average-fidelity training curves.A smaller Wasserstein distance indicates a closer QFL match; the figure’s dashed line marks the target average fidelity.Source: Appendix K · p. 48, Figure K.27 · p. 49
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