Hypergraph-product

Hypergraph-product (HGP) codes combine two binary classical parity-check matrices $H_1,H_2$ into CSS stabilizer codes. Qubits occupy variable-variable and check-check products of the two Tanner graphs, so bounded seed degrees give bounded quantum-check weight. The same construction is the total complex of two length-one chain complexes.

For $r_i\times n_i$ seed matrices $H_i$, a standard convention is

$H_X=\left[H_1\otimes I_{n_2}\mid I_{r_1}\otimes H_2^T\right],\qquad H_Z=\left[I_{n_1}\otimes H_2\mid H_1^T\otimes I_{r_2}\right].$

The Kronecker structure enforces $H_XH_Z^T=0$ over $\mathrm{GF}(2)$. Atlas uses the square product of the standard $[7,4,3]$ Hamming check matrix as its finite teaching point.

Logical classes arise from kernels and cokernels of the seed and transpose codes. The quantum distance is controlled by the relevant minimum among $d_1,d_2,d_1^T,d_2^T$; suitable classical LDPC families yield nonzero asymptotic rate and square-root distance scaling. Degeneracy means many physical errors represent the same logical coset.

With $k_i=\dim\ker H_i$ and $k_i^T=\dim\ker H_i^T$,

$N=n_1n_2+r_1r_2,\qquad K=k_1k_2+k_1^Tk_2^T.$

The Atlas Hamming square instance is $[[58,16,3]]$ with rate $8/29$ and maximum check weight seven. General binary stabilizer tables give $10\le dle15$ at the same $(n,k)$; that interval is not an HGP-family bound.

Seed-code symmetries can induce logical Clifford transformations, but an arbitrary HGP code does not inherit a universal transversal gate set. Known routes use wormhole defects and code deformation for fault-tolerant Clifford gates and state injection for non-Clifford resources; each protocol needs an explicit check schedule and circuit-distance analysis.

HGP syndromes define sparse Tanner-graph constraints. Belief propagation exploits locality but is affected by short cycles, degeneracy, and trapping sets; BP+OSD and hard/soft reliability post-processing search a structured residual space. Small-set-flip applies to suitable quantum-expander subclasses rather than every HGP seed.

Bounded stabilizer weight does not by itself provide shallow, geometrically local, fault-tolerant syndrome extraction. Ancilla reuse, gate order, and nonlocal connectivity alter fault propagation and effective circuit distance. Threshold claims must identify code-capacity, phenomenological, or circuit-level noise and the connectivity model.

HGP codes are widely used in finite-length qLDPC decoding, state-preparation, logical-gate, and hardware-mapping studies. The qLDPC software stack can construct the seed and CSS matrices, but Atlas has no verified QECirc artifact for its Hamming-square point and therefore does not present a schematic syndrome graph as a real circuit.

HGP is the length-one-chain-complex specialization of homological products, and expander seeds give quantum-expander codes. Repetition seeds recover surface/toric product presentations. Lifted, balanced, and fiber-bundle constructions reduce or quotient HGP symmetries, while bicycle relations require explicit representation data rather than parameter equality.

A reproducible HGP record should retain $H_1,H_2$, ranks, row and column weights, transpose-code dimensions, and a normalized matrix hash. QECDB currently returns one $[[58,16,3]]$ hypergraph-product record, but Atlas has not proved matrix equivalence with its Hamming-square presentation, so it exposes the family filter without claiming an exact instance match.