Frohman Symplectic Stabilizer Quantum Error-Correcting Codes, Metriplectic Syndrome Decoding, and Fault-Tolerant Logical Gate Synthesis (frohman-symplectic-qec-codes)

Benjamin Frohman

Published Aug 22, 2026• 0 citations

Abstract

PROPRIETARY COMMERCIAL SOFTWARE & HARDWARE/ALGORITHMIC IP — ALL RIGHTS RESERVED This proprietary quantum architecture specification—authored by Benjamin Stanley Frohman (ORCID: 0009-0006-7068-3718) and anchored to symplectohydronamics commit 8f7db79b5786f3a59a0696a48a675f8a1fe410ca (August 22, 2026, 12:49:03 PM CDT)—formulates the Frohman Symplectic Stabilizer Quantum Error-Correcting (QEC) Code family and Metriplectic Syndrome Decoder. By mapping n-qubit Pauli operators onto the 2n-dimensional binary/qudit symplectic vector space (F_p^2n, omega_Sp), stabilizer codes correspond to isotropic subspaces S subset S-perp, while fault-tolerant logical Clifford gates correspond to symplectic automorphisms U in Sp(2n, F_p) preserving S. We introduce a metriplectic soft-syndrome relaxation flow that suppresses trapping sets in high-rate quantum LDPC (qLDPC) codes and provides formal distance-preservingtransversal gate synthesis. IP Owner & Author: Benjamin Stanley Frohman (ORCID: 0009-0006-7068-3718) Earliest Verified Origin Provenance (America/Chicago, CDT / UTC-5): 2026-08-22 12:49:03 PM CDT (UTC-5) (symplectohydronamics commit 8f7db79b5786f3a59a0696a48a675f8a1fe410ca) Private GitHub Repository: https://github.com/BenFrohman/frohman-symplectic-qec-codes (Release v1.0.0, Commit dead18dcbd6fd2925f666fa090535a1178aa5f6d) SHA-256 Merkle Root: 7778b182a0f10243db4fd6fa8c7c7b84f036d400412ac06296c28f7d498161f1 Commercial Licensing & Federal Acquisition: Available exclusively via Metered Usage License, Annual Enterprise Site License, or Strategic Partnership Agreement countersigned by Benjamin Stanley Frohman (or his designated Texas LLC commercial licensing vehicle). Commercial Computer Software developed 100% at private expense under FAR 12.212 and DFARS 227.7202.

DOI: 10.5281/zenodo.23632773

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Frohman Symplectic Stabilizer Quantum Error-Correcting Codes, Metriplectic Syndrome Decoding, and Fault-Tolerant Logical Gate Synthesis (frohman-symplectic-qec-codes)

Benjamin Frohman

Published Aug 22, 2026• 0 citations

Publisher

Abstract

DOI: 10.5281/zenodo.23632773

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Stabilizer Formalism: The Algebraic Core of Quantum Error Correction — E8 Intelligence Research

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Aug 23, 20260 citations

\mathcal{S}

of the Pauli group

\mathcal{P}_n

; codewords are simultaneous

+1

eigenstates of all

S \in \mathcal{S}

. The code space dimension is

2^{n-k}

where

k = n - \text{rank}(\mathcal{S})

. Distance

d

corrects

t = \lfloor (d-1)/2 \rfloor

errors. Key bound:

k \le n - \text{log}_2 |\mathcal{S}|

. The Knill–Laflamme condition:

\langle \psi_i | E_a^\dagger E_b | \psi_j \rangle = \delta_{ij} C_{ab}

, where

E_a

are error operators. | CONNECTION: The stabilizer group

\mathcal{S}

is a **binary linear code** over

\mathbb{F}_2^{2n}

— a **lattice structure** in symplectic space. The symplectic inner product

\langle (a|b), (a'|b') \rangle = a \cdot b' + a' \cdot b \pmod{2}

defines a **self-d Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\mathcal{P}_n

n

qubits; code space

\mathcal{C} = \{|\psi\rangle : S_i|\psi\rangle = |\psi\rangle \ \forall i\}

, with

S_i

generators of an abelian subgroup of

\mathcal{P}_n

. - **Distance & threshold**: A

[[n,k,d]]

code corrects

t = \lfloor (d-1)/2 \rfloor

errors; fault-tolerance threshold

p_{\text{th}} \approx 10^{-3}

–

10^{-2}

per gate (surface code). - **Continuous-time QEC (CTQEC)**: Master equation

\dot{\rho} = -i[H,\rho] + \sum_k \gamma_k \mathcal{D}[L_k]\rho

, with weak measurements and feedback; error rate scales as

\gamma_k \propto (\text{measurement strength})^2 / \text{noise bandwidth}

. - **QuOps metric**: Reliable quantum operations

\sim 10^3

–

10^4

currently; tar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com