Qubit Performance Metrics

HFSS-derived qubit checks for frequency, anharmonicity, energy scales, coherence,

Purcell decay, and gate fidelity.

#	VER ID	Parameter	Severity	Design Rule / Constraint	Ideal / Optimal Value	Acceptable Range	Good	Bad	Why It Matters
26	HFSS-Q-001	Anharmonicity α	Critical	|α|/2π ≥ 150 MHz (|α| = |ω₁₂ − ω₀₁|)	−200 to −300 MHz	−100 to −500 MHz	|α| 200–300 MHz: DRAG gates < 30 ns with leakage < 0.01%; selective driving	|α| < 50 MHz: must slow gates to > 200 ns; leakage to |2⟩ > 1%	Frequency gap between 0→1 and 1→2 transitions. Must exceed pulse bandwidth for selective driving without leakage.
27	HFSS-Q-002	Qubit Frequency f_q	Critical	4.0 GHz ≤ f_q ≤ 6.0 GHz (transmon sweet spot)	4 – 6 GHz	3 – 8 GHz	4–6 GHz: kT/hf < 0.001 at 20 mK; standard microwave hardware	< 1 GHz: thermal population > 1%; > 10 GHz: substrate loss increases	Qubit transition frequency. Must be well above thermal energy (kT/h ≈ 400 MHz at 20 mK) and away from substrate TLS resonances.
28	HFSS-Q-003	Josephson Energy E_J	Critical	10 GHz ≤ E_J/h ≤ 40 GHz; E_J/E_C ≥ 50	15 – 30 GHz	5 – 60 GHz	15–30 GHz: f_q on target; charge insensitive; junction reproducible within ±5%	< 1 GHz: qubit below 2 GHz; thermally excited; E_J/E_C < 10: charge sensitive	Josephson tunneling energy sets qubit frequency and E_J/E_C ratio. Extracted in HFSS via junction inductance L_J = Φ₀²/E_J.
29	HFSS-Q-004	Charging Energy E_C	Critical	200 MHz ≤ E_C/h ≤ 350 MHz	200 – 350 MHz	100 – 500 MHz	200–350 MHz: anharmonicity ~−E_C; charge noise suppressed; qubit addressable	< 50 MHz: near-harmonic oscillator; > 1000 MHz: Cooper-pair box regime	Single-electron charging energy set by shunt capacitance. E_C = e²/2C_Σ; defines anharmonicity and charge sensitivity.
30	HFSS-Q-005	Purcell Decay Rate γ_P	Critical	γ_P/2π < 1 kHz (without filter); < 100 Hz (with)	< 500 Hz	< 10 kHz	< 500 Hz: Purcell T₁ contribution > 2 ms; does not limit qubit T₁ budget	> 100 kHz: Purcell T₁ < 10 µs; qubit lifetime dominated by readout line	Qubit decay rate into transmission line via off-resonant resonator. γ_P = (g/Δ)²κ. Limits T₁ without Purcell filter.
31	HFSS-Q-006	Predicted T₁	Critical	T₁ > 100 µs (planar 2D); > 1 ms (3D cavity)	> 500 µs (3D) / > 100 µs (2D)	50 – 500 µs	> 100 µs: supports > 1000 gate depth within coherence envelope (10 ns gates)	< 10 µs: < 100 gates within T₁; fault-tolerant computation infeasible	Predicted energy relaxation time from HFSS loss model: 1/T₁ = Σ(pᵢ × ωᵢ × tan δᵢ) + γ_Purcell + γ_radiation.
32	HFSS-Q-007	Predicted T₂	Critical	T₂ > 50 µs; ideally T₂ ≈ 2T₁ (pure dephasing limited)	> 100 µs	20 – 300 µs	T₂ ≈ 2T₁: pure dephasing negligible; charge and flux noise well-suppressed	T₂ ≪ T₁: strong 1/f dephasing; substrate charge traps or flux noise dominant	Pure dephasing time. Gap between T₂ and 2T₁ quantifies 1/f noise from TLS charge noise and flux noise in junctions.
33	HFSS-Q-008	1Q Gate Fidelity F₁Q	Critical	F₁Q ≥ 99.9% (randomised benchmarking)	> 99.9 %	99 – 99.99 %	> 99.9%: below surface-code fault-tolerance threshold (~99.4%); QEC viable	< 99%: error rate exceeds fault-tolerance threshold; errors cascade in QEC	Single-qubit gate fidelity estimated from T₁, T₂, anharmonicity, and leakage. Must exceed fault-tolerant threshold ~99.5%.
34	HFSS-Q-009	2Q Gate Fidelity F₂Q	Critical	F₂Q ≥ 99.5% (CZ or iSWAP gate)	> 99.5 %	98 – 99.9 %	> 99.5%: viable for surface code with standard overhead; ZZ residual < 10 kHz	< 97%: excessive error rate; 2Q errors dominate total circuit error budget	Two-qubit gate fidelity. More sensitive to residual ZZ coupling, leakage, coupler calibration, and neighbouring qubit crosstalk.