Coefficient Amplification Threshold

The 2-qubit sector-projected Hamiltonian H = g₀I + g₁Z₀ + g₂Z₁ + g₃Z₀Z₁ + g₄X₀X₁ + g₅Y₀Y₁ encodes a molecular electronic structure problem on two qubits. The Z-terms (g₁, g₂) are measured directly in the computational basis, while the X/Y-terms (g₄, g₅) require basis-rotation gates.

Readout errors on Z-basis measurements are the dominant noise source for shallow circuits (confirmed by our ZNE gate-folding and TREX studies). Since g₁ multiplies the Z₀ expectation value, any readout error δ on Z₀ translates to g₁·δ energy error. When |g₁| >> |g₄|, readout errors dominate over gate noise.

This "coefficient amplification" effect was first observed in our HeH+ IBM experiments (91 kcal/mol error with SamplerV2), but we lacked a quantitative framework. The threshold analysis compiles 30+ data points across two molecules, three backends, and six bond distances to test the prediction.

Data points: 30+ VQE measurements spanning H₂ (14 bond distances) and HeH+ (3 bond distances) on emulator, IBM Torino (TREX), and QI Tuna-9 (PS+REM). All use the same 2-qubit Ry-CNOT-X ansatz with 4096 shots.

Controlled comparison: H₂ at R=0.735 Å vs HeH+ at R=0.75 Å on the same hardware with the same mitigation. HeH+ has a smaller rotation angle (α=-0.127 vs -0.112), so any error difference must come from coefficient structure, not gate noise.

Within-molecule analysis: H₂ PES sweep on Tuna-9 (R=0.5 to 2.5 Å) disentangles the ratio effect from gate noise, since both change with bond distance.

Cross-molecule result (the smoking gun): IBM TREX gives H₂ 0.22 kcal/mol (PASS) vs HeH+ 4.45 kcal/mol (FAIL). Tuna-9 PS+REM gives H₂ 0.92 vs HeH+ 4.44 kcal/mol. Same circuit depth, same shot count, same mitigation. HeH+ even has a smaller rotation angle (less gate noise). The only difference: |g₁|/|g₄| = 7.8 vs 4.4.

Superlinear scaling: 1.8x ratio increase → 20x error increase (IBM). This is approximately error ∝ ratio⁵, suggesting nonlinear amplification that gets dramatically worse beyond ratio ≈ 5.

Two competing noise regimes: The H₂ PES sweep shows an error minimum at R=1.0 Å (4.12 kcal/mol), not at the lowest ratio (R=2.5, ratio=0.19) or smallest alpha (R=0.5, |α|=0.07). At small R, coefficient amplification dominates (high ratio); at large R, gate noise dominates (large |α|).

Cross-platform confirmation: IBM (4.45) and Tuna-9 (4.44) give nearly identical HeH+ errors despite completely different hardware architectures (133q heavy-hex vs 9q irregular) and mitigation strategies (TREX vs REM+PS). This proves the coefficient ratio, not hardware specifics, determines the error floor.

Practical threshold: Chemical accuracy requires |g₁|/|g₄| < ~5 with current NISQ mitigation. Above ratio ~8, no current strategy achieves chemical accuracy. Practitioners should compute this ratio before attempting hardware VQE.