Galois-qudit quantum expander code[1]
Also known as Galois-qudit Sipser-Spielman code.
Description
Galois-qudit CSS code constructed from a hypergraph product of expander codes.
Expander codes with RS inner codes contain GRM codewords because tensor products of univariate polynomials (corresponding to RS codewords) yield multivariate polynomials (corresponding to GRM codewords) [1]. This multiplication property allows for QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates while maintaining distance [1].
Magic
Hypergraph products of expander codes with RS inner codes yield \([[n,k\geq n^{1-\epsilon},d\geq n^{1/r}/\text{poly}(\log n)]]_q\) QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates [1]. This construction allows for arbitrarily small magic-state yield parameter \(\gamma\).
Transversal Gates
Hypergraph products of expander codes with RS inner codes yield \([[n,k\geq n^{1-\epsilon},d\geq n^{1/r}/\text{poly}(\log n)]]_q\) QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates [1].
Parent
Child
Cousins
- Reed-Solomon (RS) code — Hypergraph products of expander codes with RS inner codes yield \([[n,k\geq n^{1-\epsilon},d\geq n^{1/r}/\text{poly}(\log n)]]_q\) QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates [1].
- Generalized RM (GRM) code — Expander codes with RS inner codes contain GRM codewords because tensor products of univariate polynomials (corresponding to RS codewords) yield multivariate polynomials (corresponding to GRM codewords) [1].
- Balanced product (BP) code — Balanced products of expander codes with RS inner codes yield \([q^{\text{polylog}(q)},k\geq n^{1-\epsilon},n/\text{poly}(\log n)]_q\) LTCs exhibiting the multiplication property [1].
- \(q\)-ary linear LTC — Balanced products of expander codes with RS inner codes yield \([q^{\text{polylog}(q)},k\geq n^{1-\epsilon},n/\text{poly}(\log n)]_q\) LTCs exhibiting the multiplication property [1].
References
- [1]
- L. Golowich and T.-C. Lin, “Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes”, (2024) arXiv:2410.14662
Page edit log
- Victor V. Albert (2024-10-23) — most recent
Cite as:
“Galois-qudit quantum expander code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/galois_expander