Core EPR Parameters
===================

Core EPR Parameters: Primary outputs of the EPR method: participation ratios,
                zero-point fluctuations, and mode hybridization metrics

A. Junction Energy Participation Ratios

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   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - Junction Participation Ratio (transmon)
     - p_J
     - dimensionless
     - Fraction of total inductive energy stored in the Josephson junction for
                            the qubit mode. Central quantity of EPR method.
     - 0.90 – 0.99
     - 0.80 – 0.99
     - 0.50 – 0.79
     - < 0.30
     - High p_J maximises anharmonicity and qubit nonlinearity; low values
                            reduce gate speed and anharmonicity.
   * - Junction Participation Ratio (readout mode)
     - p_J^res
     - dimensionless
     - Fraction of readout resonator mode energy in the junction. Should be
                            minimised to reduce Purcell loss.
     - < 1×10⁻³
     - < 1×10⁻²
     - 1×10⁻²–5×10⁻²
     - > 0.10
     - Large p_J^res couples resonator decay channel to qubit, reducing T_1 via
                            Purcell effect.
   * - Total Junction Participation (all modes)
     - sum p_J
     - dimensionless
     - Sum of participation ratios across all simulated eigenmodes;
                            normalization check.
     - ≈ 1.00 (±0.01)
     - 0.98 – 1.02
     - 0.95 – 1.04
     - < 0.90 or > 1.10
     - Deviation from unity indicates missing modes, poor mesh, or incomplete
                            boundary conditions.
   * - Participation Ratio Asymmetry
     - Delta p_J
     - dimensionless
     - Difference in junction participation between two junctions in a
                            split-junction (SQUID) qubit.
     - < 0.01
     - < 0.05
     - 0.05–0.15
     - > 0.20
     - Asymmetry leads to flux-noise sensitivity and reduced coherence in
                            tunable qubits.

B. Zero-Point Fluctuation (ZPF) Quantities

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   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - ZPF Voltage across Junction
     - V_zpf
     - µV
     - RMS zero-point voltage fluctuation across the Josephson junction; sets
                            qubit–photon coupling strength.
     - 10 – 50 µV
     - 5 – 100 µV
     - 1 – 200 µV
     - < 0.5 µV or > 500 µV
     - Too small → weak anharmonicity; too large → unwanted multiphoton
                            transitions and leakage.
   * - ZPF Current through Junction
     - I_zpf
     - nA
     - RMS zero-point current fluctuation; related to V_zpf via junction
                            inductance.
     - 1 – 10 nA
     - 0.5 – 20 nA
     - 0.1 – 50 nA
     - < 0.05 nA
     - Determines coupling to flux noise and magnetic environment; critical for
                            flux qubits.
   * - ZPF Phase across Junction
     - varphi_zpf
     - rad
     - RMS zero-point phase fluctuation varphi_zpf = √(2eV_zpf / ℏω_q). Governs
                            perturbative expansion validity.
     - 0.1 – 0.5 rad
     - 0.05 – 0.6 rad
     - 0.6 – 0.9 rad
     - > 1.0 rad
     - Values > 1 rad invalidate the perturbative (dispersive)
                            approximation used in EPR.
   * - Hybridization Factor
     - chi
     - dimensionless
     - Degree of mode hybridization between qubit and resonator; off-diagonal
                            element in EPR Hamiltonian.
     - < 0.01 (well-dressed)
     - < 0.05
     - 0.05 – 0.15
     - > 0.20
     - Large hybridization mixes qubit and resonator, degrading single-mode
                            approximation.

C. Hamiltonian Parameters Extracted via EPR

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   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - Qubit Frequency (extracted)
     - omega_q / 2pi
     - GHz
     - Fundamental qubit transition frequency extracted from EPR eigenmode
                            simulation.
     - 4 – 6 GHz
     - 3 – 8 GHz
     - 1 – 3 or 8–12 GHz
     - < 1 GHz or > 15 GHz
     - Outside optimal window: low freq → thermal excitation; high freq →
                            limited coupling hardware.
   * - Anharmonicity (EPR-derived)
     - alpha / 2pi
     - MHz
     - Qubit anharmonicity = ω_12 - ω_01; extracted via second-order EPR
                            perturbation theory.
     - 150 – 350 MHz
     - 100 – 400 MHz
     - 50 – 99 MHz
     - < 30 MHz
     - Insufficient anharmonicity causes leakage to \|2⟩ during gates; > 400
                            MHz may indicate charge noise sensitivity.
   * - Kerr Self-Nonlinearity
     - K / 2pi
     - MHz
     - Effective Kerr coefficient (= anharmonicity for transmon); second-order
                            EPR correction term.
     - 150 – 300 MHz
     - 100 – 400 MHz
     - 50 – 99 MHz
     - < 20 MHz
     - Sets speed limit of single-qubit gates; related to DRAG pulse
                            requirements.
   * - Dispersive Shift chi/2π
     - chi/2π
     - MHz
     - Qubit-state-dependent resonator frequency shift; critical for
                            high-fidelity dispersive readout.
     - 0.5 – 3 MHz
     - 0.1 – 5 MHz
     - 0.01 – 0.09 MHz
     - < 0.01 MHz or > 10 MHz
     - Too small → insufficient readout contrast; too large →
                            measurement-induced dephasing.
   * - Cross-Kerr (qubit-qubit)
     - zeta_ij / 2pi
     - MHz
     - Always-on ZZ interaction between coupled qubits; parasitic term in
                            multi-qubit chips.
     - < 0.01 MHz
     - < 0.10 MHz
     - 0.10 – 0.50 MHz
     - > 1.0 MHz
     - Large ZZ causes always-on entanglement errors; major source of two-qubit
                            gate infidelity.