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arXiv:2611.00985

Normal Factor Graph Representations of Stabilizer Quantum Codes with Applications

July X. Li, Pascal O. Vontobel

quant-phSubmitted Oct 31, 2026• Announced Nov 3, 2026
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Abstract

We develop normal factor graph (NFG) representations of stabilizer quantum error-correcting codes and entanglement-assisted quantum error-correcting codes. We introduce two complementary representations. Quantum-code NFGs describe stabilizer and normalizer label codes and provide a graphical framework for establishing self-orthogonality, constructing codes from component codes, and identifying encoding and decoding circuits. Quantum-probability NFGs represent the quantum-mechanical probabilities associated with encoding, noise, and measurement. We also introduce a doubleedge formulation that makes the relationship between these two representations explicit. We apply the framework to convolutional and tail-biting convolutional stabilizer codes, as well as to two classes of low-density parity-check stabilizer codes. Finally, we extend the quantum-probability representation to entanglementassisted codes.

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We develop normal factor graph (NFG) representations of stabilizer quantum error-correcting codes and entanglement-assisted quantum error-correcting codes. We introduce two complementary representations. Quantum-code NFGs describe stabilizer and normalizer label codes and provide a graphical framework for establishing self-orthogonality, constructing codes from component codes, and identifying encoding and decoding circuits. Quantum-probability NFGs represent the quantum-mechanical probabilities associated with encoding, noise, and measurement. We also introduce a doubleedge formulation that makes the relationship between these two representations explicit. We apply the framework to convolutional and tail-biting convolutional stabilizer codes, as well as to two classes of low-density parity-check stabilizer codes. Finally, we extend the quantum-probability representation to entanglementassisted codes.
We study normal factor graph (NFG) representations of stabilizer quantum error-correction codes (QECCs), in particular NFG representations of the stabilizer label code and the normalizer label code associated with a stabilizer QECC. The structure of the NFGs we are using is such that the (symplectic) self-orthogonality constraint that stabilizer label codes have to satisfy can be proven rather straightforwardly by applying certain NFG reformulations. We show that a variety of well-known stabilizer QECCs can be expressed in this framework: (tail-biting) convolutional stabilizer QECCs, the toric stabilizer QECCs by Kitaev, and a class of stabilizer QECCs that was recently introduced by Tillich and Zémor. Our approach not only gives new insights into these stabilizer QECCs, but will ultimately help to formulate new classes of stabilizer QECCs and low-complexity (approximate) decoding algorithms.
Quantum error-correction codes (QECCs) are a vital ingredient of quantum computation and communication systems. In that context it is highly desirable to design QECCs that can be represented by graphical models which possess a structure that enables efficient and close-to-optimal iterative decoding. In this paper we focus on stabilizer QECCs, a class of QECCs whose construction is rendered non-trivial by the fact that the stabilizer label code, a code that is associated with a stabilizer QECC, has to satisfy a certain self-orthogonality condition. In order to design graphical models of stabilizer label codes that satisfy this condition, we extend a duality result for Forney-style factor graphs (FFGs) to the stabilizer label code framework. This allows us to formulate a simple FFG design rule for constructing stabilizer label codes, a design rule that unifies several earlier stabilizer label code constructions.
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We introduce a new graphical framework for designing quantum error correction codes based on classical principles. A key feature of this graphical language, over previous approaches, is that it is closely related to that of factor graphs or graphical models in classical information theory and machine learning. It enables us to formulate the description of the recently-introduced ‘coherent parity check’ quantum error correction codes entirely within the language of classical information theory. This makes our construction accessible without requiring background in quantum error correction or even quantum mechanics in general. More importantly, this allows for a collaborative interplay where one can design new quantum error correction codes derived from classical codes.
We introduce a new graphical framework for designing quantum error correction codes based on classical principles. A key feature of this graphical language, over previous approaches, is that it is closely related to that of factor graphs or graphical models in classical information theory and machine learning. It enables us to formulate the description of the recently-introduced `coherent parity check' quantum error correction codes entirely within the language of classical information theory. This makes our construction accessible without requiring background in quantum error correction or even quantum mechanics in general. More importantly, this allows for a collaborative interplay where one can design new quantum error correction codes derived from classical codes.
A new class of quantum stabilizer codes is presented.The problem of finding stabilizer quantum-error-correcting codes can be transformed into the problem of finding self-orthogonal codes over the Galois field GF(4) under a trace inner product.Based on this connection,the necessary and sufficient condition is proved under which classical convolutional codes over GF(4) are self-orthogonal with respect to the trace inner product,and then corresponding codes satisfying the condition are found.Tail-biting codes with a simple block structure are obtained by tail-biting transformation,which are self-orthogonal with respect to the trace inner product.Finally,relative quantum stabilizer codes are constructed.The code construction is simple,and the minimum distance of the code approaches the upper bound.