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We introduce a flexible and graphically intuitive framework that constructscomplex quantum error correction codes from simple codes or states,generalizing code concatenation. More specifically, we represent the complexcode constructions as tensor networks built from the tensors of simple codes orstates in a modular fashion. Using a set of local moves known as operatorpushing, one can derive properties of the more complex codes, such astransversal non-Clifford gates, by tracing the flow of operators in thenetwork. The framework endows a network geometry to any code it builds and isvalid for constructing stabilizer codes as well as non-stabilizer codes overqubits and qudits. For a contractible tensor network, the sequence ofcontractions also constructs a decoding/encoding circuit. To highlight theframework's range of capabilities and to provide a tutorial, we lay out someexamples where we glue together simple stabilizer codes to constructnon-trivial codes. These examples include the toric code and its variants, aholographic code with transversal non-Clifford operators, a 3d stabilizer code,and other stabilizer codes with interesting properties. Surprisingly, we findthat the surface code is equivalent to the 2d Bacon-Shor code after "dualizing"its tensor network encoding map.

Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks