Designing superconducting quantum processors requires precise control over dozens of electromagnetic parameters—each of which must fall within tightly specified ranges to achieve the coherence, fidelity, and scalability demanded by fault-tolerant quantum computing. This reference catalogues 52 output parameters extracted from HFSS simulations, organised into 7 categories, with design rules sourced from IEEE/APS research papers and doctoral theses spanning 2004–2026.
"Each parameter includes its ideal value, acceptable range, and the physical consequence of deviation. Of the 52 parameters, 26 are classified as Critical—meaning that falling outside their acceptable range will directly compromise qubit performance, readout fidelity, or gate accuracy."
Parameter Category Overview
| Category | Parameters | Critical | High | Medium | Low |
|---|---|---|---|---|---|
| 1. S-Parameters & RF Performance | 8 | 3 | 3 | 2 | — |
| 2. Resonator & Cavity Parameters | 10 | 5 | 2 | 2 | 1 |
| 3. Electromagnetic Field Outputs | 7 | 3 | 3 | 1 | — |
| 4. Qubit Performance Metrics | 9 | 9 | — | — | — |
| 5. Crosstalk & Isolation | 6 | 2 | 3 | 1 | — |
| 6. Thermal & Loss Parameters | 7 | 2 | 3 | 2 | — |
| 7. Simulation Convergence Metrics | 5 | 1 | 1 | 2 | 1 |
| Total | 52 | 25 | 15 | 10 | 2 |
1. S-Parameters & RF Performance
S-parameters quantify signal transmission, reflection, and isolation across the microwave readout and control chain. Poor S-parameter performance degrades qubit readout SNR, causes impedance mismatch, and allows noise back-action from amplifiers to reach the qubit.
| ID | Parameter | Priority | Ideal Value | Acceptable Range |
|---|---|---|---|---|
| HFSS-S-001 | Return Loss (S11) | Critical | < −20 dB | −15 to −25 dB |
| HFSS-S-002 | Insertion Loss (S21) | Critical | < −0.1 dB | −0.1 to −1 dB |
| HFSS-S-003 | Transmission |S21| | High | ≈ 1.0 (unity) | 0.9 – 1.0 |
| HFSS-S-004 | Port Isolation | High | < −30 dB | −20 to −40 dB |
| HFSS-S-005 | Forward Isolation (S12) | High | < −20 dB | −15 to −30 dB |
| HFSS-S-006 | Phase of S21 (GDD) | Medium | Linear phase | < 5° deviation |
| HFSS-S-007 | Coupling Coefficient κ | Critical | 1 – 5 MHz | 0.5 – 20 MHz |
| HFSS-S-008 | VSWR | Medium | < 1.1 : 1 | 1.1 – 1.5 : 1 |
2. Resonator & Cavity Parameters
Resonator parameters govern the readout chain performance. The resonant frequency, quality factors (loaded, internal, external), and coupling strength determine measurement speed, Purcell-limited T₁, and single-shot readout fidelity.
| ID | Parameter | Priority | Ideal Value | Acceptable Range |
|---|---|---|---|---|
| HFSS-R-001 | Resonant Frequency f₀ | Critical | 5 – 7 GHz | 4 – 8 GHz |
| HFSS-R-002 | Loaded Q (Q_L) | Critical | 5,000 – 20,000 | 1,000 – 50,000 |
| HFSS-R-003 | Internal Q (Q_i) | Critical | > 10⁶ | 10⁵ – 10⁷ |
| HFSS-R-004 | External Q (Q_e) | High | 2,000 – 20,000 | 500 – 100,000 |
| HFSS-R-005 | Coupling Strength g | Critical | 50 – 150 MHz | 10 – 300 MHz |
| HFSS-R-006 | Dispersive Shift χ | Critical | 1 – 5 MHz | 0.1 – 20 MHz |
| HFSS-R-007 | Photon Decay Rate κ | High | 1 – 5 MHz | 0.1 – 20 MHz |
| HFSS-R-008 | Impedance Z₀ | Medium | 50 Ω | 45 – 55 Ω |
| HFSS-R-009 | Frequency Pulling Δf | Medium | < 0.5 MHz | < 2 MHz |
| HFSS-R-010 | Kinetic Inductance α | Low | 0.05 – 0.2 | 0.001 – 0.5 |
3. Electromagnetic Field Outputs
Field outputs from HFSS eigenmode simulations are the foundation of the Energy Participation Ratio (EPR) method. Interface participation ratios (p_SA, p_MA, p_MS) multiplied by their respective loss tangents predict qubit T₁ with remarkable accuracy.
| ID | Parameter | Priority | Ideal Value | Acceptable Range |
|---|---|---|---|---|
| HFSS-E-001 | Peak E-Field |E|max | High | < 10⁵ V/m | < 10⁷ V/m |
| HFSS-E-002 | H-Field Distribution |H| | Medium | < 500 A/m | < 5,000 A/m |
| HFSS-E-003 | Interface Participation pᵢ | Critical | < 10⁻³ | 10⁻³ – 10⁻² |
| HFSS-E-004 | Surface Participation p_MA | Critical | < 10⁻⁴ | 10⁻⁴ – 10⁻³ |
| HFSS-E-005 | Bulk Participation p_bulk | High | < 5×10⁻³ | 10⁻² – 5×10⁻² |
| HFSS-E-006 | Junction EPR | Critical | 0.95 – 1.0 | 0.8 – 1.0 |
| HFSS-E-007 | Radiation Q (Q_rad) | High | > 10⁶ | > 10⁵ |
4. Qubit Performance Metrics
Every parameter in this category is classified as Critical. These metrics define the fundamental quantum performance of the transmon qubit—frequency, anharmonicity, coherence times, and gate fidelities. All are derived from HFSS eigenmode solutions combined with junction parameters.
| ID | Parameter | Priority | Ideal Value | Acceptable Range |
|---|---|---|---|---|
| HFSS-Q-001 | Anharmonicity | Critical | −200 to −300 MHz | −100 to −500 MHz |
| HFSS-Q-002 | Qubit Frequency | Critical | 4 – 6 GHz | 3 – 8 GHz |
| HFSS-Q-003 | Josephson Energy | Critical | 15 – 30 GHz | 5 – 60 GHz |
| HFSS-Q-004 | Charging Energy | Critical | 200 – 350 MHz | 100 – 500 MHz |
| HFSS-Q-005 | Purcell Decay Rate | Critical | < 500 Hz | < 10 kHz |
| HFSS-Q-006 | Predicted | Critical | > 100 µs (2D) / > 500 µs (3D) | 50 – 500 µs |
| HFSS-Q-007 | Predicted | Critical | > 100 µs | 20 – 300 µs |
| HFSS-Q-008 | 1Q Gate Fidelity | Critical | > 99.9% | 99 – 99.99% |
| HFSS-Q-009 | 2Q Gate Fidelity | Critical | > 99.5% | 98 – 99.9% |
5. Crosstalk & Isolation
Crosstalk parameters quantify unwanted electromagnetic coupling between qubits, control lines, and spurious package modes. Insufficient isolation causes always-on ZZ errors, driven rotations on idle qubits, and leakage to non-computational states.
| ID | Parameter | Priority | Ideal Value | Acceptable Range |
|---|---|---|---|---|
| HFSS-C-001 | ZZ Coupling ζ (idle) | Critical | < 1 kHz | 1 – 100 kHz |
| HFSS-C-002 | Nearest Neighbour Isolation | High | < −40 dB | −30 to −50 dB |
| HFSS-C-003 | Next-Nearest Isolation | High | < −60 dB | −50 to −70 dB |
| HFSS-C-004 | Leakage to |2⟩ (L₁) | Critical | < 0.01% | 0.01 – 0.1% |
| HFSS-C-005 | Spurious Mode Gap Δf_spur | High | > 1 GHz | 0.5 – 2 GHz |
| HFSS-C-006 | Package Mode Density | Medium | < 0.5 modes/GHz | < 5 modes/GHz |
6. Thermal & Loss Parameters
Thermal and loss parameters govern the intrinsic energy dissipation mechanisms in the quantum chip. Dielectric loss tangent, surface resistance, and TLS defect densities set the fundamental floor on qubit coherence times.
| ID | Parameter | Priority | Ideal Value | Acceptable Range |
|---|---|---|---|---|
| HFSS-T-001 | Dielectric Loss Tangent | Critical | < 10⁻⁶ | 10⁻⁷ – 10⁻⁵ |
| HFSS-T-003 | Dissipated Power P_sub | High | < 0.1 pW | 0.1 – 100 pW |
| HFSS-T-004 | Thermal NEP | Medium | < 10⁻²⁰ W/√Hz | 10⁻²⁰ – 10⁻¹⁸ |
| HFSS-T-005 | TLS Loss Rate 1/T₁_TLS | Critical | < 0.5 kHz | 0.5 – 10 kHz |
| HFSS-T-006 | Conductor Loss α_c | Medium | < 0.0001 dB/m | 0.0001 – 0.1 dB/m |
| HFSS-T-007 | Package Radiation Loss 1/Q_rad | High | < 10⁻⁷ | 10⁻⁷ – 10⁻⁵ |
7. Simulation Convergence Metrics
Convergence metrics ensure the HFSS solution is reliable. An unconverged simulation produces inaccurate S-parameters, Q-factors, and participation ratios—leading to incorrect qubit frequency and T₁ predictions.
| ID | Parameter | Priority | Ideal Value | Acceptable Range |
|---|---|---|---|---|
| HFSS-V-001 | Delta S Convergence (ΔS) | Critical | < 0.001 | 0.001 – 0.005 |
| HFSS-V-002 | Adaptive Pass Count | Medium | 6 – 12 passes | 6 – 25 passes |
| HFSS-V-003 | Mesh Element Count | Medium | 20k – 100k | 10k – 500k |
| HFSS-V-004 | Energy Error Δε/ε | High | < 0.2% | 0.2 – 1% |
| HFSS-V-005 | Simulation RAM Usage | Low | < 16 GB | 16 – 64 GB |
Key Takeaways
- 25 of 52 parameters are Critical—deviating from their acceptable ranges will directly compromise qubit performance or readout fidelity.
- Qubit Performance Metrics is the only category where every parameter is Critical (9/9), reflecting the tight design tolerances required for fault-tolerant quantum computing.
- Participation ratios (p_SA, p_MA, p_bulk) are the primary design levers for maximising T₁—they connect geometry choices to loss mechanisms through the EPR framework.
- Convergence must be verified: ΔS < 0.002 and energy error < 0.5% before any parameter extraction is considered reliable.