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EPJ Quantum Technology

Neural-enabled quantum information hiding with error-correcting codes: a novel framework for arbitrary quantum state embedding

Chaolong Hao, QuanGong Ma, Nianwen Si, BuYu Liu, Dan Qu

Published Jul 29, 2025• 2 citations
Publisher

Abstract

Quantum information hiding, as an extension of classical information hiding techniques into the realm of quantum information, currently focuses on embedding classical bits (0/1) within quantum carriers. This includes methods such as disguising classical secret information as channel noise and embedding it within quantum error correction codes. However, the embedding mechanism for arbitrary quantum states $\alpha |0\rangle + \beta |1\rangle $ is still in the exploratory stage. This paper proposes an innovative framework that leverages the redundant space of quantum error correction codes to construct a nonlinear decoding architecture with quantum neural networks. This approach simultaneously achieves both carrier state error correction and secret state embedding and extraction functions. Specifically, the [5,1,3] stabilizer code is used as the carrier, with secret state embedding achieved through single-qubit substitution, and a quantum autoencoder is designed for steganographic state information decoding. The proposed framework features fully quantum-based input/output systems, overcoming the limitations of traditional variational quantum circuits that rely on probabilistic measurements for output generation. By performing full ground-state measurements at the autoencoder bottleneck layer and optimizing the parallel sub-network architecture, the network achieves efficient convergence and effective extraction of single-copy quantum states. Experimental results show that under the conditions of optimized parameters and data size of 20, the training losses for the carrier and secret states are 0.03 and 0.08, respectively, with test fidelities of 0.92 and 0.93. For a data size of 50, the secret states recovery fidelity exceeds 0.87. KS test analysis indicates that the full ground-state measurement and parallel sub-network are key strategies for achieving network performance. Equivalent error analysis shows that this approach successfully utilizes the potential redundant space of quantum error correction codes, providing new research directions for quantum state information hiding.

DOI
10.1140/epjqt/s40507-025-00391-w

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Quantum information hiding, as an extension of classical information hiding techniques into the realm of quantum information, currently focuses on embedding classical bits (0/1) within quantum carriers. This includes methods such as disguising classical secret information as channel noise and embedding it within quantum error correction codes. However, the embedding mechanism for arbitrary quantum states α∣0⟩+β∣1⟩\alpha |0\rangle + \beta |1\rangle α∣0⟩+β∣1⟩ is still in the exploratory stage. This paper proposes an innovative framework that leverages the redundant space of quantum error correction codes to construct a nonlinear decoding architecture with quantum neural networks. This approach simultaneously achieves both carrier state error correction and secret state embedding and extraction functions. Specifically, the [5,1,3] stabilizer code is used as the carrier, with secret state embedding achieved through single-qubit substitution, and a quantum autoencoder is designed for steganographic state information decoding. The proposed framework features fully quantum-based input/output systems, overcoming the limitations of traditional variational quantum circuits that rely on probabilistic measurements for output generation. By performing full ground-state measurements at the autoencoder bottleneck layer and optimizing the parallel sub-network architecture, the network achieves efficient convergence and effective extraction of single-copy quantum states. Experimental results show that under the conditions of optimized parameters and data size of 20, the training losses for the carrier and secret states are 0.03 and 0.08, respectively, with test fidelities of 0.92 and 0.93. For a data size of 50, the secret states recovery fidelity exceeds 0.87. KS test analysis indicates that the full ground-state measurement and parallel sub-network are key strategies for achieving network performance. Equivalent error analysis shows that this approach successfully utilizes the potential redundant space of quantum error correction codes, providing new research directions for quantum state information hiding.
This paper introduces a novel Quantum Error Correction Embedding Algorithm designed to significantly enhance the success rate of quantum error correction. The core claim rests on leveraging the redundancy inherent in error-correcting codes to embed controllable quantum error correction (QEC) information directly into the encoding of quantum states. The fundamental mechanism involves a newly designed code that integrates error information within the superposition state of a quantum state, rather than solely focusing on error detection and correction. This approach exploits the inherent properties of superposition to perform error correction. The algorithm's potential impact lies in overcoming limitations of traditional QEC methods, particularly in scenarios where the error rate is high or the complexity of error correction is significant. This work presents a theoretical framework and outlines the key components of the algorithm, establishing a foundation for future research and development in robust quantum computation. The algorithm's success hinges on precisely controlling the superposition state to encode and decode error information effectively. Further investigation is needed to optimize the parameters and explore the algorithm's performance across various quantum systems and error models.
Quantum error correction codes (QECC) are a key component for realizing the potential of quantum computing. QECC, as its classical counterpart (ECC), enables the reduction of error rates, by distributing quantum logical information across redundant physical qubits, such that errors can be detected and corrected. In this work, we efficiently train novel end-to-end deep quantum error decoders. We resolve the quantum measurement collapse by augmenting syndrome decoding to predict an initial estimate of the system noise, which is then refined iteratively through a deep neural network. The logical error rates calculated over finite fields are directly optimized via a differentiable objective, enabling efficient decoding under the constraints imposed by the code. Finally, our architecture is extended to support faulty syndrome measurement, by efficient decoding of repeated syndrome sampling. The proposed method demonstrates the power of neural decoders for QECC by achieving state-of-the-art accuracy, outperforming for small distance topological codes, the existing end-to-end neural and classical decoders, which are often computationally prohibitive.
We investigate the use of Quantum Neural Networks for discovering and implementing quantum error-correcting codes. Our research showcases the efficacy of Quantum Neural Networks through the successful implementation of the Bit-Flip quantum error-correcting code using a Quantum Autoencoder, effectively correcting bit-flip errors in arbitrary logical qubit states. Additionally, we employ Quantum Neural Networks to restore states impacted by Amplitude Damping by utilizing an approximative 4-qubit error-correcting codeword. Our models required modification to the initially proposed Quantum Neural Network structure to avoid barren plateaus of the cost function and improve training time. Moreover, we propose a strategy that leverages Quantum Neural Networks to discover new encryption protocols tailored for specific quantum channels. This is exemplified by learning to generate logical qubits explicitly for the bit-flip channel. Our modified Quantum Neural Networks consistently outperformed the standard implementations across all tasks.
While classical steganography achieves maturity in digital media, hiding arbitrary quantum states ( α∣0⟩+β∣1⟩\alpha |0\rangle + \beta |1\rangle α∣0⟩+β∣1⟩ ) has emerged as an intriguing frontier. To address this problem, we establish a formal model of controllable random perturbation unitaries for single/multi-stego state tasks. We progressively explore Quantum Autoencoder (QAE) structures through three stages: starting from single-state scenarios without perturbation, advancing to perturbed conditions, and finally extending to multi-state tasks. We design two perturbation-based encoding schemes using Quantum Autoencoders (QAE): the simple scheme (QAE-DD) leverages the inverse application of encoding–decoding modules, while the improved scheme (QAE-OSP) incorporates orthogonal projection routing and parallel subnetworks to restructure the hidden-layer architecture. In 3-qubit entangled-state simulations with data scales n≤10n \le 10n≤10 and perturbation strengths ε∈[0,1]\varepsilon \in [0,1]ε∈[0,1] , QAE-DD performs well under low perturbation, whereas QAE-OSP maintains higher fidelity between the carrier and secret states under high perturbation conditions (e.g., n=5,ε=0.6n = 5, \varepsilon = 0.6n=5,ε=0.6 ), with fidelity values F(ρstego,ρ~stego)=0.91F\left( \rho _{\text {stego}}, \tilde{\rho }_{\text {stego}} \right) =0.91F(ρstego​,ρ~​stego​)=0.91 / F(ρS,ρ~S)=0.84F\left( \rho _{S}, \tilde{\rho }_{S} \right) =0.84F(ρS​,ρ~​S​)=0.84 providing a reference for network design. Finally, we extend the single-carrier (“ 1+11+11+1 ”) task to the multi-carrier (“ 1+N1+N1+N ”) scenario by constructing a “centroid” state training set based on the principal component of carrier-state groups and validating the applicability of both models. Under the conditions n=5n = 5n=5 and ε=0.6\varepsilon = 0.6ε=0.6 , the QAE-OSP model successfully improves the average fidelity between multiple secret states and carrier states from 0.68 to 0.90, demonstrating its capability to aggregate multiple carriers to enhance overall concealment. Although the present study covers only small-scale data and networks, it lays the groundwork for a neural network framework that covertly embeds arbitrary quantum states into high-dimensional quantum states, providing a basis for future exploration.
This paper presents a novel approach to quantum error correction (QEC) and adaptive quantum channel mapping (QECM) designed to enhance the robustness of quantum information transmission through noisy channels. The core idea is to leverage the redundancy inherent in QEC codes and combine it with a machine learning-based adaptive mapping algorithm. Unlike traditional methods that rely on pre-characterized channel models, this approach dynamically optimizes the QEC encoding and channel mapping strategies based on real-time channel feedback. The system utilizes an error correction matrix derived from the chosen QEC code to construct a quantum channel model. A neural network, specifically a variational quantum neural network (VQNN), is then trained to learn the optimal quantum channel mapping policy. This dynamic optimization addresses the limitations of fixed channel models and promises improved performance in practical quantum communication scenarios. The effectiveness of the QECM framework is demonstrated through theoretical analysis and a conceptual model, paving the way for robust long-distance quantum communication. The system's adaptability allows it to cope with unknown or time-varying channel characteristics, a critical factor for successful quantum communication implementations. The ultimate goal is to achieve efficient and reliable quantum information transfer, regardless of the underlying channel noise.
In the current thesis we first talk about the six-qubit quantum error-correcting code and show its connections to entanglement-assisted error-correcting coding theory and then to subsystem codes. This code bridges the gap between the five-qubit (perfect) and Steane codes. We discuss two methods to encode one qubit into six physical qubits. Each of the two examples corrects an arbitrary single-qubit error. The first example is a degenerate six-qubit quantum error-correcting code. We prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender (Alice) and the receiver (Bob) and corrects an arbitrary single-qubit error. In the second half of this thesis we explore the yet uncharted and relatively undiscovered area of quantum steganography. Steganography is the process of hiding secret information by embedding it in an innocent message. We present protocols for hiding quantum information in a codeword of a quantum error-correcting code passing through a channel. Using either a shared classical secret key or shared entanglement Alice disguises her information as errors in the channel. Bob can retrieve the hidden information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for certain protocols.