Hybrid Quantum-Classical Autoencoder

• HQC-AE is a hybrid framework combining parameterized quantum circuits and classical neural networks to learn compressed, task-adapted latent representations.
• It employs diverse structural motifs such as quantum bottlenecks, quantum encoders with classical decoders, and hybrid encoder-decoder pipelines for efficient data processing.
• The approach optimizes reconstruction and classification tasks using techniques like the parameter-shift rule, Adam optimization, and multi-objective loss functions.

Searching arXiv for papers on hybrid quantum-classical autoencoders to ground the article in current literature. Hybrid quantum-classical autoencoder (HQC-AE) denotes a family of autoencoder architectures in which parameterized quantum circuits are combined with classical neural networks, classical post-processing, or both. Across the literature, the hybridization appears in several forms: a quantum encoder paired with a classical decoder, a classical encoder paired with a quantum decoder, a parameterized quantum bottleneck inserted into an otherwise classical autoencoder, and fully hybrid encoder-decoder pipelines for end-to-end tasks. The common objective is to learn a compressed, task-adapted, or structurally informative latent representation while using quantum state preparation, entangling layers, expectation-value readout, and classical optimization and decoding (Tabi et al., 2024, Zhang et al., 2024).

1. Conceptual foundations

The conceptual precursor of HQC-AE is the quantum autoencoder for compression of quantum states. In that formulation, an ensemble of $(n+k)$-qubit states is processed by a parameterized unitary $U(\boldsymbol{\theta})$, the first $n$ qubits are retained as the compressed register, and the remaining $k$ qubits become a “trash” register. Perfect compression is achieved if and only if

$$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$

for all training states, so that the trash subsystem is always returned to a fixed reference state. Training is hybrid in the original sense: the cost is estimated on quantum hardware, including via the SWAP test, and the parameters are updated by classical routines such as L-BFGS-B or Basin-Hopping (Romero et al., 2016).

Later work generalized this idea from quantum-state compression to classical-data pipelines in which the latent map, the bottleneck, or the decoder is partly quantum and partly classical. In these variants, the quantum stage no longer serves only as a compression map for quantum inputs; it becomes a task-conditioned nonlinear transformation within communication systems, anomaly detectors, sequence models, reinforcement-learning agents, or tensor-network/image pipelines (Sakhnenko et al., 2021, Nagy et al., 2024, Liu et al., 2020). This suggests a broadening of the term from a single compression primitive to a general hybrid latent-space design pattern.

2. Structural patterns and interface design

A defining feature of HQC-AE is that there is no single canonical topology. One formal treatment distinguishes three end-to-end radio scenarios: quantum encoder $\rightarrow$ classical decoder, classical encoder $\rightarrow$ quantum decoder, and fully quantum encoder-decoder. In each case, a quantum map $E_Q$ or $D_Q$ is embedded inside a larger classical communication chain, and the quantum stage is linked to the classical one through measurements such as expectation values of observables (Tabi et al., 2024).

A second recurring pattern is the quantum bottleneck. In this formulation, a classical encoder first maps $x$ to a latent vector, the latent vector is encoded into a quantum state, a parameterized quantum circuit $U(\boldsymbol{\theta})$0 is applied, and measurements produce a new classical vector for the decoder. One explicit expression is

$U(\boldsymbol{\theta})$1

with reconstruction loss defined over the data distribution (Rasyidi et al., 4 Dec 2025). This architecture appears in anomaly detection and spoofing detection, where the quantum layer is inserted between classical encoder and decoder blocks (Sakhnenko et al., 2021, Enan et al., 25 Aug 2025).

A third pattern is quantum encoding followed by a classical task-specific decoder. In the molecular setting, the encoder is a quantum autoencoder with 8 physical qubits, 5 latent qubits, and 3 trash qubits, while the decoder is a 4-layer LSTM with self-attention. In wireless communication, a pair of parameterized quantum circuits replaces the classical transmitter, but the receiver remains a classical DNN (Jahin et al., 26 Aug 2025, Zhang et al., 2024).

Structural motif	Representative formulation	Representative instances
Quantum bottleneck inside a classical AE	$U(\boldsymbol{\theta})$2	anomaly detection, NIDS, GNSS spoofing detection
Quantum encoder with classical decoder	quantum state preparation and measurement feed a classical network	end-to-end communication, molecular reconstruction
Classical encoder with quantum decoder, or QC$U(\boldsymbol{\theta})$3QC	classical latent or channel output is decoded by a PQC	radio communication scenarios (b) and (c)

A distinct hybrid line replaces gate-model PQCs with tensor-network encoders. In that setting, each pixel is lifted into a “quantum-feature” vector, two Tree Tensor Network layers contract the resulting product state, and the measured latent amplitudes are decoded by a classical de-convolutional network (Liu et al., 2020).

3. Quantum circuit constructions and latent representations

Circuit design varies substantially across domains, but several motifs recur: amplitude or angle embedding, shallow repeated variational layers, explicit entangling patterns, and measurement-based quantum-to-classical interfaces.

In wireless end-to-end communication, the transmitter can be implemented as two parallel Quantum Neural Networks, one for the real part and one for the imaginary part of the transmitted symbol. Each block uses $U(\boldsymbol{\theta})$4 qubits, amplitude embedding of the $U(\boldsymbol{\theta})$5-dimensional one-hot input, and $U(\boldsymbol{\theta})$6 identical PQC layers; in the reported configuration $U(\boldsymbol{\theta})$7. Each layer applies $U(\boldsymbol{\theta})$8 rotations on every qubit with trainable angles $U(\boldsymbol{\theta})$9, followed by a nearest-neighbor CNOT chain. After the final layer, Pauli-$n$0 expectation values yield a real vector in $n$1, which is linearly rescaled to produce $n$2 or $n$3, concatenated, and normalized to satisfy $n$4 (Zhang et al., 2024).

In radio communication with a quantum decoder, the architecture instead uses 4 physical qubits and a weighted double data re-uploading ansatz. In each layer, the noisy channel features $n$5 are encoded through $n$6 rotations, followed by trainable $n$7 gates and a ring of CNOTs, and then the data-reuploading block is repeated. The paper emphasizes that generalized data re-uploading is used to meet inference-time constraints (Tabi et al., 2023).

In molecular reconstruction, the encoder employs a layered PQC with 5 identical encoding-decoding blocks. The input classical embedding is generated by the Word2Ket tensor-train method; these embeddings condition the angles of single-qubit rotation gates such as RY or RZ in the first half of each layer, while trainable variational parameters govern the remaining rotations. Controlled-RZ entangling gates are arranged in a linear chain topology. After compression into 5 latent qubits, all qubits are measured in the computational basis, and expectation values or measurement frequencies across the latent qubits form the classical vector $n$8 for the LSTM decoder (Jahin et al., 26 Aug 2025).

In GNSS spoofing detection, the classical encoder maps a standardized $n$9-dimensional feature vector to a $k$0-dimensional latent vector, which is amplitude-embedded into 4 qubits because $k$1. A 3-layer variational block from PennyLane’s BasicEntanglerLayers applies $k$2 rotations on each qubit and ring-topology entanglers, and measurement of Pauli-$k$3 expectation values yields a $k$4-dimensional quantum latent vector for the classical decoder (Enan et al., 25 Aug 2025).

Measurement strategy is therefore central to the effective latent space. Some HQC-AEs use only single-qubit expectation values; others use full computational-basis distributions, kernel matrices obtained via SWAP tests, or measured latent amplitudes reshaped into small grids (Maragkopoulos et al., 2024, Liu et al., 2020). This suggests that the dimensionality of the usable latent representation is determined not only by qubit count but also by the chosen readout map.

4. Objectives, optimization, and training regimes

The optimization problem in HQC-AE depends strongly on the application. In the simplest bottleneck autoencoder form, the loss is the mean-squared reconstruction error

$k$5

with gradients through the PQC estimated by the parameter-shift rule (Rasyidi et al., 4 Dec 2025). This reconstruction-only regime is used in anomaly detection and spoofing detection, often followed by thresholding or downstream latent-space detectors (Sakhnenko et al., 2021, Enan et al., 25 Aug 2025).

Communication-oriented HQC-AEs typically replace reconstruction loss with classification-style objectives over discrete messages. In wireless fading channels, the loss is categorical cross-entropy,

$k$6

optimized end-to-end from one-hot message through quantum circuits, channel layer, and classical decoder. The reported optimizer is Adam with learning rate $k$7 and batch size $k$8, with training SNR fixed at $k$9 dB and evaluation across $$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$0–$$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$1 dB (Zhang et al., 2024). In the radio decoder work, sparse categorical cross-entropy is used over the 16 basis outcomes, with Adam, learning rate $$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$2, batch size $$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$3, and approximately $$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$4 mini-batch updates (Tabi et al., 2023).

Sequence-oriented HQC-AEs can combine several losses simultaneously. The molecular model jointly optimizes a quantum fidelity loss, token-level cross-entropy loss, a Levenshtein-similarity-based SMILES loss, and a trash deviation loss,

$$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$5

using Adam, learning rate $$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$6, cosine annealing, batch size $$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$7, and $$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$8 epochs; teacher forcing with scheduled sampling is used to stabilize decoder training (Jahin et al., 26 Aug 2025).

In reinforcement learning, the autoencoder and the quantum policy can be trained jointly under

$$U(\boldsymbol{\theta})\,\lvert\psi_i\rangle_{AB} =\lvert\psi_i^c\rangle_A\otimes\lvert a\rangle_B$$9

so that the encoder receives gradients both from reconstruction and from policy optimization, while the QNN parameters are updated through parameter shift (Nagy et al., 2024). By contrast, other works deliberately separate the stages. In one controlled classification study, the classical AE is trained only on reconstruction loss, frozen, and then a VQC is trained only on classification loss, precisely to isolate the quantum contribution (Kölle et al., 2023). This split underscores that “hybrid training” in HQC-AE can mean either end-to-end differentiation or strict stagewise optimization.

5. Empirical behavior across domains

Reported behavior spans wireless communication, anomaly detection, sequence reconstruction, image compression, and latent-space learning. Representative results include radio-communication BLER and SER curves, molecular fidelity and sequence similarity, zero-day GNSS spoofing detection, network intrusion detection under architectural sweeps and noise, and gains in latent-space anomaly separation (Zhang et al., 2024, Tabi et al., 2023, Jahin et al., 26 Aug 2025, Enan et al., 25 Aug 2025, Rasyidi et al., 4 Dec 2025, Sakhnenko et al., 2021).

Domain	Representative configuration	Reported outcome
End-to-end wireless communication	$\rightarrow$0, $\rightarrow$1	AE = 140 048 params, QAE = 70 192 params; BLER comparable; lower steady-state BLER after ~20 epochs
Radio communication with quantum decoder	4 qubits, $\rightarrow$2	matches classical baseline across 0–30 dB; at 15 dB both achieve SER $\rightarrow$3
Molecular reconstruction	8 qubits, 5 latent, 3 trash	quantum fidelity $\rightarrow$4; classical similarity $\rightarrow$5
GNSS spoofing detection	52 $\rightarrow$6 16 classical latent $\rightarrow$7 4 qubits	average accuracy 97.71%; average FNR 0.62%; ds7 accuracy 98.23%
Network intrusion detection	best HQC configuration on UNSW-NB15	AUROC 0.901 vs 0.897 for classical AE
Classical-quantum anomaly detection	gas turbine, Circuit 10	F1 from $\rightarrow$8 to $\rightarrow$9

The communication literature is particularly explicit about parameter efficiency and convergence. For the hybrid quantum-classical autoencoder transmitter, the parameter count is

$\rightarrow$0

compared with

$\rightarrow$1

and the example $\rightarrow$2 with $\rightarrow$3 yields an approximately $\rightarrow$4 reduction in transmitter parameters while maintaining comparable BLER to classical DNN-based and conventional channel coding schemes (Zhang et al., 2024).

Beyond these cases, a tensor-network HQC-AE for image compression reported typical PSNR scores of approximately $\rightarrow$5 dB at compression ratio $\rightarrow$6, approximately $\rightarrow$7 dB at compression ratio $\rightarrow$8, and approximately $\rightarrow$9 dB at compression ratio $E_Q$0 on MNIST, with Fashion-MNIST showing a similar trend (Liu et al., 2020). In a clustering-oriented hybrid quantum autoencoder for amplitude-encoded quantum states, using the 4 smallest principal components of the latent vector gave $E_Q$1 clustering accuracy, while semi-supervised classification on the latent representation reached up to $E_Q$2 with an SVM polynomial kernel (Srikumar et al., 2021).

6. Limitations, interpretive issues, and future directions

Several limitations recur across the HQC-AE literature. First, simulation cost remains substantial. In wireless fading experiments, current quantum simulators on classical hardware incur approximately $E_Q$3 runtime overhead, and true speedups are said to await scalable quantum hardware and optimized simulation engines (Zhang et al., 2024). In radio decoding, inference time depends on circuit depth and shot count; on ibmq_belem, 16 layers correspond to depth $E_Q$4 and $E_Q$5s per shot, implying approximately $E_Q$6 ms for $E_Q$7 shots, so sub-$E_Q$8 ms budgets require fewer shots, fewer layers, or pulse-level optimization (Tabi et al., 2023).

Second, performance is highly architecture-sensitive. In unsupervised network intrusion detection, HQC-AEs can match or exceed classical performance in their best configurations, but they exhibit much larger variance across architectural choices, and simulated gate-noise experiments show early degradation: even at $E_Q$9 the AUROC drops by approximately $D_Q$0 (Rasyidi et al., 4 Dec 2025). In anomaly detection, a deeper circuit did worse than a shallower one, illustrating that more depth does not guarantee better anomaly separation (Sakhnenko et al., 2021). Likewise, in radio communication, larger qubit counts improve asymptotic BER but raise circuit-noise sensitivity, and circuit depth $D_Q$1–$D_Q$2 is reported as optimal, while beyond $D_Q$3 performance saturates or degrades under realistic gate noise (Tabi et al., 2024).

Third, interpretive attribution is nontrivial. A controlled comparison of AE+VQC, pure amplitude-embedding VQC, DQC, SEQUENT, and matched classical baselines found that AE+VQC accuracy was never significantly different from pure VQC with amplitude embedding, whereas DQC and the classical NN on uncompressed input were significantly stronger. The study argues that in DQC and SEQUENT the classical component does almost all of the classification (Kölle et al., 2023). This suggests that claims about “quantum contribution” in HQC-AE require matched ablations, parameter-count controls, and careful separation of classical pre-processing effects from quantum processing effects.

Fourth, objective alignment can be imperfect. In the molecular setting, fidelity gains do not always guarantee proportional gains in classical similarity, and the authors explicitly note a gap between preserving quantum information and decoding discrete syntax perfectly. They also note that small qubit counts, specifically 8–9 total, limit expressivity relative to future fault-tolerant devices (Jahin et al., 26 Aug 2025).

Future directions in the literature are correspondingly pragmatic: hardware-aware circuit optimizations, extensions to OFDM and MIMO systems, integration with quantum convolutional networks for large block sizes, alternative quantum-to-classical feature extraction such as kernel methods, Transformer-style or graph decoders for sequence tasks, trainable channel-state blocks, and explicit incorporation of quantum measurement noise into the loss function (Zhang et al., 2024, Jahin et al., 26 Aug 2025, Tabi et al., 2024). Across these proposals, HQC-AE is best understood not as a settled architecture, but as an active research program on how to place, train, and evaluate quantum subroutines inside autoencoding pipelines under stringent NISQ-era constraints.