Quantum Autoencoders: Compression & Feature Extraction

Updated 1 December 2025

Quantum autoencoders are parameterized circuits that compress quantum states by encoding them into lower-dimensional latent spaces.
They employ a workflow of unitary encoding, partial trace, and decoding to optimize reconstruction fidelity and reduce noise.
Applications include quantum simulation state compression, denoising, anomaly detection, and enhancing quantum communication protocols.

Quantum autoencoders (QAEs) are parameterized quantum circuits that learn to compress quantum data—typically pure or mixed quantum states—into a lower-dimensional latent subspace and reconstruct the original data with high fidelity. By generalizing the concepts of classical autoencoders to the quantum domain, QAEs have emerged as central tools for quantum data compression, feature extraction, denoising, channel coding, anomaly detection, and quantum machine learning. The structure and optimization of QAEs are intimately tied to both the representational properties of quantum circuits and the information-theoretic capacity of quantum states under the constraints of quantum mechanics.

1. Core Principles and Mathematical Structure

A quantum autoencoder operates by encoding an $n$ -qubit quantum state $\rho_{\text{in}}$ using a parameterized unitary $U_{\text{enc}}(\theta)$ into a product state on a $k$ -qubit "latent" register and an $n-k$ -qubit "trash" register. The general workflow is:

Encoding: Apply $U_{\text{enc}}(\theta)$ to the input, pushing as much information as possible into the latent subsystem.
Partial trace: Discard the trash register (i.e., trace out $n-k$ qubits), leaving a reduced state on the latent subsystem.
Decoding: Re-prepare the trash register, usually in a fixed reference state (often $|0\rangle$ ), and apply the inverse unitary $U^\dagger_{\text{enc}}(\theta)$ to recover an approximation of the original state.

The loss function is typically defined via either:

The trash occupation: measuring the probability that the trash register is found in the reference state, i.e., $F_{\text{trash}}(\theta) = \langle a| \rho_{\text{trash}}(\theta) |a\rangle$ , or
The reconstruction fidelity: $F_{\text{rec}}(\theta) = \langle \psi_{\text{in}}| \rho_{\text{out}}(\theta) | \psi_{\text{in}} \rangle$ .

A central theoretical result (Huang et al., 2019, Romero et al., 2016) states that perfect lossless quantum autoencoding is possible if and only if the number of linearly independent states in the training ensemble does not exceed the dimension of the latent space.

2. Variational Circuit Ansätze and Optimization Strategies

Quantum autoencoders rely on parameterized circuit architectures (ansätze) for $U_{\text{enc}}(\theta)$ . Common designs include layered hardware-efficient circuits consisting of single-qubit rotations and entangling gates, quantum convolutional neural network (QCNN) blocks, approximate quantum adders, and more specialized constructions based on the symmetry or structure of the input data (Bravo-Prieto, 2020, Lamata et al., 2017, Lo et al., 11 Feb 2025).

Key points:

Layered circuits: Alternating blocks of parameterized single-qubit rotations ( $R_y$ , $R_z$ ) and entangling gates (CNOTs, CZs) with shallow depth for NISQ compatibility.
Feature-dependent gates: Enhanced-Feature QAE (EF-QAE) injects a data-dependent feature vector $x$ directly into the parameterization of each rotation, enabling state-adaptive compression (Bravo-Prieto, 2020).
Quantum adders and genetic algorithms: Circuits optimized to approximate the sum of quantum states can act as autoencoder maps, with gate sequences and parameters tuned by evolutionary algorithms (Lamata et al., 2017, Ding et al., 2018).
Optimization: Classical optimizers such as Adam, BFGS, or COBYLA are employed, with cost function gradients estimated via parameter-shift rules or finite-difference methods (Bravo-Prieto, 2020, Asaoka et al., 21 Feb 2025).

To avoid barren plateaus, local cost functions—measuring only trash qubit occupations or local output fidelities—are often used (Bravo-Prieto, 2020, Wu et al., 2023).

3. Advanced Models: Mixed-State, Noise-Assisted, and Channel Autoencoders

Standard quantum autoencoders are fundamentally limited by the entropy and rank of the input states and the nature of the reference state used in reconstruction (Cao et al., 2020, Ma et al., 2023):

Mixed-reference QAEs: These introduce a reference state for the trash register that interpolates between a pure state and the maximally mixed state, parameterized by $p$ . This approach circumvents the so-called "entropy inconsistency," enabling high-fidelity compression for high-rank, high-entropy mixed states by supplying adequate entropy during decoding (Ma et al., 2023).
Noise-assisted QAEs: By measuring the trash system to estimate its mixedness and classically feeding forward this information, a noise channel is engineered on the reference such that input and output mixedness are matched, restoring fidelity for general mixed inputs (Cao et al., 2020).
Quantum Circuit Autoencoders (QCAE): These generalize the autoencoder paradigm from state channels to full quantum processes (channels/circuits), learning a compression and decompression supermap acting on quantum channels, optimized using channel Choi-state fidelities (Wu et al., 2023).
Adiabatic/annealing-based QAEs: Compression via adiabatic evolution—mapping eigenstates of a data Hamiltonian to product states in latent/trash subsystems—enables hardware-realizable implementations on quantum annealers (Cao et al., 2020).

4. Applications: Compression, Denoising, Feature Extraction, and Communication

Quantum autoencoders are applied across quantum information processing:

Compression of quantum simulation states: Molecular ground states (e.g., H $_2$ , Hubbard model) are compressed for storage or further computation, enabling resource savings in quantum simulation and chemistry (Romero et al., 2016, Huang et al., 2019).
Denoising: QAEs learn to remove noise from GHZ states and other highly entangled many-body states, restoring nearly ideal fidelities under bit-flip and depolarizing channels. This extends to denoising for quantum repeaters, metrology, and quantum secret sharing [2019.10.09169, (Achache et al., 2020, Mok et al., 2023)].
Feature map learning and anomaly detection: QAEs are used for quantum feature map generation, outperforming fixed Hamiltonian encodings on large peptide-sequence datasets (Zhuang et al., 23 Sep 2025) and exceeding classical autoencoders for anomaly detection in data-limited settings such as cybersecurity (Senthil et al., 22 Oct 2025).
Image classification and unsupervised feature extraction: Fully quantum classification via QAEs achieves competitive performance relative to classical methods—even with O(10²⁾ trainable parameters—on canonical output tasks like the MNIST digit classification and multi-class image datasets (Asaoka et al., 21 Feb 2025, Lo et al., 11 Feb 2025).
Quantum channel coding and communication: QAEs can approximate optimal encoder/decoder pairs for transmission over qubit noise channels, saturating theoretical capacity bounds for classical, quantum, and entanglement-assisted communication, and discovering explicit variational codes (Rathi et al., 2023).
Quantum data compression with group-theoretic structure: Hidden subgroup quantum autoencoders, which incorporate a variational quantum Fourier transform, can extract and leverage hidden group symmetries for exponentially efficient compression of structured classical data (Liu et al., 2023).

5. Experimental Demonstrations and Resource Considerations

QAEs have been realized on photonic chips, superconducting quantum computers (Rigetti, IBM), and solid-state spin systems (NV centers in diamond) (Pepper et al., 2018, Ding et al., 2018, Zhou et al., 2022):

Photonic implementations: Programmable optical circuits act as lossless compressors for small Hilbert spaces (e.g., qutrit→qubit), with robust performance against drift and statistical noise (Pepper et al., 2018, Huang et al., 2019).
Superconducting QPU: Demonstrations of adder-based autoencoders reach fidelities of 0.95–0.99 after gate-efficient genetic-optimization (Ding et al., 2018, Lamata et al., 2017).
Solid-state entanglement preservation: QAEs trained to compress electron-nuclear spin entanglement into the nuclear subspace in NV centers extend Bell-state lifetimes by three orders of magnitude (Zhou et al., 2022).
Scalability and parameter efficiency: QAE-based classifiers and feature maps achieve competitive accuracy with O(10^2–10³⁾ circuit parameters, as compared to O(10^3–10⁴⁾ for classical neural networks on similar tasks (Asaoka et al., 21 Feb 2025, Zhuang et al., 23 Sep 2025).
Variational resource trade-offs: Enhanced-feature QAEs (EF-QAE) or mixed-reference QAEs increase classical optimization cost, proportional to both the feature dimension and the number of circuit layers, but provide direct fidelity gains with the same quantum hardware (Bravo-Prieto, 2020, Ma et al., 2023).

6. Limitations, Open Problems, and Future Directions

Limitations of current QAE methodology and areas for further exploration include:

Entropy/rank bounds: Standard QAEs cannot perfectly compress arbitrary high-rank mixed states unless the decoder is supplied with appropriate mixedness; this is partially resolved by noise-assisted and mixed-reference variants (Cao et al., 2020, Ma et al., 2023).
Expressivity constraints: The achievable compression and reconstruction fidelity is governed by the expressiveness of the circuit ansatz and the number of layers, with complex families requiring deeper or problem-structured circuits (Romero et al., 2016, Bravo-Prieto, 2020).
Training bottlenecks: Barren plateaus, slow convergence, and local minima occur in deep or random circuits; local cost functions and adaptive training sets mitigate the issue (Bravo-Prieto, 2020, Wu et al., 2023).
Quantum channel capacity: Lossless compression or channel compression is only possible for channels of sufficient Choi-rank; bounds on reconstruction fidelity can be obtained in terms of the spectrum of the Choi-Jamiołkowski state (Wu et al., 2023).
Experimental scaling: Near-term NISQ hardware is limited by gate depth, noise, and qubit count, but shallow and hardware-efficient QAEs, as well as error mitigation strategies, are showing progress toward practically viable quantum feature learning and data compression (Zhuang et al., 23 Sep 2025, Senthil et al., 22 Oct 2025).
Theoretical open questions: Analytical characterization of the ultimate trade-off between compression ratio, input entropy, and reference entropy; adaptive and meta-learned feature encodings; and scalable quantum channel autoencoding frameworks remain active topics (Ma et al., 2023, Zhuang et al., 23 Sep 2025, Wu et al., 2023).

7. Summary Table: Key Variants and Their Features

Variant/Method	Addresses	Distinctive Mechanisms
Standard QAE	Pure, low-rank	Parameterized unitary + fixed reference
EF-QAE (Bravo-Prieto, 2020)	Data families	Feature-dependent circuit; adaptive maps
Mixed-reference QAE	High-entropy	Decoder reference with tunable mixedness
Noise-assisted QAE	High-rank mixed	Measurement-based trash feedback
QCAE (Wu et al., 2023)	Channels/circuits	Choi fidelity; local cost function
QCNN-based QAE	Images, 1D/2D	Convolutional ansatz for feature pooling
QAE-Gate (Zhu et al., 2021)	Cloud computation	Communication-efficient gate compression
Hidden subgroup QAE	Structured data	Variational QFT, group-theoretic encoding