Simulation Convergence
======================

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

A. Eigenmode Solver Convergence

.. list-table::
   :header-rows: 1

   * - 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.

B. Loss Analysis Simulation Quality

.. list-table::
   :header-rows: 1

   * - 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.