Galois-qudit expander code[1]
Alternative names: Galois-qudit Sipser-Spielman code.
Description
Galois-qudit CSS code obtained from product constructions on chain complexes associated with 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 QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates while achieving \(D\geq N^{1/r}/\operatorname{poly}(\log N)\) and \(w\leq\operatorname{poly}(\log N)\) [1].
Magic
For every integer \(r\geq 2\) and every \(\epsilon>0\), the construction yields \([[N,K\geq N^{1-\epsilon},D\geq N^{1/r}/\operatorname{poly}(\log N)]]_q\) QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates and stabilizer weight \(w\leq\operatorname{poly}(\log N)\) [1]. This construction allows for arbitrarily small magic-state yield parameter \(\gamma\).Transversal Gates
For every integer \(r\geq 2\), there are QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates and parameters \([[N,K\geq N^{1-\epsilon},D\geq N^{1/r}/\operatorname{poly}(\log N)]]_q\) [1].Cousins
- Reed-Solomon (RS) code— Products of expander codes with RS inner codes yield \([[N,K\geq N^{1-\epsilon},D\geq N^{1/r}/\operatorname{poly}(\log N)]]_q\) QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates [1]. Balanced products of expander codes with RS inner codes also yield \([n,k\geq n^{1-\epsilon},d\geq n/\operatorname{poly}(\log n)]_q\) LTCs exhibiting the multiplication property [1].
- Expander code— Products of expander codes with RS inner codes yield \([[N,K\geq N^{1-\epsilon},D\geq N^{1/r}/\operatorname{poly}(\log N)]]_q\) QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates [1]. Balanced products of expander codes with RS inner codes also yield \([n,k\geq n^{1-\epsilon},d\geq n/\operatorname{poly}(\log n)]_q\) LTCs exhibiting the multiplication property [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 \([n,k\geq n^{1-\epsilon},d\geq n/\operatorname{poly}(\log n)]_q\) LTCs exhibiting the multiplication property [1].
- \(q\)-ary linear LTC— Balanced products of expander codes with RS inner codes yield \([n,k\geq n^{1-\epsilon},d\geq n/\operatorname{poly}(\log n)]_q\) LTCs exhibiting the multiplication property [1].
Primary Hierarchy
Balanced product (BP) codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Galois-qudit expander code
Children
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 expander code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/galois_expander