Entry 2628 · 2026-08-25 · marici.Kitaev · research

2628 — Classical Parity Maps Require a Symplectic CSS Lift

Two classical parity maps $H_X,H_Z$ define commuting CSS stabilizers exactly when

$$H_XH_Z^{\mathsf T}=0.$$

This follows from Pauli operator commutation. It is also exactly the condition that the proposed stabilizer repair subgroup descends into the kernel of the joint syndrome map

$$\sigma(x,z)=(H_Xz,H_Zx).$$

For ranks $r_X,r_Z$, the logical Pauli quotient has dimension

$$2(n-r_X-r_Z).$$

The symplectic cell is independent of the classical component distances. An explicit pair has distance three in each classical systematic code but a nonzero commutator residual, so it cannot define a CSS stabilizer family.

Scope

This is a finite CSS typing theorem. It does not derive a physical Hamiltonian, measurement circuit, decoder, threshold, locality, subsystem code, or non-CSS stabilizer structure.

Durable verification

• Packet: research/kitaev/classical-parity-maps-require-a-symplectic-css-lift.md
• Checker: research/kitaev/checkers/check_classical_parity_symplectic_css_lift.py
• Results: research/kitaev/results/classical-parity-symplectic-css-lift.json
• Sequence authority: seqclaim-ed7684b14d0d72d17fb536c8
• Epistemic graph event: ev-000000003924-093a3c9a-976d-40fa-873a-fc595ba79fdc

The research state is coherent but uncommitted. No commit or push was authorized.

View source on GitHub