Simulation Convergence

Simulation Convergence & Accuracy Metrics: eigenmode simulation quality indicators for reliable EPR extraction

1. Eigenmode Solver Convergence

Parameter	Symbol	Unit	Description	Optimal / Best Value	Good Range	Acceptable Range	Poor / Worst Value	Physical Significance
Eigenfrequency Convergence ΔFreq	Delta f / f	ppm	Relative change in eigenfrequency between successive mesh refinement passes.	< 5 ppm	< 50 ppm	50 – 500 ppm	> 1000 ppm	EPR frequencies directly set qubit and resonator values; poor convergence propagates to all outputs.
Energy Error (Maxwell equations)	Delta U / U	%	Relative error in total electromagnetic energy from adaptive mesh refinement.	< 0.1%	< 0.5%	0.5 – 2%	> 5%	Energy error directly bounds participation ratio error; must be minimised for accurate loss prediction.
Tetrahedral Mesh Count	N_mesh	thousands	Number of tetrahedra in adaptive mesh; convergence criterion, not absolute target.	200 – 2000k	50 – 200k	10 – 50k	< 5k	Too few → inaccurate fields, especially at thin features (junctions, gaps); too many → slow.
Number of Eigenmode Passes	N_passes	integer	Adaptive refinement passes until convergence criterion is met.	10 – 20 passes	7 – 10 passes	4 – 6 passes	< 3 passes	Insufficient passes leave mesh unrefined in critical regions (junction vicinity, surface gaps).
Mode Participation Sum Check	sum p	dimensionless	Sum of EPR participations across all modes; should equal 1 for each junction.	0.99 – 1.01	0.97 – 1.03	0.93 – 1.07	< 0.90 or > 1.10	Deviation indicates missing eigenmodes, disconnected networks, or mesh error in junction volume.
Participation Ratio Repeatability	sigma_p / p	%	Run-to-run relative standard deviation of participation ratios across identical simulations.	< 0.5%	< 2%	2 – 5%	> 10%	Poor repeatability indicates stochastic mesh seeding issues; use fixed seed and fine mesh.

2. Loss Analysis Simulation Quality

Parameter	Symbol	Unit	Description	Optimal / Best Value	Good Range	Acceptable Range	Poor / Worst Value	Physical Significance
Surface Participation Error	Delta p_surf	%	Relative error in surface (interface) participation ratio from finite mesh at thin layers.	< 2%	< 5%	5 – 15%	> 20%	Thin oxide layers (2–5 nm) require extremely fine mesh; often estimated analytically.
Junction Volume Definition Accuracy	Delta V_J / V_J	%	Relative accuracy of the junction geometric volume used to compute junction participation.	< 1%	< 5%	5 – 10%	> 20%	Junction volume error directly maps to participation ratio error; critical CAD accuracy needed.
Number of Loss Regions Modelled	N_regions	integer	Number of distinct lossy material regions (interfaces) explicitly included in the EPR model.	= 4 (MS, SA, MA, bulk)	3	2	1 (bulk only)	Modelling only bulk misses dominant surface/interface losses in planar devices.
Frequency Band of Validity	BW_valid	GHz	Frequency range over which the extracted Hamiltonian parameters are valid (no spurious modes).	0 – 15 GHz (no spurious)	0 – 10 GHz	5 – 10 GHz only	Spurious modes present	Spurious modes can hybridize with qubit and resonator, invalidating EPR sums.