# Gate Fidelity and Errors

**Gate fidelity** measures how accurately a physical gate implements the ideal unitary. Imperfect gates accumulate errors, reducing quantum advantage.

## Fidelity Definition

Average gate fidelity: $F = \frac{1}{d+1} \text{Tr}(\rho_{\text{ideal}} \rho_{\text{actual}})$ where $d$ is dimension (2 for qubits).

**Process fidelity**: $F_p = \text{Tr}(U_{\text{ideal}}^\dagger U_{\text{actual}}) / d$ (averages over all input states).

Fidelity ranges from 0 (completely wrong) to 1 (perfect). State-of-the-art single-qubit gates: $F > 0.999$.

## Error Sources

**Systematic errors**:
- Off-resonance effects (driving at wrong frequency)
- AC Stark shift (gate strength depends on drive amplitude)
- Leakage (excitation to higher levels outside qubit subspace)

**Stochastic errors**:
- Spontaneous emission (decay during gate)
- Dephasing (random phase fluctuations)
- Charge/flux noise affecting qubit frequency

## Error Mitigation

**Before execution**:
- Calibrate gate parameters to maximize fidelity
- Use optimized pulse shapes (DRAG corrections, etc.)
- Minimize gate time (less decoherence)

**During execution**:
- Dynamical decoupling (apply pulses to refocus noise)
- Composite gates (sequence designed to cancel errors)

**After execution**:
- Readout error mitigation (measure confusion matrix, invert results)
- Zero-noise extrapolation (measure at different noise levels, extrapolate)

## Scaling

Circuit error: errors accumulate over gates. For a circuit with $n$ gates, each with fidelity $F$, total fidelity $\approx F^n$ (worst case, uncorrelated errors).

Practical example: 1000 gates at 99.9% fidelity → $0.999^{1000} \approx 37\%$ fidelity (unacceptable). Error correction is essential for large circuits.