{"ID":6023537,"CreatedAt":"2026-07-08T01:00:23.257252134Z","UpdatedAt":"2026-07-10T11:42:49.717029521Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.06177","arxiv_id":"2607.06177","title":"Using Tanner Spectral Reduction to Improve Multi-Layer Optical Lattice Routing for Hypergraph-Product and Bivariate Bicycle qLDPC Codes","abstract":"We characterize the Tanner graph spectrum of hypergraph-product (HGP) / lifted-product (LP) codes and bivariate-bicycle (BB) codes, informing qubit routing for three-dimensional reconfigurable qubit architectures. Syndrome-extraction routing depth on HGP/LP Tanner graphs reduces to a single SVD on the base parity-check matrix, using a spectral ratio $β_\\text{HGP} = (1 + β_\\text{base})/2$ where $β_\\text{base} = σ_2(H)/σ_1(H)$ for the base parity-check matrix, and a diameter identity $D_T = 2 D_\\text{base}$ where $D_\\text{base}$ is the base Tanner graph diameter. Fourier spectral reduction reveals that the BB Tanner graph spectrum equals the union, over the $l \\times m$ grid of characters of $\\mathbb{Z}_l \\times \\mathbb{Z}_m$, of the singular values of a single $2 \\times 2$ symbol matrix built from the two defining polynomials. This reduces spectral analysis from an $O((lm)^3)$ diagonalization of the $4lm$-node Tanner graph to $lm$ independent $2 \\times 2$ SVDs. These results compose into a multi-layer three-dimensional AOL routing protocol with one-time setup cost $T_\\text{Valiant} = O(\\log N)$ atom rearrangements amortizable over a memory experiment of $R$ rounds. For a Tanner graph chromatic index $χ'$ and $L_\\text{layers}$ stacked AOL planes, the per-syndrome-cycle depth is $\\lceil χ'/L_\\text{layers} \\rceil$ AOL pattern activations with no atom motion, an $8\\times$ step-count reduction at $L_\\text{layers} \\geq χ' = 8$. Contingent on multi-layer AOL hardware, this yields an estimated $\\sim50-300\\times$ per-cycle wall-clock advantage over a single-layer AOD baseline (degrading to $\\sim5-100\\times$ under AOD-crosstalk overhead), reducing to equality in the single-layer limit. This paper therefore presents a route toward practical routing improvement for future quantum hardware incorporating multi-layer reconfigurable qubit architectures.","short_abstract":"We characterize the Tanner graph spectrum of hypergraph-product (HGP) / lifted-product (LP) codes and bivariate-bicycle (BB) codes, informing qubit routing for three-dimensional reconfigurable qubit architectures. Syndrome-extraction routing depth on HGP/LP Tanner graphs reduces to a single SVD on the base parity-check...","url_abs":"https://arxiv.org/abs/2607.06177","url_pdf":"https://arxiv.org/pdf/2607.06177v1","authors":"[\"Joshua M. Courtney\"]","published":"2026-07-07T11:54:26Z","proceeding":"quant-ph","tasks":"[\"quant-ph\",\"cs.DS\"]","methods":"[]","has_code":false}