GAN decoder on a quantum toric code for noise-robust quantum teleportation

TL;DR

This work introduces a GAN-based decoder for the quantum toric code and applies it to noise-robust quantum teleportation under depolarizing noise. By training a generator–discriminator pair on stabilizer-syndrome data, the decoder learns correction paths that yield higher logical fidelity than MWPM and achieves a pseudo-threshold of $p=0.2108$, substantially above existing decoders. When integrated into a teleportation protocol, the GAN-optimized scheme shows improved fidelity across noise regimes, with thresholds $p<0.06503$ for $d=3$ and $p<0.07512$ for $d=5$, and a reported fidelity around $99.895\%$ at $p=0.05$ for $d=5$. The results indicate that GANs provide a flexible, scalable approach to decoding in topological QECCs and can enhance fault-tolerant quantum information processing in noisy environments.

Abstract

We propose a generative adversarial network (GAN)-based decoder for quantum topological codes and apply it to enhance a quantum teleportation protocol under depolarizing noise. By constructing and training the GAN's generator and discriminator networks using eigenvalue datasets from the code, we obtain a decoder with a significantly improved decoding pseudo-threshold. Simulation results show that our GAN decoder achieves a pseudo-threshold of approximately $p=0.2108$, estimated from the crossing point of logical error rate curves for code distances $d=3$ and $d=5$, nearly double that of a classical decoder under the same conditions ($p \approx 0.1099$). Moreover, at the same target logical error rate, the GAN decoder consistently achieves higher logical fidelity compared to the classical decoder. When applied to quantum teleportation, the protocol optimized using our decoder demonstrates enhanced fidelity across noise regimes. Specifically, for code distance $d=3$, fidelity improves within the depolarizing noise threshold range $P<0.06503$; for $d=5$, the range extends to $P<0.07512$. Moreover, with appropriate training, our GAN decoder can generalize to other error models. This work positions GANs as powerful tools for decoding in topological quantum error correction, offering a flexible and noise-resilient framework for fault-tolerant quantum information processing.

Figures (15)

• Figure 1: The basic structure of toric codes. X stabilizers are used to detect Z errors, while Z stabilizers are used to detect X errors. The measurement circuits are shown on the right. H gates, I gates, and CNOT gates are used. The ancilla qubits are measured to obtain the eigenvalue. For a given stabilizer, the eigenvalue is +1 if the number of errors on the neighboring data qubits is even, and -1 if it is odd.
• Figure 2: The structure of a GAN involves two components: the generator and the discriminator. The generator aims to produce fake data that approximates the target data, while the discriminator aims to distinguish between target and fake data. Through adversarial training, both the generator and discriminator are iteratively refined, leading to enhanced performance. The yellow, green, and red blocks represent different network layers of the GAN.
• Figure 3: Quantum teleportation. Particles $A$ and $B$ are the two particles of the EPR pair. The state of particle $C$, denoted as $\left| \psi_{C} \right\rangle$, is to be teleported. Alice holds particles $A$ and $C$. Bob holds particle $B$. The gray rectangles represent quantum operations.
• Figure 4: Error correction of a quantum toric code. Red data qubits indicate that an error has occurred. Blue data qubits mean they are selected as an error correction qubit. (a) A toric code with error data qubits. The eigenvalues of the stabilizers around the error qubits becomes $-1$. (b) Failed quantum toric code error correction. A error correction chain and the error chain form a nontrivial loop. (c) Successful quantum toric code error correction. A error correction chain and the error chain form a trivial loop. (d) Another successful quantum toric code error correction. This strategy forms a trivial loop, but not the optimal.
• Figure 5: The training framework for the GAN decoder. The blue dotted box illustrates the training process of the GAN, the red dotted box represents the parameter training process of the decoder, and the green dotted box depicts the testing process of the decoder.