Fidelity

A measure of how close two quantum states are, similar to a distance metric.

Fidelity quantifies how similar two quantum states are. It’s one of the most important metrics in quantum information, used to assess gate quality, state preparation, and algorithm success.

Definition

For Pure States

$$F(|\psi ⟩,|\varphi ⟩)=|⟨\psi |\varphi ⟩{|}^2$$

This equals 1 if the states are identical (up to global phase), 0 if orthogonal.

For General States (Density Matrices)

$$F(\rho ,\sigma )={\left(\text{Tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)}^2$$

Simplified when one state is pure: $$F(\rho ,|\psi ⟩)=⟨\psi |\rho |\psi ⟩$$

Properties

Fidelity in Practice

Gate Fidelity

How well does an implemented gate match the ideal?

$$F_{gate}=\frac{1}{d}\text{Tr}(U_{ideal}^{†}{U}_{actual})$$

Typical values:

• Single-qubit gates: 99.5-99.99%
• Two-qubit gates: 99-99.9%

State Fidelity

How well was a target state prepared?

$$F_{state}=⟨{\psi }_{target}|{\rho }_{actual}|{\psi }_{target}⟩$$

Process Fidelity

How well does a quantum process match the ideal, averaged over all inputs.

Average Gate Fidelity

Often reported as average over all input states:

$$\bar{F}=\frac{d\cdot F_{process}+1}{d+1}$$

where $d$ is the dimension (2 for single qubit, 4 for two qubits).

Fidelity vs Error Rate

Infidelity = $1-F$ is approximately the error rate for high-fidelity operations:

Relation to Other Metrics

• Trace distance: $D(\rho ,\sigma )\ge 1-\sqrt{F(\rho ,\sigma )}$
• Diamond distance: For processes/channels

Why It Matters

Fidelity is used everywhere in quantum computing:

• Benchmarking quantum hardware
• Comparing algorithms
• Error correction thresholds
• Quantum communication protocols

See also: Quantum State, Quantum Gate, Quantum Error Correction