Tensor-product HDX code

Tensor-product HDX code[1]

Description

A code constructed in a similar way as the HDX code, but utilizing iterated homological products of multiple Ramanujan complexes and then applying distance balancing. These improve the asymptotic code distance over the HDX codes from \(\sqrt{n}\log n\) to \(\sqrt{n}~\text{polylog}(n)\). The utility of such tensor products comes from the fact that one of the Ramanujan complexes is a collective cosystolic expander as opposed to just a cosystolic expander.

Protection

Construction yields explicit QLDPC codes with distance \(\sqrt{n}\log^c n\) using the \(c\)-tensor-product of Ramanujan complexes.

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Higher-dimensional homological product code

Parents

Higher-dimensional homological product code

Tensor-product HDX codes result from an iterated homological products of length-two chain complexes (i.e., quantum codes) based on Ramanujan complexes [2].

Tensor-product HDX code

Children

High-dimensional expander (HDX) code

Ramanujan codes result from a tensor product of a classical-code and a quantum-code chain complex.

References

[1]: T. Kaufman and R. J. Tessler, “New Cosystolic Expanders from Tensors Imply Explicit Quantum LDPC Codes with \(Ω(\sqrt{n}\log^kn)\) Distance”, (2020) arXiv:2008.09495
[2]: N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) arXiv:2103.06309 DOI

Page edit log

Victor V. Albert (2022-10-02) — most recent

Cite as:

“Tensor-product HDX code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/iterated_ramanujan

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/balanced_product/tensor/multisector/iterated_ramanujan.yml.