Quantum expander code

Quantum expander codes are an HGP/CSS subclass constructed from bounded-left- and bounded-right-degree bipartite expander graphs. The decisive hypothesis is a quantitative small-set expansion condition on the classical seed graphs, not merely sparsity of the quantum Tanner graph.

Each bipartite graph $G_i=(V_i,C_i,E_i)$ defines exactly when . The standard HGP assembly is

Suitable seed families yield constant-rate quantum codes with square-root-scale distance. Under the stated expansion inequalities, small-set-flip corrects $\Omega(\sqrt n)$ adversarial Pauli errors, a constant fraction of the distance scale. This is not a theorem for every HGP code or arbitrary correlated circuit faults.

There is no universal transversal gate set determined by the “expander” label. Logical transformations may use seed-graph automorphisms, code switching, gate teleportation, or homological maps, but each requires a finite graph and logical basis. Addressable-gate or dimensional-jump protocols must be recorded as separate execution artifacts.

Small-set-flip repeatedly selects a subset of a stabilizer support whose flip decreases residual-syndrome weight by a prescribed ratio. Expansion prevents small errors from remaining hidden and supports linear running time. Implementations must state sequential or parallel search, stopping conditions, and data- versus syndrome-noise assumptions.

Subsequent work constructs constant-overhead fault tolerance with quantum expander codes under locally stochastic noise. This is a theorem with explicit noise, decoder, and gadget assumptions, not a large plug-in experimental threshold. Two-dimensional local implementations remain subject to syndrome-circuit depth-width tradeoffs.

A reproducible record needs seed adjacency lists or check matrices, left and right degrees, expansion or spectral evidence, the HGP expansion convention, a logical basis, and small-set-flip parameters. Generic graph generators produce candidates, not verified quantum-expander instances.

It is a strict HGP subclass and therefore a length-one-chain-complex member of the homological-product lineage. Its relation to classical Sipser–Spielman expander codes is through the seed check graphs; higher-dimensional HGP, single-shot codes, and self-correcting models are extensions or applications rather than the same entity.

For lookup, verify seed degrees, expansion constants, finite graph size, whether distance is exact or bounded, and whether small-set-flip assumes adversarial or random data errors and perfect or noisy syndrome. A generic “LDPC + expander” label or matching parameters do not establish an exact instance.