IDA Quantum Computing: GF(8) as the Natural Algebra of the Steane Code and Transversal Cliord Gates
Year: 2026
Type: article
Abstract: We establish a rigorous correspondence between the IDA algebraic frame- work (Papers 3 and 5) and quantum error correction. The key result is that the Galois eld GF(8) = F2[x]/(x 3 + x + 1), identied in Paper 5 as the IDA alge- bra, is precisely the natural coecient algebra of the Steane [[7, 1, 3]] quantum error-correcting code. We prove four main theorems: (T1) the seven non-zero elements of GF(8)∗ ∼= Z7 are the columns of the parity-check matrix of the [7, 4, 3] Hamming code, giving the [[7, 1, 3]] Steane code via the CSS construc- tion; (T2) the Frobenius automorphism φ : a 7→ a 2 of GF(8) corresponds to a transversal Cliord gate on the [[7, 1, 3]] code; (T3) the IDA operations ⊕ and ⊗ map respectively to stabilizer generators (Pauli X chains) and logical operators of the Steane code; and (T4) the quantum channel capacity of the IDA wormhole (Paper 4 Corollary 4) is the quantum capacity of the associated ReedSolomon channel over GF(8). Corollaries include: the GF(64) multiverse of Paper 6 corre- sponds to the [[63, 1, 21]] quantum ReedSolomon code; fault-tolerant quantum computation via the IDA Frobenius; and the identication of dark matter (Pa- per 6) with logically undetectable quantum errors.
Author: Duc Tai, Nguyen
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221453645
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