C029 R003 plan

Claim: compared to a single MS gate and a two-loop phase-modulated MS gate based on H_A, the protocol reduces optical-qubit-frequency and Rabi-frequency errors, but provides no benefit for motional-frequency errors or fast Markovian optical-qubit dephasing.

R001/R002 used a discrete spin-only surrogate and therefore could not test the motional-frequency or continuous-time dephasing legs. R003 builds the missing motion-including model directly from Supp. Mat. S3:

• two 3-level ions, states {down, up, e}, coupled to a truncated motional Fock mode;
• Hermitian red/blue-sideband expansion of Eq. S6, reducing to the compact paper Hamiltonian when epsilon_q = 0;
• correct second-pulse Hamiltonian clock offset, so epsilon_m leaves the phase-space loop open instead of cancelling by construction;
• full spin-motion density matrix carried across 80 protocol cycles;
• discrete repump B and drive C maps applied to the full spin-motion density matrix, preserving the oscillator state;
• Lindblad dephasing leg with jump sqrt(Gamma) (sigma_z^1 + sigma_z^2).

The evidence target is qualitative C029, not a full digitized C059 plot comparison. Use a small grid over the four paper axes and compare final Bell-state infidelities for:

• single-loop MS gate targeting (|dd> - i |ee>)/sqrt(2);
• two-loop phase-modulated MS gate with the same calibrated two-loop scale used in C059;
• 80 protocol cycles targeting |Psi^->.

Verdict criterion:

• verified if the motion-including model confirms all four qualitative legs;
• partial if a leg remains model-dependent or contradictory;
• not_verified only if the reproduced trends contradict the claim.