Galois-qudit expander code[1]
Alternative names: Galois-qudit Sipser-Spielman code.
Description
Galois-qudit CSS code obtained from tensor products of chain complexes associated with an explicit family of expander codes with Reed-Solomon local checks.
In the explicit construction of [1], these expander-code complexes contain planted GRM codewords, yielding a multiplication property that 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 \(w\leq\operatorname{poly}(\log N)\).
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]. By decomposing each Galois qudit into a Kronecker product of qubits, this yields an explicit qubit CSS QLDPC family with parameters \([[N,K\geq N^{1-\epsilon},D\geq N^{1/r}/\operatorname{poly}(\log N)]]\), stabilizer weight \(\operatorname{poly}(\log N)\), and transversal \(C^{r-1}Z\) gates acting on \(N^{1-\epsilon}\) disjoint logical \(r\)-tuples.Cousins
- Reed-Solomon (RS) code— The explicit expander-code construction with Reed-Solomon local checks in [1] 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. Balanced products of the same RS-based complexes also yield \([n,k\geq n^{1-\epsilon},d\geq n/\operatorname{poly}(\log n)]_q\) LTCs exhibiting the multiplication property.
- Expander code— The explicit expander-code construction of [1] 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. Balanced products of the same expander-code complexes also yield \([n,k\geq n^{1-\epsilon},d\geq n/\operatorname{poly}(\log n)]_q\) LTCs exhibiting the multiplication property.
- Generalized RM (GRM) code— The explicit expander-code construction of [1] contains planted GRM codewords.
- Balanced product (BP) code— Balanced products of the RS-based expander-code complexes in [1] yield \([n,k\geq n^{1-\epsilon},d\geq n/\operatorname{poly}(\log n)]_q\) LTCs exhibiting the multiplication property.
- \(q\)-ary linear LTC— Balanced products of the RS-based expander-code complexes in [1] yield \([n,k\geq n^{1-\epsilon},d\geq n/\operatorname{poly}(\log n)]_q\) LTCs exhibiting the multiplication property.
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