Post-Processing Derived Outputs
===============================

Derived outputs convert Q3D matrices into qubit design metrics such as Ec, Ej, g,
                chi, ZZ, and Purcell rate.

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

   * - #
     - Parameter
     - Symbol / Unit
     - Extraction Method
     - Typical Q3D Value
     - Ideal / Optimal
     - Good Range
     - Worst Case
     - Why It Matters
     - Key Design Note
   * - 52
     - Charging Energy (Ec/h)
     - Ec / h·MHz
     - Ec = e²/(2C_Σ); C_Sigma from Q3D Maxwell matrix
     - 200 – 350 MHz
     - 200 – 350 MHz (transmon optimum)
     - 150 – 400 MHz
     - < 50 or > 1 GHz
     - Sets charge sensitivity; Ej/Ec = 50–80 ideal for transmon; deviating worsens
                        noise or anharmonicity
     - Ec/h = 200 MHz → C_Sigma = 91 fF; exact C_Sigma from Q3D is the critical
                        input to Hamiltonian model
   * - 53
     - Josephson Energy (Ej/h)
     - Ej / h·GHz
     - Ej = Φ₀²/(2L_J) = Φ₀ I_c / 2π
     - 10 – 30 GHz
     - 10 – 30 GHz (Ej/Ec ~ 50–80)
     - 5 – 50 GHz
     - < 2 or > 100 GHz
     - With Ec determines qubit frequency f₀₁ ≈ √(8EjEc)/h − Ec/h and anharmonicity
                        α = −Ec/h
     - Ej is tunable via flux in split-junction transmons; Ej/Ec spread across chip
                        sets yield
   * - 54
     - Qubit–Resonator Coupling (g / 2pi)
     - g / 2π / MHz
     - g = C_g/(C_Sigma) × √(ω_q ω_r)/2; C_g from Q3D off-diagonal
     - 50 – 150 MHz
     - 50 – 150 MHz (dispersive regime)
     - 20 – 300 MHz
     - < 5 or > 500 MHz
     - Vacuum Rabi coupling; in dispersive regime (g ≪ Δ) enables QND readout
                        without qubit decay
     - g/Δ < 0.1 ensures dispersive limit; Purcell rate Γ_P = (g/Δ)² × κ scales
                        as g²
   * - 55
     - Dispersive Shift (chi/2π)
     - chi / 2π / MHz
     - chi = g²/Δ × α/(Δ+α); Δ = ω_q − ω_r; all from Q3D + junction params
     - 1 – 5 MHz
     - 1 – 5 MHz
     - 0.5 – 10 MHz
     - < 0.1 or > 20 MHz
     - Qubit-state-dependent resonator shift; single-shot readout SNR ∝ chi/κ;
                        larger chi → better fidelity
     - chi and Purcell rate trade off via g; Purcell filter allows larger g without
                        excess Purcell loss
   * - 56
     - ZZ Coupling Rate (ζ/2π)
     - ζ / 2π / kHz
     - ζ = 2g²χ²/(Δ·α·(Δ+α)); derived from Q3D coupling capacitances
     - 10 – 100 kHz
     - < 10 kHz
     - 10 – 50 kHz
     - > 200 kHz
     - Always-on conditional phase rate between qubits; leads to leakage in
                        spectator qubits during gates
     - ZZ suppression is the central challenge of transmon scaling; tunable coupler
                        can push ζ < 1 kHz
   * - 57
     - Anharmonicity (alpha / 2pi)
     - α / 2π / MHz
     - α = −Ec/h; Ec from Q3D C_Sigma; or directly measured by two-tone
                        spectroscopy
     - −200 to −300 MHz
     - −300 to −150 MHz
     - −350 to −100 MHz
     - \|α\|/2π < 50 MHz
     - Separates \|0〉→\|1〉 from \|1〉→\|2〉 transitions; sets minimum gate duration
                        without leakage
     - Gate bandwidth BW < \|α\|/(2π) required to avoid leakage; \|α\| = 200 MHz →
                        t_gate > 5 ns
   * - 58
     - Purcell Decay Rate (Γ_P/2π)
     - Γ_P / 2π / kHz
     - Γ_P = (g/Δ)² × κ; κ = ω_r/Q_ext from Q3D; g from coupling cap
     - 1 – 10 kHz
     - < 1 kHz (with Purcell filter)
     - 1 – 10 kHz
     - > 100 kHz
     - Resonator-induced qubit relaxation limiting T₁ even with long material T₁;
                        mitigated by filter
     - Purcell filter (bandpass on resonator port) can reduce Γ_P by 10–100×
                        without affecting readout