Qubit Performance Metrics
=========================

HFSS-derived qubit checks for frequency, anharmonicity, energy scales, coherence,
                Purcell decay, and gate fidelity.

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

   * - #
     - 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.