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arXiv:2605.01258

Toward the Goldilocks blind compression of quantum states

Hyunho Cha, Chae-Yeun Park, Jungwoo Lee

quant-phSubmitted May 2, 2026• Announced May 5, 2026• 0 citations
v1 · May 2, 2026v2 · Oct 1, 2026
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Abstract

Quantum autoencoders (QAEs) are learning architectures that compress quantum data into a low-dimensional latent state while preserving the information needed for reconstruction. We study blind single-copy compression of quantum states through a $k$-qubit bottleneck and investigate the minimal circuit width required to attain the information-theoretic optimum under average infidelity. Between the conventional architecture, which is narrow but nonuniversal, and fully general \emph{completely positive and trace preserving} (CPTP) realizations, which are universal but overparameterized, we identify a \emph{Goldilocks} regime. We prove that for every distribution of pure $n$-qubit states, there exists a QAE with exactly $k$ encoder ancillas and $n$ decoder ancillas that achieves the optimal fidelity over all CPTP encoder--decoder pairs. The encoder-side statement is sharp in that we construct source families for which every optimal scheme necessarily uses at least $k$ encoder ancillas, thereby determining the universal encoder threshold exactly. On the decoder side, we show that isometric decoders are exactly optimal for several analytically tractable source families, but we also exhibit an explicit counterexample demonstrating that decoder isometry is not universally sufficient. Nevertheless, numerical experiments indicate that the performance gap is practically negligible.

Comments
60 pages, 5 figures

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Quantum autoencoders (QAEs) are learning architectures that compress quantum data into a low-dimensional latent state while preserving the information needed for reconstruction. We study blind single-copy compression of quantum states through a kkk-qubit bottleneck and investigate the minimal circuit width required to attain the information-theoretic optimum under average infidelity. Between the conventional architecture, which is narrow but nonuniversal, and fully general \emph{completely positive and trace preserving} (CPTP) realizations, which are universal but overparameterized, we identify a \emph{Goldilocks} regime. We prove that for every distribution of pure nnn-qubit states, there exists a QAE with exactly kkk encoder ancillas and nnn decoder ancillas that achieves the optimal fidelity over all CPTP encoder--decoder pairs. The encoder-side statement is sharp in that we construct source families for which every optimal scheme necessarily uses at least kkk encoder ancillas, thereby determining the universal encoder threshold exactly. On the decoder side, we show that isometric decoders are exactly optimal for several analytically tractable source families, but we also exhibit an explicit counterexample demonstrating that decoder isometry is not universally sufficient. Nevertheless, numerical experiments indicate that the performance gap is practically negligible.
Quantum compression is a cornerstone of quantum information science, promising efficient encoding of large quantum states into fewer qubits while retaining high fidelity. In this work, we compare two pivotal strategies for compressing an i.i.d. quantum source under small-scale (few-qubit) regimes. The first strategy relies on a projector-based compression method that discards low-probability components in the state’s spectral decomposition, achieving near-optimal asymptotic compression in principle. The second strategy employs a quantum autoencoder (QAE), a variational circuit that learns to embed essential information into fewer qubits. We implement both approaches in simulation, gauge their performance, and explore how many qubits can be ”saved” in practice. Our results highlight how projector-based compression provides a near-optimal yet analytically prescriptive solution, whereas autoencoders offer a machine-learning–style route that can adapt to varied input states.
Abstract Quantum autoencoder (QAE) compresses a bipartite quantum state into its subsystem using a self-checking mechanism. How to characterize and minimize the lost information in this process is essential for understanding the compression mechanism of QAE. Here, we investigate how to minimize the lost information in QAE for any input mixed state. We theoretically show that the lost information is the quantum mutual information between the remaining subsystem and the discarded one; the encoding unitary transformation is designed to minimize this mutual information. Furthermore, we show that the optimized unitary transformation can be decomposed as the product of a permutation unitary transformation and a disentanglement unitary transformation, and the permutation unitary transformation can be searched by a regular Young tableau algorithm. When the search can be made exhaustive in lower-dimensional systems, the lost information is minimized numerically, which is shown theoretically to be a global minimum. When the dimension of the system becomes larger such that an exhaustive search is impossible, we adopt an approximate search algorithm to numerically identify that our compression scheme gives lower lost information than that from the quantum variational circuit-based QAE.
As a ubiquitous aspect of modern information technology, data compression has a wide range of applications. Therefore, a quantum autoencoder which can compress quantum information into a low-dimensional space is fundamentally important to achieve automatic data compression in the field of quantum information. Such a quantum autoencoder can be implemented through training the parameters of a quantum device using classical optimization algorithms. In this paper, we demonstrate the condition of achieving a perfect quantum autoencoder and theoretically prove that a quantum autoencoder can losslessly compress high-dimensional quantum information into a low-dimensional space (also called latent space) if the number of maximum linearly independent vectors from input states is no more than the dimension of the latent space. Also, we experimentally realize a universal two-qubit unitary gate and design a quantum autoencoder device by applying a machine learning method. Experimental results demonstrate that our quantum autoencoder is able to compress two two-qubit states into two one-qubit states. Besides compressing quantum information, the quantum autoencoder is used to experimentally discriminate two groups of nonorthogonal states.
Autoencoders are fundamental tools in classical computing for unsupervised feature extraction, dimensionality reduction, and generative learning. The Quantum Autoencoder (QAE), introduced by Romero J.[2017 Quantum Sci. Technol. 2 045001], extends this concept to quantum systems and has been primarily applied to tasks like anomaly detection. Despite its potential, QAE has not been extensively evaluated in basic classification tasks such as handwritten digit classification, which could provide deeper insights into its capabilities. In this work, we systematically investigate the performance of QAE in unsupervised feature extraction and reconstruction tasks. Using various encoder and decoder architectures, we explore QAE's ability to classify MNIST and analyze its reconstruction effectiveness. Notably, with a QCNN-based encoder, QAE achieves an average accuracy of 97.59% in binary classification of MNIST (0 and 1), demonstrating the feasibility of QAE for such tasks. This result highlights the synergy between QAE and QCNN in achieving optimal feature extraction performance. However, we also identify limitations in QAE's reconstruction capability, underscoring the need for further advancements in encoder and decoder designs. Our findings provide a foundation for future research on leveraging QAE in practical quantum machine learning applications.
Qubit-efficient quantum relaxations compress classical decision variables into expectation values on substantially fewer qubits. We ask what resource tradeoffs this compression entails for quantum optimization. For the complete quadratic-Majorana encoding of m=Θ(n2)m=Θ(n^2)m=Θ(n2) binary variables on~nnn qubits, we define the universal margin as the smallest correlator magnitude that can be guaranteed with prescribed signs for every target sign (bit) assignment. We show that it is exactly ΔMaj(n)=tan⁡ ⁣(π4n)=Θ(1/n)Δ_{\rm Maj}(n)=\tan\!\left(\fracπ{4n}\right)=Θ(1/n)ΔMaj​(n)=tan(4nπ​)=Θ(1/n), whereas uniformly random sign assignments retain Θ(1/n)Θ(1/\sqrt n)Θ(1/n​) target-specific margins. Arbitrary density operators and mixed fermionic Gaussian states generate the same quadratic-Majorana covariance body, so non-Gaussian state resources cannot enlarge this two-point relaxation. More generally, standard quantum random access code bounds provide general information-theoretic baselines. For any fixed family of mmm designated binary observables on~nnn qubits, the universal margin is at most (2ln⁡2 n/m)\sqrt{(2\ln2\,n/m)}(2ln2n/m)​, and random access decoding from NNN copies with constant success probability above 1/21/21/2 requires nN=Ω(m)nN=Ω(m)nN=Ω(m). For a fixed Pauli correlation encoding required to work uniformly over all targets, maintaining a fixed nonzero decoded magnitude under smooth sign decoding therefore requires a rescaling parameter that grows as the available margin shrinks. Thus, while providing substantial qubit savings, compression can shift cost into restricted expectation value geometry, smaller expectation value magnitudes, or more demanding information recovery rather than eliminate it.
A major challenge in near-term quantum computing is its application to large real-world datasets due to scarce quantum hardware resources. One approach to enabling tractable quantum models for such datasets involves finding low-dimensional representations that preserve essential information for downstream analysis. In classical machine learning, variational autoencoders (VAEs) facilitate efficient data compression, representation learning for subsequent tasks, and novel data generation. However, no quantum model has been proposed that exactly captures all of these features for direct application to quantum data on quantum computers. Some existing quantum models for data compression lack regularization of latent representations, thus preventing direct use for generation and control of generalization. Others are hybrid models with only some internal quantum components, impeding direct training on quantum data. To address this, we present a fully quantum framework, $\ensuremath{\zeta}$-QVAE, which encompasses all the capabilities of classical VAEs and can be directly applied to map both classical and quantum data to a lower-dimensional space, while effectively reconstructing much of the original state from it. Our model utilizes regularized mixed states to attain optimal latent representations. It accommodates various divergences for reconstruction and regularization. Furthermore, by accommodating mixed states at every stage, it can utilize the full data density matrix and allow for a training objective defined on probabilistic mixtures of input data. Doing so, in turn, makes efficient optimization possible and has potential implications for private and federated learning. In addition to exploring the theoretical properties of $\ensuremath{\zeta}$-QVAE, we demonstrate its performance on representative genomics and synthetic data. Our results indicate that $\ensuremath{\zeta}$-QVAE consistently learns representations that better utilize the capacity of the latent space and exhibits similar or better performance compared with matched classical models.