Logical computation with canonical lifted product codes

• URL: http://arxiv.org/abs/2607.28605v1
• Date: Thu, 30 Jul 2026 17:55:34 GMT
• Title: Logical computation with canonical lifted product codes
• Abstract summary: Code-agnostic techniques such as code surgery and gate teleportation apply broadly, but are difficult to make modular, low-overhead, and fully certifiable on complex high-rate codes.<n>Here we overcome these obstacles by co-designing the code together with its logical instruction set for a broad family of emphcanonical lifted-product (LP) codes.<n>We show that these codes admit a emphcanonical logical basis, in which conjugate logical operators are organized into rows and columns of cyclic orbits inherited directly from the underlying classical codes.
• Score: 3.653555716265332
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: High-rate quantum low-density parity-check (qLDPC) codes encode many logical qubits with low physical-qubit overhead, but realizing efficient fault-tolerant computation on such dense encodings remains a major challenge. Generic, code-agnostic techniques such as code surgery and gate teleportation apply broadly, but are difficult to make modular, low-overhead, and fully certifiable on complex high-rate codes whose structure is left unexploited. Here we overcome these obstacles by co-designing the code together with its logical instruction set for a broad family of \emph{canonical} lifted-product (LP) codes with cyclic symmetry. We show that these codes admit a \emph{canonical logical basis}, in which conjugate logical operators are organized into rows and columns of cyclic orbits inherited directly from the underlying classical codes, analogous to the structure that makes hypergraph-product codes so tractable. This canonical basis unlocks a complete logical instruction set, including constant-depth automorphism and fold-transversal Clifford gates, modular graph code surgeries built from a constant number of reusable seed surgery gadgets or a compact canonical extractor, highly parallel logical Pauli-product measurements, and parallel magic-state injection. For example, a $[[1122,148,\leq\!20]]$ (resp. $[[4350,1224,\leq\!20]]$) LP code requires only two (resp. four) seed surgery gadgets, while arbitrary high-weight logical measurements can be implemented using a full extractor smaller than half of the data code block. These results advance the frontier of fault-tolerant quantum computation on ultra-high-rate quantum architectures.

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