Logical Spectroscopy: Lifted-Product Codes with Addressable Bases

• URL: http://arxiv.org/abs/2607.05386v2
• Date: Thu, 09 Jul 2026 17:58:26 GMT
• Title: Logical Spectroscopy: Lifted-Product Codes with Addressable Bases
• Abstract summary: LDPC memories can encode many logical qubits, but that alone does not make them usable.<n>For hypergraph-product codes this information follows from row reduction over $mathbbF$.<n>For Abelian lifted-product codes, which include prominent high-rate constructions, it does not.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum LDPC memories can encode many logical qubits, but that alone does not make them usable: applications need to know where the logical operators are supported, how they are labeled, and how conjugate $X/Z$ partners pair. For hypergraph-product codes this information follows from row reduction over $\mathbb{F}_2$. For Abelian lifted-product codes, which include prominent high-rate constructions, it does not: the entries of the defining seed matrices live in a group algebra rather than a field, so pivots need not be invertible and row reduction can fail. To address this problem, we introduce \emph{logical spectroscopy}. For an odd-order Abelian lift group, the Chinese remainder theorem splits the group algebra into finite fields, one for each Frobenius orbit of characters; we call these orbits packets. Packet by packet, we solve ordinary finite-field linear algebra, lift the answers back to the physical code with the associated packet projectors, and pair $X$ and $Z$ logicals between reciprocal packets by trace duality. The result is a complete conjugate logical basis for a finite Abelian lifted product code that is addressable: every logical coordinate carries canonical packet and Künneth-summand labels, deterministic within-block indices, a conjugate partner, and an explicit binary representative. The code's own translation symmetry then acts on these coordinates in closed form. We apply the construction to examples with up to 5000 physical qubits, including high-rate examples whose distance-witness is reported explicitly. We further extend the decomposition to even-order lifts. Logical spectroscopy thus equips Abelian lifted products with an explicit logical coordinate system and an algebraic toolkit for high-rate code search.

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