Logical Spectroscopy: Lifted-Product Codes with Addressable Bases

Quantum LDPC memories can encode many logical qubits, but dimension alone does not make them usable: applications need explicit conjugate logical operators with structured labels and physical representatives. For hypergraph-product (HGP) codes this structure is transparent, since the input matrices are binary and can be row-reduced over $\mathbb{F}_2$. Abelian lifted-product codes are subtler. Their seed entries are shifts, or sparse sums of shifts, in a group-algebra ring rather than a field, so pivot blocks need not be invertible and global row reduction can fail. We address this with \emph{logical spectroscopy}, a spectral construction that replaces global row reduction by finite-field computations in the Frobenius character packets of the Abelian lift group. The Chinese remainder theorem (CRT) decomposes the group algebra into these packets. In each packet, we compute kernels, quotients, and product-complex homology; we then lift the resulting representatives back with CRT idempotents and pair $X$ and $Z$ logicals through reciprocal trace-dual packets. This gives complete addressable conjugate logical bases for finite Abelian lifted products $\mathsf{LP}(A,B)$. The same packet data also gives design diagnostics. Packet ranks show how logical sectors split, the lifted representatives give certified upper bounds on the width of the constructed conjugate basis, and whole-orbit erasures decompose into packet-attributed erased-logical dimensions. Thus, CRT packets also serve as working coordinates: they label logical sectors, certify the constructed basis width, and attribute structured erasure failures. Under bounded seed-shape and group-basis-support assumptions, this construction gives Abelian lifted-product qLDPC families an HGP-like feature while preserving the layout freedom of group-algebra lifts.

Source: arXiv:2607.05386v1 - http://arxiv.org/abs/2607.05386v1 PDF: https://arxiv.org/pdf/2607.05386v1 Original Link: http://arxiv.org/abs/2607.05386v1