Disentangling Quantum and Classical Contributions in Hybrid Quantum

Machine Learning Architectures

Michael K

olle, Jonas Maurer, Philipp Altmann, Leo S

unkel, Jonas Stein and Claudia Linnhoff-Popien

Institute of Informatics, LMU Munich, Munich, Germany

ﬁ

Keywords:

Variational Quantum Circuits, Autoencoder, Dimensionality Reduction.

Abstract:

Quantum computing offers the potential for superior computational capabilities, particularly for data-intensive

tasks. However, the current state of quantum hardware puts heavy restrictions on input size. To address this,

hybrid transfer learning solutions have been developed, merging pre-trained classical models, capable of han-

dling extensive inputs, with variational quantum circuits. Yet, it remains unclear how much each component

– classical and quantum – contributes to the model’s results. We propose a novel hybrid architecture: instead

of utilizing a pre-trained network for compression, we employ an autoencoder to derive a compressed version

of the input data. This compressed data is then channeled through the encoder part of the autoencoder to

the quantum component. We assess our model’s classiﬁcation capabilities against two state-of-the-art hybrid

transfer learning architectures, two purely classical architectures and one quantum architecture. Their accu-

racy is compared across four datasets: Banknote Authentication, Breast Cancer Wisconsin, MNIST digits, and

AudioMNIST. Our research suggests that classical components signiﬁcantly inﬂuence classiﬁcation in hybrid

transfer learning, a contribution often mistakenly ascribed to the quantum element. The performance of our

model aligns with that of a variational quantum circuit using amplitude embedding, positioning it as a feasible

alternative.

1 INTRODUCTION

In recent years, remarkable progress has been made

in the ﬁeld of machine learning, leading to break-

throughs in various areas such as image recognition

(Dosovitskiy et al., 2021) and speech recognition

(Schneider et al., 2019). With the advancement of

technology, the scale and complexity of data continue

to increase, posing signiﬁcant challenges for classi-

cal computational methods, including the curse of di-

mensionality (Bellman, 1957). In this context, the use

of quantum computers promises performance advan-

tages.

However, we are currently in the Noisy

Intermediate-Scale Quantum (NISQ) era, char-

acterized by not only a restricted number of qubits

within the quantum circuit but also limitations on

circuit depth and the quantity of operations conducted

on the qubits (Preskill, 2018).

To overcome these limitations, approaches which

combine classical neural networks (NN) with quan-

tum circuits are subject to increased research – often

combined with transfer learning. Mari et al. (Mari

et al., 2020) propose the Dressed Quantum Circuit

(DQC), where a variational quantum circuit (VQC) is

framed by a classical pre-processing NN and a classi-

cal post-processing NN. This approach is then com-

bined with different transfer learning architectures,

whereby the most appealing one according to the au-

thors is classical to quantum transfer learning. Here a

classical pre-trained NN extracts features, which are

subsequently passed to a VQC. A major problem with

a hybrid approach like this is the uncertainty of the

actual contribution of the VQC to the classiﬁcation

performance and whether it provides any additional

beneﬁt over a purely classical NN as the VQC adds

additional calculation time and complexity.

Another approach which builds on transfer learn-

ing is Sequential Quantum Enhanced Training (SE-

QUENT) (Altmann et al., 2023). Here, the post-

processing layer is omitted and the training consists

of two steps: classical and quantum. In the classical

step, the model consists of a pre-processing layer and

a surrogate classical classiﬁer. This proxy is replaced

by a VQC after pre-training and the corresponding

quantum weights are optimized while the classical

weights are frozen. Again, the inﬂuence of the re-

spective classical and quantum parts of the model is

Kölle, M., Maurer, J., Altmann, P., Sünkel, L., Stein, J. and Linnhoff-Popien, C.

Disentangling Quantum and Classical Contributions in Hybr id Quantum Machine Learning Architectures.

DOI: 10.5220/0012381600003636

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

In Proceedings of the 16th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2024) - Volume 3, pages 649-656

ISBN: 978-989-758-680-4; ISSN: 2184-433X

649

ambiguous.

In this work, we propose an alternative approach

to address the aforementioned challenges of the NISQ

era and the limitations of the just presented models.

Instead of using transfer learning, the encoder part of

an autoencoder (AE) is utilized to compress the input

data into a lower-dimensional space. Subsequently,

this reduced input is passed to a VQC, which then

classiﬁes the data. By using an AE for the compres-

sion, we aim to provide a more transparent under-

standing of the actual classiﬁcation performance of

the VQC. The performance of our approach is then

compared to a DQC, SEQUENT, a classical NN with

the uncompressed and compressed input, and even-

tually with a pure VQC, which uses amplitude em-

bedding. The models were trained and compared

with each other on the datasets Banknote Authenti-

cation, Breast Cancer Wisconsin, MNIST, and Au-

dioMNIST – ranging from medical to image and au-

dio data. Hence, our contributions can be summarized

as follows:

• We propose an alternative approach for handling

high-dimensional input data in quantum machine

learning (QML)

• We evaluate the individual performance of classi-

cal and quantum parts in hybrid architectures

All experiment data and code can be found here

2 VARIATIONAL QUANTUM

ALGORITHMS

One of the most promising strategies for QML al-

gorithms are variational quantum algorithms (VQA)

(Cerezo et al., 2021), which can be used e.g. as clas-

siﬁers (Schuld et al., 2020; Farhi and Neven, 2018).

Generally, VQAs allow us to use quantum computing

in the NISQ era by utilizing a hybrid approach of a

quantum computer, but classical optimization strate-

gies in an iterative quantum-classical feedback loop.

A parametrized quantum circuit – a VQC – is used to

change qubit states with different gate operations and

a classical computer to optimize the parameters of the

circuit. Hybrid in this context does not mean that a

combined architecture with a classical and a quantum

part is applied, but rather that solely a quantum cir-

cuit is used in combination with classical optimiza-

tion strategies. The goal is to minimize a speciﬁed

cost function in the training process by ﬁnding the

optimal parameters for the quantum circuit (Cerezo

et al., 2021).

https://github.com/javajonny/AE-and-VQC

3 OUR APPROACH

In this chapter, we introduce our approach. First,

we will describe the architecture of the AE and the

VQC individually. Subsequently, we will illustrate

how these two components are combined to achieve

the desired reduction in dimensionality.

3.1 Autoencoder for Dimensionality

Reduction

An AE serves as a feature extractor by encoding

inputs into a smaller yet signiﬁcant representation

(Goodfellow et al., 2016), addressing VQCs’ limi-

tations in the NISQ era by minimizing input dimen-

sions to the number of output classes for VQC in-

tegration. The encoder progressively halves dimen-

sions per layer, aligning the ﬁnal layer’s neurons with

output classes, using ReLU for non-linear transfor-

mations (Nair and Hinton, 2010; Krizhevsky et al.,

2017; Ramachandran et al., 2017). A Sigmoid ac-

tivation function in the last layer ensures the output

range matches pre-processed inputs. The decoder

mirrors the encoder’s structure, reconstructing inputs

from compressed data with minimal loss, and also

employs Sigmoid after the ﬁnal layer for consistent

output range.

Due to its simplicity and reliably good perfor-

mance in NNs (Ramachandran et al., 2017), we use

ReLU in this AE architecture after the input layer and

between the hidden layers. After the last layer, a Sig-

moid function is used as an activation function, which

converts any input to the range [0, 1] and therefore

aligns with our pre-processed input data.

3.2 Variational Quantum Circuit

In the following, we will explain the architecture of

the proposed VQC, which acts as a classiﬁer. It con-

sists of the three components: state preparation, en-

tangling layers, and the measurement layer.

3.2.1 State Preparation

At ﬁrst, the provided input in the range [0, 1] is con-

verted to the range [−

] to be within the range of

typical angles used in quantum gates. Our presented

VQC then uses angle embedding to encode the fea-

ture vector of dimension N into the rotation angles of

n qubits, with N ≤ n, in the quantum Hilbert space.

In the ﬁrst step, a layer of single-qubit Hadamard

gates is applied to each qubit in the circuit and hereby

transforms the basis state |0⟩ to an equal superposition

|+⟩, making the quantum state unbiased with respect

ICAART 2024 - 16th International Conference on Agents and Artiﬁcial Intelligence

650

to |0⟩ and |1⟩. In the second step, a Ry-gate is ap-

plied to each qubit, which performs the actual angle

embedding.

3.2.2 Entangling Layers

To save time in the optimization and training pro-

cess, weight remapping is applied (K

olle et al., 2023a;

olle et al., 2023b). The entangling layers apply

a sequence of trainable operations to the prepared

states, whereby the general architecture is inspired

by the model circuit (Schuld et al., 2020) and the ar-

chitecture of SEQUENT. Each of the entangling lay-

ers consists of a CNOT ladder resulting in entangle-

ment between qubits, followed by Ry-gates that apply

parametrized rotations around the y-axis.

3.2.3 Measurement Layer

The last component performs the measurements of

each wire in the computational basis. More speciﬁ-

cally, the expectation value of the Pauli-Z operator is

calculated and returned for each wire.

3.3 Integration of the Autoencoder and

the Variational Quantum Circuit

The ﬁrst step consists of data pre-processing, which is

followed by the initialization and training of the AE.

We selected the Mean Square Error (MSE) as the loss

function and the Adam optimizer algorithm (Kingma

and Ba, 2015) as the optimizer. After training the AE,

only the optimized encoder is used by passing the in-

put data to it and compressing the input dimension to

the number of labels.

In order to classify the reduced features, the VQC

must then be initialized, which is again followed by

the training of the same, for which classical optimiza-

tion techniques can be used in an iterative process

with a quantum-classical feedback loop. The Cross

Entropy Loss is taken as the criterion and the Stochas-

tic Gradient Descent for the optimization. Subse-

quently, the trained VQC is used for classiﬁcation

tasks. An illustration of our proposed architecture can

be found in Fig. 1.

4 EXPERIMENTAL SETUP

In this chapter, the datasets, baselines, and the hyper-

parameter optimization are presented. For the AE, the

MSE as the reconstruction loss is the main evaluation

metric, and for our model and the baselines, the accu-

racy is most relevant.

4.1 Datasets and Pre-Processing

The four different datasets Banknote Authentication

(Lohweg, 2013), Breast Cancer Wisconsin (Wolberg

et al., 1995), MNIST (LeCun et al., 1998) and Au-

dioMNIST (Becker et al., 2019) were chosen to

demonstrate the versatility of the suggested approach.

Each dataset was split into a training set, a validation

set and a test set. The input values were then nor-

malized to the range [0, 1] and the output labels were

one-hot encoded. The images of the MNIST dataset

were ﬁrst ﬂattened to one-dimensional tensors. For

AudioMNIST, the initial audio ﬁles were converted

to Mel spectrograms. These were then converted to

images and subsequently ﬂattened.

4.2 Baselines

Our presented model will be compared against the

following baselines: a VQC with amplitude embed-

ding, a DQC, SEQUENT, a classical NN, and an AE

in combination with a classical NN.

4.2.1 Variational Quantum Classiﬁer

(Amplitude Embedding)

The VQC with amplitude embedding just differs in

the state preparation. Amplitude embedding encodes

the features of the input vector into the amplitudes of

the qubits and makes superposition its advantage: an

exponential number of features, 2

, can be mapped

into the amplitude vector of n qubits (Schuld and

Petruccione, 2018). It’s important to note that, for

cases where n is less than log

N, where N represents

the number of classical features, padding is applied

to the original input, i.e. ﬁll the vector with zeros.

An additional requirement is the normalization of the

padded input to unit length: x

padded

· x

padded

= 1.

4.2.2 Dressed Quantum Circuit

The DQC as suggested by Mari et al. (Mari et al.,

2020) consists of three parts. A classical pre-

processing part, a VQC, and a post-processing part.

The pre-processing part is a NN, consisting of one

layer, which in our case reduces the input vector di-

rectly to the number of output classes, followed by

applying a Sigmoid activation function. This reduced

input is then passed to a VQC which has the same

architecture and number of layers as our reference

model. The post-processing part also consists of one

layer and maps the data from the dimension of the

VQC width to the number of output classes. In our

case, both dimensions are the same.

Disentangling Quantum and Classical Contributions in Hybrid Quantum Machine Learning Architectures

651

(Φ

1,1

)

𝕏

3) Measurement2)Entangling Layers1) State Preparation

Encoder

VQC

)

(Φ

2,1

)

(Φ

n,1

)

(Φ

2,δ

)

(Φ

n,δ

)

(Φ

1,δ

)

Figure 1: Architecture of our approach, which consists of an encoder and a VQC parametrized with φ. The input data is given

by X and the prediction targets by ˆy. The VQC has n qubits and consists of δ entangling layers.

The training consists of two stages. In the classical

stage, just the weights of the pre- and post-processing

are optimized. In the second stage – the quantum

stage – these classical weights are frozen and just the

parameters in the VQC are optimized.

We use this model as a baseline since the pre-

processing layer also reduces the input. Here the clas-

sical pre-processing layer already classiﬁes. The dis-

advantage of the DQC is that the effect of the classical

and quantum parts is not separately assessable.

4.2.3 SEQUENT

The third baseline SEQUENT consists of a classical

compression layer and a classiﬁcation part. The train-

ing takes place in two stages. In the classical stage,

the classiﬁcation part is a classical surrogate feed for-

ward NN. We reduce the input directly to the number

of output classes in the classical compression layer

and then apply a Sigmoid activation function. This

reduced input is then passed to the second NN for

classiﬁcation. The two parts are both trained and the

weights are optimized.

In the quantum training step, the classical weights

are frozen and the classical surrogate classiﬁcation

network is replaced by a VQC. For the training, just

the quantum parameters are optimized. The VQC

is deﬁned as in the DQC. Altmann et al. (Altmann

et al., 2023) argue that this two-step procedure in SE-

QUENT and DQC can be seen as transfer learning

because it is transferred from classical to quantum

weights. This model also has the same disadvantage

as the DQC in that the effect of the classical and quan-

tum components are not separately assessable.

4.2.4 Classical Feed Forward Neural Network

on Uncompressed Input

Another baseline is a classical feed forward NN,

which is introduced to verify if our model achieves

a quantum speedup. The NN for this paper consists

of one input layer, followed by a hidden layer and an

output layer. The ReLU activation function is applied

after the input layer. The number of neurons in the

hidden layer is chosen to match the number of train-

able parameters in our proposed approach.

4.2.5 Classical Feed Forward Neural Network

on Compressed Input

The last baseline is an AE, which compressed the in-

put data, in combination with a NN. The AE is the

same as for our introduced model. The classical NN

has the same architecture as the one just presented in

Section 4.2.4. The difference is the number of neu-

rons in the hidden layer – only the number of train-

able parameters of our VQC is relevant since the AE

is the same.

4.3 Optimization, Training and

Hyperparameters

The applied optimization technique was grid search

and all experiments were performed several times

for multiple seeds to obtain a more reliable and ro-

bust result. First the AE was optimized and then

the different models. The corresponding results can

be seen in Table 1 and Table 2. Python (v3.8.10)

with the frameworks PyTorch (v1.9.0+cpu) and Pen-

nyLane (v0.27.0) were used for all of our experi-

ments. For all plots in this paper, the exponential

moving average with a smoothing factor α = 0.6 was

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652

used to display the curves.

Table 1: AE optimization with the test reconstruction loss

and 95% conﬁdence interval for the best hyperparameter

combination per dataset..

Dataset Epochs Learning Rate Batch Size Test Loss

Banknote Authentication 500 0.1 128 0.0046 ± 0.0002

Breast Cancer Wisconsin 500 0.01 32 0.0077 ± 0.0034

MNIST 500 0.001 64 0.0198 ± 0.0012

AudioMNIST 500 0.001 128 0.0006 ± 0.0001

Table 2: Optimal learning rate for each model and dataset

obtained from the grid search.

Model AE+VQC (angle) VQC (amplitude) DQC SEQUENT AE+NN NN

Dataset

Banknote Authentication 0.01 0.01 0.1 0.1 0.1 0.1

Breast Cancer Wisconsin 0.1 0.01 0.1 0.1 0.1 0.1

MNIST 0.01 0.01 0.01 0.001 0.01 0.1

AudioMNIST 0.001 0.1 0.1 0.1 0.01 0.1

5 RESULTS

We selected optimal hyperparameters from our op-

timization process for each model and dataset (Ta-

ble 2), running ten experiments with varying seeds

(0-9). The VQC maintained six layers, and the batch

size was set at ﬁve, training for the same epochs as

optimization.

This section presents model performances per

dataset and overall results averaged across datasets.

Validation accuracies and test accuracies are shown in

Fig. 3 and Table 3, respectively. We veriﬁed the AE’s

non-classiﬁcation assumption with tests. Addition-

ally, statistical analyses determined signiﬁcant perfor-

mance differences between models, with follow-up

tests for pairwise analyses, assuming a signiﬁcance

level of α = 0.05. Results focus on pairwise differ-

ences involving our model.

5.1 Banknote Authentication

For the Banknote Authentication dataset, the DQC

performed the best with an accuracy of 0.994, fol-

lowed by the classical NN on the uncompressed in-

put with 0.991, and SEQUENT with 0.979. The

VQC with amplitude embedding yielded an accuracy

of 0.847 and therefore performed better than our ap-

proach with an accuracy of 0.787. The classical NN

on the compressed input was the worst of the models

with 0.699.

The Friedman test indicates that the performance

differences of the approaches are signiﬁcant, χ

(5) =

45.88, p < 0.001. The Wilcoxon signed-rank tests

show that our approach yielded signiﬁcantly worse re-

sults than DQC (p < 0.001), SEQUENT (p = 0.028),

and the NN with uncompressed input (p = 0.001).

The differences of our model to the VQC with am-

plitude embedding and to the NN on the compressed

input are not signiﬁcant.

5.2 Breast Cancer Wisconsin

For the Breast Cancer Wisconsin dataset, the classical

NN on the uncompressed input performed best with

an accuracy of 0.974. This performance is closely

followed by DQC and SEQUENT with an accuracy

of 0.972 for each. VQC with amplitude embedding

yielded a performance of 0.849, followed by the clas-

sical NN on the compressed input with 0.833. Our

approach achieved an accuracy of 0.816 and therefore

performed slightly worse than the VQC with ampli-

tude embedding.

The Friedman test indicates that the performance

differences of the approaches are signiﬁcant, χ

(5) =

31.72, p < 0.001. The Wilcoxon signed-rank tests

show that our model performed signiﬁcantly worse

than DQC (p = 0.042) and SEQUENT (p = 0.042).

The differences between our approach and the VQC

with amplitude embedding, the NN on the com-

pressed input, and the NN on the uncompressed input

are not signiﬁcant.

5.3 MNIST

Also for dataset MNIST, the NN on the uncompressed

input obtained the best accuracy with 0.985. This is

followed by DQC with 0.896 and the classical NN

on the compressed input with 0.831. SEQUENT

achieved an accuracy of 0.508, marginally better than

our approach with 0.507. The worst accuracy of 0.444

was reached by the VQC with amplitude embedding.

The repeated measures ANOVA with

Greenhouse-Geiser corrected values (ε < 0.75)

indicates that there is a statistically signiﬁcant

performance difference between the approaches,

F(2.77, 25.92) = 260.67, p < 0.001. Our model per-

formed signiﬁcantly worse than DQC (p < 0.001),

the classical NN on the compressed input (p < 0.001),

and the NN on the uncompressed input (p < 0.001).

The difference between our approach and the VQC

with amplitude embedding and between SEQUENT

is not signiﬁcant.

5.4 AudioMNIST

For the AudioMNIST dataset, the classical NN on the

uncompressed input performed the best with an ac-

curacy of 0.879, followed by SEQUENT with 0.385

Disentangling Quantum and Classical Contributions in Hybrid Quantum Machine Learning Architectures

653

and DQC with 0.356. The classical NN on the com-

pressed input achieved an accuracy of 0.298. The

VQC with amplitude embedding yielded a result of

0.257 and was therefore better than our approach with

an accuracy of 0.240. For DQC and SEQUENT, it

can be seen in Fig. 2 that the classical training step

achieved validation accuracies of about 0.8. In the

quantum training step, ﬁrst an expected drop occurred

and then only a slight improvement of accuracy can

be observed visually for SEQUENT. The validation

accuracy of DQC deteriorates.

The Friedman test indicates that the performance

differences between the approaches are signiﬁcant,

(5) = 33.03, p < 0.001. The Wilcoxon signed-rank

tests show that our model performed signiﬁcantly

worse than SEQUENT (p = 0.028) and the classi-

cal NN on the uncompressed input (p < 0.001). No

statistically signiﬁcant difference can be obtained be-

tween our model and VQC with amplitude embed-

ding, DQC, and the classical NN on the compressed

input.

Figure 2: Validation accuracies for AudioMNIST. The clas-

sical and quantum validation accuracies are shown for the

two-step hybrid transfer learning of DQC and SEQUENT.

5.5 Overall Comparison

Averaged over all datasets, the classical NN on the

uncompressed input achieved an accuracy of 0.957,

followed by the DQC with 0.805 and SEQUENT with

0.711. The NN on the compressed input gives an ac-

curacy of 0.665. The VQC with amplitude embedding

had a marginally better performance with 0.599 than

our approach with 0.588.

The Kruskal-Wallis test indicates that the perfor-

mance differences of the approaches are signiﬁcant,

(5) = 80.75, p < 0.001. The Mann-Whitney tests

show that our approach performed signiﬁcantly worse

than DQC (p < 0.001) and the NN on the uncom-

pressed input (p < 0.001). The observed difference

between our approach and SEQUENT falls just short

of achieving statistical signiﬁcance (p = 0.052). Be-

tween our approach and the VQC with amplitude em-

bedding and the NN on the compressed input no sig-

niﬁcant difference can be obtained.

5.6 Discussion

The NN on the uncompressed input, DQC, and SE-

QUENT achieved better results than our approach,

where the former was superior. Across DQC, SE-

QUENT, and our model the VQC shared the same ar-

chitecture and numbers of layers. In contrast to DQC

and SEQUENT, our compression part (AE) did not

classify which can be seen in Table 3. This leads to

the assumption that the classical compression layer

plays a pivotal role in the overall performance of these

hybrid transfer learning approaches and the role of the

VQC itself may be questioned. This assumption is

also supported by Fig. 2, which shows the respective

validation accuracies for the two-stage training pro-

cess of DQC and SEQUENT, exemplary for AudioM-

NIST. Already after the classical training step, a very

high validation accuracy can be seen for both models

over both datasets. After this training stage, a drop

in the accuracy can be observed because the quantum

weights were randomly initialized. The validation ac-

curacy subsequently did not increase any further in

the quantum training step, where just the weights of

the VQC were optimized – the validation accuracies

for DQC and SEQUENT after the complete training

are worse than the accuracies after the classical train-

ing stage. A possible explanation could be that the

extracted features of the pre-processing or compres-

sion layer do not contain enough information for fur-

ther classiﬁcation, or that the VQC lacks the required

power or complexity to maintain the desired results.

The comparison between the VQC with amplitude

embedding and our model did not show any statisti-

cally signiﬁcant difference in performance – making

our approach a valid alternative. To test the perfor-

mance of the AE, a classical NN that uses the reduced

input of the AE was introduced. This model achieved

good results for Breast Cancer Wisconsin and for

MNIST. However, especially for AudioMNIST an ac-

curacy of just 0.298 was achieved – compared to the

NN on the uncompressed input with 0.879. This sug-

gests that the proposed AE may not have been able

to adequately extract the essential information within

the latent space. It is hence possible that the architec-

ture is not able to adapt well to speciﬁc characteristics

of a dataset. Further research is needed to enhance

the effectiveness of the AE and come up with spe-

cialized architectures for datasets. Additionally, joint

training of the AE and the VQC in contrast to sequen-

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654

Table 3: Test accuracy and 95% conﬁdence interval for all models.

Model AE+VQC (angle) VQC (amplitude) DQC SEQUENT AE+NN NN AE

Dataset

Banknote Authentication 0.787± 0.023 0.847 ± 0.036 0.994 ± 0.004 0.979 ± 0.009 0.699 ± 0.084 0.991 ± 0.008 0.172 ± 0.030

Breast Cancer Wisconsin 0.816 ± 0.114 0.849 ± 0.021 0.972 ± 0.018 0.972 ± 0.016 0.833 ± 0.121 0.974 ± 0.022 0.016 ±0.024

MNIST 0.507 ± 0.036 0.444 ± 0.033 0.896 ± 0.005 0.508 ± 0.046 0.831 ± 0.042 0.985 ± 0.001 < 0.001± < 0.001

AudioMNIST 0.240 ± 0.028 0.257 ± 0.025 0.356 ± 0.188 0.385 ± 0.036 0.298 ± 0.028 0.879 ± 0.039 < 0.001± < 0.001

Average 0.588 ± 0.430 0.599 ± 0.473 0.805 ± 0.480 0.711 ± 0.492 0.665 ± 0.402 0.957 ± 0.084 0.047 ± 0.133

(a) Banknote Authentication (b) Breast Cancer Wisconsin

Figure 3: Plots for the validation accuracies for all models over the ﬁrst 30 epochs. The datasets Banknote Authentication,

Breast Cancer Wisconsin, MNIST, and AudioMNIST are displayed.

tial training should be considered. Interestingly, the

other models (except for the NN on the uncompressed

input) also did not exceed 40% accuracy either. This

indicates that the pre-processing of the AudioMNIST

dataset may not have been effective. Improving this

process could also be a promising direction for future

research.

Another limitation of the AE in our approach is

the additional effort for the training of the AE, es-

pecially compared to the VQC with amplitude em-

bedding. It is also worthwhile to consider other data

compression techniques, e.g. the principal component

analysis. As already mentioned, the architecture of

our VQC could pose a limitation. It can be beneﬁ-

cial to allow for rotations across all three axes and to

increase the number of layers. Other techniques to

ﬁnd the optimal hyperparameters of a model should

be considered.

6 CONCLUSION

We introduced an approach to tackle the issues of

the current NISQ era by using the encoder of an AE

to reduce the input dimension of a dataset. This

compressed input is fed to a VQC, which uses an-

gle embedding to map the data from the classical

to the Hilbert space. The performance was mea-

sured across the four datasets Banknote Authentica-

tion, Breast Cancer Wisconsin, MNIST, and AudioM-

NIST – ranging from medical to image and audio

samples. We then compared our model to other de-

signs: SEQUENT and DQC from the topic of clas-

sical to quantum transfer learning, VQC with ampli-

Disentangling Quantum and Classical Contributions in Hybrid Quantum Machine Learning Architectures

655

tude embedding as a purely quantum architecture and

a purely classical NN on the compressed and uncom-

pressed input.

Our results suggest that the classiﬁcation perfor-

mance in hybrid transfer learning is mainly inﬂuenced

by the classifying compression layer and that the ac-

tual contribution of the VQC may be doubted. Ad-

ditionally, these approaches yield better results than

models where solely the VQC classiﬁes.

Even though our model performs worse on aver-

age than the hybrid transfer learning models DQC and

SEQUENT, it allows for a more transparent and inter-

pretable analysis of the quantum circuit’s role in the

machine learning task because of the clear distinction

between the components. Additionally, our research

indicates that our approach with angle embedding on

the compressed input is a valid alternative to a VQC

with amplitude embedding on the original input.

ACKNOWLEDGEMENTS

This work is part of the Munich Quantum Valley,

which is supported by the Bavarian state government

with funds from the Hightech Agenda Bayern Plus.

This paper was partially funded by the German Fed-

eral Ministry for Economic Affairs and Climate Ac-

tion through the funding program ”Quantum Comput-

ing – Applications for the industry” based on the al-

lowance ”Development of digital technologies” (con-

tract number: 01MQ22008A).

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