Short answer: Hypergraph product codes are quantum low-density parity-check (LDPC) codes first constructed by Tillich and Zemor, and they have a constant encoding rate, which means less qubit overhead than topological codes in the long run. A 2019 Physical Review Letters paper generalizes them, a 2021 Physical Review X paper shows how to do Clifford gates on them by code deformation, and a 2024 arXiv preprint proposes a variant with a constant rate of 0.2. These are theory and simulation results.
Why these codes
The 2021 Physical Review X abstract notes that current approaches to quantum error correction focus on topological codes, which it says will likely lead to a first proof of principle of fault tolerance in the next five years, but which in the long run need a very large number of physical qubits to run interesting algorithms. The authors propose quantum LDPC codes as an architecture that avoids this problem. They call hypergraph product codes, discovered by Tillich and Zemor, the best candidate class, with a constant encoding rate and, as shown recently, a constant fault-tolerant error threshold. They say that asymptotically these give smaller overhead than topological codes.
2019: a generalization
The 2019 Physical Review Letters abstract describes a family of quantum error-correcting codes that generalizes both the Tillich-Zemor hypergraph-product codes and all families of toric codes on hypercubic lattices of any dimension. The parameters of the constructed codes, including minimum distances, are given explicitly in terms of those of binary codes associated with the matrices used in the construction.
2021: gates by code deformation
The 2021 authors show how to perform Clifford gates on hypergraph product codes using code deformation, for which they introduce wormhole defects. Together with state injection, they say a universal set of gates can be performed within a single block of this class of codes.
2024: a higher-rate variant
The 2024 arXiv preprint begins from the low coding rate of quantum stabilizer codes, which it says causes a formidable physical qubit overhead. The authors propose a new hypergraph-product variant they call the TGRE-hypergraph-product code, with a constant coding rate of 0.2, which they say, to the best of their knowledge, is the highest constant rate among quantum stabilizer codes. Their simulations find a code-capacity noise threshold of around 0.096 in the depolarizing noise channel. This is a preprint and the claim of highest rate is the authors' own.
Limits
A code-capacity threshold assumes ideal syndrome measurement, so it is not a threshold for a full circuit-level implementation. The 2021 paper's statement about overhead is an asymptotic claim, and these codes need long-range connectivity that the abstracts do not discuss. None of the three papers reports a hardware demonstration.
FAQ
What is a quantum LDPC code?
A quantum code defined by sparse parity checks. The 2021 paper proposes them as an alternative to topological codes with lower asymptotic overhead.
What does constant rate mean?
The ratio of logical to physical qubits stays constant as the code grows, per the 2021 abstract.
Can these codes do universal computation?
The 2021 authors show Clifford gates by code deformation and, with state injection, a universal gate set within one block.
What is the 0.2 rate?
The 2024 preprint's claimed constant rate for its TGRE variant, which it says is the highest among stabilizer codes to its authors' knowledge.
Has this been built?
None of the three papers reports hardware.
All field notes