Investigating Parameter-Efﬁciency of Hybrid QuGANs Based on

Geometric Properties of Generated Sea Route Graphs

Tobias Rohe , Florian Burger, Michael K

olle , Sebastian W

olckert, Maximilian Zorn

and Claudia Linnhoff-Popien

Mobile and Distributed Systems Group, LMU Munich, Germany

ﬁ

Keywords:

Quantum Computing, Generative Adversarial Networks (GANs), Graph Generation, Quantum-Classical

Hybrid Algorithms, Geometric Data Analysis.

Abstract:

The demand for artiﬁcially generated data for the development, training and testing of new algorithms is om-

nipresent. Quantum computing (QC), does offer the hope that its inherent probabilistic functionality can be

utilised in this ﬁeld of generative artiﬁcial intelligence. In this study, we use quantum-classical hybrid gener-

ative adversarial networks (QuGANs) to artiﬁcially generate graphs of shipping routes. We create a training

dataset based on real shipping data and investigate to what extent QuGANs are able to learn and reproduce

inherent distributions and geometric features of this data. We compare hybrid QuGANs with classical Gen-

erative Adversarial Networks (GANs), with a special focus on their parameter efﬁciency. Our results indicate

that QuGANs are indeed able to quickly learn and represent underlying geometric properties and distributions,

although they seem to have difﬁculties in introducing variance into the sampled data. Compared to classical

GANs of greater size, measured in the number of parameters used, some QuGANs show similar result quality.

Our reference to concrete use cases, such as the generation of shipping data, provides an illustrative example

and demonstrate the potential and diversity in which QC can be used.

1 INTRODUCTION

QC with its probabilistic character and generative AI

are two topics of our time that go together surpris-

ingly well. In the event that Moore’s Law comes to

an end, QC could be a real alternative - assuming im-

proved QC hardware - as many methods of classical

generative AI can be transferred to the ﬁeld of QC.

In this study, we look at the successful concept

of GANs and how these can be implemented in hy-

brid form on both classical and QC hardware. Al-

though literature on classical GANs is already ex-

tensive (Goodfellow et al., 2014; Wang et al., 2017;

Creswell et al., 2018; Goodfellow et al., 2020), and

their use cases are diverse, examples are the gener-

ation of brain images (Islam and Zhang, 2020), the

enhancement of image quality (Chen et al., 2018), or

the estimation of road-trafﬁc (Xu et al., 2020), there

is a lack of literature that looks at the functionalities

and abilities of its quantum-classical hybrid version,

so-called QuGANs. Like classical GANs, QuGANs

consist of a discriminator and a generator, but at least

one of the two components is implemented using QC.

In our study, we implement the generator of the

GAN on a QC-simulator, which then subsequently

interacts with a classically implemented discrimina-

tor. We learn and generate artiﬁcial shipping routes

that are geometrically subject to the triangle inequal-

ity, which must never be broken. Our primary focus is

on the question of how well and how efﬁciently such

hybrid QuGANs can recognise the underlying, sim-

plest geometric structures and reproduce them in the

generated data, while approximating with the sam-

pled port distances a bimodal distribution. To learn

and represent the underlying geometric structures in

the generated data is thereby an ability which is cru-

cial for the successfully execution of many generative

tasks (De Cao and Kipf, 2018). Artiﬁcial generated

data in general can be used as training and / or test

instances to develop new tools and algorithms (Islam

and Zhang, 2020). Here, for example, a future use

of our artiﬁcially created realistic shipping routes as

problem instances for the travelling salesman prob-

lem is conceivable. We evaluate the efﬁciency of

our QuGAN implementations with that of a classical

GAN with similar, even slightly higher, number of pa-

rameters, while we also benchmark the results against

a random baseline. This research should help to shed

724

Rohe, T., Burger, F., Kölle, M., Wölckert, S., Zorn, M. and Linnhoff-Popien, C.

Investigating Parameter-Efﬁciency of Hybrid QuGANs Based on Geometric Properties of Generated Sea Route Graphs.

DOI: 10.5220/0013350200003890

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 1, pages 724-730

ISBN: 978-989-758-737-5; ISSN: 2184-433X

more light on the inherent functioning and capabili-

ties of hybrid QuGANs - motivating further research

into this area.

Our work is structured as follows: In Sec. 2 we

provide theoretical background for the work at hand.

In Sec. 3 we present related work, followed by Sec. 4

and the methodology of our paper. Sec. 5 presents

the results of this work. The paper ends with Sec. 6

Conclusion.

2 BACKGROUND

2.1 Generative Adversarial Networks

(GANs)

GANs represent a framework in the ﬁeld of machine

learning where two neural network models, the gen-

erator (G) and the discriminator (D), respectively pa-

rameterised by sets of neural network weights θ

, are trained adversarially in a zero-sum game.

(z) = x

f ake

aims to synthesise fake data x

f ake

from

the latent space z, that is indistinguishable from a set

of real training data x

real

which the model is trying to

replicate. The purpose of training G is thus to newly

generate samples that the discriminator mistakes for

increasingly realistic data. D

(x) evaluates the au-

thenticity of both the received real data from the train-

ing set and the fake data generated by G with the aim

of accurately labeling samples as “real” (in the orig-

inal data set) or not. The training process involves

G and D simultaneously adjusting their parameters

and θ

. The loss of the discriminator D that is

incurred in judging the data generated by G is gen-

erally quantiﬁed via the binary cross-entropy (BCE)

loss function:

BCE(y, d) = −(y∗log(d)+(1−y)∗ log(1− d)) (1)

where d ∈ [0; 1] represents the probability estimated

by the discriminator that the sample is real, while y is

the associated ground-truth label (1, real or 0, fake).

Training continues until a Nash equilibrium is

reached where neither G nor D can trivially improve

their strategies, meaning that D is not able to discern

between real and generated data anymore. The gen-

eral dynamic of the mini-max game can be described

by the following value function:

min

max

V (D, G) = E

x∼p

data

(x)

[logD(x)]

+ E

z∼p

(z)

[log(1 − D(G(z)))]

(2)

where z ∼ N (0, 1) is randomly sampled noisy input

and x = G(z) is the latent feature vector generated by

One of the main issues with the GAN architecture is

mode collapse, where G learns to produce a limited

variety of outputs, thus not capturing the full variety

of input data. Here, the training process of GANs is

unstable, which is characterised by oscillations and

non-convergence during the training process. This

instability is due to the balance required between G

and D, where disproportionate gradient updates, pro-

duced by e.g. imbalanced learning rates or model

sizes,cause one to overpower the other and undermine

the training process.

2.2 Quantum Generative Adversarial

Networks (QuGANs)

In a QuGAN, either D and / or G can be replaced

with a variational quantum circuit (VQC) (Sim et al.,

2019). In this setup, classical data is mapped to a

quantum state using a feature map f : R

→ H

⊗n

, al-

lowing for the application of parameterised quantum

gates to generate a desired output state Ψ. In the end,

we obtain the classical data by measuring the quan-

tum system, for instance, in the computational basis.

Similar to classical GANs, the parameters are trained

against the BCE loss and the minimising direction is

calculated via gradient based methods.

3 RELATED WORK

In case of the realisation of only one component

of a GAN by means of QC, it is referred to as a

hybrid QuGAN (Ngo et al., 2023). Already early

work has speculated that QuGANs will have more di-

verse representation power than their classical coun-

terparts, particularly for very high-dimensional data

(Dallaire-Demers and Killoran, 2018; Lloyd and

Weedbrook, 2018). Over time, various realisations

of QuGANs such as Tensor-Network-Based GANs

(Huggins et al., 2019), Quantum Conditional GANs

(Dallaire-Demers and Killoran, 2018) and Quan-

tum Wasserstein GANs (Chakrabarti et al., 2019)

have been developed. The application of QuGANs

can be diverse, from approximating quantum states

(Chakrabarti et al., 2019), to generating discrete dis-

tributions (Situ et al., 2020), to learn and load random

distributions (Zoufal et al., 2019), to generating im-

ages (Stein et al., 2021).

Comprehensive studies have already been carried

out on the efﬁciency, especially the parameter ef-

ﬁciency, of QuGANs. These studies have investi-

gated how QuGANs perform in comparison to clas-

sic GANs with the same number of parameters. Kao

Investigating Parameter-Efﬁciency of Hybrid QuGANs Based on Geometric Properties of Generated Sea Route Graphs

725

et al. (2023) (Kao et al., 2023) and Stein et al.

(2021) (Stein et al., 2021) demonstrate that QuGANs

achieve comparable or superior performance with

substantially fewer parameters. Kao et al. highlight

ongoing challenges in generating unique and valid

molecules. Li et al. (2021) (Li et al., 2021) further

emphasise the efﬁciency of QuGANs, showing that

their models can learn molecular distributions effec-

tively with a reduced parameter count and improved

training processes through the use of multiple quan-

tum sub-circuits. Additionally, Anand et al. (2021)

(Anand et al., 2021) explore noise resistance in hybrid

quantum-classical setups, ﬁnding that QuGANs can

maintain performance despite moderate noise levels,

which is pivotal for practical applications.

However, to the best of our knowledge, there is no

research into the creation of graphs using QuGANs,

although some models do have connections to graphs,

especially the underlying graph structures of their ap-

plication, such as for generative chemistry (De Cao

and Kipf, 2018). In the ﬁeld of classical GANs, on

the other hand, there is already literature and various

adapted models. A survey by Zhu et al. (2022) (Zhu

et al., 2022) describes the current state of research on

graph generation via deep learning models, also look-

ing at how classical GANs can be used for this pur-

pose.

4 METHODOLOGY

Our methodology builds on a synthesis of concepts

from two foundational pipelines: a pipeline for quan-

tum machine learning (Gabor et al., 2020) and a

pipeline for quantum optimization (Rohe et al., 2024),

ensuring a comprehensive approach to the problem.

4.1 Data Preprocessing

We employed the Python package searoute to cal-

culate realistic sea-routes between different ports

(Halili, 2024). For the underlying training dataset,

we randomly sampled four ports from the 3, 669 ports

of the searoute package and computed the distances

between all pairs sampled, creating a fully connected

graph, where each node represents a port, and the

edge weights denotes the minimum distances between

those ports. To avoid extreme cases where two ports

are directly next to each other, all graphs where two

ports were closer than 100 nautical miles to each other

were excluded. This follows from the logic that the

triangle inequality can be broken particularly easily

in those cases. We sampled a total of 1, 000 graphs

for our training dataset and pre-processed our data by

normalising the sum of edges to one, while preserving

the integrity of the distance relationships.

4.2 Classical GAN

The discriminator used for our classical GAN, but

also for the later described QuGANs, consists of a

fully connected neural network with three linear lay-

ers: an input layer, a hidden layer with 16 neurons,

and an output layer. The network processes tensors

representing graph data, with an input size of 6, cor-

responding to the edges in our fully connected four-

port graphs. Each layer, except for the output layer, is

activated by a LeakyReLU function to mitigate the is-

sue of dying gradients (Maas et al., 2013). The output

layer utilises a sigmoid activation function to classify

the inputs as real or fake. The discriminator has a

total of 129 trainable parameters. The classical gen-

erator mirrors this architecture but includes a hidden

layer of 10 neurons and receives a noise vector sam-

pled from a normal distribution as input. The output is

a tensor matching the dimensions of real graph data,

activated by a ReLU function to ensure non-negative

edge weights. The generator has a total of 136 train-

able parameters.

4.3 QuGAN

For each of the four hyrbid QuGAN models tested, we

employed the classical discriminator described above

and constructed four different quantum generators.

For the quantum generator, the latent variable was

sampled from a standard normal distribution and then

encoded into a quantum state using angle embedding,

leading us to utilise six qubits. We test two differ-

ent quantum circuits (Ans

atze) based on QISKIT’s

efﬁcient SU(2) 2-local circuit, each one with 5 and

with 10 layers, giving us four generator models. The

ﬁrst circuit consists of layers of one ladder rotational-

X gates and one ladder Pauli-Y gates, followed by

a circular CNOT entanglement. The second circuit

is less restrictive and consists of layers of two lad-

ders of rotational-X and rotational-Y gates, and again

a circular CNOT entanglement. The Ans

atze utiliz-

ing the Pauli-Y gates have 36 and 66 parameters for

the different numbers of layers, while the Ans

atze

with rotational-Y gates have 72 and 132 parameters.

We will refer to them as QuGAN followed by their

respective number of parameters, e.g. QuGAN(36).

The output of these circuits calculate the probability

that each qubit is in the state 0, which represents an

edge weight, therefor the distance between two ports.

We re-normalised the measurement results to the sum

of one, so that the edge-ratios remain the same, but the

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726

measurement data do not differ from the real training

data by their sum. Our numerical simulations were

performed without noise.

4.4 Training & Evaluation

All models were trained using the Adam optimiser

over 1, 000 epochs, with a batch size of 32. The learn-

ing rates were set at 0.3 for the discriminator and

0.001 for the generator, optimising against the BCE

loss. This training process was repeated across ﬁve

different seeds, and the average outcomes were re-

ported.

To assess the validity of the generated graphs, we

applied the triangle inequality test. Valid graphs were

those where the post-processed edge weights comply

with the triangle inequality. For each vertice triple

A, B, C ∈ V it hold that d(A, B) ≤ d(A, C) + d(C, B),

with a distance function d : R × R → R.

As a further additional baseline, we sampled

1, 000 graphs from the training data’s post-processed

weight distribution using kernel density estimation,

providing a reference for generating random graphs

based on the training dataset. This should help us to

estimate how easy or hard it is to violate the triangle

inequality if the edges are blindly sampled from the

distribution without considering their geometric rela-

tions to each other.

5 RESULT

In the following, we will take a closer look at our re-

sults presented in Fig. 1. We ﬁrst look at the results

in the light of valid graphs (Fig. 1a), then at the stan-

dard deviation (Fig. 1b) and the density functions of

the sampled edge weights (Fig. 1c), and ﬁnally at the

losses achieved by the different generators (Fig. 1d).

At the beginning, all implementations start with

approximately 1000 valid graphs, which then rela-

tively fast decline within the ﬁrst 100 epochs, but then

recover again. In general, it can be noted that the

QuGANs keep up with the classical implementation

in terms of valid graphs as evaluation metric. The

QuGAN(36) and QuGAN(132) implementation show

very stable and good (approx. 80% valid graphs) re-

sults. The QuGAN(66) and QuGAN(72) implemen-

tations are less stable and show signiﬁcantly less valid

graphs (approx. 60% valid graphs). The classical

GAN, on the other hand, takes much longer to reach a

solid level and ultimately ﬂuctuates between 60% and

80% valid graphs. However, it should be noted that all

(Qu)GAN implementations clearly outperform ran-

dom sampling with its 27.9% valid graphs baseline.

The standard deviation of the edge weights in the

training dataset was 0.0826, which also should be

observed in our results. However, all implementa-

tions start with a very low standard deviation (close

to zero), and increase it over the subsequent training

process. These results explain the observation of the

development of the valid graphs in Fig. 1a, where

all (Qu)GANs start at around 100% valid graphs and

then drop sharply. At the beginning of the training

process, all (Qu)GAN implementations produce very

monotonous edge weigths without strong variances

and deviate from this behaviour later in the train-

ing process. While the generation of similar edge

weights ensures that the triangle inequality is not bro-

ken, it makes it for the discriminator a simple task

to distinguish between real and fake data. With the

increase of variance of edge weights, the differen-

tiation becomes more difﬁcult, but the triangle in-

equality is also increasingly broken. Our results in-

dicate, that the QuGANs in general seem to have

a harder time in terms of generating variance, with

only implementation QuGAN(66) showing a similar,

even higher standard deviation compared to the clas-

sical GAN towards the end of training. Implemen-

tations QuGAN(36), QuGAN(132) and QuGAN(72)

are clearly behind here.

Looking at the edge-weight distribution, the clas-

sical GAN and the hybrid QuGAN(66) implementa-

tion approximate the bimodal nature of the underly-

ing training data (dashed-red line) more effectively

than other models, although they do not sharply repli-

cate the two peaks. These models show a more pro-

nounced but smoother rise towards the second peak,

with a centre slightly offset from the training data. Al-

though these implementations do not match the ex-

act height of the training data peaks, as well as the

variability, they come closest to capturing the over-

all trend, despite the evident shift and reduced peak

magnitude in the ﬁrst peak. Conversely, the mod-

els represented by the QuGAN(132) and QuGAN(72)

curves depict a unimodal distribution with a peak sit-

uated between the two actual peaks of the training

data, suggesting a simpler but less accurate modeling

of the dataset. The QuGAN(36) model also follows

this trend but with a more pronounced central peak,

indicating a tighter concentration around the median

weigh.

In all our implemented (Qu)GAN versions, we use

the same discriminator, which is why we attribute a

certain signiﬁcance to the comparison of the average

losses of the generator compared to the other imple-

mentations. The classical GAN performs best here.

This can be interpreted with caution as meaning that

the classical generator is best able to build artiﬁcial

Investigating Parameter-Efﬁciency of Hybrid QuGANs Based on Geometric Properties of Generated Sea Route Graphs

727

(a) Average Number of Valid Graphs (b) Average Standard Deviation of Edge-Weights

Figure 1: Subgraph a) shows the development of the number of valid graphs over the epochs. Subgraph b) illustrates the

development of the calculated standard deviation of the weights generated over the epochs. Subgraph c) shows the density of

the edge weights, while ﬁnally, subgraph d) visualises the loss of the generators.

graphs that cannot be distinguished by the discrimi-

nator, although the feedback loop of the training ef-

fect of D by G must also be mentioned here. The

best quantum implementation is QuGAN(66), closely

followed by QuGAN(72) and QuGAN(132). The

QuGAN(36) implementation performs the worst, al-

though we had the best results here in terms of evalu-

ating the triangle inequality.

6 CONCLUSION

In our, to our knowledge, unique problem setting, we

examined QuGANs for their parameter efﬁciency. In

particular, we chose a setup where not only the mod-

elling of a density function was in the foreground, but

in each instance several values were sampled from

the distribution, which were all directly connected

and dependent on each other via the triangle inequal-

ity. With the setting applied, not only the sampling,

but also the underlying geometric properties are in-

cluded in the evaluation and comparison. The best-

performing quantum generators were QuGAN(66)

and QuGAN(132), which showed a clear difference

in the learned weight distribution. Both Ans

atze are

built using the same generators, but QuGAN(66) has

ﬁxed Pauli-Y gates. This reduced expressibility al-

lows for easier training while still being capable of ap-

proximating the bimodal distribution of the real data.

Overall, our results suggest that QuGANs struggle to

reﬂect variance in their generated data. The normali-

sation of the generated graphs plays an important role

in this context. While it reduces the variance of the

generated weights, it also decreases the variance of

the training data. Although GANs are generally dif-

ﬁcult to train, this approach enabled a more stable

training process. At the same time, they can still gen-

QAIO 2025 - Workshop on Quantum Artiﬁcial Intelligence and Optimization 2025

728

erate balanced results in the area of tension between

the variation in the edge weights and the fulﬁlment of

the triangle inequality. For example, the QuGAN(66)

implementation shows equally good results in terms

of variance and valid graphs compared to the clas-

sical GAN, although less than half as many param-

eters were used. This is in line with previous work

on QuGANs and parameter efﬁciency, although our

work extends this with underlying geometric proper-

ties. In particular, the underlying task chosen here has

great potential for future research. The search for cir-

cuits and implementations that can better reﬂect the

variance of the training data as well as their geometric

properties should be the main focus of future research.

ACKNOWLEDGEMENTS

This paper was partially funded by the German Fed-

eral Ministry of Education and Research through the

funding program ”quantum technologies - from basic

research to market” (contract number: 13N16196).

Furthermore, this paper was also partially funded

by the German Federal Ministry for Economic Af-

fairs and Climate Action through the funding program

”Quantum Computing – Applications for the indus-

try” (contract number: 01MQ22008A).

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