
A measure of how close a quantum state or operation is to the ideal target.Fidelity is a quantitative measure of how closely a real quantum state or operation matches its ideal, error-free target. In quantum computing, where operations are probabilistic and noise is ever-present, fidelity tells you how 'true' your qubit is after a gate or measurement. A fidelity of 1 means perfect agreement; anything less indicates the presence of errors, decoherence, or imperfect control. It is the central metric for benchmarking quantum hardware, from single-qubit gates to multi-qubit entangling operations.
The concept was formalized in the 1990s as quantum error correction and benchmarking matured, with key contributions from Richard Jozsa and others who adapted classical fidelity measures to quantum mechanics.
For a quantum state, fidelity is defined as the squared overlap between the actual density matrix and the target pure state: F = ⟨ψ_target|ρ_actual|ψ_target⟩. For quantum operations, it is often computed via the average fidelity over all possible input states, or via the entanglement fidelity that accounts for correlations with an ancilla. In practice, researchers estimate fidelity by preparing known states, applying the operation, and performing tomography or randomized benchmarking to extract the error rate.
Fidelity is used to calibrate and validate quantum processors, guiding error-correction protocols and comparing the performance of different qubit technologies, as seen in recent records with 120 entangled superconducting qubits.
A fidelity of 99.9% sounds impressive, but for a large quantum computation with millions of gates, even that small error accumulates and must be corrected by fault-tolerant schemes.
| F | variable | Quantum fidelity between states — fidelity |
| ρ | variable | Density matrix of first quantum state — density matrix |
| σ | variable | Density matrix of second quantum state — density matrix |
| Tr | operator | Sum of diagonal elements of a matrix — trace operator |
| √ | operator | Principal square root of a positive semidefinite matrix — matrix square root |














