Using Tanner Spectral Reduction to Improve Multi Layer Optical Lattice Routing for Hypergraph Product and Bivariate Bicycle qLDPC Codes

Jul 16, 2026

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Paper characterizes the Tanner graph spectra of hypergraph-product (HGP)/lifted-product (LP) and bivariate-bicycle (BB) qLDPC codes to inform qubit syndrome-extraction routing in 3D reconfigurable architectures. For HGP/LP, syndrome-extraction routing depth reduces to a single SVD on the base parity-check matrix via spectral ratio βHGP=(1+βbase)/2 and diameter identity DT=2Dbase. For BB codes, Fourier spectral reduction turns the Tanner spectrum into lm independent 2×2 SVDs, cutting analysis cost from O((lm)^3) to lm. Results assemble a multi-layer AOL routing protocol with per-cycle depth ⌈χ′/Llayers⌉, yielding sizable wall-clock speedups under multi-layer conditions.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages24

Reading Time1 h 0 min

Introduction

Tanner spectral characterization for HGP/LP and BB codes

Syndrome-extraction routing protocol and depth bound

Numerical comparison and routing regime evaluation

arXiv :2607 .06177v1 [ quant-ph] 7 Jul 2026 Using Tanner Spectral Reduction to Improve MultiLayer Optical Lattice Routing for HypergraphProduct and Bivariate Bicycle qLDPC Codes Joshua M. Courtney University of Georgia, Department of Physics and Astronomy July 6, 2026 We characterize the Tanner graph spectrum of hypergraph-product (HGP) / liftedproduct (LP) codes and bivariate-bicycle (BB) codes, informing qubit routing for threedimensional 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 βHGP = (1 + βbase)/2 where βbase = σ2 (H)/σ 1 (H) for the base parity-check matrix, and a diameter identity DT = 2Dbase where Dbase is the base Tanner graph diameter. Fourier spectral reduction reveals that the BB Tanner graph spectrum equals the union, over the l × m grid of characters of Zl × Zm , of the singular values of a single 2 × 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 × 2 SVDs. These results compose into a multi-layer three-dimensional AOL routing protocol with one-time setup cost TValiant = O(log N ) atom rearrangements amortizable over a memory experiment of R rounds. For a Tanner graph chromatic index χ′ and Llayers stacked AOL planes, the per-syndrome-cycle depth is ⌈χ′ /Llayers⌉ AOL pattern activations with no atom motion, an 8 × step-count reduction at Llayers ≥ χ′ = 8 . Contingent on multi-layer AOL hardware, this yields an estimated ∼ 50–300× per-cycle wall-clock advantage over a single-layer AOD baseline (degrading to ∼ 5–100× 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. 1 Introduction Recent demonstrations of quantum low-density parity check (qLDPC) code architectures [1–3] substantially reduce physical-qubit overhead for fault-tolerant quantum computation compared to surface codes, realizing a constant overhead [4] . These works use routing primitives moving qubits between data and ancilla positions across each syndrome extraction cycle. While syndromeextraction circuit depth (number of CNOT layers) is well-understood for these codes [5, 6], atomrearrangement depth required to support each circuit layer has yet to be characterized in terms of the underlying Tanner graph spectrum. Impetus for this characterization stems from both hardware runtime bottlenecks in atom/ion reconfigurations and assessing the potential advantage of incorporating an effective third dimension to reduce routing overhead in neutral atom and trapped ion qubit architectures. We give a closed-form formula βHGP = (1+βbase )/2 and an exact diameter identity DT = 2Dbase for hypergraph-product/lifted-product (HGP/LP) code Tanner graphs (Sec. 3) . Using Fourier diagonalization (Theorem 3.6), we reduce the bivariate bicycle (BB) Tanner graph spectrum to lm independent 2 × 2 singular-value decompositions, and prove that no scalar (1 + βbase )/2-style Joshua M. Courtney: [Joshua.Courtney1@uga.edu](Joshua.Courtney1@uga.edu) one-liner holds for BB codes. We give a syndrome-extraction protocol and depth bound for a 2D acousto-optic deflector (AOD) atom array augmented with Llayers stacked 3D acousto-optic lens (AOL) planes, applicable uniformly to HGP, LP, and BB codes (Sec. 4) . Section 5 gives a numerical comparison to Xu et al. [2]’s scheme on the published HGP code family, under matched amortization assumptions, with BB codes (Sec. 6.1) measured in the same framework. We connect to two recent qLDPC architectures, situated among a growing body of neutral-atom qLDPC layout and routing work [7–10] . Xu et al. [2] implement HGP/LP codes on reconfigurable atom arrays with 2D divide-and-conquer scra

FAQs

What problem does the paper address in qLDPC quantum hardware routing?

It addresses how atom/ion rearrangement depth for syndrome-extraction depends on the underlying Tanner graph spectrum, beyond what is already well understood for CNOT-layer circuit depth.

How does the paper simplify the routing-depth analysis for HGP/LP Tanner graphs?

Syndrome-extraction routing depth reduces to a single SVD on the base parity-check matrix, using βHGP=(1+βbase)/2 and the diameter identity DT=2Dbase.

What computational improvement is achieved for BB Tanner graph spectral analysis?

Fourier spectral reduction expresses the BB spectrum as a union over the l×m grid of Zl×Zm, enabling analysis through lm independent 2×2 SVDs instead of full diagonalization of the full Tanner graph.

Jul 16, 2026

Upload by Rowan