S-Parameters & RF Performance
=============================

RF checks for matching, transmission, isolation, coupling, phase response, and
                standing-wave behavior.

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

   * - #
     - VER ID
     - Parameter
     - Severity
     - Design Rule / Constraint
     - Ideal / Optimal Value
     - Acceptable Range
     - Good
     - Bad
     - Why It Matters
   * - 1
     - HFSS-S-001
     - Return Loss
     - Critical
     - S11 ≤ −20 dB at operating frequency
     - < −20 dB
     - −15 to −25 dB
     - < −20 dB: excellent impedance match; full power into resonator/qubit
     - > −10 dB: >10% power reflected; readout chain SNR degraded
     - Reflected power from input port. Poor return loss = impedance mismatch →
                        signal reflections degrade qubit readout fidelity.
   * - 2
     - HFSS-S-002
     - Insertion Loss
     - Critical
     - S21 ≥ −0.1 dB in passband
     - < −0.1 dB
     - −0.1 to −1 dB
     - < −0.1 dB: near-lossless transmission; signal integrity preserved
     - > −3 dB: half power lost; readout SNR < 3 dB, fidelity severely
                        impacted
     - Signal transmission efficiency. High insertion loss reduces readout SNR,
                        requiring higher drive power that heats the device.
   * - 3
     - HFSS-S-003
     - Transmission \|S21\|
     - High
     - \|S21\| ≥ 0.95 (linear) in passband
     - ≈ 1.0 (unity)
     - 0.9 – 1.0
     - ≥ 0.95: >90% amplitude transmission; strong coupling confirmed
     - < 0.5: >50% amplitude loss; readout inefficient
     - Linear magnitude of S21. Near-unity confirms full coupling efficiency; used
                        in EPR extraction and resonator characterisation.
   * - 4
     - HFSS-S-004
     - Port Isolation
     - High
     - Sij ≤ −30 dB between non-coupled ports
     - < −30 dB
     - −20 to −40 dB
     - < −30 dB: crosstalk negligible; simultaneous multi-qubit readout viable
     - > −15 dB: strong port coupling; driven rotations on idle qubits
     - Cross-port electromagnetic isolation. Insufficient isolation causes
                        simultaneous readout errors and qubit–qubit cross-drive.
   * - 5
     - HFSS-S-005
     - Forward Isolation (S12)
     - High
     - S12 ≤ −20 dB (Purcell filter context)
     - < −20 dB
     - −15 to −30 dB
     - < −20 dB: amplifier backaction blocked; qubit protected from output noise
     - > −10 dB: HEMT noise reaches qubit; excess excitation and T₁ degradation
     - Reverse isolation prevents HEMT amplifier noise photons from reaching qubit.
                        Critical in Purcell filter and circulator design.
   * - 6
     - HFSS-S-006
     - Phase of S21 (GDD)
     - Medium
     - Group delay deviation < 1 ns across qubit bandwidth
     - Linear phase
     - < 5° deviation
     - Linear phase: negligible pulse distortion; gate calibration stable
     - Non-linear phase: pulse distortion → systematic gate errors
     - Non-linear phase response causes group delay dispersion that distorts shaped
                        control pulses, increasing gate error.
   * - 7
     - HFSS-S-007
     - Coupling Coefficient κ
     - Critical
     - 1 MHz ≤ κ/2π ≤ 10 MHz (readout resonator)
     - 1 – 5 MHz
     - 0.5 – 20 MHz
     - 1–5 MHz: fast readout (< 1 µs) with Purcell rate < 1 kHz; near quantum
                        limit
     - < 0.1 MHz: readout > 10 µs; > 100 MHz: Purcell collapse of T₁
     - External coupling rate of readout resonator. Sets fundamental trade-off
                        between measurement speed and Purcell-induced qubit decay.
   * - 8
     - HFSS-S-008
     - VSWR
     - Medium
     - VSWR ≤ 1.1 : 1 at operating frequency
     - < 1.1 : 1
     - 1.1 – 1.5 : 1
     - < 1.1:1: >99% power transfer; standing waves negligible
     - > 2.0:1: standing waves cause frequency-dependent errors
     - Voltage standing wave ratio quantifies impedance mismatch. High VSWR
                        degrades power delivery to qubit and readout resonator.