Qubit Performance
=================

Qubit Performance Metrics: Coherence times, gate fidelities, and spectral properties
                derived from EPR loss analysis

A. Coherence Times

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

   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - Energy Relaxation Time T_1
     - T_1
     - µs
     - Time for qubit to decay from \|1⟩ to \|0⟩; bounded by all loss channels
                            weighted by EPR participation.
     - > 500 µs
     - 100 – 500 µs
     - 10 – 99 µs
     - < 1 µs
     - T_1 is the hard ceiling on gate fidelity; EPR identifies dominant loss
                            channel for improvement.
   * - Pure Dephasing Time T_f
     - T_f
     - µs
     - Dephasing time due to low-frequency noise (flux, charge, 1/f); not
                            directly from EPR but informed by participation.
     - > 200 µs
     - 50 – 200 µs
     - 10 – 49 µs
     - < 5 µs
     - Limits T2; EPR participation at surfaces informs TLS dephasing
                            contribution.
   * - Coherence Time T2 (Ramsey)
     - T2*
     - µs
     - Total dephasing time including low-frequency noise; T2* = 2T1.
     - > 300 µs
     - 100 – 300 µs
     - 20 – 99 µs
     - < 10 µs
     - Practical coherence limit; T2*/2T1 ≈ 1 indicates pure-dephasing free
                            regime.
   * - Coherence Time T2 (Echo)
     - T_2^E
     - µs
     - Echo coherence time; removes low-frequency noise contributions; T_2^E =
                            2T1.
     - > 500 µs
     - 200 – 500 µs
     - 50 – 199 µs
     - < 20 µs
     - Ratio T_2^E/T2* quantifies 1/f noise power; EPR participations guide
                            substrate/surface optimization.
   * - Quality Factor Q_qubit
     - Q_q
     - dimensionless
     - Qubit quality factor Q = ω_q·T_1; dimensionless figure of merit across
                            frequencies.
     - > 10⁷
     - 10⁶ – 10⁷
     - 10⁵ – 10⁶
     - < 10⁴
     - Universal metric independent of frequency; Q > 10⁷ represents
                            state-of-the-art performance.

B. Gate Performance

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

   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - Single-Qubit Gate Fidelity
     - F_1Q
     - %
     - Average fidelity of single-qubit Clifford gates; limited by T_1, T2,
                            leakage (anharmonicity).
     - > 99.9%
     - 99.5 – 99.9%
     - 99.0 – 99.4%
     - < 98%
     - < 99.9% limits surface-code error correction threshold; leakage tied
                            to anharmonicity from EPR.
   * - Two-Qubit Gate Fidelity
     - F_2Q
     - %
     - Average fidelity of two-qubit entangling gates (CZ, iSWAP); limited by
                            ZZ, T_1, T2.
     - > 99.5%
     - 99.0 – 99.5%
     - 97.0 – 98.9%
     - < 95%
     - ZZ coupling (cross-Kerr from EPR) is primary source of two-qubit gate
                            error on fixed-frequency chips.
   * - Leakage Rate
     - L_1
     - % per gate
     - Probability of leaking to non-computational \|2⟩ state per gate
                            operation.
     - < 0.01%
     - < 0.1%
     - 0.1 – 0.5%
     - > 1.0%
     - Leakage non-destructively accumulates; requires active reset. Minimized
                            by maximising anharmonicity.
   * - Readout Fidelity
     - F_RO
     - %
     - Assignment fidelity for single-shot qubit state discrimination.
     - > 99%
     - 97 – 99%
     - 90 – 96%
     - < 85%
     - Limited by chi (must be large), T_1 during readout, photon number. chi
                            extracted directly via EPR.

C. Spectral Properties

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

   * - Parameter
     - Symbol
     - Unit
     - Description
     - Optimal / Best Value
     - Good Range
     - Acceptable Range
     - Poor / Worst Value
     - Physical Significance
   * - Charge Dispersion
     - varepsilon_q
     - MHz
     - Sensitivity of qubit frequency to offset charge; exponentially
                            suppressed in transmon regime.
     - < 0.01 MHz
     - < 0.1 MHz
     - 0.1 – 1 MHz
     - > 5 MHz
     - Large dispersion → charge noise dephasing. EPR ratio EJ/EC must be > 
                            50 for transmon.
   * - E_J / E_C Ratio
     - E_J / E_C
     - dimensionless
     - Josephson to charging energy ratio; governs charge noise sensitivity vs.
                            anharmonicity trade-off.
     - 50 – 100
     - 40 – 120
     - 20 – 39
     - < 10
     - < 20: charge qubit regime with high sensitivity; > 150:
                            anharmonicity too small for fast gates.
   * - Flux Sensitivity (tunable qubits)
     - domega / dPhi
     - GHz/F0
     - Sensitivity of qubit frequency to external flux; relevant for
                            flux-tunable transmons and SQUID qubits.
     - < 0.1 GHz/F0 at sweet spot
     - < 0.5 GHz/F0
     - 0.5 – 2 GHz/F0
     - > 5 GHz/F0
     - High flux sensitivity amplifies flux noise dephasing; biasing at sweet
                            spot minimizes first-order sensitivity.
   * - Frequency Spread (fabrication)
     - sigma_omega / 2pi
     - MHz
     - Standard deviation of qubit frequencies across a chip due to junction
                            fabrication variation.
     - < 5 MHz
     - < 20 MHz
     - 20 – 50 MHz
     - > 100 MHz
     - Large spread causes frequency collisions; EPR helps identify geometry
                            sensitivities to dimension variation.