Structure of logical operators in lifted quantum Tanner codes

Characterize the structure of logical operators in quantum Tanner codes constructed from left–right Cayley complexes by lifting local product codes along a finite group G. In particular, ascertain which logical operators are supported on single vertical slices corresponding to the A- or B-side classical Tanner codes and describe the forms and support patterns of the remaining logical operators that are not confined to single vertical slices.

Background

Quantum Tanner codes considered in the paper are CSS codes built on left–right Cayley complexes defined by a finite group G and multisets A and B, with stabilizer generators supported on horizontal slices and their translates in the G-fiber. Logical operators of the base (unlifted) code are supported on single rows or columns, and in certain cases (e.g., odd |G| with equal lifted and base dimensions) these replicate across the fiber to give logicals in the lifted code.

The authors observe that many logical operators in the lifted codes appear within single vertical slices and correspond to codewords of associated classical Tanner codes on the Cayley graphs. However, they also report that, in general, one cannot find a logical basis where each logical operator is confined to a single vertical slice, motivating a precise characterization of the logical-operator structure beyond slice-supported cases.