Quantum Fidelity

For two pure states |psi> and |phi>, the fidelity is simply F = |<psi|phi>|^2, the squared magnitude of their inner product. This equals 1 when the states are identical up to a global phase and 0 when they are orthogonal. The fidelity has a direct probabilistic interpretation: if a system is prepared in |phi> and measured in a basis that includes |psi>, then F is the probability of obtaining the |psi> outcome. This makes pure-state fidelity intuitive and easy to estimate experimentally by state tomography followed by projection.

For mixed states described by density matrices rho and sigma, the generalization is the Uhlmann fidelity F(rho, sigma) = (Tr sqrt(sqrt(rho) sigma sqrt(rho)))^2. This reduces to |<psi|phi>|^2 when both states are pure and to Tr(rho sigma) when one of them is pure. The Uhlmann fidelity is symmetric, satisfies 0 <= F <= 1, and is preserved under unitary transformations. Computing it requires matrix square roots, which is straightforward for small systems but can be costly for large density matrices, motivating approximations such as the Hilbert-Schmidt inner product Tr(rho sigma) when one state is nearly pure.

Process fidelity extends the concept to quantum channels by comparing an implemented channel E to an ideal channel U via the Choi-Jamiolkowski isomorphism. The average gate fidelity F_avg integrates the fidelity F(U|psi><psi|U†, E(|psi><psi|)) over the Haar measure on pure states and relates to process fidelity by F_avg = (d F_process + 1)/(d + 1) for a d-dimensional system. Worst-case (minimum) fidelity and average fidelity can differ substantially: a gate that fails badly on a small set of inputs may still achieve high average fidelity, so security-critical applications often demand worst-case guarantees rather than averages.

Hardware vendors report fidelities in several ways that are not always directly comparable. IBM reports two-qubit gate error rates derived from randomized benchmarking, which measures average gate fidelity efficiently without full process tomography. IonQ reports algorithmic qubit counts based on circuit-level fidelity thresholds. Quantinuum uses cross-entropy benchmarking for system-level comparisons. Because conventions differ, fidelity numbers across vendors should be interpreted with care, paying attention to whether the figure reflects a single gate, a full circuit, or an average over random circuits. Direct comparison requires running equivalent circuits on each platform under the same noise conditions.