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Quantum Error-Correcting Codes (QECCs) are vital for ensuring the reliabilityof quantum computing and quantum communication systems. Among QECCs, stabilizercodes, particularly graph codes, have attracted considerable attention due totheir unique properties and potential applications. Concatenated codes, whichcombine multiple layers of quantum codes, offer a powerful technique forachieving high levels of error correction with a relatively low resourceoverhead. In this paper, we examine the concatenation of graph codes using thepowerful and versatile graphical language of ZX-calculus. We establish acorrespondence between the encoding map and ZX-diagrams, and provide a simpleproof of the equivalence between encoding maps in the Pauli X basis and thegraphic operation "generalized local complementation" (GLC) as previouslydemonstrated in [J. Math. Phys. 52, 022201]. Our analysis reveals that theresulting concatenated code remains a graph code only when the encoding qubitsof the same inner code are not directly connected. When they are directlyconnected, additional Clifford operations are necessary to transform theconcatenated code into a graphcode, thus generalizing the results in [J. Math.Phys. 52, 022201]. We further explore concatenated graph codesin differentbases, including the examination of holographic codes as concatenated graphcodes. Our findings showcase the potential of ZX-calculus in advancing thefield of quantum error correction.

A ZX-Calculus Approach to Concatenated Graph Codes