The Energy Participation Ratio (EPR) method is a semi-classical technique for characterizing superconducting qubit circuits. It extracts quantum Hamiltonian parameters from classical EM simulations (HFSS/FEM), bridging the gap between physical layout and quantum behavior.
EPR Core Parameters
| Parameter | Symbol (Unit) | Optimal | Acceptable | Poor | Physical Meaning |
|---|---|---|---|---|---|
| Energy Participation Ratio | (Dimensionless) | 0.90 – 0.99 | 0.70 – 0.84 | < 0.70 or === 1.0 | Fraction of mode energy stored in junction . A value close to 1 means the junction dominates the mode (strong nonlinearity). A value exactly equal to 1 is unphysical. This is the central quantity in the EPR method — it links classical EM simulation results to quantum circuit parameters. |
| Josephson Inductance | (nH) | 8 – 15 nH | 3–6 or 18–25 nH | < 3 nH or > 25 nH | Kinetic inductance of the Josephson junction. Sets the qubit frequency together with geometric capacitance. Too small → high frequency, hard to control; too large → low frequency, susceptible to thermal errors. Typical range 5–20 nH for transmon qubits. |
| Josephson Energy | (GHz, h units) | 15 – 40 GHz | 5–12 or 45–60 GHz | < 5 GHz or > 60 GHz | Energy scale of Josephson coupling between the two superconductors in the junction. The ratio determines the qubit regime — for transmon operation we need . Too small → charge sensitive; too large → reduced anharmonicity. |
| Charging Energy | (GHz, h units) | 0.15 – 0.30 GHz | 0.05–0.12 or 0.35–0.5 GHz | < 0.05 or > 0.5 GHz | Electrostatic energy scale — energy cost to add one Cooper pair to the island. Controls charge sensitivity and anharmonicity. In transmon regime, small reduces charge noise but also reduces anharmonicity. Must balance noise immunity vs. qubit addressability. |
| E_J / E_C Ratio | (Dimensionless) | 80 – 120 | 40–60 or 130–200 | < 40 or > 200 | The key regime parameter for transmon qubits. Values → transmon regime (charge insensitive). Values > 150 → very low anharmonicity with leakage risk. This ratio must be carefully tuned to balance charge noise suppression against sufficient anharmonicity for gate operations. |
| Plasma Frequency | (GHz) | 5 – 8 GHz | 3–4.5 or 9–12 GHz | < 3 GHz or > 12 GHz | Small-oscillation frequency of the Josephson junction. Related to qubit frequency in the linear limit. Sets the frequency hierarchy of the circuit — must be above qubit frequency and below parasitic modes. Determines the cut-off for junction dynamics. |
| Participation Loss Rate | (kHz) | < 10 kHz | 50 – 200 kHz | > 200 kHz | Contribution of each lossy element to mode decay rate, weighted by its EPR. This is the EPR method's way of attributing loss: each element contributes proportionally to how much energy it stores. Lower values mean that element is not a dominant loss channel for the mode. |
| Cross-Mode EPR | p_cross (Dimensionless) | < 0.01 | 0.02 – 0.05 | > 0.05 | Energy leakage of a mode into unintended junctions. High values indicate unwanted mode hybridization — a mode that should be purely a qubit mode has significant overlap with other junctions. This can cause unexpected coupling and frequency shifts in multi-junction circuits. |
| Inductive Participation (Linear) | p_L (Dimensionless) | < 0.10 | 0.20 – 0.40 | > 0.40 | Fraction of mode energy stored in linear inductors (e.g. geometric/kinetic inductance of wiring). Higher values mean the mode is more distributed in linear elements and has weaker nonlinearity per mode. Ideally most energy should be in the Josephson junction, not in linear inductors. |
| EPR Convergence Error | Δp_mj (Dimensionless) | < 0.001 | 0.005 – 0.02 | > 0.02 | Simulation convergence criterion for the EPR extraction. Poor convergence leads to unreliable Hamiltonian parameters. The energy sum must satisfy Σ p_mj ≤ 1. This is a quality-control metric for your HFSS simulation — if this is large, your mesh or simulation settings need improvement. |
Qubit Parameters
| Parameter | Symbol (Unit) | Optimal | Acceptable | Poor | Physical Meaning |
|---|---|---|---|---|---|
| Qubit Transition Frequency | (GHz) | 5 – 6 GHz | 3–4.5 or 7–9 GHz | < 3 GHz or > 9 GHz | The transition frequency. Must be well below the readout resonator and above thermal energy (). The sweet spot of 5–6 GHz is ideal for dilution fridge operation (~20 mK). Frequencies below 3 GHz risk thermal excitation; above 9 GHz control electronics become challenging. |
| Anharmonicity | (MHz) | −300 to −200 MHz | −150 to −180 or −320 to −400 MHz | < −100 MHz or > +50 MHz | Frequency difference between the and transitions. Negative in transmon (straddling regime). Must be large enough in magnitude for selective qubit addressing — must exceed gate bandwidth. Too small → leakage to |2⟩ during gates. Too large → charge sensitivity increases. |
| Relative Anharmonicity | (%) | −4.5% to −3.5% | −6% to −5% or −3% to −2% | < −7% or > −1.5% | Normalized anharmonicity as a fraction of the qubit frequency. Provides a dimensionless figure of merit that is independent of qubit frequency. Too small → leakage to |2⟩ state during gate pulses; too large → return to charge sensitivity regime. Target: −4.5% to −3.5%. |
| Energy Relaxation Time | T1 (μs (microseconds)) | > 300 μs | 50 – 100 μs | < 10 μs | Time for qubit to decay from excited |1⟩ to ground |0⟩ state. Primary coherence limit — sets the maximum circuit depth. Related to total quality factor: T1 = Q/ω_01. State-of-the-art transmons now exceed 500 μs. The revised excellent target from recent literature is > 300 μs. |
| Pure Dephasing Time | T_φ (μs (microseconds)) | > 200 μs | 10 – 100 μs | < 10 μs | Dephasing from low-frequency noise sources (flux noise, charge noise) without the contribution of energy relaxation. Echo sequences can extend the effective dephasing time by refocusing quasi-static noise. A long T_φ indicates good magnetic shielding and stable fabrication. |
| Ramsey / T2* Time | T2* (μs) | > 100 μs | 20 – 50 μs | < 5 μs | Free-precession coherence time including effects of low-frequency noise. Satisfies . Often limited by flux noise or two-level system (TLS) defects. Directly impacts algorithm performance as gates must complete within . The revised excellent target is > 100 μs. |
| Echo Coherence Time | T2_echo (μs (microseconds)) | > 100 μs | 10 – 50 μs | < 10 μs | Hahn-echo decoupled coherence time. Dynamical decoupling removes quasi-static noise contributions. Should approach 2T1 in well-designed qubits. The gap between T2_echo and 2T1 reveals the magnitude of non-Markovian (quasi-static) noise in your device. |
| Quality Factor | (Dimensionless) | > 10⁶ | 10⁴ – 5×10⁵ | < 10⁴ | Dimensionless coherence figure of merit combining frequency and . Includes all loss channels weighted by EPR: . This EPR-weighted sum tells you which physical element limits your coherence. Higher Q means more oscillations before decoherence. |
| Thermal Occupation | (Photons) | < 0.005 | 0.01 – 0.05 | > 0.05 | Mean thermal photon number in the qubit mode. Calculated as . Causes state preparation errors — a thermally excited qubit starts in mixed state. Requires temperature for a 5 GHz qubit to keep . Sensitive to infrared radiation leakage. |
| Sweet Spot Sensitivity (∂f/∂Φ) | (MHz/mΦ₀) | 0 (at sweet spot) | 1 – 5 MHz/mΦ₀ | > 10 MHz/mΦ₀ | Flux sensitivity at the operating point. Zero at the sweet spot ( or ) → first-order insensitive to flux noise. This is why transmons are operated at the flux sweet spot. Flux qubits operated away from sweet spot trade coherence for tunability. Lower value = better noise immunity. |
Coupling Parameters
| Parameter | Symbol (Unit) | Optimal | Acceptable | Poor | Physical Meaning |
|---|---|---|---|---|---|
| Resonator Frequency | (GHz) | 7 – 8 GHz | 5–6.5 or 8.5–10 GHz | < 5 GHz or > 10 GHz | Readout/coupling resonator frequency. Must be detuned from the qubit by Δ >> g to remain in the dispersive regime. Higher frequency enables faster readout but requires more demanding fabrication. The resonator acts as the 'antenna' for reading out qubit state. |
| Qubit-Resonator Coupling | (MHz) | 100 – 200 MHz | 30–80 or 250–400 MHz | < 30 MHz or > 400 MHz | Vacuum Rabi coupling strength between the qubit and resonator. Determines hybridization and the dispersive shift magnitude. Must satisfy for dispersive regime validity. Too small → weak dispersive shift, slow readout; too large → strong hybridization, increased Purcell loss. |
| Qubit-Resonator Detuning | (GHz) | 1.0 – 2.0 GHz | 0.3–0.8 or 2.5–4 GHz | < 0.3 GHz (resonant!) | Frequency detuning between qubit and resonator. Must be >> g for dispersive regime. Too small → hybridization (resonant regime — catastrophic for readout!); too large → dispersive shift becomes negligible and readout SNR suffers. Optimal 1–2 GHz balances both constraints. |
| Dispersive Shift (χ) | (MHz) | 1 – 5 MHz | 0.1–0.5 or 8–15 MHz | < 0.1 MHz or > 15 MHz | State-dependent resonator frequency shift — the resonator frequency shifts by depending on qubit state. Sets readout SNR. Must satisfy for quantum non-demolition (QND) readout. Larger → faster readout but also more Purcell-induced qubit decay. Balance is critical. |
| Coupling Ratio (g/Δ) | (Dimensionless) | 0.05 – 0.10 | 0.12 – 0.20 | > 0.20 (non-dispersive!) | Ratio of coupling strength to detuning. Must be ≪ 1 for dispersive approximation validity. Values > 0.1 introduce significant higher-order corrections to the dispersive Hamiltonian. This is the single most important check for whether your circuit is truly in the dispersive regime. |
| ZZ Coupling (always-on) | (kHz) | < 10 kHz | 20 – 50 kHz | > 100 kHz | Residual always-on qubit-qubit interaction causing correlated errors. A major source of crosstalk in multi-qubit processors. . This interaction cannot be turned off and causes phase errors on neighboring qubits during single-qubit gates. The revised acceptable target is < 10 kHz. |
| Qubit-Qubit Exchange Coupling | (MHz) | 5–50 MHz (tunable range) | 0.5–2 MHz (off) / 100–200 MHz (on) | < 100 kHz off or uncontrollable | Direct qubit-qubit coupling used for 2-qubit gates. Tunable couplers allow on/off ratio > 1000. When 'off', static coupling should be < 1 MHz for low crosstalk. When 'on', 5–50 MHz enables fast gates. The ability to switch this coupling is key to scalable multi-qubit architectures. |
| Purcell Decay Rate | (kHz) | < 100 Hz | 0.5 – 5 kHz | > 5 kHz | Qubit relaxation rate via photon emission through the resonator into the transmission line. . This is loss caused by the coupling itself — the resonator acts as a decay channel. Mitigated by Purcell filter (a bandpass filter blocking qubit frequency from reaching the line). |
| Readout Resonator Linewidth | (MHz) | 1 – 5 MHz | 0.1–0.5 or 8–20 MHz | < 0.1 MHz or > 20 MHz | Resonator photon decay rate — determines readout speed vs. Purcell limit trade-off. Fast readout needs large (ring-up and ring-down quickly). But large increases Purcell loss. High-fidelity readout requires . Optimal design uses a Purcell filter to decouple these constraints. |
| Critical Photon Number | (Photons) | > 100 photons | 10 – 50 photons | < 10 photons | Maximum resonator photon number before the dispersive approximation breaks down. Readout pulses must use far fewer than photons. Larger provides a more robust readout window. Below , you're safely in the linear dispersive regime; above it, nonlinear effects corrupt readout. |
Loss & Dissipation
| Parameter | Symbol (Unit) | Optimal | Acceptable | Poor | Physical Meaning |
|---|---|---|---|---|---|
| Dielectric Loss Tangent (Bulk) | tan δ_bulk (Dimensionless) | < 10⁻⁷ | 10⁻⁶ – 10⁻⁵ | > 10⁻⁵ | Bulk substrate dielectric loss. Sapphire: ~10⁻⁸; high-resistivity silicon: ~10⁻⁷; standard silicon: ~10⁻⁶. The EPR method weights this loss by the fraction of energy in the substrate. Sapphire is preferred for highest-coherence devices. Substrate choice has a large impact on T1. |
| Surface Dielectric Loss Tangent | tan δ_surf (Dimensionless) | < 10⁻⁴ | 5×10⁻⁴ – 10⁻³ | > 10⁻³ | Surface/interface oxide loss at the metal-air (MA), metal-substrate (MS), and substrate-air (SA) interfaces. Dominant loss mechanism at low drive power due to two-level system (TLS) defects in native oxides. Reduced by surface cleaning, encapsulation, and geometric redesign (trenching). |
| Surface Participation Ratio | p_SA, p_MS, p_MA (Dimensionless) | < 10⁻⁴ | 5×10⁻⁴ – 10⁻³ | > 10⁻³ | Fraction of electric field energy at each interface layer. Lower values mean less field is concentrated at lossy surfaces. Reduced by geometry optimization: larger electrode gap, thicker ground plane, and surface trenching all reduce surface participation. This is where EPR simulation guides fabrication choices. |
| Conductor (Resistive) Loss | 1/Q_cond (Dimensionless) | < 10⁻⁸ | 10⁻⁷ – 10⁻⁶ | > 10⁻⁶ | Ohmic losses in the superconducting film caused by quasiparticles (vortices, sub-gap states). Sensitive to film quality, film thickness, and magnetic shielding. Magnetic vortices trapped in the superconductor during cool-down are a major source. Requires careful magnetic shielding and field-free cooling. |
| Radiation Loss Rate | 1/Q_rad (Dimensionless) | < 10⁻⁸ | 10⁻⁷ – 10⁻⁵ | > 10⁻⁵ | Electromagnetic radiation from open structures — energy escaping the circuit as EM waves. Minimized by full 3D enclosure, ground vias, and controlled impedance environments. Particularly important for planar circuits where radiation into the substrate or free space can be significant. |
| Internal Quality Factor (Resonator) | Q_int (Dimensionless) | > 3×10⁶ | 10⁵ – 10⁶ | < 10⁵ | Resonator quality factor excluding coupling to external transmission line. Reflects material and fabrication quality. Q_int = ω_r · T1_resonator. Best 3D aluminum cavities exceed 10⁹. Planar resonators typically 10⁶–10⁷. This metric tells you about your material quality independent of circuit design. |
| Coupled (Loaded) Quality Factor | Q_c (Dimensionless) | 10³ – 10⁴ | 100–500 or 3×10⁴–10⁵ | < 100 or > 10⁵ | Coupling quality factor to external transmission line. Determines readout bandwidth κ = ω_r/Q_c. Must be designed so Q_c << Q_int to minimize readout backaction. This is the intentional coupling designed into the circuit — it sets how strongly the resonator is coupled to the readout amplifier chain. |
| Two-Level System (TLS) Loss | F_TLS · tan δ_TLS (Dimensionless) | < 10⁻⁶ | 5×10⁻⁶ – 10⁻⁵ | > 10⁻⁵ | Loss from amorphous oxide TLS defects — quantum two-level systems in surface oxides that absorb energy. Saturates at high drive power (advantageous for readout at high power). Filling factor F_TLS from EPR simulation, tan δ from measurement. Dominant loss at single-photon level. |
| Quasiparticle Loss Rate | Γ_qp (kHz) | < 1 kHz | 10 – 100 kHz | > 100 kHz | Qubit relaxation from non-equilibrium quasiparticles tunneling across the junction. Even at millikelvin temperatures, stray radiation generates excess quasiparticles above the BCS gap. Requires careful infrared shielding and phonon trapping to suppress. One of the harder loss channels to eliminate. |
| Flux Noise Spectral Density | √S_Φ(1Hz) (μΦ₀/√Hz) | < 1 μΦ₀/√Hz | 2 – 5 μΦ₀/√Hz | > 5 μΦ₀/√Hz | 1/f flux noise amplitude — low-frequency magnetic flux fluctuations that dephase flux-sensitive qubits. Limits T2 for qubits operated away from the flux sweet spot. Reduced by larger junction loop area (more flux per noise unit) and better magnetic shielding of the dilution refrigerator. |
Gate Fidelity & Readout
| Parameter | Symbol (Unit) | Optimal | Acceptable | Poor | Physical Meaning |
|---|---|---|---|---|---|
| Single-Qubit Gate Fidelity | F_1Q (%) | > 99.9% | 99 – 99.5% | < 99% | Average single-qubit gate fidelity via randomized benchmarking. Error budget: T1/T2 decoherence errors + leakage to |2⟩ + control errors. NISQ threshold: > 99%. Fault-tolerant threshold (surface code): > 99.9%. State-of-the-art demonstrations now exceed 99.99%. |
| Two-Qubit Gate Fidelity | F_2Q (%) | > 99.5% | 97 – 99% | < 97% | Average two-qubit (CZ/CNOT) gate fidelity. Limited by ZZ coupling, leakage to higher levels, and finite coherence. Fault-tolerant threshold for surface code: > 99%. The most challenging performance metric to achieve — 2-qubit gates are typically 10–50× slower than single-qubit gates. |
| Single-Qubit Gate Duration | t_gate_1Q (ns (nanoseconds)) | 10 – 20 ns | 30 – 80 ns | > 100 ns | Gaussian pulse duration for π rotation. Must satisfy t_gate >> 1/|α| to avoid leakage to |2⟩. DRAG (Derivative Removal via Adiabatic Gate) pulse shapes allow shorter durations while suppressing leakage. Faster gates → deeper circuits within coherence time. |
| Two-Qubit Gate Duration | t_gate_2Q (ns (nanoseconds)) | 20 – 50 ns | 80 – 200 ns | > 200 ns | Duration for CZ/iSWAP gate. Limited by coupling strength J and the need for adiabatic evolution. Fast gates require strong tunable coupling. Longer duration → more decoherence error. The product of gate duration and decoherence rate sets the gate error floor. |
| Leakage Rate (per gate) | L1 (%) | < 0.05% | 0.1 – 0.5% | > 0.5% | Probability of leaving the computational subspace {|0⟩, |1⟩} per gate — landing in |2⟩ or higher. Caused by insufficient anharmonicity or excessively fast gate pulses. DRAG pulses can suppress leakage to < 0.01%. Leakage is particularly dangerous as it doesn't decay quickly and can spread. |
| Readout Assignment Fidelity | F_readout (%) | > 99% | 95 – 98% | < 95% | Probability of correctly assigning the qubit state (0→0 and 1→1). Limited by dispersive shift χ, resonator SNR, T1 decay during readout integration, and amplifier noise. State-of-the-art uses parametric amplifiers (JPA/TWPA) near quantum noise limit. Fast high-fidelity readout is essential for error correction. |
| Readout Duration | t_readout (ns (nanoseconds)) | 50 – 150 ns | 200 – 500 ns | > 500 ns | Time for resonator to ring up, integrate signal, and return sufficient SNR for state discrimination. Trade-off between speed and fidelity — faster readout means less signal but less T1 decay during measurement. Purcell filter enables faster readout by allowing large κ without Purcell loss penalty. |
| State Preparation Fidelity | F_prep (%) | > 99.5% | 98 – 99% | < 98% | Fidelity of preparing the |0⟩ ground state before algorithm execution. Limited by thermal occupation n_th and measurement-induced transitions. Active reset (conditional π pulse based on measurement result) can improve state preparation significantly and reduce cycle time compared to passive T1 decay. |
| Reset Time (Active Reset) | t_reset (ns (nanoseconds)) | < 100 ns | 200 – 500 ns | > 500 ns | Time to return qubit to |0⟩ after measurement. Passive reset is limited by T1 (microseconds). Active reset using a conditional π pulse can achieve < 200 ns — a 100–1000× speedup. Fast reset is critical for quantum error correction where measurement and re-initialization happen every error correction cycle. |
| Circuit Depth Limit (T2-limited) | (Gates) | > 1000 gates | 100 – 500 gates | < 100 gates | Maximum coherent circuit depth . The most important practical figure of merit for NISQ algorithms. Longer coherence and faster gates multiplicatively increase this. is needed for meaningful quantum advantage. This is the ultimate metric that EPR analysis helps optimize. |