Lifted-Product Codes
Lifting a classical protograph over a group, then taking a product, yields quantum LDPC codes with better distance than the plain hypergraph product.
Protographs and lifts
A lifted code starts from a small base graph, a protograph, whose edges are labeled by elements of a group, often a cyclic group. Lifting replaces each labeled edge with a permutation matrix, expanding the small template into a large, structured code. This is the quantum analog of quasi-cyclic classical LDPC codes, which power modern communication standards.
The lifted product
Applying the CSS/hypergraph product to lifted classical codes gives the lifted-product construction. The added algebraic structure lets the distance grow faster than in the plain hypergraph product, in the best cases nearly linearly, while keeping the checks sparse and the code highly structured.
- The group label turns a tiny protograph into a large regular code.
- Structure aids both analysis of distance and hardware scheduling.
- Lifted products were among the first to break the square-root distance barrier.
- Quasi-cyclic structure can simplify the syndrome-extraction schedule.
Lifted-product codes are attractive for engineering because their regularity translates into repeatable wiring and measurement patterns. Recent hardware proposals for high-rate memories on neutral-atom and superconducting-with-couplers platforms often use lifted-product or closely related bicycle codes precisely for this structured connectivity.
They sit on the path from the hypergraph product to fully good codes: enough structure to reason about and build, enough distance to be a serious alternative to the surface code for the memory portion of a machine, where high rate matters most and logical gates can be handled separately.