Core EPR Parameters¶

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

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.

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.

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.